https://mywikibiz.com/api.php?action=feedcontributions&user=Jon+Awbrey&feedformat=atomMyWikiBiz - User contributions [en]2024-03-29T07:50:44ZUser contributionsMediaWiki 1.35.3https://mywikibiz.com/index.php?title=Pragmatic_Semiotic_Information&diff=480723Pragmatic Semiotic Information2024-03-02T15:16:04Z<p>Jon Awbrey: /* What is information that a sign may bear it? */ {| align="center" style="text-align:center; width:50%"</p>
<hr />
<div>'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''<br />
<br />
'''Semiotic information''' is the information content of signs as conceived within the [[semeiotic]] or [[sign-relational]] framework developed by [[Charles Sanders Peirce]].<br />
<br />
==Once over quickly==<br />
<br />
===What's it good for?===<br />
<br />
The good of information is its use in reducing our uncertainty about some issue that comes before us. Generally speaking, uncertainty comes in several flavors, and so the information that serves to reduce uncertainty can be applied in several different ways. The situations of uncertainty that human agents commonly find themselves facing have been investigated under many headings, literally for ages, and the classifications that subtle thinkers arrived at long before the dawn of modern information theory still have their uses in setting the stage of an introduction.<br />
<br />
Picking an example of a subtle thinker almost at random, the philosopher-scientist Immanuel Kant divided the principal questions of human existence into three parts:<br />
<br />
:* What's true?<br />
:* What's to do?<br />
:* What's to hope?<br />
<br />
The third question is a bit too subtle for the present frame of discussion, but the first and second are easily recognizable as staking out the two main axes of information theory, namely, the dual dimensions of ''information'' and ''control''. Roughly the same space of concerns is elsewhere spanned by the dual axes of ''competence'' and ''performance'', ''specification'' and ''optimization'', or just plain ''knowledge'' and ''skill''.<br />
<br />
A question of ''what's true'' is a ''descriptive question'', and there exist what are called ''[[descriptive science]]s'' devoted to answering descriptive questions about any domain of phenomena that one might care to name.<br />
<br />
A question of ''what's to do'', in other words, what must be done by way of achieving a given aim, is a ''normative question'', and there exist what are called ''[[normative science]]s'' devoted to answering normative questions about any domain of problems that one might care to address.<br />
<br />
Since information plays its role on a stage set by uncertainty, a big part of saying what information is will necessarily involve saying what uncertainty is. There is little chance that the vagueness of a word like ''uncertainty'', given the nuances of its ordinary, poetic, and technical uses, can be corralled by a single pen, but there do exist established models and formal theories that address definable aspects of uncertainty, and these have enough uses to make them worth looking into.<br />
<br />
===What is information that a sign may bear it?===<br />
<br />
Three more questions arise at this juncture:<br />
# How is a sign empowered to contain information?<br />
# What is the practical context of communication?<br />
# Why do we care about these bits of information?<br />
<br />
A very rough answer to these questions might begin as follows:<br />
<br />
Human beings are initially concerned solely with their own lives, but then a world obtrudes on their subjective existence, and so they find themselves forced to take an interest in the objective realities of its nature.<br />
<br />
In pragmatic terms our initial aim, concern, interest, object, or ''pragma'' is expressed by the verbal infinitive ''to live'', but the infinitive is soon reified into the derivative substantial forms of ''nature'', ''reality'', ''the world'', and so on. Against this backdrop we find ourselves cast as the protagonists on a ''scene of uncertainty''. The situation may be pictured as a juncture from which a manifold of options fan out before us. It may be an issue of ''truth'', ''duty'', or ''hope'', the last codifying a special type of uncertainty as to ''what regulative principle has any chance of success'', but the chief uncertainty is that we are called on to make a choice and find that we all too often have almost no clue as to which of the options is most fit to pick.<br />
<br />
Just to make up a discrete example, let us suppose that the cardinality of this choice is a finite ''n'', and just to make it fully concrete let us say that ''n''&nbsp;=&nbsp;5. Figure 1 affords a rough picture of the situation.<br />
<br />
{| align="center" style="text-align:center; width:50%"<br />
|<br />
<pre><br />
o-------------------------------------------------o<br />
| |<br />
| ? ? ? ? ? |<br />
| o o o o o |<br />
| |<br />
| o o o o o |<br />
| |<br />
| o o o o o |<br />
| |<br />
| o o o o o |<br />
| |<br />
| o o o o o |<br />
| |<br />
| ooooo |<br />
| |<br />
| O n = 5 |<br />
| |<br />
o-------------------------------------------------o<br />
Figure 1. Juncture of Degree 5<br />
</pre><br />
|}<br />
<br />
This pictures a juncture, represented by &ldquo;O&rdquo;, where there are ''n'' options for the outcome of a conduct, and we have no clue as to which it must be. In a sense, the degree of this node, in this case ''n''&nbsp;=&nbsp;5, measures the uncertainty that we have at this point.<br />
<br />
This is the minimal sort of setting in which a sign can make any sense at all. A sign has significance for an agent, interpreter, or observer because its actualization, its being given or its being present, serves to reduce the uncertainty of a decision that the agent has to make, whether it concerns the actions that the agent ought to take in order to achieve some objective of interest, or whether it concerns the predicates that the agent ought to treat as being true of some object in the world.<br />
<br />
The way that signs enter the scene is shown in Figure 2.<br />
<br />
{| align="center" style="text-align:center; width:50%"<br />
|<br />
<pre><br />
o-------------------------------------------------o<br />
| |<br />
| k_1 = 3 k_2 = 2 |<br />
| o-----o-----o o-----o |<br />
| "A" "B" |<br />
| o----o----o o----o |<br />
| |<br />
| o---o---o o---o |<br />
| |<br />
| o--o--o o--o |<br />
| |<br />
| o-o-o o-o |<br />
| |<br />
| ooooo |<br />
| |<br />
| O n = 5 |<br />
| |<br />
o-------------------------------------------------o<br />
Figure 2. Partition of Degrees 3 and 2<br />
</pre><br />
|}<br />
<br />
This illustrates a situation of uncertainty that has been augmented by a classification.<br />
<br />
In the particular pattern of classification that is shown here, the first three outcomes fall under the sign &ldquo;A&rdquo;, and the next two outcomes fall under the sign &ldquo;B&rdquo;. If the outcomes make up a set of ''things that might be true about an object'', then the signs could be read as nomens (terms) or notions (concepts) of a relevant empirical, ontological, taxonomical, or theoretical scheme, that is, as predicates and predictions of the outcomes. If the outcomes make up a set of ''things that might be good to do in order to achieve an objective'', then the signs could be read as bits of advice or other sorts of indicators that tell us what to do in the situation, relative to our active goals.<br />
<br />
This is the basic framework for talking about information and signs in regard to communication, decision, and the uncertainties thereof.<br />
<br />
Just to unpack some of the many things that may be getting glossed over in this little word ''sign'', it encompasses all of the ''data of the senses'' (DOTS) that we take as informing us about inner and outer worlds, plus all of the concepts and terms that we use to argue, to communicate, to inquire, or even to speculate, both about our ontologies for beings in the worlds and about our policies for action in the world.<br />
<br />
Here is one of the places where it is tempting to try to collapse the 3-adic sign relation into a 2-adic relation. For if these DOTS are so closely identified with objects that we can scarcely imagine how they might be discrepant, then it will appear to us that one role of beings can be eliminated from our picture of the world. In this event, the only things that we are required to inform ourselves about, via the inspection of these DOTS, are yet more DOTS, whether past, or present, or prospective, just more DOTS. This is the special form to which we frequently find the idea of an information channel being reduced, namely, to a ''source'' that has nothing more to tell us about than its own conceivable conducts or its own potential issues.<br />
<br />
As a matter of fact, at least in this discrete type of case, it would be possible to use the degree of the node as a measure of uncertainty, but it would operate as a multiplicative measure rather than the sort of additive measure that we would normally prefer. To illustrate how this would work out, let us consider an easier example, one where the degree of the choice point is 4.<br />
<br />
{| align="center" style="text-align:center; width:50%"<br />
|<br />
<pre><br />
o-------------------------------------------------o<br />
| |<br />
| ? ? ? ? |<br />
| o o o o |<br />
| |<br />
| o o o o |<br />
| |<br />
| o o o o |<br />
| |<br />
| o o o o |<br />
| |<br />
| o o o o |<br />
| |<br />
| oo oo |<br />
| |<br />
| O n = 4 |<br />
| |<br />
o-------------------------------------------------o<br />
Figure 3. Juncture of Degree 4<br />
</pre><br />
|}<br />
<br />
Suppose that we contemplate making another decision after the present issue has been decided, one that has a degree of 2 in every case. The compound situation is depicted in Figure&nbsp;4.<br />
<br />
{| align="center" style="text-align:center; width:50%"<br />
|<br />
<pre><br />
o-------------------------------------------------o<br />
| |<br />
| o o o o o o o o |<br />
| \ / \ / \ / \ / |<br />
| o o o o n_2 = 2 |<br />
| |<br />
| o o o o |<br />
| |<br />
| o o o o |<br />
| |<br />
| o o o o |<br />
| |<br />
| o o o o |<br />
| |<br />
| oo oo |<br />
| |<br />
| O n_1 = 4 |<br />
| |<br />
o-------------------------------------------------o<br />
Figure 4. Compound Junctures of Degrees 4 and 2<br />
</pre><br />
|}<br />
<br />
This illustrates the fact that the compound uncertainty, 8, is the product of the two component uncertainties, 4&nbsp;times&nbsp;2. To convert this to an additive measure, one simply takes the logarithms to a convenient base, say 2, and thus arrives at the not too astounding fact that the uncertainty of the first choice is 2 bits, the uncertainty of the next choice is 1 bit, and the compound uncertainty is 2&nbsp;+&nbsp;1&nbsp;=&nbsp;3 bits.<br />
<br />
In many ways, the provision of information, a process that reduces uncertainty, is the inverse process to the kind of uncertainty augmentation that occurs in compound decisions. By way of illustrating this relationship, let us return to our initial example.<br />
<br />
A set of signs enters on a setup like this as a system of ''middle terms'', a collection of signs that one may regard, aptly enough, as constellating a ''medium''.<br />
<br />
{| align="center" style="text-align:center; width:50%"<br />
|<br />
<pre><br />
o-------------------------------------------------o<br />
| |<br />
| k_1 = 3 k_2 = 2 |<br />
| o-----o-----o o-----o |<br />
| "A" "B" |<br />
| o----o----o o----o |<br />
| |<br />
| o---o---o o---o |<br />
| |<br />
| o--o--o o--o |<br />
| |<br />
| o-o-o o-o |<br />
| |<br />
| ooooo |<br />
| |<br />
| O n = 5 |<br />
| |<br />
o-------------------------------------------------o<br />
Figure 5. Partition of Degrees 3 and 2<br />
</pre><br />
|}<br />
<br />
The ''language'' or ''medium'' here is the set of signs {&ldquo;A&rdquo;,&nbsp;&ldquo;B&rdquo;}. On the assumption that the initial 5 outcomes are equally likely, one may associate a frequency distribution (''k''<sub>1</sub>, ''k''<sub>2</sub>) = (3, 2) and thus a probability distribution (''p''<sub>1</sub>, ''p''<sub>2</sub>) = (3/5, 2/5) = (0.6, 0.4) with this language, and thus define a communication ''channel''.<br />
<br />
The most important thing here is really just to get a handle on the ''conditions for the possibility of signs making sense'', but once we have this much of a setup we find that we can begin to construct some rough and ready bits of information-theoretic furniture, like measures of uncertainty, channel capacity, and the amount of information that can be associated with the reception or the recognition of a single sign. Still, before we get into all of this, it needs to be emphasized that, even when these measures are too ad&nbsp;hoc and insufficient to be of much use per&nbsp;se, the significance of the setup that it takes to support them is not at all diminished.<br />
<br />
Consider the classification-augmented or sign-enhanced situation of uncertainty that was depicted above. What happens if one or the other of the two signs, &ldquo;A&rdquo; or &ldquo;B&rdquo;, is observed or received, on the constant assumption that its significance is recognized on receipt?<br />
<br />
:* If we receive &ldquo;A&rdquo; our uncertainty is reduced from <math>\log 5</math> to <math>\log 3.</math><br />
<br />
:* If we receive &ldquo;B&rdquo; our uncertainty is reduced from <math>\log 5</math> to <math>\log 2.</math><br />
<br />
It is from these characteristics that the ''information capacity'' of a communication channel can be defined, specifically, as the ''average uncertainty reduction on receiving a sign'', a formula with the splendid mnemonic &ldquo;AURORAS&rdquo;.<br />
<br />
:* On receiving the message &ldquo;A&rdquo;, the additive measure of uncertainty is reduced from <math>\log 5</math> to <math>\log 3</math>, so the net reduction is <math>(\log 5 - \log 3).</math><br />
<br />
:* On receiving the message &ldquo;B&rdquo;, the additive measure of uncertainty is reduced from <math>\log 5</math> to <math>\log 2</math>, so the net reduction is <math>(\log 5 - \log 2).</math><br />
<br />
The average uncertainty reduction per sign of the language is computed by taking a ''weighted average'' of the reductions that occur in the channel, where the weight of each reduction is the number of options or outcomes that fall under the associated sign.<br />
<br />
:* The uncertainty reduction of <math>(\log 5 - \log 3)\!</math> gets a weight of 3.<br />
<br />
:* The uncertainty reduction of <math>(\log 5 - \log 2)\!</math> gets a weight of 2.<br />
<br />
Finally, the weighted average of these two reductions is:<br />
<br />
: <math>{1 \over {2 + 3}}(3(\log 5 - \log 3) + 2(\log 5 - \log 2))\!</math><br />
<br />
Extracting the general pattern of this calculation yields the following worksheet for computing the capacity of a 2-symbol channel with frequencies that partition as <math>n = k_1 + k_2.\!</math><br />
<br />
{| cellspacing="6" <br />
| Capacity<br />
| of a channel {&ldquo;A&rdquo;, &ldquo;B&rdquo;} that bears the odds of 60 &ldquo;A&rdquo; to 40 &ldquo;B&rdquo;<br />
|-<br />
| &nbsp;<br />
| <math>=\quad {1 \over n}(k_1(\log n - \log k_1) + k_2(\log n - \log k_2))\!</math><br />
|-<br />
| &nbsp;<br />
| <math>=\quad {k_1 \over n}(\log n - \log k_1) + {k_2 \over n}(\log n - \log k_2)\!</math><br />
|-<br />
| &nbsp;<br />
| <math>=\quad -{k_1 \over n}(\log k_1 - \log n) -{k_2 \over n}(\log k_2 - \log n)\!</math><br />
|-<br />
| &nbsp;<br />
| <math>=\quad -{k_1 \over n}(\log {k_1 \over n}) - {k_2 \over n}(\log {k_2 \over n})\!</math><br />
|-<br />
| &nbsp;<br />
| <math>=\quad -(p_1 \log p_1 + p_2 \log p_2)\!</math><br />
|-<br />
| &nbsp;<br />
| <math>=\quad - (0.6 \log 0.6 + 0.4 \log 0.4)\!</math><br />
|-<br />
| &nbsp;<br />
| <math>=\quad 0.971\!</math><br />
|}<br />
<br />
In other words, the capacity of this channel is slightly under 1 bit. This makes intuitive sense, since 3 against 2 is a near-even split of 5, and the measure of the channel capacity or the ''entropy'' is supposed to attain its maximum of 1 bit whenever a two-way partition is 50-50, that is to say, when it's as ''uniform'' a distribution as it can be.<br />
<br />
==Bibliography==<br />
<br />
* [[Charles Sanders Peirce (Bibliography)]]<br />
<br />
* Peirce, C.S. (1867), &ldquo;Upon Logical Comprehension and Extension&rdquo;, [http://www.iupui.edu/~peirce/writings/v2/w2/w2_06/v2_06.htm Online].<br />
<br />
==See also==<br />
<br />
===Related topics===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [[Ampheck]]<br />
* [[Boolean domain]]<br />
* [[Boolean function]]<br />
* [[Boolean-valued function]]<br />
{{col-break}}<br />
* [[Logical graph]]<br />
* [[Logical matrix]]<br />
* [[Logical NAND]]<br />
* [[Logical NNOR]]<br />
{{col-break}}<br />
* [[Minimal negation operator]]<br />
* [[Peirce's law]]<br />
* [[Propositional calculus]]<br />
* [[Semeiotic]]<br />
{{col-break}}<br />
* [[Sign relation]]<br />
* [[Triadic relation]]<br />
* [[Truth table]]<br />
* [[Zeroth order logic]]<br />
{{col-end}}<br />
<br />
===Related articles===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Peirce%27s_Logic_Of_Information Peirce's Logic Of Information]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Information_%3D_Comprehension_%C3%97_Extension Information = Comprehension × Extension]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]<br />
{{col-break}}<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]<br />
{{col-end}}<br />
<br />
[[Category:Charles Sanders Peirce]]<br />
[[Category:Information Theory]]<br />
[[Category:Inquiry]]<br />
[[Category:Inquiry Driven Systems]]<br />
[[Category:Logic]]<br />
[[Category:Pragmatism]]<br />
[[Category:Scientific Method]]<br />
[[Category:Semiotics]]<br />
[[Category:Sign Relations]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Pragmatic_Semiotic_Information&diff=480722Pragmatic Semiotic Information2024-03-02T15:10:06Z<p>Jon Awbrey: + Pragmatic Semiotic Information</p>
<hr />
<div>'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''<br />
<br />
'''Semiotic information''' is the information content of signs as conceived within the [[semeiotic]] or [[sign-relational]] framework developed by [[Charles Sanders Peirce]].<br />
<br />
==Once over quickly==<br />
<br />
===What's it good for?===<br />
<br />
The good of information is its use in reducing our uncertainty about some issue that comes before us. Generally speaking, uncertainty comes in several flavors, and so the information that serves to reduce uncertainty can be applied in several different ways. The situations of uncertainty that human agents commonly find themselves facing have been investigated under many headings, literally for ages, and the classifications that subtle thinkers arrived at long before the dawn of modern information theory still have their uses in setting the stage of an introduction.<br />
<br />
Picking an example of a subtle thinker almost at random, the philosopher-scientist Immanuel Kant divided the principal questions of human existence into three parts:<br />
<br />
:* What's true?<br />
:* What's to do?<br />
:* What's to hope?<br />
<br />
The third question is a bit too subtle for the present frame of discussion, but the first and second are easily recognizable as staking out the two main axes of information theory, namely, the dual dimensions of ''information'' and ''control''. Roughly the same space of concerns is elsewhere spanned by the dual axes of ''competence'' and ''performance'', ''specification'' and ''optimization'', or just plain ''knowledge'' and ''skill''.<br />
<br />
A question of ''what's true'' is a ''descriptive question'', and there exist what are called ''[[descriptive science]]s'' devoted to answering descriptive questions about any domain of phenomena that one might care to name.<br />
<br />
A question of ''what's to do'', in other words, what must be done by way of achieving a given aim, is a ''normative question'', and there exist what are called ''[[normative science]]s'' devoted to answering normative questions about any domain of problems that one might care to address.<br />
<br />
Since information plays its role on a stage set by uncertainty, a big part of saying what information is will necessarily involve saying what uncertainty is. There is little chance that the vagueness of a word like ''uncertainty'', given the nuances of its ordinary, poetic, and technical uses, can be corralled by a single pen, but there do exist established models and formal theories that address definable aspects of uncertainty, and these have enough uses to make them worth looking into.<br />
<br />
===What is information that a sign may bear it?===<br />
<br />
Three more questions arise at this juncture:<br />
# How is a sign empowered to contain information?<br />
# What is the practical context of communication?<br />
# Why do we care about these bits of information?<br />
<br />
A very rough answer to these questions might begin as follows:<br />
<br />
Human beings are initially concerned solely with their own lives, but then a world obtrudes on their subjective existence, and so they find themselves forced to take an interest in the objective realities of its nature.<br />
<br />
In pragmatic terms our initial aim, concern, interest, object, or ''pragma'' is expressed by the verbal infinitive ''to live'', but the infinitive is soon reified into the derivative substantial forms of ''nature'', ''reality'', ''the world'', and so on. Against this backdrop we find ourselves cast as the protagonists on a ''scene of uncertainty''. The situation may be pictured as a juncture from which a manifold of options fan out before us. It may be an issue of ''truth'', ''duty'', or ''hope'', the last codifying a special type of uncertainty as to ''what regulative principle has any chance of success'', but the chief uncertainty is that we are called on to make a choice and find that we all too often have almost no clue as to which of the options is most fit to pick.<br />
<br />
Just to make up a discrete example, let us suppose that the cardinality of this choice is a finite ''n'', and just to make it fully concrete let us say that ''n''&nbsp;=&nbsp;5. Figure 1 affords a rough picture of the situation.<br />
<br />
{| align="center" cellspacing="6" style="text-align:center; width:60%"<br />
|<br />
<pre><br />
o-------------------------------------------------o<br />
| |<br />
| ? ? ? ? ? |<br />
| o o o o o |<br />
| |<br />
| o o o o o |<br />
| |<br />
| o o o o o |<br />
| |<br />
| o o o o o |<br />
| |<br />
| o o o o o |<br />
| |<br />
| ooooo |<br />
| |<br />
| O n = 5 |<br />
| |<br />
o-------------------------------------------------o<br />
Figure 1. Juncture of Degree 5<br />
</pre><br />
|}<br />
<br />
This pictures a juncture, represented by &ldquo;O&rdquo;, where there are ''n'' options for the outcome of a conduct, and we have no clue as to which it must be. In a sense, the degree of this node, in this case ''n''&nbsp;=&nbsp;5, measures the uncertainty that we have at this point.<br />
<br />
This is the minimal sort of setting in which a sign can make any sense at all. A sign has significance for an agent, interpreter, or observer because its actualization, its being given or its being present, serves to reduce the uncertainty of a decision that the agent has to make, whether it concerns the actions that the agent ought to take in order to achieve some objective of interest, or whether it concerns the predicates that the agent ought to treat as being true of some object in the world.<br />
<br />
The way that signs enter the scene is shown in Figure 2.<br />
<br />
{| align="center" cellspacing="6" style="text-align:center; width:60%"<br />
|<br />
<pre><br />
o-------------------------------------------------o<br />
| |<br />
| k_1 = 3 k_2 = 2 |<br />
| o-----o-----o o-----o |<br />
| "A" "B" |<br />
| o----o----o o----o |<br />
| |<br />
| o---o---o o---o |<br />
| |<br />
| o--o--o o--o |<br />
| |<br />
| o-o-o o-o |<br />
| |<br />
| ooooo |<br />
| |<br />
| O n = 5 |<br />
| |<br />
o-------------------------------------------------o<br />
Figure 2. Partition of Degrees 3 and 2<br />
</pre><br />
|}<br />
<br />
This illustrates a situation of uncertainty that has been augmented by a classification.<br />
<br />
In the particular pattern of classification that is shown here, the first three outcomes fall under the sign &ldquo;A&rdquo;, and the next two outcomes fall under the sign &ldquo;B&rdquo;. If the outcomes make up a set of ''things that might be true about an object'', then the signs could be read as nomens (terms) or notions (concepts) of a relevant empirical, ontological, taxonomical, or theoretical scheme, that is, as predicates and predictions of the outcomes. If the outcomes make up a set of ''things that might be good to do in order to achieve an objective'', then the signs could be read as bits of advice or other sorts of indicators that tell us what to do in the situation, relative to our active goals.<br />
<br />
This is the basic framework for talking about information and signs in regard to communication, decision, and the uncertainties thereof.<br />
<br />
Just to unpack some of the many things that may be getting glossed over in this little word ''sign'', it encompasses all of the ''data of the senses'' (DOTS) that we take as informing us about inner and outer worlds, plus all of the concepts and terms that we use to argue, to communicate, to inquire, or even to speculate, both about our ontologies for beings in the worlds and about our policies for action in the world.<br />
<br />
Here is one of the places where it is tempting to try to collapse the 3-adic sign relation into a 2-adic relation. For if these DOTS are so closely identified with objects that we can scarcely imagine how they might be discrepant, then it will appear to us that one role of beings can be eliminated from our picture of the world. In this event, the only things that we are required to inform ourselves about, via the inspection of these DOTS, are yet more DOTS, whether past, or present, or prospective, just more DOTS. This is the special form to which we frequently find the idea of an information channel being reduced, namely, to a ''source'' that has nothing more to tell us about than its own conceivable conducts or its own potential issues.<br />
<br />
As a matter of fact, at least in this discrete type of case, it would be possible to use the degree of the node as a measure of uncertainty, but it would operate as a multiplicative measure rather than the sort of additive measure that we would normally prefer. To illustrate how this would work out, let us consider an easier example, one where the degree of the choice point is 4.<br />
<br />
{| align="center" cellspacing="6" style="text-align:center; width:60%"<br />
|<br />
<pre><br />
o-------------------------------------------------o<br />
| |<br />
| ? ? ? ? |<br />
| o o o o |<br />
| |<br />
| o o o o |<br />
| |<br />
| o o o o |<br />
| |<br />
| o o o o |<br />
| |<br />
| o o o o |<br />
| |<br />
| oo oo |<br />
| |<br />
| O n = 4 |<br />
| |<br />
o-------------------------------------------------o<br />
Figure 3. Juncture of Degree 4<br />
</pre><br />
|}<br />
<br />
Suppose that we contemplate making another decision after the present issue has been decided, one that has a degree of 2 in every case. The compound situation is depicted in Figure&nbsp;4.<br />
<br />
{| align="center" cellspacing="6" style="text-align:center; width:60%"<br />
|<br />
<pre><br />
o-------------------------------------------------o<br />
| |<br />
| o o o o o o o o |<br />
| \ / \ / \ / \ / |<br />
| o o o o n_2 = 2 |<br />
| |<br />
| o o o o |<br />
| |<br />
| o o o o |<br />
| |<br />
| o o o o |<br />
| |<br />
| o o o o |<br />
| |<br />
| oo oo |<br />
| |<br />
| O n_1 = 4 |<br />
| |<br />
o-------------------------------------------------o<br />
Figure 4. Compound Junctures of Degrees 4 and 2<br />
</pre><br />
|}<br />
<br />
This illustrates the fact that the compound uncertainty, 8, is the product of the two component uncertainties, 4&nbsp;times&nbsp;2. To convert this to an additive measure, one simply takes the logarithms to a convenient base, say 2, and thus arrives at the not too astounding fact that the uncertainty of the first choice is 2 bits, the uncertainty of the next choice is 1 bit, and the compound uncertainty is 2&nbsp;+&nbsp;1&nbsp;=&nbsp;3 bits.<br />
<br />
In many ways, the provision of information, a process that reduces uncertainty, is the inverse process to the kind of uncertainty augmentation that occurs in compound decisions. By way of illustrating this relationship, let us return to our initial example.<br />
<br />
A set of signs enters on a setup like this as a system of ''middle terms'', a collection of signs that one may regard, aptly enough, as constellating a ''medium''.<br />
<br />
{| align="center" cellspacing="6" style="text-align:center; width:60%"<br />
|<br />
<pre><br />
o-------------------------------------------------o<br />
| |<br />
| k_1 = 3 k_2 = 2 |<br />
| o-----o-----o o-----o |<br />
| "A" "B" |<br />
| o----o----o o----o |<br />
| |<br />
| o---o---o o---o |<br />
| |<br />
| o--o--o o--o |<br />
| |<br />
| o-o-o o-o |<br />
| |<br />
| ooooo |<br />
| |<br />
| O n = 5 |<br />
| |<br />
o-------------------------------------------------o<br />
Figure 5. Partition of Degrees 3 and 2<br />
</pre><br />
|}<br />
<br />
The ''language'' or ''medium'' here is the set of signs {&ldquo;A&rdquo;,&nbsp;&ldquo;B&rdquo;}. On the assumption that the initial 5 outcomes are equally likely, one may associate a frequency distribution (''k''<sub>1</sub>, ''k''<sub>2</sub>) = (3, 2) and thus a probability distribution (''p''<sub>1</sub>, ''p''<sub>2</sub>) = (3/5, 2/5) = (0.6, 0.4) with this language, and thus define a communication ''channel''.<br />
<br />
The most important thing here is really just to get a handle on the ''conditions for the possibility of signs making sense'', but once we have this much of a setup we find that we can begin to construct some rough and ready bits of information-theoretic furniture, like measures of uncertainty, channel capacity, and the amount of information that can be associated with the reception or the recognition of a single sign. Still, before we get into all of this, it needs to be emphasized that, even when these measures are too ad&nbsp;hoc and insufficient to be of much use per&nbsp;se, the significance of the setup that it takes to support them is not at all diminished.<br />
<br />
Consider the classification-augmented or sign-enhanced situation of uncertainty that was depicted above. What happens if one or the other of the two signs, &ldquo;A&rdquo; or &ldquo;B&rdquo;, is observed or received, on the constant assumption that its significance is recognized on receipt?<br />
<br />
:* If we receive &ldquo;A&rdquo; our uncertainty is reduced from <math>\log 5</math> to <math>\log 3.</math><br />
<br />
:* If we receive &ldquo;B&rdquo; our uncertainty is reduced from <math>\log 5</math> to <math>\log 2.</math><br />
<br />
It is from these characteristics that the ''information capacity'' of a communication channel can be defined, specifically, as the ''average uncertainty reduction on receiving a sign'', a formula with the splendid mnemonic &ldquo;AURORAS&rdquo;.<br />
<br />
:* On receiving the message &ldquo;A&rdquo;, the additive measure of uncertainty is reduced from <math>\log 5</math> to <math>\log 3</math>, so the net reduction is <math>(\log 5 - \log 3).</math><br />
<br />
:* On receiving the message &ldquo;B&rdquo;, the additive measure of uncertainty is reduced from <math>\log 5</math> to <math>\log 2</math>, so the net reduction is <math>(\log 5 - \log 2).</math><br />
<br />
The average uncertainty reduction per sign of the language is computed by taking a ''weighted average'' of the reductions that occur in the channel, where the weight of each reduction is the number of options or outcomes that fall under the associated sign.<br />
<br />
:* The uncertainty reduction of <math>(\log 5 - \log 3)\!</math> gets a weight of 3.<br />
<br />
:* The uncertainty reduction of <math>(\log 5 - \log 2)\!</math> gets a weight of 2.<br />
<br />
Finally, the weighted average of these two reductions is:<br />
<br />
: <math>{1 \over {2 + 3}}(3(\log 5 - \log 3) + 2(\log 5 - \log 2))\!</math><br />
<br />
Extracting the general pattern of this calculation yields the following worksheet for computing the capacity of a 2-symbol channel with frequencies that partition as <math>n = k_1 + k_2.\!</math><br />
<br />
{| cellspacing="6" <br />
| Capacity<br />
| of a channel {&ldquo;A&rdquo;, &ldquo;B&rdquo;} that bears the odds of 60 &ldquo;A&rdquo; to 40 &ldquo;B&rdquo;<br />
|-<br />
| &nbsp;<br />
| <math>=\quad {1 \over n}(k_1(\log n - \log k_1) + k_2(\log n - \log k_2))\!</math><br />
|-<br />
| &nbsp;<br />
| <math>=\quad {k_1 \over n}(\log n - \log k_1) + {k_2 \over n}(\log n - \log k_2)\!</math><br />
|-<br />
| &nbsp;<br />
| <math>=\quad -{k_1 \over n}(\log k_1 - \log n) -{k_2 \over n}(\log k_2 - \log n)\!</math><br />
|-<br />
| &nbsp;<br />
| <math>=\quad -{k_1 \over n}(\log {k_1 \over n}) - {k_2 \over n}(\log {k_2 \over n})\!</math><br />
|-<br />
| &nbsp;<br />
| <math>=\quad -(p_1 \log p_1 + p_2 \log p_2)\!</math><br />
|-<br />
| &nbsp;<br />
| <math>=\quad - (0.6 \log 0.6 + 0.4 \log 0.4)\!</math><br />
|-<br />
| &nbsp;<br />
| <math>=\quad 0.971\!</math><br />
|}<br />
<br />
In other words, the capacity of this channel is slightly under 1 bit. This makes intuitive sense, since 3 against 2 is a near-even split of 5, and the measure of the channel capacity or the ''entropy'' is supposed to attain its maximum of 1 bit whenever a two-way partition is 50-50, that is to say, when it's as ''uniform'' a distribution as it can be.<br />
<br />
==Bibliography==<br />
<br />
* [[Charles Sanders Peirce (Bibliography)]]<br />
<br />
* Peirce, C.S. (1867), &ldquo;Upon Logical Comprehension and Extension&rdquo;, [http://www.iupui.edu/~peirce/writings/v2/w2/w2_06/v2_06.htm Online].<br />
<br />
==See also==<br />
<br />
===Related topics===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [[Ampheck]]<br />
* [[Boolean domain]]<br />
* [[Boolean function]]<br />
* [[Boolean-valued function]]<br />
{{col-break}}<br />
* [[Logical graph]]<br />
* [[Logical matrix]]<br />
* [[Logical NAND]]<br />
* [[Logical NNOR]]<br />
{{col-break}}<br />
* [[Minimal negation operator]]<br />
* [[Peirce's law]]<br />
* [[Propositional calculus]]<br />
* [[Semeiotic]]<br />
{{col-break}}<br />
* [[Sign relation]]<br />
* [[Triadic relation]]<br />
* [[Truth table]]<br />
* [[Zeroth order logic]]<br />
{{col-end}}<br />
<br />
===Related articles===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Peirce%27s_Logic_Of_Information Peirce's Logic Of Information]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Information_%3D_Comprehension_%C3%97_Extension Information = Comprehension × Extension]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]<br />
{{col-break}}<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]<br />
{{col-end}}<br />
<br />
[[Category:Charles Sanders Peirce]]<br />
[[Category:Information Theory]]<br />
[[Category:Inquiry]]<br />
[[Category:Inquiry Driven Systems]]<br />
[[Category:Logic]]<br />
[[Category:Pragmatism]]<br />
[[Category:Scientific Method]]<br />
[[Category:Semiotics]]<br />
[[Category:Sign Relations]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Directory:Jon_Awbrey/Differential_Logic_and_Dynamic_Systems_2.0&diff=480650Directory:Jon Awbrey/Differential Logic and Dynamic Systems 2.02023-03-03T14:32:16Z<p>Jon Awbrey: /* Document History */</p>
<hr />
<div>{{DISPLAYTITLE:Differential Logic and Dynamic Systems 2.0}}<br />
'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''<br />
<br />
{| align="center" cellpadding="10"<br />
| [[Image:Tangent_Functor_Ferris_Wheel.gif]]<br />
|}<br />
<br />
{| style="height:36px; width:100%"<br />
| align="left" | ''Stand and unfold yourself.''<br />
| align="right" | Hamlet: Francsico&mdash;1.1.2<br />
|}<br />
<br />
This article develops a differential extension of propositional calculus and applies it to a context of problems arising in dynamic systems. The work pursued here is coordinated with a parallel application that focuses on neural network systems, but the dependencies are arranged to make the present article the main and the more self-contained work, to serve as a conceptual frame and a technical background for the network project.<br />
<br />
==Review and Transition==<br />
<br />
This note continues a previous discussion on the problem of dealing with change and diversity in logic-based intelligent systems. It is useful to begin by summarizing essential material from previous reports.<br />
<br />
Table 1 outlines a notation for propositional calculus based on two types of logical connectives, both of variable <math>k\!</math>-ary scope.<br />
<br />
* A bracketed list of propositional expressions in the form <math>\texttt{(} e_1, e_2, \ldots, e_{k-1}, e_k \texttt{)}\!</math> indicates that exactly one of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> is false.<br />
<br />
* A concatenation of propositional expressions in the form <math>e_1 ~ e_2 ~ \ldots ~ e_{k-1} ~ e_k</math> indicates that all of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> are true, in other words, that their [[logical conjunction]] is true.<br />
<br />
All other propositional connectives can be obtained in a very efficient style of representation through combinations of these two forms. Strictly speaking, the concatenation form is dispensable in light of the bracketed form, but it is convenient to maintain it as an abbreviation of more complicated bracket expressions.<br />
<br />
This treatment of propositional logic is derived from the work of C.S. Peirce [P1, P2], who gave this approach an extensive development in his graphical systems of predicate, relational, and modal logic [Rob]. More recently, these ideas were revived and supplemented in an alternative interpretation by George Spencer-Brown [SpB]. Both of these authors used other forms of enclosure where I use parentheses, but the structural topologies of expression and the functional varieties of interpretation are fundamentally the same.<br />
<br />
While working with expressions solely in propositional calculus, it is easiest to use plain parentheses for logical connectives. In contexts where parentheses are needed for other purposes &ldquo;teletype&rdquo; parentheses <math>\texttt{(} \ldots \texttt{)}\!</math> or barred parentheses <math>(\!| \ldots |\!)</math> may be used for logical operators.<br />
<br />
The briefest expression for logical truth is the empty word, usually denoted by <math>{}^{\backprime\backprime} \boldsymbol\varepsilon {}^{\prime\prime}\!</math> or <math>{}^{\backprime\backprime} \boldsymbol\lambda {}^{\prime\prime}\!</math> in formal languages, where it forms the identity element for concatenation. To make it visible in this text, it may be denoted by the equivalent expression <math>{}^{\backprime\backprime} \texttt{((~))} {}^{\prime\prime},\!</math> or, especially if operating in an algebraic context, by a simple <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}.\!</math> Also when working in an algebraic mode, the plus sign <math>{}^{\backprime\backprime} + {}^{\prime\prime}\!</math> may be used for [[exclusive disjunction]]. For example, we have the following paraphrases of algebraic expressions by bracket expressions:<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
|<br />
<math>\begin{matrix}<br />
x + y ~=~ \texttt{(} x, y \texttt{)}<br />
\\[6pt]<br />
x + y + z ~=~ \texttt{((} x, y \texttt{)}, z \texttt{)} ~=~ \texttt{(} x, \texttt{(} y, z \texttt{))}<br />
\end{matrix}</math><br />
|}<br />
<br />
It is important to note that the last expressions are not equivalent to the triple bracket <math>\texttt{(} x, y, z \texttt{)}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 1.} ~~ \text{Syntax and Semantics of a Calculus for Propositional Logic}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Expression}~\!</math><br />
| <math>\text{Interpretation}\!</math><br />
| <math>\text{Other Notations}\!</math><br />
|-<br />
| &nbsp;<br />
| <math>\text{True}\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{False}\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>x\!</math><br />
| <math>x\!</math><br />
| <math>x\!</math><br />
|-<br />
| <math>\texttt{(} x \texttt{)}\!</math><br />
| <math>\text{Not}~ x\!</math><br />
|<br />
<math>\begin{matrix}<br />
x'<br />
\\<br />
\tilde{x}<br />
\\<br />
\lnot x<br />
\end{matrix}\!</math><br />
|-<br />
| <math>x~y~z\!</math><br />
| <math>x ~\text{and}~ y ~\text{and}~ z\!</math><br />
| <math>x \land y \land z\!</math><br />
|-<br />
| <math>\texttt{((} x \texttt{)(} y \texttt{)(} z \texttt{))}\!</math><br />
| <math>x ~\text{or}~ y ~\text{or}~ z\!</math><br />
| <math>x \lor y \lor z\!</math><br />
|-<br />
| <math>\texttt{(} x ~ \texttt{(} y \texttt{))}\!</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{implies}~ y<br />
\\<br />
\mathrm{If}~ x ~\text{then}~ y<br />
\end{matrix}</math><br />
| <math>x \Rightarrow y\!</math><br />
|-<br />
| <math>\texttt{(} x \texttt{,} y \texttt{)}\!</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{not equal to}~ y<br />
\\<br />
x ~\text{exclusive or}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x \ne y<br />
\\<br />
x + y<br />
\end{matrix}</math><br />
|-<br />
| <math>\texttt{((} x \texttt{,} y \texttt{))}\!</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{is equal to}~ y<br />
\\<br />
x ~\text{if and only if}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x = y<br />
\\<br />
x \Leftrightarrow y<br />
\end{matrix}</math><br />
|-<br />
| <math>\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Just one of}<br />
\\<br />
x, y, z<br />
\\<br />
\text{is false}.<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x'y~z~ & \lor<br />
\\<br />
x~y'z~ & \lor<br />
\\<br />
x~y~z' &<br />
\end{matrix}</math><br />
|-<br />
| <math>\texttt{((} x \texttt{),(} y \texttt{),(} z \texttt{))}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Just one of}<br />
\\<br />
x, y, z<br />
\\<br />
\text{is true}.<br />
\\<br />
&<br />
\\<br />
\text{Partition all}<br />
\\<br />
\text{into}~ x, y, z.<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x~y'z' & \lor<br />
\\<br />
x'y~z' & \lor<br />
\\<br />
x'y'z~ &<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,} y \texttt{),} z \texttt{)}<br />
\\<br />
&<br />
\\<br />
\texttt{(} x \texttt{,(} y \texttt{,} z \texttt{))}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Oddly many of}<br />
\\<br />
x, y, z<br />
\\<br />
\text{are true}.<br />
\end{matrix}\!</math><br />
|<br />
<p><math>x + y + z\!</math></p><br />
<br><br />
<p><math>\begin{matrix}<br />
x~y~z~ & \lor<br />
\\<br />
x~y'z' & \lor<br />
\\<br />
x'y~z' & \lor<br />
\\<br />
x'y'z~ &<br />
\end{matrix}\!</math></p><br />
|-<br />
| <math>\texttt{(} w \texttt{,(} x \texttt{),(} y \texttt{),(} z \texttt{))}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Partition}~ w<br />
\\<br />
\text{into}~ x, y, z.<br />
\\<br />
&<br />
\\<br />
\text{Genus}~ w ~\text{comprises}<br />
\\<br />
\text{species}~ x, y, z.<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
w'x'y'z' & \lor<br />
\\<br />
w~x~y'z' & \lor<br />
\\<br />
w~x'y~z' & \lor<br />
\\<br />
w~x'y'z~ &<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
'''Note.''' The usage that one often sees, of a plus sign "<math>+\!</math>" to represent inclusive disjunction, and the reference to this operation as ''boolean addition'', is a misnomer on at least two counts. Boole used the plus sign to represent exclusive disjunction (at any rate, an operation of aggregation restricted in its logical interpretation to cases where the represented sets are disjoint (Boole, 32)), as any mathematician with a sensitivity to the ring and field properties of algebra would do:<br />
<br />
<blockquote><br />
The expression <math>x + y\!</math> seems indeed uninterpretable, unless it be assumed that the things represented by <math>x\!</math> and the things represented by <math>y\!</math> are entirely separate; that they embrace no individuals in common. (Boole, 66).<br />
</blockquote><br />
<br />
It was only later that Peirce and Jevons treated inclusive disjunction as a fundamental operation, but these authors, with a respect for the algebraic properties that were already associated with the plus sign, used a variety of other symbols for inclusive disjunction (Sty, 177, 189). It seems to have been Schröder who later reassigned the plus sign to inclusive disjunction (Sty, 208). Additional information, discussion, and references can be found in (Boole) and (Sty, 177&ndash;263). Aside from these historical points, which never really count against a current practice that has gained a life of its own, this usage does have a further disadvantage of cutting or confounding the lines of communication between algebra and logic. For this reason, it will be avoided here.<br />
<br />
==A Functional Conception of Propositional Calculus==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Out of the dimness opposite equals advance . . . .<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Always substance and increase,<br><br />
Always a knit of identity . . . . always distinction . . . .<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;always a breed of life.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]<br />
|}<br />
<br />
In the general case, we start with a set of logical features <math>\{a_1, \ldots, a_n\}</math> that represent properties of objects or propositions about the world. In concrete examples the features <math>\{a_i\!\}</math> commonly appear as capital letters from an ''alphabet'' like <math>\{A, B, C, \ldots\}</math> or as meaningful words from a linguistic ''vocabulary'' of codes. This language can be drawn from any sources, whether natural, technical, or artificial in character and interpretation. In the application to dynamic systems we tend to use the letters <math>\{x_1, \ldots, x_n\}</math> as our coordinate propositions, and to interpret them as denoting properties of a system's ''state'', that is, as propositions about its location in configuration space. Because I have to consider non-deterministic systems from the outset, I often use the word ''state'' in a loose sense, to denote the position or configuration component of a contemplated state vector, whether or not it ever gets a deterministic completion.<br />
<br />
The set of logical features <math>\{a_1, \ldots, a_n\}</math> provides a basis for generating an <math>n\!</math>-dimensional ''[[universe of discourse]]'' that I denote as <math>[a_1, \ldots, a_n].</math> It is useful to consider each universe of discourse as a unified categorical object that incorporates both the set of points <math>\langle a_1, \ldots, a_n \rangle</math> and the set of propositions <math>f : \langle a_1, \ldots, a_n \rangle \to \mathbb{B}</math> that are implicit with the ordinary picture of a venn diagram on <math>n\!</math> features. Thus, we may regard the universe of discourse <math>[a_1, \ldots, a_n]</math> as an ordered pair having the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}),</math> and we may abbreviate this last type designation as <math>\mathbb{B}^n\ +\!\to \mathbb{B},</math> or even more succinctly as <math>[\mathbb{B}^n].</math> (Used this way, the angle brackets <math>\langle\ldots\rangle</math> are referred to as ''generator brackets''.)<br />
<br />
Table 2 exhibits the scheme of notation I use to formalize the domain of propositional calculus, corresponding to the logical content of truth tables and venn diagrams. Although it overworks the square brackets a bit, I also use either one of the equivalent notations <math>[n]\!</math> or <math>\mathbf{n}</math> to denote the data type of a finite set on <math>n\!</math> elements.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 2.} ~~ \text{Propositional Calculus : Basic Notation}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Symbol}\!</math><br />
| <math>\text{Notation}\!</math><br />
| <math>\text{Description}\!</math><br />
| <math>\text{Type}\!</math><br />
|-<br />
| <math>\mathfrak{A}\!</math><br />
| <math>\{ {}^{\backprime\backprime} a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} a_n {}^{\prime\prime} \}\!</math><br />
| <math>\text{Alphabet}\!</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>\mathcal{A}\!</math><br />
| <math>\{ a_1, \ldots, a_n \}\!</math><br />
| <math>\text{Basis}\!</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>A_i\!</math><br />
| <math>\{ \texttt{(} a_i \texttt{)}, a_i \}\!</math><br />
| <math>\text{Dimension}~ i\!</math><br />
| <math>\mathbb{B}\!</math><br />
|-<br />
| <math>A\!</math><br />
|<br />
<math>\begin{matrix}<br />
\langle \mathcal{A} \rangle<br />
\\[2pt]<br />
\langle a_1, \ldots, a_n \rangle<br />
\\[2pt]<br />
\{ (a_1, \ldots, a_n) \}<br />
\\[2pt]<br />
A_1 \times \ldots \times A_n<br />
\\[2pt]<br />
\textstyle \prod_{i=1}^n A_i<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Set of cells},<br />
\\[2pt]<br />
\text{coordinate tuples},<br />
\\[2pt]<br />
\text{points, or vectors}<br />
\\[2pt]<br />
\text{in the universe}<br />
\\[2pt]<br />
\text{of discourse}<br />
\end{matrix}</math><br />
| <math>\mathbb{B}^n\!</math><br />
|-<br />
| <math>A^*\!</math><br />
| <math>(\mathrm{hom} : A \to \mathbb{B})\!</math><br />
| <math>\text{Linear functions}\!</math><br />
| <math>(\mathbb{B}^n)^* \cong \mathbb{B}^n\!</math><br />
|-<br />
| <math>A^\uparrow\!</math><br />
| <math>(A \to \mathbb{B})\!</math><br />
| <math>\text{Boolean functions}\!</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}\!</math><br />
|-<br />
| <math>A^\bullet\!</math><br />
|<br />
<math>\begin{matrix}<br />
[\mathcal{A}]<br />
\\[2pt]<br />
(A, A^\uparrow)<br />
\\[2pt]<br />
(A ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
(A, (A \to \mathbb{B}))<br />
\\[2pt]<br />
[a_1, \ldots, a_n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Universe of discourse}<br />
\\[2pt]<br />
\text{based on the features}<br />
\\[2pt]<br />
\{ a_1, \ldots, a_n \}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))<br />
\\[2pt]<br />
(\mathbb{B}^n ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
[\mathbb{B}^n]<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
===Qualitative Logic and Quantitative Analogy===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
''Logical'', however, is used in a third sense, which is at once more vital and more practical; to denote, namely, the systematic care, negative and positive, taken to safeguard reflection so that it may yield the best results under the given conditions.<br />
| width="4%" | &nbsp;<br />
|- <br />
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 56]<br />
|}<br />
<br />
These concepts and notations may now be explained in greater detail. In order to begin as simply as possible, let us distinguish two levels of analysis and set out initially on the easier path. On the first level of analysis we take spaces like <math>\mathbb{B},</math> <math>\mathbb{B}^n,</math> and <math>(\mathbb{B}^n \to \mathbb{B})</math> at face value and treat them as the primary objects of interest. On the second level of analysis we use these spaces as coordinate charts for talking about points and functions in more fundamental spaces.<br />
<br />
A pair of spaces, of types <math>\mathbb{B}^n</math> and <math>(\mathbb{B}^n \to \mathbb{B}),</math> give typical expression to everything that we commonly associate with the ordinary picture of a venn diagram. The dimension, <math>n,\!</math> counts the number of "circles" or simple closed curves that are inscribed in the universe of discourse, corresponding to its relevant logical features or basic propositions. Elements of type <math>\mathbb{B}^n</math> correspond to what are often called propositional ''interpretations'' in logic, that is, the different assignments of truth values to sentence letters. Relative to a given universe of discourse, these interpretations are visualized as its ''cells'', in other words, the smallest enclosed areas or undivided regions of the venn diagram. The functions <math>f : \mathbb{B}^n \to \mathbb{B}</math> correspond to the different ways of shading the venn diagram to indicate arbitrary propositions, regions, or sets. Regions included under a shading indicate the ''models'', and regions excluded represent the ''non-models'' of a proposition. To recognize and formalize the natural cohesion of these two layers of concepts into a single universe of discourse, we introduce the type notations <math>[\mathbb{B}^n] = \mathbb{B}^n\ +\!\to \mathbb{B}</math> to stand for the pair of types <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})).</math> The resulting "stereotype" serves to frame the universe of discourse as a unified categorical object, and makes it subject to prescribed sets of evaluations and transformations (categorical morphisms or ''arrows'') that affect the universe of discourse as an integrated whole.<br />
<br />
Most of the time we can serve the algebraic, geometric, and logical interests of our study without worrying about their occasional conflicts and incidental divergences. The conventions and definitions already set down will continue to cover most of the algebraic and functional aspects of our discussion, but to handle the logical and qualitative aspects we will need to add a few more. In general, abstract sets may be denoted by gothic, greek, or script capital variants of <math>A, B, C,\!</math> and so on, with their elements being denoted by a corresponding set of subscripted letters in plain lower case, for example, <math>\mathcal{A} = \{a_i\}.</math> Most of the time, a set such as <math>\mathcal{A} = \{a_i\}\!</math> will be employed as the ''alphabet'' of a [[formal language]]. These alphabet letters serve to name the logical features (properties or propositions) that generate a particular universe of discourse. When we want to discuss the particular features of a universe of discourse, beyond the abstract designation of a type like <math>(\mathbb{B}^n\ +\!\to \mathbb{B}),</math> then we may use the following notations. If <math>\mathcal{A} = \{a_1, \ldots, a_n\}</math> is an alphabet of logical features, then <math>A = \langle \mathcal{A} \rangle = \langle a_1, \ldots, a_n \rangle</math> is the set of interpretations, <math>A^\uparrow = (A \to \mathbb{B})</math> is the set of propositions, and <math>A^\bullet = [\mathcal{A}] = [a_1, \ldots, a_n]</math> is the combination of these interpretations and propositions into the universe of discourse that is based on the features <math>\{a_1, \ldots, a_n\}.</math><br />
<br />
As always, especially in concrete examples, these rules may be dropped whenever necessary, reverting to a free assortment of feature labels. However, when we need to talk about the logical aspects of a space that is already named as a vector space, it will be necessary to make special provisions. At any rate, these elaborations can be deferred until actually needed.<br />
<br />
===Philosophy of Notation : Formal Terms and Flexible Types===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Where number is irrelevant, regimented mathematical technique has hitherto tended to be lacking. Thus it is that the progress of natural science has depended so largely upon the discernment of measurable quantity of one sort or another.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]<br />
|}<br />
<br />
For much of our discussion propositions and boolean functions are treated as the same formal objects, or as different interpretations of the same formal calculus. This rule of interpretation has exceptions, though. There is a distinctively logical interest in the use of propositional calculus that is not exhausted by its functional interpretation. It is part of our task in this study to deal with these uniquely logical characteristics as they present themselves both in our subject matter and in our formal calculus. Just to provide a hint of what's at stake: In logic, as opposed to the more imaginative realms of mathematics, we consider it a good thing to always know what we are talking about. Where mathematics encourages tolerance for uninterpreted symbols as intermediate terms, logic exerts a keener effort to interpret directly each oblique carrier of meaning, no matter how slight, and to unfold the complicities of every indirection in the flow of information. Translated into functional terms, this means that we want to maintain a continual, immediate, and persistent sense of both the converse relation <math>f^{-1} \subseteq \mathbb{B} \times \mathbb{B}^n,</math> or what is the same thing, <math>f^{-1} : \mathbb{B} \to \mathcal{P}(\mathbb{B}^n),</math> and the ''fibers'' or inverse images <math>f^{-1}(0)\!</math> and <math>f^{-1}(1),\!</math> associated with each boolean function <math>f : \mathbb{B}^n \to \mathbb{B}</math> that we use. In practical terms, the desired implementation of a propositional interpreter should incorporate our intuitive recognition that the induced partition of the functional domain into level sets <math>f^{-1}(b),\!</math> for <math>b \in \mathbb{B},</math> is part and parcel of understanding the denotative uses of each propositional function <math>f.\!</math><br />
<br />
===Special Classes of Propositions===<br />
<br />
It is important to remember that the coordinate propositions <math>\{a_i\},\!</math> besides being projection maps <math>a_i : \mathbb{B}^n \to \mathbb{B},</math> are propositions on an equal footing with all others, even though employed as a basis in a particular moment. This set of <math>n\!</math> propositions may sometimes be referred to as the ''basic propositions'', the ''coordinate propositions'', or the ''simple propositions'' that found a universe of discourse. Either one of the equivalent notations, <math>\{a_i : \mathbb{B}^n \to \mathbb{B}\}</math> or <math>(\mathbb{B}^n \xrightarrow{i} \mathbb{B}),</math> may be used to indicate the adoption of the propositions <math>a_i\!</math> as a basis for describing a universe of discourse.<br />
<br />
Among the <math>2^{2^n}</math> propositions in <math>[a_1, \ldots, a_n]</math> are several families of <math>2^n\!</math> propositions each that take on special forms with respect to the basis <math>\{ a_1, \ldots, a_n \}.</math> Three of these families are especially prominent in the present context, the ''linear'', the ''positive'', and the ''singular'' propositions. Each family is naturally parameterized by the coordinate <math>n\!</math>-tuples in <math>\mathbb{B}^n</math> and falls into <math>n + 1\!</math> ranks, with a binomial coefficient <math>\tbinom{n}{k}</math> giving the number of propositions that have rank or weight <math>k.\!</math><br />
<br />
<ul><br />
<br />
<li><br />
<p>The ''linear propositions'', <math>\{ \ell : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{\ell} \mathbb{B}),\!</math> may be written as sums:</p><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\sum_{i=1}^n e_i ~=~ e_1 + \ldots + e_n<br />
~\text{where}~<br />
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 0 \end{matrix}\right\}<br />
~\text{for}~ i = 1 ~\text{to}~ n.\!</math><br />
|}<br />
</li><br />
<br />
<li><br />
<p>The ''positive propositions'', <math>\{ p : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{p} \mathbb{B}),\!</math> may be written as products:</p><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\prod_{i=1}^n e_i ~=~ e_1 \cdot \ldots \cdot e_n<br />
~\text{where}~<br />
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 1 \end{matrix}\right\}<br />
~\text{for}~ i = 1 ~\text{to}~ n.\!</math><br />
|}<br />
</li><br />
<br />
<li><br />
<p>The ''singular propositions'', <math>\{ \mathbf{x} : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{s} \mathbb{B}),\!</math> may be written as products:</p><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\prod_{i=1}^n e_i ~=~ e_1 \cdot \ldots \cdot e_n<br />
~\text{where}~<br />
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = \texttt{(} a_i \texttt{)} \end{matrix}\right\}<br />
~\text{for}~ i = 1 ~\text{to}~ n.\!</math><br />
|}<br />
</li><br />
<br />
</ul><br />
<br />
In each case the rank <math>k\!</math> ranges from <math>0\!</math> to <math>n\!</math> and counts the number of positive appearances of the coordinate propositions <math>a_1, \ldots, a_n\!</math> in the resulting expression. For example, for <math>{n = 3},\!</math> the linear proposition of rank <math>0\!</math> is <math>0,\!</math> the positive proposition of rank <math>0\!</math> is <math>1,\!</math> and the singular proposition of rank <math>0\!</math> is <math>\texttt{(} a_1 \texttt{)(} a_2 \texttt{)(} a_3\texttt{)}.\!</math><br />
<br />
The basic propositions <math>a_i : \mathbb{B}^n \to \mathbb{B}</math> are both linear and positive. So these two kinds of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions.<br />
<br />
Linear propositions and positive propositions are generated by taking boolean sums and products, respectively, over selected subsets of basic propositions, so both families of propositions are parameterized by the powerset <math>\mathcal{P}(\mathcal{I}),</math> that is, the set of all subsets <math>J\!</math> of the basic index set <math>\mathcal{I} = \{1, \ldots, n\}.\!</math><br />
<br />
Let us define <math>\mathcal{A}_J</math> as the subset of <math>\mathcal{A}</math> that is given by <math>\{a_i : i \in J\}.\!</math> Then we may comprehend the action of the linear and the positive propositions in the following terms:<br />
<br />
* The linear proposition <math>\ell_J : \mathbb{B}^n \to \mathbb{B}</math> evaluates each cell <math>\mathbf{x}</math> of <math>\mathbb{B}^n</math> by looking at the coefficients of <math>\mathbf{x}</math> with respect to the features that <math>\ell_J</math> "likes", namely those in <math>\mathcal{A}_J,</math> and then adds them up in <math>\mathbb{B}.</math> Thus, <math>\ell_J(\mathbf{x})</math> computes the parity of the number of features that <math>\mathbf{x}</math> has in <math>\mathcal{A}_J,</math> yielding one for odd and zero for even. Expressed in this idiom, <math>\ell_J(\mathbf{x}) = 1</math> says that <math>\mathbf{x}</math> seems ''odd'' (or ''oddly true'') to <math>\mathcal{A}_J,</math> whereas <math>\ell_J(\mathbf{x}) = 0</math> says that <math>\mathbf{x}</math> seems ''even'' (or ''evenly true'') to <math>\mathcal{A}_J,</math> so long as we recall that ''zero times'' is evenly often, too.<br />
<br />
* The positive proposition <math>p_J : \mathbb{B}^n \to \mathbb{B}</math> evaluates each cell <math>\mathbf{x}</math> of <math>\mathbb{B}^n</math> by looking at the coefficients of <math>\mathbf{x}</math> with regard to the features that <math>p_J\!</math> "likes", namely those in <math>\mathcal{A}_J,</math> and then takes their product in <math>\mathbb{B}.</math> Thus, <math>p_J(\mathbf{x})</math> assesses the unanimity of the multitude of features that <math>\mathbf{x}</math> has in <math>\mathcal{A}_J,</math> yielding one for all and aught for else. In these consensual or contractual terms, <math>p_J(\mathbf{x}) = 1</math> means that <math>\mathbf{x}</math> is ''AOK'' or congruent with all of the conditions of <math>\mathcal{A}_J,</math> while <math>p_J(\mathbf{x}) = 0</math> means that <math>\mathbf{x}</math> defaults or dissents from some condition of <math>\mathcal{A}_J.</math><br />
<br />
===Basis Relativity and Type Ambiguity===<br />
<br />
Finally, two things are important to keep in mind with regard to the simplicity, linearity, positivity, and singularity of propositions.<br />
<br />
First, all of these properties are relative to a particular basis. For example, a singular proposition with respect to a basis <math>\mathcal{A}</math> will not remain singular if <math>\mathcal{A}</math> is extended by a number of new and independent features. Even if we stick to the original set of pairwise options <math>\{a_i\} \cup \{ \texttt{(} a_i \texttt{)} \}\!</math> to select a new basis, the sets of linear and positive propositions are determined by the choice of simple propositions, and this determination is tantamount to the conventional choice of a cell as origin.<br />
<br />
Second, the singular propositions <math>\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B},</math> picking out as they do a single cell or a coordinate tuple <math>\mathbf{x}</math> of <math>\mathbb{B}^n,</math> become the carriers or the vehicles of a certain type-ambiguity that vacillates between the dual forms <math>\mathbb{B}^n</math> and <math>(\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B})</math> and infects the whole hierarchy of types built on them. In other words, the terms that signify the interpretations <math>\mathbf{x} : \mathbb{B}^n</math> and the singular propositions <math>\mathbf{x} : \mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B}</math> are fully equivalent in information, and this means that every token of the type <math>\mathbb{B}^n</math> can be reinterpreted as an appearance of the subtype <math>\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B}.</math> And vice versa, the two types can be exchanged with each other everywhere that they turn up. In practical terms, this allows the use of singular propositions as a way of denoting points, forming an alternative to coordinate tuples.<br />
<br />
For example, relative to the universe of discourse <math>[a_1, a_2, a_3]\!</math> the singular proposition <math>a_1 a_2 a_3 : \mathbb{B}^3 \xrightarrow{s} \mathbb{B}</math> could be explicitly retyped as <math>a_1 a_2 a_3 : \mathbb{B}^3</math> to indicate the point <math>(1, 1, 1)\!</math> but in most cases the proper interpretation could be gathered from context. Both notations remain dependent on a particular basis, but the code that is generated under the singular option has the advantage in its self-commenting features, in other words, it constantly reminds us of its basis in the process of denoting points. When the time comes to put a multiplicity of different bases into play, and to search for objects and properties that remain invariant under the transformations between them, this infinitesimal potential advantage may well evolve into an overwhelming practical necessity.<br />
<br />
===The Analogy Between Real and Boolean Types===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Measurement consists in correlating our subject matter with the series of real numbers; and such correlations are desirable because, once they are set up, all the well-worked theory of numerical mathematics lies ready at hand as a tool for our further reasoning.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]<br />
|}<br />
<br />
There are two further reasons why it useful to spend time on a careful treatment of types, and they both have to do with our being able to take full computational advantage of certain dimensions of flexibility in the types that apply to terms. First, the domains of differential geometry and logic programming are connected by analogies between real and boolean types of the same pattern. Second, the types involved in these patterns have important isomorphisms connecting them that apply on both the real and the boolean sides of the picture.<br />
<br />
Amazingly enough, these isomorphisms are themselves schematized by the axioms and theorems of propositional logic. This fact is known as the ''propositions as types'' analogy or the Curry&ndash;Howard isomorphism [How]. In another formulation it says that terms are to types as proofs are to propositions. See [LaS, 42&ndash;46] and [SeH] for a good discussion and further references. To anticipate the bearing of these issues on our immediate topic, Table&nbsp;3 sketches a partial overview of the Real to Boolean analogy that may serve to illustrate the paradigm in question.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 3.} ~~ \text{Analogy Between Real and Boolean Types}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Real Domain} ~ \mathbb{R}\!</math><br />
| <math>\longleftrightarrow\!</math><br />
| <math>\text{Boolean Domain} ~ \mathbb{B}\!</math><br />
|-<br />
| <math>\mathbb{R}^n\!</math><br />
| <math>\text{Basic Space}\!</math><br />
| <math>\mathbb{B}^n\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}\!</math><br />
| <math>\text{Function Space}\!</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R}\!</math><br />
| <math>\text{Tangent Vector}\!</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to ((\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R})\!</math><br />
| <math>\text{Vector Field}\!</math><br />
| <math>\mathbb{B}^n \to ((\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B})\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \times (\mathbb{R}^n \to \mathbb{R})) \to \mathbb{R}\!</math><br />
| '''<font size="4">"</font>'''<br />
| <math>(\mathbb{B}^n \times (\mathbb{B}^n \to \mathbb{B})) \to \mathbb{B}\!</math><br />
|-<br />
| <math>((\mathbb{R}^n \to \mathbb{R}) \times \mathbb{R}^n) \to \mathbb{R}\!</math><br />
| '''<font size="4">"</font>'''<br />
| <math>((\mathbb{B}^n \to \mathbb{B}) \times \mathbb{B}^n) \to \mathbb{B}\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^n \to \mathbb{R})~\!</math><br />
| <math>\text{Derivation}\!</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^n \to \mathbb{B})\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}^m\!</math><br />
| <math>\begin{matrix}\text{Basic}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}^m\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^m \to \mathbb{R})\!</math><br />
| <math>\begin{matrix}\text{Function}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^m \to \mathbb{B})\!</math><br />
|}<br />
<br />
<br><br />
<br />
The Table exhibits a sample of likely parallels between the real and boolean domains. The central column gives a selection of terminology that is borrowed from differential geometry and extended in its meaning to the logical side of the Table. These are the varieties of spaces that come up when we turn to analyzing the dynamics of processes that pursue their courses through the states of an arbitrary space <math>X.\!</math> Moreover, when it becomes necessary to approach situations of overwhelming dynamic complexity in a succession of qualitative reaches, then the methods of logic that are afforded by the boolean domains, with their declarative means of synthesis and deductive modes of analysis, supply a natural battery of tools for the task.<br />
<br />
It is usually expedient to take these spaces two at a time, in dual pairs of the form <math>X\!</math> and <math>(X \to \mathbb{K}).</math> In general, one creates pairs of type schemas by replacing any space <math>X\!</math> with its dual <math>(X \to \mathbb{K}),</math> for example, pairing the type <math>X \to Y</math> with the type <math>(X \to \mathbb{K}) \to (Y \to \mathbb{K}),</math> and <math>X \times Y</math> with <math>(X \to \mathbb{K}) \times (Y \to \mathbb{K}).</math> The word ''dual'' is used here in its broader sense to mean all of the functionals, not just the linear ones. Given any function <math>f : X \to \mathbb{K},</math> the ''converse'' or inverse relation corresponding to <math>f\!</math> is denoted <math>f^{-1},\!</math> and the subsets of <math>X\!</math> that are defined by <math>f^{-1}(k),\!</math> taken over <math>k\!</math> in <math>\mathbb{K},</math> are called the ''fibers'' or the ''level sets'' of the function <math>f.\!</math><br />
<br />
===Theory of Control and Control of Theory===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
You will hardly know who I am or what I mean,<br><br />
But I shall be good health to you nevertheless,<br><br />
And filter and fibre your blood.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]<br />
|}<br />
<br />
In the boolean context a function <math>f : X \to \mathbb{B}</math> is tantamount to a ''proposition'' about elements of <math>X,\!</math> and the elements of <math>X\!</math> constitute the ''interpretations'' of that proposition. The fiber <math>f^{-1}(1)\!</math> comprises the set of ''models'' of <math>f,\!</math> or examples of elements in <math>X\!</math> satisfying the proposition <math>f.\!</math> The fiber <math>f^{-1}(0)\!</math> collects the complementary set of ''anti-models'', or the exceptions to the proposition <math>f\!</math> that exist in <math>X.\!</math> Of course, the space of functions <math>(X \to \mathbb{B})\!</math> is isomorphic to the set of all subsets of <math>X,\!</math> called the ''power set'' of <math>X,\!</math> and often denoted <math>\mathcal{P}(X)</math> or <math>2^X.\!</math><br />
<br />
The operation of replacing <math>X\!</math> by <math>(X \to \mathbb{B})\!</math> in a type schema corresponds to a certain shift of attitude towards the space <math>X,\!</math> in which one passes from a focus on the ostensibly individual elements of <math>X\!</math> to a concern with the states of information and uncertainty that one possesses about objects and situations in <math>X.\!</math> The conceptual obstacles in the path of this transition can be smoothed over by using singular functions <math>(\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B})</math> as stepping stones. First of all, it's an easy step from an element <math>\mathbf{x}</math> of type <math>\mathbb{B}^n</math> to the equivalent information of a singular proposition <math>\mathbf{x} : X \xrightarrow{s} \mathbb{B}, </math> and then only a small jump of generalization remains to reach the type of an arbitrary proposition <math>f : X \to \mathbb{B},</math> perhaps understood to indicate a relaxed constraint on the singularity of points or a neighborhood circumscribing the original <math>\mathbf{x}.</math> This is frequently a useful transformation, communicating between the ''objective'' and the ''intentional'' perspectives, in spite perhaps of the open objection that this distinction is transient in the mean time and ultimately superficial.<br />
<br />
It is hoped that this measure of flexibility, allowing us to stretch a point into a proposition, can be useful in the examination of inquiry driven systems, where the differences between empirical, intentional, and theoretical propositions constitute the discrepancies and the distributions that drive experimental activity. I can give this model of inquiry a cybernetic cast by realizing that theory change and theory evolution, as well as the choice and the evaluation of experiments, are actions that are taken by a system or its agent in response to the differences that are detected between observational contents and theoretical coverage.<br />
<br />
All of the above notwithstanding, there are several points that distinguish these two tasks, namely, the ''theory of control'' and the ''control of theory'', features that are often obscured by too much precipitation in the quickness with which we understand their similarities. In the control of uncertainty through inquiry, some of the actuators that we need to be concerned with are axiom changers and theory modifiers, operators with the power to compile and to revise the theories that generate expectations and predictions, effectors that form and edit our grammars for the languages of observational data, and agencies that rework the proposed model to fit the actual sequences of events and the realized relationships of values that are observed in the environment. Moreover, when steps must be taken to carry out an experimental action, there must be something about the particular shape of our uncertainty that guides us in choosing what directions to explore, and this impression is more than likely influenced by previous accumulations of experience. Thus it must be anticipated that much of what goes into scientific progress, or any sustainable effort toward a goal of knowledge, is necessarily predicated on long term observation and modal expectations, not only on the more local or short term prediction and correction.<br />
<br />
===Propositions as Types and Higher Order Types===<br />
<br />
The types collected in Table&nbsp;3 (repeated below) serve to illustrate the themes of ''higher order propositional expressions'' and the ''propositions as types'' (PAT) analogy.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 3.} ~~ \text{Analogy Between Real and Boolean Types}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Real Domain} ~ \mathbb{R}\!</math><br />
| <math>\longleftrightarrow\!</math><br />
| <math>\text{Boolean Domain} ~ \mathbb{B}\!</math><br />
|-<br />
| <math>\mathbb{R}^n\!</math><br />
| <math>\text{Basic Space}\!</math><br />
| <math>\mathbb{B}^n\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}\!</math><br />
| <math>\text{Function Space}\!</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R}\!</math><br />
| <math>\text{Tangent Vector}\!</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to ((\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R})\!</math><br />
| <math>\text{Vector Field}\!</math><br />
| <math>\mathbb{B}^n \to ((\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B})\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \times (\mathbb{R}^n \to \mathbb{R})) \to \mathbb{R}\!</math><br />
| '''<font size="4">"</font>'''<br />
| <math>(\mathbb{B}^n \times (\mathbb{B}^n \to \mathbb{B})) \to \mathbb{B}\!</math><br />
|-<br />
| <math>((\mathbb{R}^n \to \mathbb{R}) \times \mathbb{R}^n) \to \mathbb{R}\!</math><br />
| '''<font size="4">"</font>'''<br />
| <math>((\mathbb{B}^n \to \mathbb{B}) \times \mathbb{B}^n) \to \mathbb{B}\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^n \to \mathbb{R})~\!</math><br />
| <math>\text{Derivation}\!</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^n \to \mathbb{B})\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}^m\!</math><br />
| <math>\begin{matrix}\text{Basic}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}^m\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^m \to \mathbb{R})\!</math><br />
| <math>\begin{matrix}\text{Function}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^m \to \mathbb{B})\!</math><br />
|}<br />
<br />
<br><br />
<br />
First, observe that the type of a ''tangent vector at a point'', also known as a ''directional derivative'' at that point, has the form <math>(\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K},</math> where <math>\mathbb{K}</math> is the chosen ground field, in the present case either <math>\mathbb{R}</math> or <math>\mathbb{B}.</math> At a point in a space of type <math>\mathbb{K}^n,</math> a directional derivative operator <math>\vartheta\!</math> takes a function on that space, an <math>f\!</math> of type <math>(\mathbb{K}^n \to \mathbb{K}),</math> and maps it to a ground field value of type <math>\mathbb{K}.</math> This value is known as the ''derivative'' of <math>f\!</math> in the direction <math>\vartheta\!</math> [Che46, 76&ndash;77]. In the boolean case <math>\vartheta : (\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math> has the form of a proposition about propositions, in other words, a proposition of the next higher type.<br />
<br />
Next, by way of illustrating the propositions as types idea, consider a proposition of the form <math>X \Rightarrow (Y \Rightarrow Z).</math> One knows from propositional calculus that this is logically equivalent to a proposition of the form <math>(X \land Y) \Rightarrow Z.</math> But this equivalence should remind us of the functional isomorphism that exists between a construction of the type <math>X \to (Y \to Z)</math> and a construction of the type <math>(X \times Y) \to Z.</math> The propositions as types analogy permits us to take a functional type like this and, under the right conditions, replace the functional arrows &ldquo;<math>\to~\!</math>&rdquo; and products &ldquo;<math>\times\!</math>&rdquo; with the respective logical arrows &ldquo;<math>\Rightarrow\!</math>&rdquo; and products &ldquo;<math>\land\!</math>&rdquo;. Accordingly, viewing the result as a proposition, we can employ axioms and theorems of propositional calculus to suggest appropriate isomorphisms among the categorical and functional constructions.<br />
<br />
Finally, examine the middle four rows of Table&nbsp;3. These display a series of isomorphic types that stretch from the categories that are labeled ''Vector Field'' to those that are labeled ''Derivation''. A ''vector field'', also known as an ''infinitesimal transformation'', associates a tangent vector at a point with each point of a space. In symbols, a vector field is a function of the form <math>\textstyle \xi : X \to \bigcup_{x \in X} \xi_x\!</math> that assigns to each point <math>x\!</math> of the space <math>X\!</math> a tangent vector to <math>X\!</math> at that point, namely, the tangent vector <math>\xi_x\!</math> [Che46, 82&ndash;83]. If <math>X\!</math> is of the type <math>\mathbb{K}^n,</math> then <math>\xi\!</math> is of the type <math>\mathbb{K}^n \to ((\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K}).</math> This has the pattern <math>X \to (Y \to Z),</math> with <math>X = \mathbb{K}^n,</math> <math>Y = (\mathbb{K}^n \to \mathbb{K}),</math> and <math>Z = \mathbb{K}.</math><br />
<br />
Applying the propositions as types analogy, one can follow this pattern through a series of metamorphoses from the type of a vector field to the type of a derivation, as traced out in Table&nbsp;4. Observe how the function <math>f : X \to \mathbb{K},</math> associated with the place of <math>Y\!</math> in the pattern, moves through its paces from the second to the first position. In this way, the vector field <math>\xi,\!</math> initially viewed as attaching each tangent vector <math>\xi_x\!</math> to the site <math>x\!</math> where it acts in <math>X,\!</math> now comes to be seen as acting on each scalar potential <math>f : X \to \mathbb{K}</math> like a generalized species of differentiation, producing another function <math>\xi f : X \to \mathbb{K}</math> of the same type.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 4.} ~~ \text{An Equivalence Based on the Propositions as Types Analogy}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Pattern}\!</math><br />
| <math>\text{Construct}\!</math><br />
| <math>\text{Instance}\!</math><br />
|-<br />
| <math>X \to (Y \to Z)\!</math><br />
| <math>\text{Vector Field}\!</math><br />
| <math>\mathbb{K}^n \to ((\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K})\!</math><br />
|-<br />
| <math>(X \times Y) \to Z\!</math><br />
| <math>\Uparrow\!</math><br />
| <math>(\mathbb{K}^n \times (\mathbb{K}^n \to \mathbb{K})) \to \mathbb{K}\!</math><br />
|-<br />
| <math>(Y \times X) \to Z\!</math><br />
| <math>\Downarrow\!</math><br />
| <math>((\mathbb{K}^n \to \mathbb{K}) \times \mathbb{K}^n) \to \mathbb{K}\!</math><br />
|-<br />
| <math>Y \to (X \to Z)\!</math><br />
| <math>\text{Derivation}\!</math><br />
| <math>(\mathbb{K}^n \to \mathbb{K}) \to (\mathbb{K}^n \to \mathbb{K})\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Reality at the Threshold of Logic===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
But no science can rest entirely on measurement, and many scientific investigations are quite out of reach of that device. To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]<br />
|}<br />
<br />
Table 5 accumulates an array of notation that I hope will not be too distracting. Some of it is rarely needed, but has been filled in for the sake of completeness. Its purpose is simple, to give literal expression to the visual intuitions that come with venn diagrams, and to help build a bridge between our qualitative and quantitative outlooks on dynamic systems.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 5.} ~~ \text{A Bridge Over Troubled Waters}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| align="center" | <math>\text{Linear Space}\!</math><br />
| align="center" | <math>\text{Liminal Space}\!</math><br />
| align="center" | <math>\text{Logical Space}\!</math><br />
|-<br />
| <math>\begin{matrix}\mathcal{X} & = & \{ x_1, \ldots, x_n \}\end{matrix}</math><br />
| <math>\begin{matrix}\underline{\mathcal{X}} & = & \{ \underline{x}_1, \ldots, \underline{x}_n \}\end{matrix}</math><br />
| <math>\begin{matrix}\mathcal{A} & = & \{ a_1, \ldots, a_n \}\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X_i & = & \langle x_i \rangle<br />
\\<br />
& \cong & \mathbb{K}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}_i & = & \{ \texttt{(} \underline{x}_i \texttt{)}, \underline{x}_i \}<br />
\\<br />
& \cong & \mathbb{B}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A_i & = & \{ \texttt{(} a_i \texttt{)}, a_i \}<br />
\\<br />
& \cong & \mathbb{B}<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X<br />
\\<br />
= & \langle \mathcal{X} \rangle<br />
\\<br />
= & \langle x_1, \ldots, x_n \rangle<br />
\\<br />
= & X_1 \times \ldots \times X_n<br />
\\<br />
= & \prod_{i=1}^n X_i<br />
\\<br />
\cong & \mathbb{K}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}<br />
\\<br />
= & \langle \underline{\mathcal{X}} \rangle<br />
\\<br />
= & \langle \underline{x}_1, \ldots, \underline{x}_n \rangle<br />
\\<br />
= & \underline{X}_1 \times \ldots \times \underline{X}_n<br />
\\<br />
= & \prod_{i=1}^n \underline{X}_i<br />
\\<br />
\cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A<br />
\\<br />
= & \langle \mathcal{A} \rangle<br />
\\<br />
= & \langle a_1, \ldots, a_n \rangle<br />
\\<br />
= & A_1 \times \ldots \times A_n<br />
\\<br />
= & \prod_{i=1}^n A_i<br />
\\<br />
\cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X^* & = & (\ell : X \to \mathbb{K})<br />
\\<br />
& \cong & \mathbb{K}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}^* & = & (\ell : \underline{X} \to \mathbb{B})<br />
\\<br />
& \cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A^* & = & (\ell : A \to \mathbb{B})<br />
\\<br />
& \cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X^\uparrow & = & (X \to \mathbb{K})<br />
\\<br />
& \cong & (\mathbb{K}^n \to \mathbb{K})<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}^\uparrow & = & (\underline{X} \to \mathbb{B})<br />
\\<br />
& \cong & (\mathbb{B}^n \to \mathbb{B})<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A^\uparrow & = & (A \to \mathbb{B})<br />
\\<br />
& \cong & (\mathbb{B}^n \to \mathbb{B})<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X^\bullet<br />
\\<br />
= & [\mathcal{X}]<br />
\\<br />
= & [x_1, \ldots, x_n]<br />
\\<br />
= & (X, X^\uparrow)<br />
\\<br />
= & (X ~+\!\to \mathbb{K})<br />
\\<br />
= & (X, (X \to \mathbb{K}))<br />
\\<br />
\cong & (\mathbb{K}^n, (\mathbb{K}^n \to \mathbb{K}))<br />
\\<br />
= & (\mathbb{K}^n ~+\!\to \mathbb{K})<br />
\\<br />
= & [\mathbb{K}^n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}^\bullet<br />
\\<br />
= & [\underline{\mathcal{X}}]<br />
\\<br />
= & [\underline{x}_1, \ldots, \underline{x}_n]<br />
\\<br />
= & (\underline{X}, \underline{X}^\uparrow)<br />
\\<br />
= & (\underline{X} ~+\!\to \mathbb{B})<br />
\\<br />
= & (\underline{X}, (\underline{X} \to \mathbb{B}))<br />
\\<br />
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))<br />
\\<br />
= & (\mathbb{B}^n ~+\!\to \mathbb{B})<br />
\\<br />
= & [\mathbb{B}^n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A^\bullet<br />
\\<br />
= & [\mathcal{A}]<br />
\\<br />
= & [a_1, \ldots, a_n]<br />
\\<br />
= & (A, A^\uparrow)<br />
\\<br />
= & (A ~+\!\to \mathbb{B})<br />
\\<br />
= & (A, (A \to \mathbb{B}))<br />
\\<br />
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))<br />
\\<br />
= & (\mathbb{B}^n ~+\!\to \mathbb{B})<br />
\\<br />
= & [\mathbb{B}^n]<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
The left side of the Table collects mostly standard notation for an <math>n\!</math>-dimensional vector space over a field <math>\mathbb{K}.</math> The right side of the table repeats the first elements of a notation that I sketched above, to be used in further developments of propositional calculus. (I plan to use this notation in the logical analysis of neural network systems.) The middle column of the table is designed as a transitional step from the case of an arbitrary field <math>\mathbb{K},</math> with a special interest in the continuous line <math>\mathbb{R},</math> to the qualitative and discrete situations that are instanced and typified by <math>\mathbb{B}.</math><br />
<br />
I now proceed to explain these concepts in more detail. The most important ideas developed in Table&nbsp;5 are these:<br />
<br />
* The idea of a universe of discourse, which includes both a space of ''points'' and a space of ''maps'' on those points.<br />
<br />
* The idea of passing from a more complex universe to a simpler universe by a process of ''thresholding'' each dimension of variation down to a single bit of information.<br />
<br />
For the sake of concreteness, let us suppose that we start with a continuous <math>n\!</math>-dimensional vector space like <math>X = \langle x_1, \ldots, x_n \rangle \cong \mathbb{R}^n.</math> The coordinate system <math>\mathcal{X} = \{x_i\}</math> is a set of maps <math>x_i : \mathbb{R}^n \to \mathbb{R},</math> also known as the ''coordinate projections''. Given a "dataset" of points <math>\mathbf{x}</math> in <math>\mathbb{R}^n,</math> there are numerous ways of sensibly reducing the data down to one bit for each dimension. One strategy that is general enough for our present purposes is as follows. For each <math>i\!</math> we choose an <math>n\!</math>-ary relation <math>L_i\!</math> on <math>\mathbb{R}^n,</math> that is, a subset of <math>\mathbb{R}^n,</math> and then we define the <math>i^\mathrm{th}\!</math> threshold map, or ''limen'' <math>\underline{x}_i</math> as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\underline{x}_i : \mathbb{R}^n \to \mathbb{B}\ \text{such that:}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
\underline{x}_i(\mathbf{x}) = 1 & \text{if} & \mathbf{x} \in L_i,<br />
\\[4pt]<br />
\underline{x}_i(\mathbf{x}) = 0 & \text{if} & \mathbf{x} \not\in L_i.<br />
\end{matrix}</math><br />
|}<br />
<br />
In other notations that are sometimes used, the operator <math>\chi (\ldots)</math> or the corner brackets <math>\lceil\ldots\rceil</math> can be used to denote a ''characteristic function'', that is, a mapping from statements to their truth values in <math>\mathbb{B}.</math> Finally, it is not uncommon to use the name of the relation itself as a predicate that maps <math>n\!</math>-tuples into truth values. Thus we have the following notational variants of the above definition:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
\underline{x}_i (\mathbf{x}) & = & \chi (\mathbf{x} \in L_i) & = & \lceil \mathbf{x} \in L_i \rceil & = & L_i (\mathbf{x}).<br />
\end{matrix}</math><br />
|}<br />
<br />
Notice that, as defined here, there need be no actual relation between the <math>n\!</math>-dimensional subsets <math>\{L_i\}\!</math> and the coordinate axes corresponding to <math>\{x_i\},\!</math> aside from the circumstance that the two sets have the same cardinality. In concrete cases, though, one usually has some reason for associating these "volumes" with these "lines", for instance, <math>L_i\!</math> is bounded by some hyperplane that intersects the <math>i^\text{th}\!</math> axis at a unique threshold value <math>r_i \in \mathbb{R}.\!</math> Often, the hyperplane is chosen normal to the axis. In recognition of this motive, let us make the following convention. When the set <math>L_i\!</math> has points on the <math>i^\text{th}\!</math> axis, that is, points of the form <math>(0, \ldots, 0, r_i, 0, \ldots, 0)</math> where only the <math>x_i\!</math> coordinate is possibly non-zero, we may pick any one of these coordinate values as a parametric index of the relation. In this case we say that the indexing is ''real'', otherwise the indexing is ''imaginary''. For a knowledge based system <math>X,\!</math> this should serve once again to mark the distinction between ''acquaintance'' and ''opinion''.<br />
<br />
States of knowledge about the location of a system or about the distribution of a population of systems in a state space <math>X = \mathbb{R}^n</math> can now be expressed by taking the set <math>\underline{\mathcal{X}} = \{\underline{x}_i\}</math> as a basis of logical features. In picturesque terms, one may think of the underscore and the subscript as combining to form a subtextual spelling for the <math>i^\text{th}\!</math> threshold map. This can help to remind us that the ''threshold operator'' <math>(\underline{~})_i</math> acts on <math>\mathbf{x}</math> by setting up a kind of a &ldquo;hurdle&rdquo; for it. In this interpretation the coordinate proposition <math>\underline{x}_i</math> asserts that the representative point <math>\mathbf{x}</math> resides ''above'' the <math>i^\mathrm{th}\!</math> threshold.<br />
<br />
Primitive assertions of the form <math>\underline{x}_i (\mathbf{x})</math> may then be negated and joined by means of propositional connectives in the usual ways to provide information about the state <math>\mathbf{x}</math> of a contemplated system or a statistical ensemble of systems. Parentheses <math>\texttt{(} \ldots \texttt{)}\!</math> may be used to indicate logical negation. Eventually one discovers the usefulness of the <math>k\!</math>-ary ''just one false'' operators of the form <math>\texttt{(} a_1 \texttt{,} \ldots \texttt{,} a_k \texttt{)},\!</math> as treated in earlier reports. This much tackle generates a space of points (cells, interpretations), <math>\underline{X} \cong \mathbb{B}^n,</math> and a space of functions (regions, propositions), <math>\underline{X}^\uparrow \cong (\mathbb{B}^n \to \mathbb{B}).</math> Together these form a new universe of discourse <math>\underline{X}^\bullet</math> of the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})),</math> which we may abbreviate as <math>\mathbb{B}^n\ +\!\to \mathbb{B}</math> or most succinctly as <math>[\mathbb{B}^n].</math><br />
<br />
The square brackets have been chosen to recall the rectangular frame of a venn diagram. In thinking about a universe of discourse it is a good idea to keep this picture in mind, graphically illustrating the links among the elementary cells <math>\underline{\mathbf{x}},</math> the defining features <math>\underline{x}_i,</math> and the potential shadings <math>f : \underline{X} \to \mathbb{B}</math> all at the same time, not to mention the arbitrariness of the way we choose to inscribe our distinctions in the medium of a continuous space.<br />
<br />
Finally, let <math>X^*\!</math> denote the space of linear functions, <math>(\ell : X \to \mathbb{K}),</math> which has in the finite case the same dimensionality as <math>X,\!</math> and let the same notation be extended across the Table.<br />
<br />
We have just gone through a lot of work, apparently doing nothing more substantial than spinning a complex spell of notational devices through a labyrinth of baffled spaces and baffling maps. The reason for doing this was to bind together and to constitute the intuitive concept of a universe of discourse into a coherent categorical object, the kind of thing, once grasped, that can be turned over in the mind and considered in all its manifold changes and facets. The effort invested in these preliminary measures is intended to pay off later, when we need to consider the state transformations and the time evolution of neural network systems.<br />
<br />
===Tables of Propositional Forms===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope. It provides explicit techniques for manipulating the most basic ingredients of discourse.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7&ndash;8]<br />
|}<br />
<br />
To prepare for the next phase of discussion, Tables&nbsp;6 and 7 collect and summarize all of the propositional forms on one and two variables. These propositional forms are represented over bases of boolean variables as complete sets of boolean-valued functions. Adjacent to their names and specifications are listed what are roughly the simplest expressions in the ''[[cactus language]]'', the particular syntax for propositional calculus that I use in formal and computational contexts. For the sake of orientation, the English paraphrases and the more common notations are listed in the last two columns. As simple and circumscribed as these low-dimensional universes may appear to be, a careful exploration of their differential extensions will involve us in complexities sufficient to demand our attention for some time to come.<br />
<br />
Propositional forms on one variable correspond to boolean functions <math>f : \mathbb{B}^1 \to \mathbb{B}.</math> In Table&nbsp;6 these functions are listed in a variant form of [[truth table]], one in which the axes of the usual arrangement are rotated through a right angle. Each function <math>f_i\!</math> is indexed by the string of values that it takes on the points of the universe <math>X^\bullet = [x] \cong \mathbb{B}^1.</math> The binary index generated in this way is converted to its decimal equivalent and these are used as conventional names for the <math>f_i,\!</math> as shown in the first column of the Table. In their own right the <math>2^1\!</math> points of the universe <math>X^\bullet</math> are coordinated as a space of type <math>\mathbb{B}^1,</math> this in light of the universe <math>X^\bullet</math> being a functional domain where the coordinate projection <math>x\!</math> takes on its values in <math>\mathbb{B}.</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 6.} ~~ \text{Propositional Forms on One Variable}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_0\!</math><br />
| <math>f_{00}\!</math><br />
| <math>0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>f_1\!</math><br />
| <math>f_{01}\!</math><br />
| <math>0~1\!</math><br />
| <math>\texttt{(} x \texttt{)}\!</math><br />
| <math>\text{not}~ x\!</math><br />
| <math>\lnot x\!</math><br />
|-<br />
| <math>f_2\!</math><br />
| <math>f_{10}\!</math><br />
| <math>1~0\!</math><br />
| <math>x\!</math><br />
| <math>x\!</math><br />
| <math>x\!</math><br />
|-<br />
| <math>f_3\!</math><br />
| <math>f_{11}\!</math><br />
| <math>1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
Propositional forms on two variables correspond to boolean functions <math>f : \mathbb{B}^2 \to \mathbb{B}.</math> In Table&nbsp;7 each function <math>f_i\!</math> is indexed by the values that it takes on the points of the universe <math>X^\bullet = [x, y] \cong \mathbb{B}^2.</math> Converting the binary index thus generated to a decimal equivalent, we obtain the functional nicknames that are listed in the first column. The <math>2^2\!</math> points of the universe <math>X^\bullet</math> are coordinated as a space of type <math>\mathbb{B}^2,</math> as indicated under the heading of the Table, where the coordinate projections <math>x\!</math> and <math>y\!</math> run through the various combinations of their values in <math>\mathbb{B}.</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 7-a.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}\!</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}\!</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}\!</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}\!</math><br />
| style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}\!</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}\!</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[4pt]<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\\[4pt]<br />
f_{3}<br />
\\[4pt]<br />
f_{4}<br />
\\[4pt]<br />
f_{5}<br />
\\[4pt]<br />
f_{6}<br />
\\[4pt]<br />
f_{7}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0000}<br />
\\[4pt]<br />
f_{0001}<br />
\\[4pt]<br />
f_{0010}<br />
\\[4pt]<br />
f_{0011}<br />
\\[4pt]<br />
f_{0100}<br />
\\[4pt]<br />
f_{0101}<br />
\\[4pt]<br />
f_{0110}<br />
\\[4pt]<br />
f_{0111}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~0<br />
\\[4pt]<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~0~1~1<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
0~1~0~1<br />
\\[4pt]<br />
0~1~1~0<br />
\\[4pt]<br />
0~1~1~1<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{)} ~ y ~<br />
\\[4pt]<br />
\texttt{(} x \texttt{)}<br />
\\[4pt]<br />
~ x ~ \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{,} ~ y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x ~ y \texttt{)}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{false}<br />
\\[4pt]<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\[4pt]<br />
y ~\text{without}~ x<br />
\\[4pt]<br />
\text{not}~ x<br />
\\[4pt]<br />
x ~\text{without}~ y<br />
\\[4pt]<br />
\text{not}~ y<br />
\\[4pt]<br />
x ~\text{not equal to}~ y<br />
\\[4pt]<br />
\text{not both}~ x ~\text{and}~ y<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
\lnot x \land \lnot y<br />
\\[4pt]<br />
\lnot x \land y<br />
\\[4pt]<br />
\lnot x<br />
\\[4pt]<br />
x \land \lnot y<br />
\\[4pt]<br />
\lnot y<br />
\\[4pt]<br />
x \ne y<br />
\\[4pt]<br />
\lnot x \lor \lnot y<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{8}<br />
\\[4pt]<br />
f_{9}<br />
\\[4pt]<br />
f_{10}<br />
\\[4pt]<br />
f_{11}<br />
\\[4pt]<br />
f_{12}<br />
\\[4pt]<br />
f_{13}<br />
\\[4pt]<br />
f_{14}<br />
\\[4pt]<br />
f_{15}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1000}<br />
\\[4pt]<br />
f_{1001}<br />
\\[4pt]<br />
f_{1010}<br />
\\[4pt]<br />
f_{1011}<br />
\\[4pt]<br />
f_{1100}<br />
\\[4pt]<br />
f_{1101}<br />
\\[4pt]<br />
f_{1110}<br />
\\[4pt]<br />
f_{1111}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
1~0~0~0<br />
\\[4pt]<br />
1~0~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\\[4pt]<br />
1~1~1~1<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x ~ y<br />
\\[4pt]<br />
\texttt{((} x \texttt{,} ~ y \texttt{))}<br />
\\[4pt]<br />
y<br />
\\[4pt]<br />
\texttt{(} x ~ \texttt{(} y \texttt{))}<br />
\\[4pt]<br />
x<br />
\\[4pt]<br />
\texttt{((} x \texttt{)} ~ y \texttt{)}<br />
\\[4pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
\texttt{((~))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x ~\text{and}~ y<br />
\\[4pt]<br />
x ~\text{equal to}~ y<br />
\\[4pt]<br />
y<br />
\\[4pt]<br />
\text{not}~ x ~\text{without}~ y<br />
\\[4pt]<br />
x<br />
\\[4pt]<br />
\text{not}~ y ~\text{without}~ x<br />
\\[4pt]<br />
x ~\text{or}~ y<br />
\\[4pt]<br />
\text{true}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x \land y<br />
\\[4pt]<br />
x = y<br />
\\[4pt]<br />
y<br />
\\[4pt]<br />
x \Rightarrow y<br />
\\[4pt]<br />
x<br />
\\[4pt]<br />
x \Leftarrow y<br />
\\[4pt]<br />
x \lor y<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 7-b.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}</math><br />
| style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>f_{0000}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\\[4pt]<br />
f_{4}<br />
\\[4pt]<br />
f_{8}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0001}<br />
\\[4pt]<br />
f_{0010}<br />
\\[4pt]<br />
f_{0100}<br />
\\[4pt]<br />
f_{1000}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
1~0~0~0<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{)} ~ y ~<br />
\\[4pt]<br />
~ x ~ \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
~ x ~~ y ~<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\[4pt]<br />
y ~\text{without}~ x<br />
\\[4pt]<br />
x ~\text{without}~ y<br />
\\[4pt]<br />
x ~\text{and}~ y<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot x \land \lnot y<br />
\\[4pt]<br />
\lnot x \land y<br />
\\[4pt]<br />
x \land \lnot y<br />
\\[4pt]<br />
x \land y<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{3}<br />
\\[4pt]<br />
f_{12}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0011}<br />
\\[4pt]<br />
f_{1100}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\[4pt]<br />
x<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{not}~ x<br />
\\[4pt]<br />
x<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot x<br />
\\[4pt]<br />
x<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[4pt]<br />
f_{9}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0110}<br />
\\[4pt]<br />
f_{1001}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[4pt]<br />
1~0~0~1<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{,} y \texttt{)}<br />
\\[4pt]<br />
\texttt{((} x \texttt{,} y \texttt{))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x ~\text{not equal to}~ y<br />
\\[4pt]<br />
x ~\text{equal to}~ y<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x \ne y<br />
\\[4pt]<br />
x = y<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{5}<br />
\\[4pt]<br />
f_{10}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0101}<br />
\\[4pt]<br />
f_{1010}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\[4pt]<br />
y<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{not}~ y<br />
\\[4pt]<br />
y<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot y<br />
\\[4pt]<br />
y<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[4pt]<br />
f_{11}<br />
\\[4pt]<br />
f_{13}<br />
\\[4pt]<br />
f_{14}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0111}<br />
\\[4pt]<br />
f_{1011}<br />
\\[4pt]<br />
f_{1101}<br />
\\[4pt]<br />
f_{1110}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} ~ x ~~ y ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{(} ~ x ~ \texttt{(} y \texttt{))}<br />
\\[4pt]<br />
\texttt{((} x \texttt{)} ~ y ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{not both}~ x ~\text{and}~ y<br />
\\[4pt]<br />
\text{not}~ x ~\text{without}~ y<br />
\\[4pt]<br />
\text{not}~ y ~\text{without}~ x<br />
\\[4pt]<br />
x ~\text{or}~ y<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot x \lor \lnot y<br />
\\[4pt]<br />
x \Rightarrow y<br />
\\[4pt]<br />
x \Leftarrow y<br />
\\[4pt]<br />
x \lor y<br />
\end{matrix}\!</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>f_{1111}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
==A Differential Extension of Propositional Calculus==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Fire over water:<br><br />
The image of the condition before transition.<br><br />
Thus the superior man is careful<br><br />
In the differentiation of things,<br><br />
So that each finds its place.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; ''I Ching'', Hexagram 64, [Wil, 249]<br />
|}<br />
<br />
This much preparation is enough to begin introducing my subject, if I excuse myself from giving full arguments for my definitional choices until some later stage. I am trying to develop a ''differential theory of qualitative equations'' that parallels the application of differential geometry to dynamic systems. The idea of a tangent vector is key to this work and a major goal is to find the right logical analogues of tangent spaces, bundles, and functors. The strategy is taken of looking for the simplest versions of these constructions that can be discovered within the realm of propositional calculus, so long as they serve to fill out the general theme.<br />
<br />
===Differential Propositions : Qualitative Analogues of Differential Equations===<br />
<br />
In order to define the differential extension of a universe of discourse <math>[\mathcal{A}],</math> the initial alphabet <math>\mathcal{A}</math> must be extended to include a collection of symbols for ''differential features'', or basic ''changes'' that are capable of occurring in <math>[\mathcal{A}].</math> Intuitively, these symbols may be construed as denoting primitive features of change, qualitative attributes of motion, or propositions about how things or points in the universe may be changing or moving with respect to the features that are noted in the initial alphabet.<br />
<br />
Therefore, let us define the corresponding ''differential alphabet'' or ''tangent alphabet'' as <math>\mathrm{d}\mathcal{A}\!</math> <math>=\!</math> <math>\{\mathrm{d}a_1, \ldots, \mathrm{d}a_n\},</math> in principle just an arbitrary alphabet of symbols, disjoint from the initial alphabet <math>\mathcal{A}\!</math> <math>=\!</math> <math>\{ a_1, \ldots, a_n \},\!</math> that is intended to be interpreted in the way just indicated. It only remains to be understood that the precise interpretation of the symbols in <math>\mathrm{d}\mathcal{A}\!</math> is often conceived to be changeable from point to point of the underlying space <math>A.\!</math> Indeed, for all we know, the state space <math>A\!</math> might well be the state space of a language interpreter, one that is concerned, among other things, with the idiomatic meanings of the dialect generated by <math>\mathcal{A}</math> and <math>\mathrm{d}\mathcal{A}.\!</math><br />
<br />
The ''tangent space'' to <math>A\!</math> at one of its points <math>x,\!</math> sometimes written <math>\mathrm{T}_x(A),</math> takes the form <math>\mathrm{d}A</math> <math>=\!</math> <math>\langle \mathrm{d}\mathcal{A} \rangle</math> <math>=\!</math> <math>\langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle.\!</math> Strictly speaking, the name ''cotangent space'' is probably more correct for this construction, but the fact that we take up spaces and their duals in pairs to form our universes of discourse allows our language to be pliable here.<br />
<br />
Proceeding as we did with the base space <math>A,\!</math> the tangent space <math>\mathrm{d}A</math> at a point of <math>A\!</math> can be analyzed as a product of distinct and independent factors:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\mathrm{d}A ~=~ \prod_{i=1}^n \mathrm{d}A_i ~=~ \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n.\!</math><br />
|}<br />
<br />
Here, <math>\mathrm{d}A_i\!</math> is a set of two differential propositions, <math>\mathrm{d}A_i = \{ (\mathrm{d}a_i), \mathrm{d}a_i \},\!</math> where <math>\texttt{(} \mathrm{d}a_i \texttt{)}\!</math> is a proposition with the logical value of <math>\text{not} ~ \mathrm{d}a_i.\!</math> Each component <math>\mathrm{d}A_i\!</math> has the type <math>\mathbb{B},\!</math> operating under the ordered correspondence <math>\{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \} \cong \{ 0, 1 \}.\!</math> However, clarity is often served by acknowledging this differential usage with a superficially distinct type <math>\mathbb{D},\!</math> whose intension may be indicated as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\mathbb{D} = \{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \} = \{ \text{same}, \text{different} \} = \{ \text{stay}, \text{change} \} = \{ \text{stop}, \text{step} \}.\!</math><br />
|}<br />
<br />
Viewed within a coordinate representation, spaces of type <math>\mathbb{B}^n\!</math> and <math>\mathbb{D}^n\!</math> may appear to be identical sets of binary vectors, but taking a view at this level of abstraction would be like ignoring the qualitative units and the diverse dimensions that distinguish position and momentum, or the different roles of quantity and impulse.<br />
<br />
===An Interlude on the Path===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
There would have been no beginnings: instead, speech would proceed from me, while I stood in its path &ndash; a slender gap &ndash; the point of its possible disappearance.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
A sense of the relation between <math>\mathbb{B}</math> and <math>\mathbb{D}</math> may be obtained by considering the ''path classifier'' (or the ''equivalence class of curves'') approach to tangent vectors. Consider a universe <math>[\mathcal{X}].\!</math> Given the boolean value system, a path in the space <math>X = \langle \mathcal{X} \rangle</math> is a map <math>q : \mathbb{B} \to X.</math> In this context the set of paths <math>(\mathbb{B} \to X)</math> is isomorphic to the cartesian square <math>X^2 = X \times X,</math> or the set of ordered pairs chosen from <math>X.\!</math><br />
<br />
We may analyze <math>X^2 = \{ (u, v) : u, v \in X \}</math> into two parts, specifically, the ordered pairs <math>(u, v)\!</math> that lie on and off the diagonal:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}X^2 & = & \{ (u, v) : u = v \} & \cup & \{ (u, v) : u \ne v \}.\end{matrix}</math><br />
|}<br />
<br />
This partition may also be expressed in the following symbolic form:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}X^2 & \cong & \operatorname{diag} (X) & + & 2 \binom{X}{2}.\end{matrix}</math><br />
|}<br />
<br />
The separate terms of this formula are defined as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}\operatorname{diag} (X) & = & \{ (x, x) : x \in X \}.\end{matrix}\!</math><br />
|}<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}\binom{X}{k} & = & X ~\text{choose}~ k & = & \{ k\text{-sets from}~ X \}.\end{matrix}\!</math><br />
|}<br />
<br />
Thus we have:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}\binom{X}{2} & = & \{ \{ u, v \} : u, v \in X \}.\end{matrix}</math><br />
|}<br />
<br />
We may now use the features in <math>\mathrm{d}\mathcal{X} = \{ \mathrm{d}x_i \} = \{ \mathrm{d}x_1, \ldots, \mathrm{d}x_n \}</math> to classify the paths of <math>(\mathbb{B} \to X)</math> by way of the pairs in <math>X^2.\!</math> If <math>X \cong \mathbb{B}^n,</math> then a path <math>q\!</math> in <math>X\!</math> has the following form:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
q : (\mathbb{B} \to \mathbb{B}^n) & \cong & \mathbb{B}^n \times \mathbb{B}^n & \cong & \mathbb{B}^{2n} & \cong & (\mathbb{B}^2)^n.<br />
\end{matrix}</math><br />
|}<br />
<br />
Intuitively, we want to map this <math>(\mathbb{B}^2)^n</math> onto <math>\mathbb{D}^n</math> by mapping each component <math>\mathbb{B}^2</math> onto a copy of <math>\mathbb{D}.</math> But in the presenting context <math>{}^{\backprime\backprime} \mathbb{D} {}^{\prime\prime}</math> is just a name associated with, or an incidental quality attributed to, coefficient values in <math>\mathbb{B}</math> when they are attached to features in <math>\mathrm{d}\mathcal{X}.</math><br />
<br />
Taking these intentions into account, define <math>\mathrm{d}x_i : X^2 \to \mathbb{B}</math> in the following manner:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcrcl}<br />
\mathrm{d}x_i(u, v)<br />
& = & \texttt{(} ~ x_i(u) & \texttt{,} & x_i(v) ~ \texttt{)}<br />
\\<br />
& = & x_i(u) & + & x_i(v)<br />
\\<br />
& = & x_i(v) & - & x_i(u).<br />
\end{array}</math><br />
|}<br />
<br />
In the above transcription, the operator bracket of the form <math>\texttt{(} \ldots \texttt{,} \ldots \texttt{)}\!</math> is a ''cactus lobe'', in general signifying that just one of the arguments listed is false. In the case of two arguments this is the same thing as saying that the arguments are not equal. The plus sign signifies boolean addition, in the sense of addition in <math>\mathrm{GF}(2),</math> and thus means the same thing in this context as the minus sign, in the sense of adding the additive inverse.<br />
<br />
The above definition of <math>\mathrm{d}x_i : X^2 \to \mathbb{B}</math> is equivalent to defining <math>\mathrm{d}x_i : (\mathbb{B} \to X) \to \mathbb{B}\!</math> in the following way:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcrcl}<br />
\mathrm{d}x_i (q)<br />
& = & \texttt{(} ~ x_i(q_0) & \texttt{,} & x_i(q_1) ~ \texttt{)}<br />
\\<br />
& = & x_i(q_0) & + & x_i(q_1)<br />
\\<br />
& = & x_i(q_1) & - & x_i(q_0).<br />
\end{array}</math><br />
|}<br />
<br />
In this definition <math>q_b = q(b),\!</math> for each <math>b\!</math> in <math>\mathbb{B}.</math> Thus, the proposition <math>\mathrm{d}x_i</math> is true of the path <math>q = (u, v)\!</math> exactly if the terms of <math>q,\!</math> the endpoints <math>u\!</math> and <math>v,\!</math> lie on different sides of the question <math>x_i.\!</math><br />
<br />
The language of features in <math>\langle \mathrm{d}\mathcal{X} \rangle,</math> indeed the whole calculus of propositions in <math>[\mathrm{d}\mathcal{X}],</math> may now be used to classify paths and sets of paths. In other words, the paths can be taken as models of the propositions <math>g : \mathrm{d}X \to \mathbb{B}.</math> For example, the paths corresponding to <math>\mathrm{diag}(X)</math> fall under the description <math>\texttt{(} \mathrm{d}x_1 \texttt{)} \cdots \texttt{(} \mathrm{d}x_n \texttt{)},\!</math> which says that nothing changes against the backdrop of the coordinate frame <math>\{ x_1, \ldots, x_n \}.\!</math><br />
<br />
Finally, a few words of explanation may be in order. If this concept of a path appears to be described in a roundabout fashion, it is because I am trying to avoid using any assumption of vector space properties for the space <math>X\!</math> that contains its range. In many ways the treatment is still unsatisfactory, but improvements will have to wait for the introduction of substitution operators acting on singular propositions.<br />
<br />
===The Extended Universe of Discourse===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
At the moment of speaking, I would like to have perceived a nameless voice, long preceding me, leaving me merely to enmesh myself in it, taking up its cadence, and to lodge myself, when no one was looking, in its interstices as if it had paused an instant, in suspense, to beckon to me.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
Next we define the ''extended alphabet'' or ''bundled alphabet'' <math>\mathrm{E}\mathcal{A}</math> as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lclcl}<br />
\mathrm{E}\mathcal{A}<br />
& = & \mathcal{A} \cup \mathrm{d}\mathcal{A}<br />
& = & \{ a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}.<br />
\end{array}</math><br />
|}<br />
<br />
This supplies enough material to construct the ''differential extension'' <math>\mathrm{E}A,</math> or the ''tangent bundle'' over the initial space <math>A,\!</math> in the following fashion:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcl}<br />
\mathrm{E}A<br />
& = & \langle \mathrm{E}\mathcal{A} \rangle<br />
\\[4pt]<br />
& = & \langle \mathcal{A} \cup \mathrm{d}\mathcal{A} \rangle<br />
\\[4pt]<br />
& = & \langle a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle,<br />
\end{array}</math><br />
|}<br />
<br />
and also:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcl}<br />
\mathrm{E}A<br />
& = & A \times \mathrm{d}A<br />
\\[4pt]<br />
& = & A_1 \times \ldots \times A_n \times \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n.<br />
\end{array}</math><br />
|}<br />
<br />
This gives <math>\mathrm{E}A</math> the type <math>\mathbb{B}^n \times \mathbb{D}^n.</math><br />
<br />
Finally, the tangent universe <math>\mathrm{E}A^\bullet = [\mathrm{E}\mathcal{A}]\!</math> is constituted from the totality of points and maps, or interpretations and propositions, that are based on the extended set of features <math>\mathrm{E}\mathcal{A},</math> and this fact is summed up in the following notation:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lclcl}<br />
\mathrm{E}A^\bullet<br />
& = & [\mathrm{E}\mathcal{A}]<br />
& = & [a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n].<br />
\end{array}</math><br />
|}<br />
<br />
This gives the tangent universe <math>\mathrm{E}A^\bullet\!</math> the type:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcl}<br />
(\mathbb{B}^n \times \mathbb{D}^n\ +\!\to \mathbb{B})<br />
& = & (\mathbb{B}^n \times \mathbb{D}^n, (\mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B})).<br />
\end{array}</math><br />
|}<br />
<br />
A proposition in the tangent universe <math>[\mathrm{E}\mathcal{A}]</math> is called a ''differential proposition'' and forms the analogue of a system of differential equations, constraints, or relations in ordinary calculus.<br />
<br />
With these constructions, the differential extension <math>\mathrm{E}A</math> and the space of differential propositions <math>(\mathrm{E}A \to \mathbb{B}),\!</math> we have arrived, in main outline, at one of the major subgoals of this study. Table&nbsp;8 summarizes the concepts that have been introduced for working with differentially extended universes of discourse.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 8.} ~~ \text{Differential Extension : Basic Notation}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Symbol}\!</math><br />
| <math>\text{Notation}\!</math><br />
| <math>\text{Description}\!</math><br />
| <math>\text{Type}\!</math><br />
|-<br />
| <math>\mathrm{d}\mathfrak{A}\!</math><br />
| <math>\{ {}^{\backprime\backprime} \mathrm{d}a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} \mathrm{d}a_n {}^{\prime\prime} \}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Alphabet of}<br />
\\[2pt]<br />
\text{differential symbols}<br />
\end{matrix}</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>\mathrm{d}\mathcal{A}\!</math><br />
| <math>\{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Basis of}<br />
\\[2pt]<br />
\text{differential features}<br />
\end{matrix}</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>\mathrm{d}A_i\!</math><br />
| <math>\{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \}\!</math><br />
| <math>\text{Differential dimension}~ i\!</math><br />
| <math>\mathbb{D}\!</math><br />
|-<br />
| <math>\mathrm{d}A\!</math><br />
|<br />
<math>\begin{matrix}<br />
\langle \mathrm{d}\mathcal{A} \rangle<br />
\\[2pt]<br />
\langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle<br />
\\[2pt]<br />
\{ (\mathrm{d}a_1, \ldots, \mathrm{d}a_n) \}<br />
\\[2pt]<br />
\mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n<br />
\\[2pt]<br />
\textstyle \prod_i \mathrm{d}A_i<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Tangent space at a point:}<br />
\\[2pt]<br />
\text{Set of changes, motions,}<br />
\\[2pt]<br />
\text{steps, tangent vectors}<br />
\\[2pt]<br />
\text{at a point}<br />
\end{matrix}</math><br />
| <math>\mathbb{D}^n\!</math><br />
|-<br />
| <math>\mathrm{d}A^*\!</math><br />
| <math>(\mathrm{hom} : \mathrm{d}A \to \mathbb{B})\!</math><br />
| <math>\text{Linear functions on}~ \mathrm{d}A\!</math><br />
| <math>(\mathbb{D}^n)^* \cong \mathbb{D}^n\!</math><br />
|-<br />
| <math>\mathrm{d}A^\uparrow\!</math><br />
| <math>(\mathrm{d}A \to \mathbb{B})\!</math><br />
| <math>\text{Boolean functions on}~ \mathrm{d}A\!</math><br />
| <math>\mathbb{D}^n \to \mathbb{B}\!</math><br />
|-<br />
| <math>\mathrm{d}A^\bullet\!</math><br />
|<br />
<math>\begin{matrix}<br />
[\mathrm{d}\mathcal{A}]<br />
\\[2pt]<br />
(\mathrm{d}A, \mathrm{d}A^\uparrow)<br />
\\[2pt]<br />
(\mathrm{d}A ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
(\mathrm{d}A, (\mathrm{d}A \to \mathbb{B}))<br />
\\[2pt]<br />
[\mathrm{d}a_1, \ldots, \mathrm{d}a_n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Tangent universe at a point of}~ A^\bullet,<br />
\\[2pt]<br />
\text{based on the tangent features}<br />
\\[2pt]<br />
\{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
(\mathbb{D}^n, (\mathbb{D}^n \to \mathbb{B}))<br />
\\[2pt]<br />
(\mathbb{D}^n ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
[\mathbb{D}^n]<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
The adjectives ''differential'' or ''tangent'' are systematically attached to every construct based on the differential alphabet <math>\mathrm{d}\mathfrak{A},</math> taken by itself. Strictly speaking, we probably ought to call <math>\mathrm{d}\mathcal{A}</math> the set of ''cotangent'' features derived from <math>\mathcal{A},</math> but the only time this distinction really seems to matter is when we need to distinguish the tangent vectors as maps of type <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math> from cotangent vectors as elements of type <math>\mathbb{D}^n.</math> In like fashion, having defined <math>\mathrm{E}\mathcal{A} = \mathcal{A} \cup \mathrm{d}\mathcal{A},</math> we can systematically attach the adjective ''extended'' or the substantive ''bundle'' to all of the constructs associated with this full complement of <math>{2n}\!</math> features.<br />
<br />
It eventually becomes necessary to extend the initial alphabet even further, to allow for the discussion of higher order differential expressions. Table&nbsp;9 provides a suggestion of how these further extensions can be carried out.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 9.} ~~ \text{Higher Order Differential Features}\!</math><br />
|<br />
<p><math>\begin{array}{lllll}<br />
\mathrm{d}^0 \mathcal{A} & = & \{ a_1, \ldots, a_n \} & = & \mathcal{A}<br />
\\<br />
\mathrm{d}^1 \mathcal{A} & = & \{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \} & = & \mathrm{d}\mathcal{A}<br />
\end{array}</math></p><br />
<br />
<p><math>\begin{array}{lll}<br />
\mathrm{d}^k \mathcal{A} & = & \{ \mathrm{d}^k a_1, \ldots, \mathrm{d}^k a_n \}<br />
\\<br />
\mathrm{d}^* \mathcal{A} & = & \{ \mathrm{d}^0 \mathcal{A}, \ldots, \mathrm{d}^k \mathcal{A}, \ldots \}<br />
\end{array}</math></p><br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}^0 \mathcal{A} & = & \mathrm{d}^0 \mathcal{A}<br />
\\<br />
\mathrm{E}^1 \mathcal{A} & = & \mathrm{d}^0 \mathcal{A} ~\cup~ \mathrm{d}^1 \mathcal{A}<br />
\\<br />
\mathrm{E}^k \mathcal{A} & = & \mathrm{d}^0 \mathcal{A} ~\cup~ \ldots ~\cup~ \mathrm{d}^k \mathcal{A}<br />
\\<br />
\mathrm{E}^\infty \mathcal{A} & = & \bigcup~ \mathrm{d}^* \mathcal{A}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
===Intentional Propositions===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Do you guess I have some intricate purpose?<br><br />
Well I have . . . . for the April rain has, and the mica on<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the side of a rock has.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 45]<br />
|}<br />
<br />
In order to analyze the behavior of a system at successive moments in time, while staying within the limitations of propositional logic, it is necessary to create independent alphabets of logical features for each moment of time that we contemplate using in our discussion. These moments have reference to typical instances and relative intervals, not actual or absolute times. For example, to discuss ''velocities'' (first order rates of change) we need to consider points of time in pairs. There are a number of natural ways of doing this. Given an initial alphabet, we could use its symbols as a lexical basis to generate successive alphabets of compound symbols, say, with temporal markers appended as suffixes.<br />
<br />
As a standard way of dealing with these situations, the following scheme of notation suggests a way of extending any alphabet of logical features through as many temporal moments as a particular order of analysis may demand. The lexical operators <math>\mathrm{p}^k</math> and <math>\mathrm{Q}^k</math> are convenient in many contexts where the accumulation of prime symbols and union symbols would otherwise be cumbersome.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 10.} ~~ \text{A Realm of Intentional Features}\!</math><br />
|<br />
<p><math>\begin{array}{lllll}<br />
\mathrm{p}^0 \mathcal{A} & = & \{ a_1, \ldots, a_n \} & = & \mathcal{A}<br />
\\<br />
\mathrm{p}^1 \mathcal{A} & = & \{ a_1^\prime, \ldots, a_n^\prime \} & = & \mathcal{A}^\prime<br />
\\<br />
\mathrm{p}^2 \mathcal{A} & = & \{ a_1^{\prime\prime}, \ldots, a_n^{\prime\prime} \} & = & \mathcal{A}^{\prime\prime}<br />
\\<br />
\cdots & & \cdots &<br />
\end{array}</math></p><br />
<br />
<p><math>\begin{array}{lll}<br />
\mathrm{p}^k \mathcal{A} & = & \{ \mathrm{p}^k a_1, \ldots, \mathrm{p}^k a_n \}<br />
\end{array}</math></p><br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{Q}^0 \mathcal{A} & = & \mathcal{A}<br />
\\<br />
\mathrm{Q}^1 \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}'<br />
\\<br />
\mathrm{Q}^2 \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}' \cup \mathcal{A}''<br />
\\<br />
\cdots & & \cdots<br />
\\<br />
\mathrm{Q}^k \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}' \cup \ldots \cup \mathrm{p}^k \mathcal{A}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
The resulting augmentations of our logical basis determine a series of discursive universes that may be called the ''intentional extension'' of propositional calculus. This extension follows a pattern analogous to the differential extension, which was developed in terms of the operators <math>\mathrm{d}^k</math> and <math>\mathrm{E}^k,</math> and there is a natural relation between these two extensions that bears further examination. In contexts displaying this pattern, where a sequence of domains stretches from an anchoring domain <math>X\!</math> through an indefinite number of higher reaches, a particular collection of domains based on <math>X\!</math> will be referred to as a ''realm'' of <math>X,\!</math> and when the succession exhibits a temporal aspect, as a ''reign'' of <math>X.\!</math><br />
<br />
For the purposes of this discussion, an ''intentional proposition'' is defined as a proposition in the universe of discourse <math>\mathrm{Q}X^\bullet = [\mathrm{Q}\mathcal{X}],</math> in other words, a map <math>q : \mathrm{Q}X \to \mathbb{B}.</math> The sense of this definition may be seen if we consider the following facts. First, the equivalence <math>\mathrm{Q}X = X \times X'</math> motivates the following chain of isomorphisms between spaces:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lllcl}<br />
(\mathrm{Q}X \to \mathbb{B})<br />
& \cong & (X & \times & ~X' \to \mathbb{B})<br />
\\[4pt]<br />
& \cong & (X & \to & (X' \to \mathbb{B}))<br />
\\[4pt]<br />
& \cong & (X' & \to & (X~ \to \mathbb{B})).<br />
\end{array}</math><br />
|}<br />
<br />
Viewed in this light, an intentional proposition <math>q\!</math> may be rephrased as a map <math>q : X \times X' \to \mathbb{B},</math> which judges the juxtaposition of states in <math>X\!</math> from one moment to the next. Alternatively, <math>q\!</math> may be parsed in two stages in two different ways, as <math>q : X \to (X' \to \mathbb{B})</math> and as <math>q : X' \to (X \to \mathbb{B}),</math> which associate to each point of <math>X\!</math> or <math>X'\!</math> a proposition about states in <math>X'\!</math> or <math>X,\!</math> respectively. In this way, an intentional proposition embodies a type of value system, in effect, a proposal that places a value on a collection of ends-in-view, or a project that evaluates a set of goals as regarded from each point of view in the state space of a system.<br />
<br />
In sum, the intentional proposition <math>q\!</math> indicates a method for the systematic selection of local goals. As a general form of description, a map of the type <math>q : \mathrm{Q}^i X \to \mathbb{B}\!</math> may be referred to as an "<math>i^\text{th}</math> order intentional proposition". Naturally, when we speak of intentional propositions without qualification, we usually mean first order intentions.<br />
<br />
Many different realms of discourse have the same structure as the extensions that have been indicated here. From a strictly logical point of view, each new layer of terms is composed of independent logical variables that are no different in kind from those that go before, and each further course of logical atoms is treated like so many individual, but otherwise indifferent bricks by the prototype computer program that I use as a propositional interpreter. Thus, the names that I use to single out the differential and the intentional extensions, and the lexical paradigms that I follow to construct them, are meant to suggest the interpretations that I have in mind, but they can only hint at the extra meanings that human communicators may pack into their terms and inflections.<br />
<br />
As applied here, the word ''intentional'' is drawn from common use and may have little bearing on its technical use in other, more properly philosophical, contexts. I am merely using the complex of intentional concepts &mdash; aims, ends, goals, objectives, purposes, and so on &mdash; metaphorically to flesh out and vividly to represent any situation where one needs to contemplate a system in multiple aspects of state and destination, that is, its being in certain states and at the same time acting as if headed through certain states. If confusion arises, more neutral words like ''conative'', ''contingent'', ''discretionary'', ''experimental'', ''kinetic'', ''progressive'', ''tentative'', or ''trial'' would probably serve as well.<br />
<br />
===Life on Easy Street===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Failing to fetch me at first keep encouraged,<br><br />
Missing me one place search another,<br><br />
I stop some where waiting for you<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]<br />
|}<br />
<br />
The finite character of the extended universe <math>[\mathrm{E}\mathcal{A}]</math> makes the problem of solving differential propositions relatively straightforward, at least, in principle. The solution set of the differential proposition <math>q : \mathrm{E}A \to \mathbb{B}</math> is the set of models <math>q^{-1}(1)\!</math> in <math>\mathrm{E}A.</math> Finding all the models of <math>q,\!</math> the extended interpretations in <math>\mathrm{E}A</math> that satisfy <math>q,\!</math> can be carried out by a finite search. Being in possession of complete algorithms for propositional calculus modeling, theorem checking, or theorem proving makes the analytic task fairly simple in principle, though the question of efficiency in the face of arbitrary complexity may always remain another matter entirely. While the fact that propositional satisfiability is NP-complete may be discouraging for the prospects of a single efficient algorithm that covers the whole space <math>[\mathrm{E}\mathcal{A}]</math> with equal facility, there appears to be much room for improvement in classifying special forms and in developing algorithms that are tailored to their practical processing.<br />
<br />
In view of these constraints and contingencies, our focus shifts to the tasks of approximation and interpretation that support intuition, especially in dealing with the natural kinds of differential propositions that arise in applications, and in the effort to understand, in succinct and adaptive forms, their dynamic implications. In the absence of direct insights, these tasks are partially carried out by forging analogies with the familiar situations and customary routines of ordinary calculus. But the indirect approach, going by way of specious analogy and intuitive habit, forces us to remain on guard against the circumstance that occurs when the word ''forging'' takes on its shadier nuance, indicting the constant risk of a counterfeit in the proportion.<br />
<br />
==Back to the Beginning : Exemplary Universes==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
I would have preferred to be enveloped in words, borne way beyond all possible beginnings.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
To anchor our understanding of differential logic, let us look at how the various concepts apply in the simplest possible concrete cases, where the initial dimension is only 1 or 2. In spite of the obvious simplicity of these cases, it is possible to observe how central difficulties of the subject begin to arise already at this stage.<br />
<br />
===A One-Dimensional Universe===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
There was never any more inception than there is now,<br><br />
Nor any more youth or age than there is now;<br><br />
And will never be any more perfection than there is now,<br><br />
Nor any more heaven or hell than there is now.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]<br />
|}<br />
<br />
Let <math>\mathcal{X} = \{ x_1 \} = \{ A \}</math> be an alphabet that represents one boolean variable or a single logical feature. In this example the capital letter <math>{}^{\backprime\backprime} A {}^{\prime\prime}\!</math> is used usual informally, to name a feature and not a space, in departure from our formerly stated formal conventions. At any rate, the basis element <math>A = x_1\!</math> may be interpreted as a simple proposition or a coordinate projection <math>A = x_1 : \mathbb{B} \xrightarrow{i} \mathbb{B}.</math> The space <math>X = \langle A \rangle = \{ \texttt{(} A \texttt{)}, A \}</math> of points (cells, vectors, interpretations) has cardinality <math>2^n = 2^1 = 2\!</math> and is isomorphic to <math>\mathbb{B} = \{ 0, 1 \}.</math> Moreover, <math>X\!</math> may be identified with the set of singular propositions <math>\{ x : \mathbb{B} \xrightarrow{s} \mathbb{B} \}.</math> The space of linear propositions <math>X^* = \{ \mathrm{hom} : \mathbb{B} \xrightarrow{\ell} \mathbb{B} \} = \{ 0, A \}</math> is algebraically dual to <math>X\!</math> and also has cardinality <math>2.\!</math> Here, <math>{}^{\backprime\backprime} 0 {}^{\prime\prime}\!</math> is interpreted as denoting the constant function <math>0 : \mathbb{B} \to \mathbb{B},</math> amounting to the linear proposition of rank <math>0,\!</math> while <math>A\!</math> is the linear proposition of rank <math>1.\!</math> Last but not least we have the positive propositions <math>\{ \mathrm{pos} : \mathbb{B} \xrightarrow{p} \mathbb{B} \} = \{ A, 1 \},\!</math> of rank <math>1\!</math> and <math>0,\!</math> respectively, where <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}\!</math> is understood as denoting the constant function <math>1 : \mathbb{B} \to \mathbb{B}.</math> In sum, there are <math>2^{2^n} = 2^{2^1} = 4</math> propositions altogether in the universe of discourse, comprising the set <math>X^\uparrow = \{ f : X \to \mathbb{B} \} = \{ 0, \texttt{(} A \texttt{)}, A, 1 \} \cong (\mathbb{B} \to \mathbb{B}).</math><br />
<br />
The first order differential extension of <math>\mathcal{X}</math> is <math>\mathrm{E}\mathcal{X} = \{ x_1, \mathrm{d}x_1 \} = \{ A, \mathrm{d}A \}.</math> If the feature <math>A\!</math> is understood as applying to some object or state, then the feature <math>\mathrm{d}A</math> may be interpreted as an attribute of the same object or state that says that it is changing ''significantly'' with respect to the property <math>A,\!</math> or that it has an ''escape velocity'' with respect to the state <math>A.\!</math> In practice, differential features acquire their logical meaning through a class of ''temporal inference rules''.<br />
<br />
For example, relative to a frame of observation that is left implicit for now, one is permitted to make the following sorts of inference: From the fact that <math>A\!</math> and <math>\mathrm{d}A</math> are true at a given moment one may infer that <math>\texttt{(} A \texttt{)}\!</math> will be true in the next moment of observation. Altogether in the present instance, there is the fourfold scheme of inference that is shown below:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:50%"<br />
|<br />
<math>\begin{matrix}<br />
\text{From} & \texttt{(} A \texttt{)}<br />
& \text{and} & \texttt{(} \mathrm{d}A \texttt{)}<br />
& \text{infer} & \texttt{(} A \texttt{)} & \text{next.}<br />
\\[8pt]<br />
\text{From} & \texttt{(} A \texttt{)}<br />
& \text{and} & \mathrm{d}A<br />
& \text{infer} & A & \text{next.}<br />
\\[8pt]<br />
\text{From} & A<br />
& \text{and} & \texttt{(} \mathrm{d}A \texttt{)}<br />
& \text{infer} & A & \text{next.}<br />
\\[8pt]<br />
\text{From} & A<br />
& \text{and} & \mathrm{d}A<br />
& \text{infer} & \texttt{(} A \texttt{)} & \text{next.}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
It might be thought that an independent time variable needs to be brought in at this point, but it is an insight of fundamental importance that the idea of process is logically prior to the notion of time. A time variable is a reference to a ''clock'' &mdash; a canonical, conventional process that is accepted or established as a standard of measurement, but in essence no different than any other process. This raises the question of how different subsystems in a more global process can be brought into comparison, and what it means for one process to serve the function of a local standard for others. But these inquiries only wrap up puzzles in further riddles, and are obviously too involved to be handled at our current level of approximation.<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
The clock indicates the moment . . . . but what does<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;eternity indicate?<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 79]<br />
|}<br />
<br />
Observe that the secular inference rules, used by themselves, involve a loss of information, since nothing in them can tell us whether the momenta <math>\{ \texttt{(} \mathrm{d}A \texttt{)}, \mathrm{d}A \}\!</math> are changed or unchanged in the next instance. In order to know this, one would have to determine <math>\mathrm{d}^2 A,\!</math> and so on, pursuing an infinite regress. Ultimately, in order to rest with a finitely determinate system, it is necessary to make an infinite assumption, for example, that <math>\mathrm{d}^k A = 0\!</math> for all <math>k\!</math> greater than some fixed value <math>M.\!</math> Another way to escape the regress is through the provision of a dynamic law, in typical form making higher order differentials dependent on lower degrees and estates.<br />
<br />
===Example 1. A Square Rigging===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Urge and urge and urge,<br><br />
Always the procreant urge of the world.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]<br />
|}<br />
<br />
By way of example, suppose that we are given the initial condition <math>A = \mathrm{d}A\!</math> and the law <math>\mathrm{d}^2 A = \texttt{(} A \texttt{)}.\!</math> Since the equation <math>A = \mathrm{d}A\!</math> is logically equivalent to the disjunction <math>A ~ \mathrm{d}A ~\text{or}~ \texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)},\!</math> we may infer two possible trajectories, as displayed in Table&nbsp;11. In either case the state <math>A ~ \texttt{(} \mathrm{d}A \texttt{)(} \mathrm{d}^2 A \texttt{)}\!</math> is a stable attractor or a terminal condition for both starting points.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 11.} ~~ \text{A Pair of Commodious Trajectories}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Time}\!</math><br />
| <math>\text{Trajectory 1}\!</math><br />
| <math>\text{Trajectory 2}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
2<br />
\\[4pt]<br />
3<br />
\\[4pt]<br />
4<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A & \mathrm{d}A & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
\texttt{(} A \texttt{)} & \mathrm{d}A & \mathrm{d}^2 A<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
{}^{\shortparallel} & {}^{\shortparallel} & {}^{\shortparallel}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{)} & \texttt{(} \mathrm{d}A \texttt{)} & \mathrm{d}^2 A<br />
\\[4pt]<br />
\texttt{(} A \texttt{)} & \mathrm{d}A & \mathrm{d}^2 A<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
{}^{\shortparallel} & {}^{\shortparallel} & {}^{\shortparallel}<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Because the initial space <math>X = \langle A \rangle\!</math> is one-dimensional, we can easily fit the second order extension <math>\mathrm{E}^2 X = \langle A, \mathrm{d}A, \mathrm{d}^2 A \rangle\!</math> within the compass of a single venn diagram, charting the couple of converging trajectories as shown in Figure&nbsp;12.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 12 -- The Anchor.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 12.} ~~ \text{The Anchor}\!</math><br />
|}<br />
<br />
If we eliminate from view the regions of <math>\mathrm{E}^2 X\!</math> that are ruled out by the dynamic law <math>\mathrm{d}^2 A = \texttt{(} A \texttt{)},\!</math> then what remains is the quotient structure that is shown in Figure&nbsp;13. This picture makes it easy to see that the dynamically allowable portion of the universe is partitioned between the properties <math>A\!</math> and <math>\mathrm{d}^2 A\!.</math> As it happens, this fact might have been expressed &ldquo;right off the bat&rdquo; by an equivalent formulation of the differential law, one that uses the exclusive disjunction to state the law as <math>\texttt{(} A \texttt{,} \mathrm{d}^2 A \texttt{)}\!.</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 13 -- The Tiller.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 13.} ~~ \text{The Tiller}\!</math><br />
|}<br />
<br />
What we have achieved in this example is to give a differential description of a simple dynamic process. In effect, we did this by embedding a directed graph, which can be taken to represent the state transitions of a finite automaton, in a dynamically allotted quotient structure that is created from a boolean lattice or an <math>n\!</math>-cube by nullifying all of the regions that the dynamics outlaws. With growth in the dimensions of our contemplated universes, it becomes essential, both for human comprehension and for computer implementation, that the dynamic structures of interest to us be represented not actually, by acquaintance, but virtually, by description. In our present study, we are using the language of propositional calculus to express the relevant descriptions, and to comprehend the structure that is implicit in the subsets of a <math>n\!</math>-cube without necessarily being forced to actualize all of its points.<br />
<br />
One of the reasons for engaging in this kind of extremely reduced, but explicitly controlled case study is to throw light on the general study of languages, formal and natural, in their full array of syntactic, semantic, and pragmatic aspects. Propositional calculus is one of the last points of departure where we can view these three aspects interacting in a non-trivial way without being immediately and totally overwhelmed by the complexity they generate. Often this complexity causes investigators of formal and natural languages to adopt the strategy of focusing on a single aspect and to abandon all hope of understanding the whole, whether it's the still living natural language or the dynamics of inquiry that lies crystallized in formal logic.<br />
<br />
From the perspective that I find most useful here, a language is a syntactic system that is designed or evolved in part to express a set of descriptions. When the explicit symbols of a language have extensions in its object world that are actually infinite, or when the implicit categories and generative devices of a linguistic theory have extensions in its subject matter that are potentially infinite, then the finite characters of terms, statements, arguments, grammars, logics, and rhetorics force an excess of intension to reside in all these symbols and functions, across the spectrum from the object language to the metalinguistic uses. In the aphorism from W. von Humboldt that Chomsky often cites, for example, in [Cho86, 30] and [Cho93, 49], language requires &ldquo;the infinite use of finite means&rdquo;. This is necessarily true when the extensions are infinite, when the referential symbols and grammatical categories of a language possess infinite sets of models and instances. But it also voices a practical truth when the extensions, though finite at every stage, tend to grow at exponential rates.<br />
<br />
This consequence of dealing with extensions that are &ldquo;practically infinite&rdquo; becomes crucial when one tries to build neural network systems that learn, since the learning competence of any intelligent system is limited to the objects and domains that it is able to represent. If we want to design systems that operate intelligently with the full deck of propositions dealt by intact universes of discourse, then we must supply them with succinct representations and efficient transformations in this domain. Furthermore, in the project of constructing inquiry driven systems, we find ourselves forced to contemplate the level of generality that is embodied in propositions, because the dynamic evolution of these systems is driven by the measurable discrepancies that occur among their expectations, intentions, and observations, and because each of these subsystems or components of knowledge constitutes a propositional modality that can take on the fully generic character of an empirical summary or an axiomatic theory.<br />
<br />
A compression scheme by any other name is a symbolic representation, and this is what the differential extension of propositional calculus, through all of its many universes of discourse, is intended to supply. Why is this particular program of mental calisthenics worth carrying out in general? By providing a uniform logical medium for describing dynamic systems we can make the task of understanding complex systems much easier, both in looking for invariant representations of individual cases and in finding points of comparison among diverse structures that would otherwise appear as isolated systems. All of this goes to facilitate the search for compact knowledge and to adapt what is learned from individual cases to the general realm.<br />
<br />
===Back to the Feature===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
I guess it must be the flag of my disposition, out of hopeful<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;green stuff woven.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 31]<br />
|}<br />
<br />
Let us assume that the sense intended for differential features is well enough established in the intuition, for now, that we may continue with outlining the structure of the differential extension <math>[\mathrm{E}\mathcal{X}] = [A, \mathrm{d}A].\!</math> Over the extended alphabet <math>\mathrm{E}\mathcal{X} = \{ x_1, \mathrm{d}x_1 \} = \{ A, \mathrm{d}A \}\!</math> of cardinality <math>2^n = 2\!</math> we generate the set of points <math>\mathrm{E}X\!</math> of cardinality <math>2^{2n} = 4\!</math> that bears the following chain of equivalent descriptions:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}X & = & \langle A, \mathrm{d}A \rangle<br />
\\[4pt]<br />
& = & \{ \texttt{(} A \texttt{)}, A \} ~\times~ \{ \texttt{(} \mathrm{d}A \texttt{)}, \mathrm{d}A \}<br />
\\[4pt]<br />
& = &<br />
\{<br />
\texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)},~<br />
\texttt{(} A \texttt{)} \mathrm{d}A,~<br />
A \texttt{(} \mathrm{d}A \texttt{)},~<br />
A ~ \mathrm{d}A<br />
\}.<br />
\end{array}</math><br />
|}<br />
<br />
The space <math>\mathrm{E}X\!</math> may be assigned the mnemonic type <math>\mathbb{B} \times \mathbb{D},\!</math> which is really no different than <math>\mathbb{B} \times \mathbb{B} = \mathbb{B}^2.\!</math> An individual element of <math>\mathrm{E}X\!</math> may be regarded as a ''disposition at a point'' or a ''situated direction'', in effect, a singular mode of change occurring at a single point in the universe of discourse. In applications, the modality of this change can be interpreted in various ways, for example, as an expectation, an intention, or an observation with respect to the behavior of a system.<br />
<br />
To complete the construction of the extended universe of discourse <math>\mathrm{E}X^\bullet = [x_1, \mathrm{d}x_1] = [A, \mathrm{d}A]\!</math> one must add the set of differential propositions <math>\mathrm{E}X^\uparrow = \{ g : \mathrm{E}X \to \mathbb{B} \} \cong (\mathbb{B} \times \mathbb{D} \to \mathbb{B})\!</math> to the set of dispositions in <math>\mathrm{E}X.\!</math> There are <math>2^{2^{2n}} = 16\!</math> propositions in <math>\mathrm{E}X^\uparrow,\!</math> as detailed in Table&nbsp;14.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 14.} ~~ \text{Differential Propositions}\!</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>A\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>\mathrm{d}A\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>g_{0}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{1}<br />
\\[4pt]<br />
g_{2}<br />
\\[4pt]<br />
g_{4}<br />
\\[4pt]<br />
g_{8}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
1~0~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
\texttt{(} A \texttt{)} ~ \mathrm{d}A ~<br />
\\[4pt]<br />
~ A ~ \texttt{(} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
~ A ~~ \mathrm{d}A ~<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{neither}~ A ~\text{nor}~ \mathrm{d}A<br />
\\[4pt]<br />
\mathrm{d}A ~\text{and not}~ A<br />
\\[4pt]<br />
A ~\text{and not}~ \mathrm{d}A<br />
\\[4pt]<br />
A ~\text{and}~ \mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot A \land \lnot \mathrm{d}A<br />
\\[4pt]<br />
\lnot A \land \mathrm{d}A<br />
\\[4pt]<br />
A \land \lnot \mathrm{d}A<br />
\\[4pt]<br />
A \land \mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
g_{3}<br />
\\[4pt]<br />
g_{12}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{)}<br />
\\[4pt]<br />
A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ A<br />
\\[4pt]<br />
A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot A<br />
\\[4pt]<br />
A<br />
\end{matrix}\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{6}<br />
\\[4pt]<br />
g_{9}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[4pt]<br />
1~0~0~1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{,} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
\texttt{((} A \texttt{,} \mathrm{d}A \texttt{))}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
A ~\text{not equal to}~ \mathrm{d}A<br />
\\[4pt]<br />
A ~\text{equal to}~ \mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
A \ne \mathrm{d}A<br />
\\[4pt]<br />
A = \mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{5}<br />
\\[4pt]<br />
g_{10}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
\mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ \mathrm{d}A<br />
\\[4pt]<br />
\mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot \mathrm{d}A<br />
\\[4pt]<br />
\mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{7}<br />
\\[4pt]<br />
g_{11}<br />
\\[4pt]<br />
g_{13}<br />
\\[4pt]<br />
g_{14}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} ~ A ~~ \mathrm{d}A ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{(} ~ A ~ \texttt{(} \mathrm{d}A \texttt{))}<br />
\\[4pt]<br />
\texttt{((} A \texttt{)} ~ \mathrm{d}A ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{((} A \texttt{)(} \mathrm{d}A \texttt{))}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not both}~ A ~\text{and}~ \mathrm{d}A<br />
\\[4pt]<br />
\text{not}~ A ~\text{without}~ \mathrm{d}A<br />
\\[4pt]<br />
\text{not}~ \mathrm{d}A ~\text{without}~ A<br />
\\[4pt]<br />
A ~\text{or}~ \mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot A \lor \lnot \mathrm{d}A<br />
\\[4pt]<br />
A \Rightarrow \mathrm{d}A<br />
\\[4pt]<br />
A \Leftarrow \mathrm{d}A<br />
\\[4pt]<br />
A \lor \mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
| <math>f_{3}\!</math><br />
| <math>g_{15}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
Aside from changing the names of variables and shuffling the order of rows, this Table follows the format that was used previously for boolean functions of two variables. The rows are grouped to reflect natural similarity classes among the propositions. In a future discussion, these classes will be given additional explanation and motivation as the orbits of a certain transformation group acting on the set of 16 propositions. Notice that four of the propositions, in their logical expressions, resemble those given in the table for <math>X^\uparrow.\!</math> Thus the first set of propositions <math>\{ f_i \}\!</math> is automatically embedded in the present set <math>\{ g_j \}\!</math> and the corresponding inclusions are indicated at the far left margin of the Table.<br />
<br />
===Tacit Extensions===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
I would really like to have slipped imperceptibly into this lecture, as into all the others I shall be delivering, perhaps over the years ahead.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
Strictly speaking, however, there is a subtle distinction in type between the function <math>f_i : X \to \mathbb{B}</math> and the corresponding function <math>g_j : \mathrm{E}X \to \mathbb{B},</math> even though they share the same logical expression. Naturally, we want to maintain the logical equivalence of expressions that represent the same proposition while appreciating the full diversity of that proposition's functional and typical representatives. Both perspectives, and all the levels of abstraction extending through them, have their reasons, as will develop in time.<br />
<br />
Because this special circumstance points up an important general theme, it is a good idea to discuss it more carefully. Whenever there arises a situation like this, where one alphabet <math>\mathcal{X}</math> is a subset of another alphabet <math>\mathcal{Y},</math> then we say that any proposition <math>f : \langle \mathcal{X} \rangle \to \mathbb{B}</math> has a ''tacit extension'' to a proposition <math>\boldsymbol\varepsilon f : \langle \mathcal{Y} \rangle \to \mathbb{B},\!</math> and that the space <math>(\langle \mathcal{X} \rangle \to \mathbb{B})</math> has an ''automatic embedding'' within the space <math>(\langle \mathcal{Y} \rangle \to \mathbb{B}).</math> The extension is defined in such a way that <math>\boldsymbol\varepsilon f\!</math> puts the same constraint on the variables of <math>\mathcal{X}</math> that are contained in <math>\mathcal{Y}</math> as the proposition <math>f\!</math> initially did, while it puts no constraint on the variables of <math>\mathcal{Y}</math> outside of <math>\mathcal{X},</math> in effect, conjoining the two constraints.<br />
<br />
If the variables in question are indexed as <math>\mathcal{X} = \{ x_1, \ldots, x_n \}</math> and <math>\mathcal{Y} = \{ x_1, \ldots, x_n, \ldots, x_{n+k} \},</math> then the definition of the tacit extension from <math>\mathcal{X}</math> to <math>\mathcal{Y}</math> may be expressed in the form of an equation:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\boldsymbol\varepsilon f(x_1, \ldots, x_n, \ldots, x_{n+k}) ~=~ f(x_1, \ldots, x_n).\!</math><br />
|}<br />
<br />
On formal occasions, such as the present context of definition, the tacit extension from <math>\mathcal{X}</math> to <math>\mathcal{Y}</math> is explicitly symbolized by the operator <math>\boldsymbol\varepsilon : (\langle \mathcal{X} \rangle \to \mathbb{B}) \to (\langle \mathcal{Y} \rangle \to \mathbb{B}),</math> where the appropriate alphabets <math>\mathcal{X}</math> and <math>\mathcal{Y}</math> are understood from context, but normally one may leave the "<math>\boldsymbol\varepsilon\!</math>" silent.<br />
<br />
Let's explore what this means for the present Example. Here, <math>\mathcal{X} = \{ A \}</math> and <math>\mathcal{Y} = \mathrm{E}\mathcal{X} = \{ A, \mathrm{d}A \}.</math> For each of the propositions <math>f_i\!</math> over <math>X\!,</math> specifically, those whose expression <math>e_i\!</math> lies in the collection <math>\{ 0, \texttt{(} A \texttt{)}, A, 1 \},\!</math> the tacit extension <math>\boldsymbol\varepsilon f\!</math> of <math>f\!</math> to <math>\mathrm{E}X</math> can be phrased as a logical conjunction of two factors, <math>f_i = e_i \cdot \tau ~ ,\!</math> where <math>\tau\!</math> is a logical tautology that uses all the variables of <math>\mathcal{Y} - \mathcal{X}.</math> Working in these terms, the tacit extensions <math>\boldsymbol\varepsilon f\!</math> of <math>f\!</math> to <math>\mathrm{E}X</math> may be explicated as shown in Table&nbsp;15.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:60%"<br />
|+ style="height:30px" | <math>\text{Table 15.} ~~ \text{Tacit Extension of}~ [A] ~\text{to}~ [A, \mathrm{d}A]\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0<br />
& = & 0 & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))} & = & & 0<br />
\\[8pt]<br />
\texttt{(} A \texttt{)}<br />
& = & \texttt{(} A \texttt{)} & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))}<br />
& = & \texttt{(} A \texttt{)} \, \mathrm{d}A ~ & + & \texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)}<br />
\\[8pt]<br />
A<br />
& = & ~A~ & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))}<br />
& = & ~A~ ~\mathrm{d}A~ & + & ~A~ \texttt{(} \mathrm{d}A \texttt{)}<br />
\\[8pt]<br />
1<br />
& = & 1 & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))} & = & & 1<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
In its effect on the singular propositions over <math>X,\!</math> this analysis has an interesting interpretation. The tacit extension takes us from thinking about a particular state, like <math>A\!</math> or <math>\texttt{(} A \texttt{)},\!</math> to considering the collection of outcomes, the outgoing changes or the singular dispositions, that spring from that state.<br />
<br />
===Example 2. Drives and Their Vicissitudes===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
I open my scuttle at night and see the far-sprinkled systems,<br><br />
And all I see, multiplied as high as I can cipher, edge but<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the rim of the farther systems.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 81]<br />
|}<br />
<br />
Before we leave the one-feature case let's look at a more substantial example, one that illustrates a general class of curves that can be charted through the extended feature spaces and that provides an opportunity to discuss a number of important themes concerning their structure and dynamics.<br />
<br />
Again, let <math>\mathcal{X} = \{ x_1 \} = \{ A \}.\!</math> In the discussion that follows we will consider a class of trajectories having the property that <math>\mathrm{d}^k A = 0\!</math> for all <math>k\!</math> greater than some fixed <math>m\!</math> and we may indulge in the use of some picturesque terms that describe salient classes of such curves. Given the finite order condition, there is a highest order non-zero difference <math>\mathrm{d}^m A\!</math> exhibited at each point in the course of any determinate trajectory that one may wish to consider. With respect to any point of the corresponding orbit or curve let us call this highest order differential feature <math>\mathrm{d}^m A\!</math> the ''drive'' at that point. Curves of constant drive <math>\mathrm{d}^m A\!</math> are then referred to as ''<math>m^\text{th}\!</math>-gear curves''.<br />
<br />
* '''Scholium.''' The fact that a difference calculus can be developed for boolean functions is well known [Fuji], [Koh, &sect; 8-4] and was probably familiar to Boole, who was an expert in difference equations before he turned to logic. And of course there is the strange but true story of how the Turin machines of the 1840s prefigured the Turing machines of the 1940s [Men, 225-297]. At the very outset of general purpose, mechanized computing we find that the motive power driving the Analytical Engine of Babbage, the kernel of an idea behind all of his wheels, was exactly his notion that difference operations, suitably trained, can serve as universal joints for any conceivable computation [M&M], [Mel, ch. 4].<br />
<br />
Given this language, the Example we take up here can be described as the family of <math>4^\text{th}\!</math>-gear curves through <math>\mathrm{E}^4 X\!</math> <math>=\!</math> <math>\langle A, ~\mathrm{d}A, ~\mathrm{d}^2\!A, ~\mathrm{d}^3\!A, ~\mathrm{d}^4\!A \rangle.</math> These are the trajectories generated subject to the dynamic law <math>\mathrm{d}^4 A = 1,\!</math> where it is understood in such a statement that all higher order differences are equal to <math>0.\!</math> Since <math>\mathrm{d}^4 A\!</math> and all higher <math>\mathrm{d}^k A\!</math> are fixed, the temporal or transitional conditions (initial, mediate, terminal &mdash; transient or stable states) vary only with respect to their projections as points of <math>\mathrm{E}^3 X = \langle A, ~\mathrm{d}A, ~\mathrm{d}^2\!A, ~\mathrm{d}^3\!A \rangle.</math> Thus, there is just enough space in a planar venn diagram to plot all of these orbits and to show how they partition the points of <math>\mathrm{E}^3 X.\!</math> It turns out that there are exactly two possible orbits, of eight points each, as illustrated in Figure&nbsp;16.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 16 -- A Couple of Fourth Gear Orbits.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 16.} ~~ \text{A Couple of Fourth Gear Orbits}\!</math><br />
|}<br />
<br />
With a little thought it is possible to devise an indexing scheme for the general run of dynamic states that allows for comparing universes of discourse that weigh in on different scales of observation. With this end in sight, let us index the states <math>q \in \mathrm{E}^m X\!</math> with the dyadic rationals (or the binary fractions) in the half-open interval <math>[0, 2).\!</math> Formally and canonically, a state <math>q_r\!</math> is indexed by a fraction <math>r = \tfrac{s}{t}\!</math> whose denominator is the power of two <math>t = 2^m\!</math> and whose numerator is a binary numeral formed from the coefficients of state in a manner to be described next. The ''differential coefficients'' of the state <math>q\!</math> are just the values <math>\mathrm{d}^k\!A(q)</math> for <math>k = 0 ~\text{to}~ m,\!</math> where <math>\mathrm{d}^0\!A</math> is defined as being identical to <math>A.\!</math> To form the binary index <math>d_0.d_1 \ldots d_m\!</math> of the state <math>q\!</math> the coefficient <math>\mathrm{d}^k\!A(q)</math> is read off as the binary digit <math>d_k\!</math> associated with the place value <math>2^{-k}.\!</math> Expressed by way of algebraic formulas, the rational index <math>r\!</math> of the state <math>q\!</math> can be given by the following equivalent formulations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
r(q)<br />
& = &<br />
\displaystyle\sum_k d_k \cdot 2^{-k}<br />
& = &<br />
\displaystyle\sum_k \text{d}^k A(q) \cdot 2^{-k}<br />
\\[8pt]<br />
=<br />
\\[8pt]<br />
\displaystyle\frac{s(q)}{t}<br />
& = &<br />
\displaystyle\frac{\sum_k d_k \cdot 2^{(m-k)}}{2^m}<br />
& = &<br />
\displaystyle\frac{\sum_k \text{d}^k A(q) \cdot 2^{(m-k)}}{2^m}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Applied to the example of <math>4^\text{th}\!</math>-gear curves, this scheme results in the data of Tables&nbsp;17-a and 17-b, which exhibit one period for each orbit. The states in each orbit are listed as ordered pairs <math>(p_i, q_j),\!</math> where <math>p_i\!</math> may be read as a temporal parameter that indicates the present time of the state and where <math>j\!</math> is the decimal equivalent of the binary numeral <math>s.\!</math> Informally and more casually, the Tables exhibit the states <math>q_s\!</math> as subscripted with the numerators of their rational indices, taking for granted the constant denominators of <math>2^m\! = 2^4 = 16.\!</math> In this set-up the temporal successions of states can be reckoned as given by a kind of ''parallel round-up rule''. That is, if <math>(d_k, d_{k+1})\!</math> is any pair of adjacent digits in the state index <math>r,\!</math> then the value of <math>d_k\!</math> in the next state is <math>{d_k}' = d_k + d_{k+1}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:55%"<br />
|+ style="height:30px" | <math>\text{Table 17-a.} ~~ \text{A Couple of Orbits in Fourth Gear : Orbit 1}\!</math><br />
|- style="background:ghostwhite"<br />
| <math>\text{Time}\!</math><br />
| <math>\text{State}\!</math><br />
| <math>A\!</math><br />
| <math>\mathrm{d}A\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| <math>p_i\!</math><br />
| <math>q_j\!</math><br />
| <math>\mathrm{d}^0\!A</math><br />
| <math>\mathrm{d}^1\!A</math><br />
| <math>\mathrm{d}^2\!A</math><br />
| <math>\mathrm{d}^3\!A</math><br />
| <math>\mathrm{d}^4\!A</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
p_0<br />
\\[4pt]<br />
p_1<br />
\\[4pt]<br />
p_2<br />
\\[4pt]<br />
p_3<br />
\\[4pt]<br />
p_4<br />
\\[4pt]<br />
p_5<br />
\\[4pt]<br />
p_6<br />
\\[4pt]<br />
p_7<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
q_{01}<br />
\\[4pt]<br />
q_{03}<br />
\\[4pt]<br />
q_{05}<br />
\\[4pt]<br />
q_{15}<br />
\\[4pt]<br />
q_{17}<br />
\\[4pt]<br />
q_{19}<br />
\\[4pt]<br />
q_{21}<br />
\\[4pt]<br />
q_{31}<br />
\end{matrix}\!</math><br />
| colspan="5" |<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="text-align:center; width:100%"<br />
|<br />
<math>\begin{matrix}<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
1.<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:55%"<br />
|+ style="height:30px" | <math>\text{Table 17-b.} ~~ \text{A Couple of Orbits in Fourth Gear : Orbit 2}\!</math><br />
|- style="background:ghostwhite"<br />
| <math>\text{Time}\!</math><br />
| <math>\text{State}\!</math><br />
| <math>A\!</math><br />
| <math>\mathrm{d}A\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| <math>p_i\!</math><br />
| <math>q_j\!</math><br />
| <math>\mathrm{d}^0\!A</math><br />
| <math>\mathrm{d}^1\!A</math><br />
| <math>\mathrm{d}^2\!A</math><br />
| <math>\mathrm{d}^3\!A</math><br />
| <math>\mathrm{d}^4\!A</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
p_0<br />
\\[4pt]<br />
p_1<br />
\\[4pt]<br />
p_2<br />
\\[4pt]<br />
p_3<br />
\\[4pt]<br />
p_4<br />
\\[4pt]<br />
p_5<br />
\\[4pt]<br />
p_6<br />
\\[4pt]<br />
p_7<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
q_{25}<br />
\\[4pt]<br />
q_{11}<br />
\\[4pt]<br />
q_{29}<br />
\\[4pt]<br />
q_{07}<br />
\\[4pt]<br />
q_{09}<br />
\\[4pt]<br />
q_{27}<br />
\\[4pt]<br />
q_{13}<br />
\\[4pt]<br />
q_{23}<br />
\end{matrix}\!</math><br />
| colspan="5" |<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="text-align:center; width:100%"<br />
|<br />
<math>\begin{matrix}<br />
1.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
==Transformations of Discourse==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
It is understandable that an engineer should be completely absorbed in his speciality, instead of pouring himself out into the freedom and vastness of the world of thought, even though his machines are being sent off to the ends of the earth; for he no more needs to be capable of applying to his own personal soul what is daring and new in the soul of his subject than a machine is in fact capable of applying to itself the differential calculus on which it is based. The same thing cannot, however, be said about mathematics; for here we have the new method of thought, pure intellect, the very well-spring of the times, the ''fons et origo'' of an unfathomable transformation.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 39]<br />
|}<br />
<br />
In this section we take up the general study of logical transformations, or maps that relate one universe of discourse to another. In many ways, and especially as applied to the subject of intelligent dynamic systems, my argument develops the antithesis of the statement just quoted. Along the way, if incidental to my ends, I hope this essay can pose a fittingly irenic epitaph to the frankly ironic epigraph inscribed at its head.<br />
<br />
My goal in this section is to answer a single question: What is a propositional tangent functor? In other words, my aim is to develop a clear conception of what manner of thing would pass in the logical realm for a genuine analogue of the tangent functor, an object conceived to generalize as far as possible in the abstract terms of category theory the ordinary notions of functional differentiation and the all too familiar operations of taking derivatives.<br />
<br />
As a first step I discuss the kinds of transformations that we already know as ''extensions'' and ''projections'', and I use these special cases to illustrate several different styles of logical and visual representation that will figure heavily in the sequel.<br />
<br />
===Foreshadowing Transformations : Extensions and Projections of Discourse===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
And, despite the care which she took to look behind her at every moment, she failed to see a shadow which followed her like her own shadow, which stopped when she stopped, which started again when she did, and which made no more noise than a well-conducted shadow should.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 126]<br />
|}<br />
<br />
Many times in our discussion we have occasion to place one universe of discourse in the context of a larger universe of discourse. An embedding of the general type <math>[\mathcal{X}] \to [\mathcal{Y}]\!</math> is implied any time that we make use of one alphabet <math>[\mathcal{X}]\!</math> that happens to be included in another alphabet <math>[\mathcal{Y}].\!</math> When we are discussing differential issues we usually have in mind that the extended alphabet <math>[\mathcal{Y}]\!</math> has a special construction or a specific lexical relation with respect to the initial alphabet <math>[\mathcal{X}],\!</math> one that is marked by characteristic types of accents, indices, or inflected forms.<br />
<br />
====Extension from 1 to 2 Dimensions====<br />
<br />
Figure 18-a lays out the ''angular form'' of venn diagram for universes of 1 and 2 dimensions, indicating the embedding map of type <math>\mathbb{B}^1 \to \mathbb{B}^2\!</math> and detailing the coordinates that are associated with individual cells. Because all points, cells, or logical interpretations are represented as connected geometric areas, we can say that these pictures provide us with an ''areal view'' of each universe of discourse.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-a -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-a.} ~~ \text{Extension from 1 to 2 Dimensions : Areal}\!</math><br />
|}<br />
<br />
Figure 18-b shows the differential extension from <math>X^\bullet = [x]\!</math> to <math>\mathrm{E}X^\bullet = [x, \mathrm{d}x]\!</math> in a ''bundle of boxes'' form of venn diagram. As awkward as it may seem at first, this type of picture is often the most natural and the most easily available representation when we want to conceptualize the localized information or momentary knowledge of an intelligent dynamic system. It gives a ready picture of a ''proposition at a point'', in the present instance, of a proposition about changing states which is itself associated with a particular dynamic state of a system. It is easy to see how this application might be extended to conceive of more general types of instantaneous knowledge that are possessed by a system.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-b -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-b.} ~~ \text{Extension from 1 to 2 Dimensions : Bundle}\!</math><br />
|}<br />
<br />
Figure 18-c shows the same extension in a ''compact'' style of venn diagram, where the differential features at each position are represented by arrows extending from that position that cross or do not cross, as the case may be, the corresponding feature boundaries.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-c -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-c.} ~~ \text{Extension from 1 to 2 Dimensions : Compact}\!</math><br />
|}<br />
<br />
Figure 18-d compresses the picture of the differential extension even further, yielding a directed graph or ''digraph'' form of representation. (Notice that my definition of a digraph allows for loops or ''slings'' at individual points, in addition to arcs or ''arrows'' between the points.)<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-d -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-d.} ~~ \text{Extension from 1 to 2 Dimensions : Digraph}\!</math><br />
|}<br />
<br />
====Extension from 2 to 4 Dimensions====<br />
<br />
Figure 19-a lays out the ''areal view'' or the ''angular form'' of venn diagram for universes of 2 and 4 dimensions, indicating the embedding map of type <math>\mathbb{B}^2 \to \mathbb{B}^4.\!</math> In many ways these pictures are the best kind there is, giving full canvass to an ideal vista. Their style allows the clearest, the fairest, and the plainest view that we can form of a universe of discourse, affording equal representation to all dispositions and maintaining a balance with respect to ordinary and differential features. If only we could extend this view! Unluckily, an obvious difficulty beclouds this prospect, and that is how precipitately we run into the limits of our plane and visual intuitions. Even within the scope of the spare few dimensions that we have scanned up to this point subtle discrepancies have crept in already. The circumstances that bind us and the frameworks that block us, the flat distortion of the planar projection and the inevitable ineffability that precludes us from wrapping its rhomb figure into rings around a torus, all of these factors disguise the underlying but true connectivity of the universe of discourse.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-a -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-a.} ~~ \text{Extension from 2 to 4 Dimensions : Areal}\!</math><br />
|}<br />
<br />
Figure 19-b shows the differential extension from <math>U^\bullet = [u, v]\!</math> to <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v]\!</math> in the ''bundle of boxes'' form of venn diagram.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-b -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-b.} ~~ \text{Extension from 2 to 4 Dimensions : Bundle}\!</math><br />
|}<br />
<br />
As dimensions increase, this factorization of the extended universe along the lines that are marked out by the bundle picture begins to look more and more like a practical necessity. But whenever we use a propositional model to address a real situation in the context of nature we need to remain aware that this articulation into factors, affecting our description, may be wholly artificial in nature and cleave to nothing, no joint in nature, nor any juncture in time to be in or out of joint.<br />
<br />
Figure 19-c illustrates the extension from 2 to 4 dimensions in the ''compact'' style of venn diagram. Here, just the changes with respect to the center cell are shown.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-c -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-c.} ~~ \text{Extension from 2 to 4 Dimensions : Compact}\!</math><br />
|}<br />
<br />
Figure 19-d gives the ''digraph'' form of representation for the differential extension <math>U^\bullet \to \mathrm{E}U^\bullet,\!</math> where the 4 nodes marked with a circle <math>{}^{\bigcirc}\!</math> are the cells <math>uv,\, u \texttt{(} v \texttt{)},\, \texttt{(} u \texttt{)} v,\, \texttt{(} u \texttt{)(} v \texttt{)},\!</math> respectively, and where a 2-headed arc counts as 2 arcs of the differential digraph.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-d -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-d.} ~~ \text{Extension from 2 to 4 Dimensions : Digraph}\!</math><br />
|}<br />
<br />
===Thematization of Functions : And a Declaration of Independence for Variables===<br />
<br />
{| width="100%"<br />
| align="left" |<br />
''And as imagination bodies forth''<br><br />
''The forms of things unknown, the poet's pen''<br><br />
''Turns them to shapes, and gives to airy nothing''<br><br />
''A local habitation and a name.''<br />
| align="right" valign="bottom" | A Midsummer Night's Dream, 5.1.18<br />
|}<br />
<br />
In the representation of propositions as functions it is possible to notice different degrees of explicitness in the way their functional character is symbolized. To indicate what I mean by this, the next series of Figures illustrates a set of graphic conventions that will be put to frequent use in the remainder of this discussion, both to mark the relevant distinctions and to help us convert between related expressions at different levels of explicitness in their functionality.<br />
<br />
====Thematization : Venn Diagrams====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
The known universe has one complete lover and that is the greatest poet. He consumes an eternal passion and is indifferent which chance happens and which possible contingency of fortune or misfortune and persuades daily and hourly his delicious pay.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 11&ndash;12]<br />
|}<br />
<br />
Figure 20-i traces the first couple of steps in this order of ''thematic'' progression, that will gradually run the gamut through a complete series of degrees of functional explicitness in the expression of logical propositions. The first venn diagram represents a situation where the function is indicated by a shaded figure and a logical expression. At this stage one may be thinking of the proposition only as expressed by a formula in a particular language and its content only as a subset of the universe of discourse, as when considering the proposition <math>u\!\cdot\!v</math> in the universe <math>[u, v].\!</math> The second venn diagram depicts a situation in which two significant steps have been taken. First, one has taken the trouble to give the proposition <math>u\!\cdot\!v</math> a distinctive functional name <math>{}^{\backprime\backprime} J {}^{\prime\prime}.\!</math> Second, one has come to think explicitly about the target domain that contains the functional values of <math>J,\!</math> as when writing <math>J : \langle u, v \rangle \to \mathbb{B}.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 20-i -- Thematization of Conjunction (Stage 1).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 20-i.} ~~ \text{Thematization of Conjunction (Stage 1)}\!</math><br />
|}<br />
<br />
In Figure 20-ii the proposition <math>J\!</math> is viewed explicitly as a transformation from one universe of discourse to another.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 20-ii -- Thematization of Conjunction (Stage 2).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 20-ii.} ~~ \text{Thematization of Conjunction (Stage 2)}\!</math><br />
|}<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
o-------------------------------o o-------------------------------o<br />
| | | |<br />
| o-----o o-----o | | o-----o o-----o |<br />
| / \ / \ | | / \ / \ |<br />
| / o \ | | / o \ |<br />
| / /`\ \ | | / /`\ \ |<br />
| o o```o o | | o o```o o |<br />
| | u |```| v | | | | u |```| v | |<br />
| o o```o o | | o o```o o |<br />
| \ \`/ / | | \ \`/ / |<br />
| \ o / | | \ o / |<br />
| \ / \ / | | \ / \ / |<br />
| o-----o o-----o | | o-----o o-----o |<br />
| | | |<br />
o-------------------------------o o-------------------------------o<br />
\ / \ /<br />
\ / \ /<br />
\ / \ J /<br />
\ / \ /<br />
\ / \ /<br />
o----------\---------/----------o o----------\---------/----------o<br />
| \ / | | \ / |<br />
| \ / | | \ / |<br />
| o-----@-----o | | o-----@-----o |<br />
| /`````````````\ | | /`````````````\ |<br />
| /```````````````\ | | /```````````````\ |<br />
| /`````````````````\ | | /`````````````````\ |<br />
| o```````````````````o | | o```````````````````o |<br />
| |```````````````````| | | |```````````````````| |<br />
| |```````` J ````````| | | |```````` x ````````| |<br />
| |```````````````````| | | |```````````````````| |<br />
| o```````````````````o | | o```````````````````o |<br />
| \`````````````````/ | | \`````````````````/ |<br />
| \```````````````/ | | \```````````````/ |<br />
| \`````````````/ | | \`````````````/ |<br />
| o-----------o | | o-----------o |<br />
| | | |<br />
| | | |<br />
o-------------------------------o o-------------------------------o<br />
J = u v x = J<u, v><br />
<br />
Figure 20-ii. Thematization of Conjunction (Stage 2)<br />
</pre><br />
|}<br />
<br />
In the first venn diagram the name that is assigned to a composite proposition, function, or region in the source universe is delegated to a simple feature in the target universe. This can result in a single character or term exceeding the responsibilities it can carry off well. Allowing the name of a function <math>J : \langle u, v \rangle \to \mathbb{B}\!</math> to serve as the name of its dependent variable <math>J : \mathbb{B}\!</math> does not mean that one has to confuse a function with any of its values, but it does put one at risk for a number of obvious problems, and we should not be surprised, on numerous and limiting occasions, when quibbling arises from the attempts of a too original syntax to serve these two masters.<br />
<br />
The second venn diagram circumvents these difficulties by introducing a new variable name for each basic feature of the target universe, as when writing <math>J : \langle u, v \rangle \to \langle x \rangle,\!</math> and thereby assigns a concrete type <math>\langle x \rangle</math> to the abstract codomain <math>\mathbb{B}.\!</math> To make this induction of variables more formal one can append subscripts, as in <math>x_J,\!</math> to indicate the origin or derivation of the new characters. Or we may use a lexical modifier to convert function names into variable names, for example, associating the function name <math>J\!</math> with the variable name <math>\check{J}.\!</math> Thus we may think of <math>x = x_J = \check{J}\!</math> as the ''cache variable'' corresponding to the function <math>J\!</math> or the symbol <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> considered as a contingent variable.<br />
<br />
In Figure 20-iii we arrive at a stage where the functional equations <math>J = u\!\cdot\!v</math> and <math>x = u\!\cdot\!v</math> are regarded as propositions in their own right, reigning in and ruling over the 3-feature universes of discourse <math>[u, v, J]~\!</math> and <math>[u, v, x],\!</math> respectively. Subject to the cautions already noted, the function name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> can be reinterpreted as the name of a feature <math>\check{J}</math> and the equation <math>J = u\!\cdot\!v</math> can be read as the logical equivalence <math>\texttt{((} J, u ~ v \texttt{))}.\!</math> To give it a generic name let us call this newly expressed, collateral proposition the ''thematization'' or the ''thematic extension'' of the original proposition <math>J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 20-iii -- Thematization of Conjunction (Stage 3).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 20-iii.} ~~ \text{Thematization of Conjunction (Stage 3)}\!</math><br />
|}<br />
<br />
The first venn diagram represents the thematization of the conjunction <math>J\!</math> with shading in the appropriate regions of the universe <math>[u, v, J].\!</math> Also, it illustrates a quick way of constructing a thematic extension. First, draw a line, in practice or the imagination, that bisects every cell of the original universe, placing half of each cell under the aegis of the thematized proposition and the other half under its antithesis. Next, on the scene where the theme applies leave the shade wherever it lies, and off the stage, where it plays otherwise, stagger the pattern in a harlequin guise.<br />
<br />
In the final venn diagram of this sequence the thematic progression comes full circle and completes one round of its development. The ambiguities that were occasioned by the changing role of the name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> are resolved by introducing a new variable name <math>{}^{\backprime\backprime} x {}^{\prime\prime}</math> to take the place of <math>\check{J},\!</math> and the region that represents this fresh featured <math>x\!</math> is circumscribed in a more conventional symmetry of form and placement. Just as we once gave the name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> to the proposition <math>u\!\cdot\!v,</math> we now give the name <math>{}^{\backprime\backprime} \iota {}^{\prime\prime}</math> to its thematization <math>\texttt{((} x, u ~ v \texttt{))}.\!</math> Already, again, at this culminating stage of reflection, we begin to think of the newly named proposition as a distinctive individual, a particular function <math>\iota : \langle u, v, x \rangle \to \mathbb{B}.\!</math><br />
<br />
From now on, the terms ''thematic extension'' and ''thematization'' will be used to describe both the process and degree of explication that progresses through this series of pictures, both the operation of increasingly explicit symbolization and the dimension of variation that is swept out by it. To speak of this change in general, that takes us in our current example from <math>J\!</math> to <math>\iota,\!</math> we introduce a class of operators symbolized by the Greek letter <math>\theta,\!</math> writing <math>\iota = \theta J\!</math> in the present instance. The operator <math>\theta,\!</math> in the present situation bearing the type <math>\theta : [u, v] \to [u, v, x],\!</math> provides us with a convenient way of recapitulating and summarizing the complete cycle of thematic developments.<br />
<br />
Figure 21 shows how the thematic extension operator <math>\theta\!</math> acts on two further examples, the disjunction <math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math> and the equality <math>\texttt{((} u, v \texttt{))}.\!</math> Referring to the disjunction as <math>f(u, v)\!</math> and the equality as <math>f(u, v),\!</math> we may express the thematic extensions as <math>\varphi = \theta f\!</math> and <math>\gamma = \theta g.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 21 -- Thematization of Disjunction and Equality.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 21.} ~~ \text{Thematization of Disjunction and Equality}\!</math><br />
|}<br />
<br />
====Thematization : Truth Tables====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
That which distorts honest shapes or which creates unearthly beings or places or contingencies is a nuisance and a revolt.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 19]<br />
|}<br />
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Tables 22 through 25 outline a method for computing the thematic extensions of propositions in terms of their coordinate values.<br />
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A preliminary step, as illustrated in Table&nbsp;22, is to write out the truth table representations of the propositional forms whose thematic extensions one wants to compute, in the present instance, the functions <math>f(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math> and <math>g(u, v) = \texttt{((} u, v \texttt{))}.\!</math><br />
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{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:50%"<br />
|+ style="height:30px" | <math>\text{Table 22.} ~~ \text{Disjunction}~ f ~\text{and Equality}~ g\!</math><br />
|- style="height:35px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black" | <math>f\!</math><br />
| <math>g\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
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Next, each propositional form is individually represented in the fashion shown in Tables&nbsp;23-i and 23-ii, using <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} g {}^{\prime\prime}\!</math> as function names and creating new variables <math>x\!</math> and <math>y\!</math> to hold the associated functional values. This pair of Tables outlines the first stage in the transition from the <math>2\!</math>-dimensional universes of <math>f\!</math> and <math>g\!</math> to the <math>3\!</math>-dimensional universes of <math>\theta f\!</math> and <math>\theta g.\!</math> The top halves of the Tables replicate the truth table patterns for <math>f\!</math> and <math>g\!</math> in the form <math>f : [u, v] \to [x]\!</math> and <math>g : [u, v] \to [y].\!</math> The bottom halves of the tables print the negatives of these pictures, as it were, and paste the truth tables for <math>\texttt{(} f \texttt{)}\!</math> and <math>\texttt{(} g \texttt{)}\!</math> under the copies for <math>f\!</math> and <math>g.\!</math> At this stage, the columns for <math>\theta f\!</math> and <math>\theta g\!</math> are appended almost as afterthoughts, amounting to indicator functions for the sets of ordered triples that make up the functions <math>f\!</math> and <math>g.\!</math><br />
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{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 23-i and 23-ii.} ~~ \text{Thematics of Disjunction and Equality (1)}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 23-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black" | <math>f\!</math><br />
| <math>x\!</math><br />
| <math>\varphi\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |<br />
<math>\begin{matrix}\to\\\to\\\to\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" | &nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 23-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black" | <math>g\!</math><br />
| <math>y\!</math><br />
| <math>\gamma\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |<br />
<math>\begin{matrix}\to\\\to\\\to\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" | &nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|}<br />
|}<br />
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All the data are now in place to give the truth tables for <math>\theta f\!</math> and <math>\theta g.\!</math> All that remains to be done is to permute the rows and change the roles of <math>x\!</math> and <math>y\!</math> from dependent to independent variables. In Tables&nbsp;24-i and 24-ii the rows are arranged in such a way as to put the 3-tuples <math>(u, v, x)\!</math> and <math>(u, v, y)\!</math> in binary numerical order, suitable for viewing as the arguments of the maps <math>\theta f = \varphi : [u, v, x] \to \mathbb{B}\!</math> and <math>\theta g = \gamma : [u, v, y] \to \mathbb{B}.\!</math> Moreover, the structure of the tables is altered slightly, allowing the now vestigial functions <math>\theta f\!</math> and <math>\theta g\!</math> to be passed over without further attention and shifting the heavy vertical bars a notch to the right. In effect, this clinches the fact that the thematic variables <math>x := \check{f}\!</math> and <math>y := \check{g}\!</math> are now treated as independent variables.<br />
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{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 24-i and 24-ii.} ~~ \text{Thematics of Disjunction and Equality (2)}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 24-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>f\!</math><br />
| <math>x\!</math><br />
| style="border-left:1px solid black" | <math>\varphi\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <font size="+2">&nbsp;<br></font><math>\begin{matrix}\to\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 24-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>g\!</math><br />
| <math>y\!</math><br />
| style="border-left:1px solid black" | <math>\gamma\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | &nbsp;<br><math>\begin{matrix}\to\\\to\end{matrix}\!</math><br>&nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
|}<br />
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An optional reshuffling of the rows brings additional features of the thematic extensions to light. Leaving the columns in place for the sake of comparison, Tables&nbsp;25-i and 25-ii sort the rows in a different order, in effect treating <math>x\!</math> and <math>y\!</math> as the primary variables in their respective 3-tuples. Regarding the thematic extensions in the form <math>\varphi : [x, u, v] \to \mathbb{B}\!</math> and <math>\gamma : [y, u, v] \to \mathbb{B}\!</math> makes it easier to see in this tabular setting a property that was graphically obvious in the venn diagrams above. Specifically, when the thematic variable <math>\check{F}\!</math> is true then <math>\theta F\!</math> exhibits the pattern of the original <math>F,\!</math> and when <math>\check{F}\!</math> is false then <math>\theta F\!</math> exhibits the pattern of its negation <math>\texttt{(} F \texttt{)}.\!</math><br />
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{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 25-i and 25-ii.} ~~ \text{Thematics of Disjunction and Equality (3)}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 25-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>f\!</math><br />
| <math>x\!</math><br />
| style="border-left:1px solid black" | <math>\varphi\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>{\to}\!</math><br><font size="+2">&nbsp;<br>&nbsp;<br>&nbsp;<br></font><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <font size="+2">&nbsp;<br></font><math>\begin{matrix}\to\\\to\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 25-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>g\!</math><br />
| <math>y\!</math><br />
| style="border-left:1px solid black" | <math>\gamma\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | &nbsp;<br><math>\begin{matrix}\to\\\to\end{matrix}\!</math><br>&nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
|}<br />
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Finally, Tables&nbsp;26-i and 26-ii compare the tacit extensions <math>\boldsymbol\varepsilon : [u, v] \to [u, v, x]\!</math> and <math>\boldsymbol\varepsilon : [u, v] \to [u, v, y]\!</math> with the thematic extensions of the same types, as applied to the propositions <math>f\!</math> and <math>g,\!</math> respectively.<br />
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{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 26-i and 26-ii.} ~~ \text{Tacit Extension and Thematization}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 26-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>x\!</math><br />
| style="border-left:1px solid black" | <math>\boldsymbol\varepsilon f\!</math><br />
| <math>\theta f\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 26-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>y\!</math><br />
| style="border-left:1px solid black" | <math>\boldsymbol\varepsilon g\!</math><br />
| <math>\theta g\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
|}<br />
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Table 27 summarizes the thematic extensions of all propositions on two variables. Column&nbsp;4 lists the equations of form <math>\texttt{((} \check{f_i}, f_i (u, v) \texttt{))}\!</math> and Column&nbsp;5 simplifies these equations into the form of algebraic expressions. As always, <math>{}^{\backprime\backprime} + {}^{\prime\prime}\!</math> refers to exclusive disjunction and each <math>{}^{\backprime\backprime} \check{f} {}^{\prime\prime}\!</math> appearing in the last two Columns refers to the corresponding variable name <math>{}^{\backprime\backprime} \check{f_i} {}^{\prime\prime}.~\!</math><br />
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{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 27.} ~~ \text{Thematization of Bivariate Propositions}\!</math><br />
|- style="height:30px; background:ghostwhite"<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>{f}\!</math><br />
| <math>\theta f\!</math><br />
| <math>\theta f\!</math><br />
|- style="background:ghostwhite"<br />
| align="right" | <math>u\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| align="right" | <math>v\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| align="left" | <math>\texttt{((} \check{f} \texttt{,~(~)~))}\!</math><br />
| align="left" | <math>\check{f} + 1\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\\[4pt]<br />
f_{4}<br />
\\[4pt]<br />
f_{8}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
1~0~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} u \texttt{)~} v \texttt{~}<br />
\\[4pt]<br />
\texttt{~} u \texttt{~(} v \texttt{)}<br />
\\[4pt]<br />
\texttt{~} u \texttt{~~} v \texttt{~}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(u)(v)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~(u)~v~~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~u~(v)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~u~~v~~))}<br />
\end{array}</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + u + v + uv<br />
\\[4pt]<br />
\check{f} + v + uv + 1<br />
\\[4pt]<br />
\check{f} + u + uv + 1<br />
\\[4pt]<br />
\check{f} + uv + 1<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{3}<br />
\\[4pt]<br />
f_{12}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{)}<br />
\\[4pt]<br />
\texttt{~} u \texttt{~}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(u)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~u~~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + u<br />
\\[4pt]<br />
\check{f} + u + 1<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[4pt]<br />
f_{9}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[4pt]<br />
1~0~0~1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{,} v \texttt{)}<br />
\\[4pt]<br />
\texttt{((} u \texttt{,} v \texttt{))}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~~(} u \texttt{,} v \texttt{)~~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~((} u \texttt{,} v \texttt{))~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + u + v + 1<br />
\\[4pt]<br />
\check{f} + u + v<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{5}<br />
\\[4pt]<br />
f_{10}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} v \texttt{)}<br />
\\[4pt]<br />
\texttt{~} v \texttt{~}<br />
\end{matrix}</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(} v \texttt{)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~} v \texttt{~~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + v<br />
\\[4pt]<br />
\check{f} + v + 1<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[4pt]<br />
f_{11}<br />
\\[4pt]<br />
f_{13}<br />
\\[4pt]<br />
f_{14}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(~} u \texttt{~~} v \texttt{~)}<br />
\\[4pt]<br />
\texttt{(~} u \texttt{~(} v \texttt{))}<br />
\\[4pt]<br />
\texttt{((} u \texttt{)~} v \texttt{~)}<br />
\\[4pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(~} u \texttt{~~} v \texttt{~)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~(~} u \texttt{~(} v \texttt{))~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~((} u \texttt{)~} v \texttt{~)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~((} u \texttt{)(} v \texttt{))~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + uv<br />
\\[4pt]<br />
\check{f} + u + uv<br />
\\[4pt]<br />
\check{f} + v + uv<br />
\\[4pt]<br />
\check{f} + u + v + uv + 1<br />
\end{array}\!</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| align="left" | <math>\texttt{((} \check{f} \texttt{,~((~))~))}\!</math><br />
| align="left" | <math>\check{f}\!</math><br />
|}<br />
<br />
<br><br />
<br />
In order to show what all of the thematic extensions from two dimensions to three dimensions look like in terms of coordinates, Tables&nbsp;28 and 29 present ordinary truth tables for the functions <math>f_i : \mathbb{B}^2 \to \mathbb{B}\!</math> and for the corresponding thematizations <math>\theta f_i = \varphi_i : \mathbb{B}^3 \to \mathbb{B}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 28.} ~~ \text{Propositions on Two Variables}\!</math><br />
|- style="height:35px; background:ghostwhite"<br />
| style="width:5%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:5%; border-bottom:1px solid black; border-left:1px solid black" | <math>f_{0}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{1}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{2}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{3}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{4}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{5}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{6}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{7}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{8}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{9}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{10}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{11}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{12}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{13}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{14}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{15}\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 29.} ~~ \text{Thematic Extensions of Bivariate Propositions}\!</math><br />
|- style="height:35px; background:ghostwhite"<br />
| style="width:5%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\check{f}\!</math><br />
| style="width:5%; border-bottom:1px solid black; border-left:1px solid black" | <math>\varphi_{0}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{1}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{2}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{3}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{4}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{5}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{6}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{7}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{8}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{9}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{10}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{11}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{12}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{13}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{14}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{15}\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
|-<br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Propositional Transformations===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
If only the word &lsquo;artificial&rsquo; were associated with the idea of ''art'', or expert skill gained through voluntary apprenticeship (instead of suggesting the factitious and unreal), we might say that ''logical'' refers to artificial thought.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 56&ndash;57]<br />
|}<br />
<br />
In this section we develop a comprehensive set of concepts for dealing with transformations between universes of discourse. In this most general setting the source and target universes of a transformation are allowed to be different, but may be the same. When we apply these concepts to dynamic systems we focus on the important special case of transformations that map a universe into itself, regarding them as the state transitions of a discrete dynamical process and placing them among the myriad ways that a universe of discourse might change, and by that change turn into itself.<br />
<br />
====Alias and Alibi Transformations====<br />
<br />
There are customarily two modes of understanding a transformation, at least, when we try to interpret its relevance to something in reality. A transformation always refers to a changing prospect, to say it in a unified but equivocal way, but this can be taken to mean either a subjective change in the interpreting observer's point of view or an objective change in the systematic subject of discussion. In practice these variant uses of the transformation concept are distinguished in the following terms:<br />
<br />
# A ''perspectival'' or ''alias'' transformation refers to a shift in perspective or a change in language that takes place in the observer's frame of reference.<br />
# A ''transitional'' or ''alibi'' transformation refers to a change of position or an alteration of state that occurs in the object system as it falls under study.<br />
<br />
(For a recent discussion of the alias vs. alibi issue, as it relates to linear transformations in vector spaces and to other issues of an algebraic nature, see [MaB, 256, 582-4].)<br />
<br />
Naturally, when we are concerned with the dynamical properties of a system, the transitional aspect of transformation is the factor that comes to the fore, and this involves us in contemplating all of the ways of changing a universe into itself while remaining under the rule of established dynamical laws. In the prospective application to dynamic systems, and to neural networks viewed in this light, our interest lies chiefly with the transformations of a state space into itself that constitute the state transitions of a discrete dynamic process. Nevertheless, many important properties of these transformations, and some constructions that we need to see most clearly, are independent of the transitional interpretation and are likely to be confounded with irrelevant features if presented first and only in that association.<br />
<br />
In addition, and in partial contrast, intelligent systems are exactly that species of dynamic agents that have the capacity to have a point of view, and we cannot do justice to their peculiar properties without examining their ability to form and transform their own frames of reference in exposure to the elements of their own experience. In this setting, the perspectival aspect of transformation is the facet that shines most brightly, perhaps too often leaving us fascinated with mere glimmerings of its actual potential. It needs to be emphasized that nothing of the ordinary sort needs be moved in carrying out a transformation under the alias interpretation, that it may only involve a change in the forms of address, an amendment of the terms which are customed to approach and fashioned to describe the very same things in the very same world. But again, working within a discipline of realistic computation, we know how formidably complex and resource-consuming such transformations of perspective can be to implement in practice, much less to endow in the self-governed form of a nascently intelligent dynamical system.<br />
<br />
====Transformations of General Type====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
''Es ist passiert'', &ldquo;it just sort of happened&rdquo;, people said there when other people in other places thought heaven knows what had occurred. It was a peculiar phrase, not known in this sense to the Germans and with no equivalent in other languages, the very breath of it transforming facts and the bludgeonings of fate into something light as eiderdown, as thought itself.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 34]<br />
|}<br />
<br />
Consider the situation illustrated in Figure&nbsp;30, where the alphabets <math>\mathcal{U} = \{ u, v \}\!</math> and <math>\mathcal{X} = \{ x, y, z \}\!</math> are used to label basic features in two different logical universes, <math>U^\bullet = [u, v]\!</math> and <math>X^\bullet = [x, y, z].\!</math><br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
o-------------------------------------------------------o<br />
| U |<br />
| |<br />
| o-----------o o-----------o |<br />
| / \ / \ |<br />
| / o \ |<br />
| / / \ \ |<br />
| / / \ \ |<br />
| o o o o |<br />
| | | | | |<br />
| | u | | v | |<br />
| | | | | |<br />
| o o o o |<br />
| \ \ / / |<br />
| \ \ / / |<br />
| \ o / |<br />
| \ / \ / |<br />
| o-----------o o-----------o |<br />
| |<br />
| |<br />
o---------------------------o---------------------------o<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
o-------------------------o o-------------------------o o-------------------------o<br />
| U | | U | | U |<br />
| o---o o---o | | o---o o---o | | o---o o---o |<br />
| / \ / \ | | / \ / \ | | / \ / \ |<br />
| / o \ | | / o \ | | / o \ |<br />
| / / \ \ | | / / \ \ | | / / \ \ |<br />
| o o o o | | o o o o | | o o o o |<br />
| | u | | v | | | | u | | v | | | | u | | v | |<br />
| o o o o | | o o o o | | o o o o |<br />
| \ \ / / | | \ \ / / | | \ \ / / |<br />
| \ o / | | \ o / | | \ o / |<br />
| \ / \ / | | \ / \ / | | \ / \ / |<br />
| o---o o---o | | o---o o---o | | o---o o---o |<br />
| | | | | |<br />
o-------------------------o o-------------------------o o-------------------------o<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ g | \ f / | h /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ o----------|-----------\-----/-----------|----------o /<br />
\ | X | \ / | | /<br />
\ | | \ / | | /<br />
\ | | o-----o-----o | | /<br />
\| | / \ | |/<br />
\ | / \ | /<br />
|\ | / \ | /|<br />
| \ | / \ | / |<br />
| \ | / \ | / |<br />
| \ | o x o | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \| | | |/ |<br />
| o--o--------o o--------o--o |<br />
| / \ \ / / \ |<br />
| / \ \ / / \ |<br />
| / \ o / \ |<br />
| / \ / \ / \ |<br />
| / \ / \ / \ |<br />
| o o--o-----o--o o |<br />
| | | | | |<br />
| | | | | |<br />
| | | | | |<br />
| | y | | z | |<br />
| | | | | |<br />
| | | | | |<br />
| o o o o |<br />
| \ \ / / |<br />
| \ \ / / |<br />
| \ o / |<br />
| \ / \ / |<br />
| \ / \ / |<br />
| o-----------o o-----------o |<br />
| |<br />
| |<br />
o---------------------------------------------------o<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ p , q /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
o<br />
<br />
Figure 30. Generic Frame of a Logical Transformation<br />
</pre><br />
|}<br />
<br />
Enter the picture, as we usually do, in the middle of things, with features like <math>x, y , z\!</math> that present themselves to be simple enough in their own right and that form a satisfactory, if temporary foundation to provide a basis for discussion. In this universe and on these terms we find expression for various propositions and questions of principal interest to ourselves, as indicated by the maps <math>p, q : X \to \mathbb{B}.\!</math> Then we discover that the simple features <math>\{ x, y, z \}\!</math> are really more complex than we thought at first, and it becomes useful to regard them as functions <math>\{ f, g, h \}\!</math> of other features <math>\{ u, v \}\!</math> that we place in a preface to our original discourse, or suppose as topics of a preliminary universe of discourse <math>U^\bullet = [u, v].\!</math> It may happen that these late-blooming but pre-ambling features are found to lie closer, in a sense that may be our job to determine, to the central nature of the situation of interest, in which case they earn our regard as being more fundamental, but these functions and features are only required to supply a critical stance on the universe of discourse or an alternate perspective on the nature of things in order to be preserved as useful.<br />
<br />
A particular transformation <math>F : [u, v] \to [x, y, z]\!</math> may be expressed by a system of equations, as shown below. Here, <math>F\!</math> is defined by its component maps <math>F = (F_1, F_2, F_3) = (f, g, h),\!</math> where each component map in <math>\{ f, g, h \}\!</math> is a proposition of type <math>\mathbb{B}^n \to \mathbb{B}^1.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:50%"<br />
|<br />
<math>\begin{matrix}<br />
x & = & f(u, v)<br />
\\[10pt]<br />
y & = & g(u, v)<br />
\\[10pt]<br />
z & = & h(u, v)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Regarded as a logical statement, this system of equations expresses a relation between a collection of freely chosen propositions <math>\{ f, g, h \}\!</math> in one universe of discourse and the special collection of simple propositions <math>\{ x, y, z \}\!</math> on which is founded another universe of discourse. Growing familiarity with a particular transformation of discourse, and the desire to achieve a ready understanding of its implications, requires that we be able to convert this information about generals and simples into information about all the main subtypes of propositions, including the linear and singular propositions.<br />
<br />
===Analytic Expansions : Operators and Functors===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Consider what effects that might ''conceivably'' have practical bearings you ''conceive'' the objects of your ''conception'' to have. Then, your ''conception'' of those effects is the whole of your ''conception'' of the object.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; C.S. Peirce, &ldquo;The Maxim of Pragmatism&rdquo;, CP 5.438<br />
|}<br />
<br />
Given the barest idea of a logical transformation, as suggested by the sketch in Figure&nbsp;30, and having conceptualized the universe of discourse, with all of its points and propositions, as a beginning object of discussion, we are ready to enter the next phase of our investigation.<br />
<br />
====Operators on Propositions and Transformations====<br />
<br />
The next step is naturally inclined toward objects of the next higher order, namely, with operators that take in argument lists of logical transformations and that give back specified types of logical transformations as their results. For our present aims, we do not need to consider the most general class of such operators, nor any one of them for its own sake. Rather, we are interested in the special sorts of operators that arise in the study and analysis of logical transformations. Figuratively speaking, these operators serve as instruments for the live tomography (and hopefully not the vivisection) of the forms of change under view. Beyond that, they open up ways to implement the changes of view that we need to grasp all the variations on a transformational theme, or to appreciate enough of its significant features to &ldquo;get the drift&rdquo; of the change occurring, to form a passing acquaintance or a synthetic comprehension of its general character and disposition.<br />
<br />
The simplest type of operator is one that takes a single transformation as an argument and returns a single transformation as a result, and most of the operators explicitly considered in our discussion will be of this kind. Figure&nbsp;31 illustrates the typical situation.<br />
<br />
{| align="center" border="0" cellpadding="20"<br />
|<br />
<pre><br />
o---------------------------------------o<br />
| |<br />
| |<br />
| U% F X% |<br />
| o------------------>o |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| !W! | | !W! |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| v v |<br />
| o------------------>o |<br />
| !W!U% !W!F !W!X% |<br />
| |<br />
| |<br />
o---------------------------------------o<br />
Figure 31. Operator Diagram (1)<br />
</pre><br />
|}<br />
<br />
In this Figure <math>{}^{\backprime\backprime} \mathsf{W} {}^{\prime\prime}\!</math> stands for a generic operator <math>\mathsf{W},\!</math> in this case one that takes a logical transformation <math>F\!</math> of type <math>(U^\bullet \to X^\bullet)\!</math> into a logical transformation <math>\mathsf{W}F\!</math> of type <math>(\mathsf{W}U^\bullet \to \mathsf{W}X^\bullet).\!</math> Thus, the operator <math>\mathsf{W}\!</math> must be viewed as making assignments for both families of objects we have previously considered, that is, for universes of discourse like <math>{U^\bullet}\!</math> and <math>{X^\bullet}\!</math> and for logical transformations like <math>F.\!</math><br />
<br />
'''Note.''' Strictly speaking, an operator like <math>\mathsf{W}\!</math> works between two whole categories of universes and transformations, which we call the ''source'' and the ''target'' categories of <math>\mathsf{W}.\!</math> Given this setting, <math>\mathsf{W}\!</math> specifies for each universe <math>U^\bullet\!</math> in its source category a definite universe <math>\mathsf{W}U^\bullet\!</math> in its target category, and to each transformation <math>F\!</math> in its source category it assigns a unique transformation <math>\mathsf{W}F\!</math> in its target category. Naturally, this only works if <math>\mathsf{W}\!</math> takes the source <math>U^\bullet</math> and the target <math>X^\bullet</math> of the map <math>F\!</math> over to the source <math>\mathsf{W}U^\bullet\!</math> and the target <math>\mathsf{W}X^\bullet\!</math> of the map <math>\mathsf{W}F.\!</math> With luck or care enough, we can avoid ever having to put anything like that in words again, letting diagrams do the work. In the situations of present concern we are usually focused on a single transformation <math>F,\!</math> and thus we can take it for granted that the assignment of universes under <math>\mathsf{W}\!</math> is defined appropriately at the source and target ends of <math>F.\!</math> It is not always the case, though, that we need to use the particular names (like <math>{}^{\backprime\backprime} \mathsf{W}U^\bullet {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \mathsf{W}X^\bullet {}^{\prime\prime}\!</math>) that <math>\mathsf{W}\!</math> assigns by default to its operative image universes. In most contexts we will usually have a prior acquaintance with these universes under other names and it is necessary only that we can tell from the information associated with an operator <math>\mathsf{W}\!</math> what universes they are.<br />
<br />
In Figure&nbsp;31 the maps <math>F\!</math> and <math>\mathsf{W}F\!</math> are displayed horizontally, the way one normally orients functional arrows in a written text, and <math>\mathsf{W}\!</math> rolls the map <math>F\!</math> downward into the images that are associated with <math>\mathsf{W}F.\!</math> In Figure&nbsp;32 the same information is redrawn so that the maps <math>F\!</math> and <math>\mathsf{W}F\!</math> flow down the page, and <math>\mathsf{W}\!</math> unfurls the map <math>F\!</math> rightward into domains that are the eminent purview of <math>\mathsf{W}F.\!</math><br />
<br />
{| align="center" border="0" cellpadding="20"<br />
|<br />
<pre><br />
o---------------------------------------o<br />
| |<br />
| |<br />
| U% !W! !W!U% |<br />
| o------------------>o |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| F | | !W!F |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| v v |<br />
| o------------------>o |<br />
| X% !W! !W!X% |<br />
| |<br />
| |<br />
o---------------------------------------o<br />
Figure 32. Operator Diagram (2)<br />
</pre><br />
|}<br />
<br />
The latter arrangement, as exhibited in Figure&nbsp;32, is more congruent with the thinking about operators that we shall do in the rest of this discussion, since all logical transformations from here on out will be pictured vertically, after the fashion of Figure&nbsp;30.<br />
<br />
====Differential Analysis of Propositions and Transformations====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" | The resultant metaphysical problem now is this: ''Does the man go round the squirrel or not?''<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 43]<br />
|}<br />
<br />
The approach to the differential analysis of logical propositions and transformations of discourse to be pursued here is carried out in terms of particular operators <math>\mathsf{W}\!</math> that act on propositions <math>F\!</math> or on transformations <math>F\!</math> to yield the corresponding operator maps <math>\mathsf{W}F.\!</math> The operator results then become the subject of a series of further stages of analysis, which take them apart into their propositional components, rendering them as a set of purely logical constituents. After this is done, all the parts are then re-integrated to reconstruct the original object in the light of a more complete understanding, at least in ways that enable one to appreciate certain aspects of it with fresh insight.<br />
<br />
* '''Remark on Strategy.''' At this point we run into a set of conceptual difficulties that force us to make a strategic choice in how we proceed. Part of the problem can be remedied by extending our discussion of tacit extensions to the transformational context. But the troubles that remain are much more obstinate and lead us to try two different types of solution. The approach that we develop first makes use of a variant type of extension operator, the ''trope extension'', to be defined below. This method is more conservative and requires less preparation, but has features which make it seem unsatisfactory in the long run. A more radical approach, but one with a better hope of long term success, makes use of the notion of ''contingency spaces''. These are an even more generous type of extended universe than the kind we currently use, but are defined subject to certain internal constraints. The extra work needed to set up this method forces us to put it off to a later stage. However, as a compromise, and to prepare the ground for the next pass, we call attention to the various conceptual difficulties as they arise along the way and try to give an honest estimate of how well our first approach deals with them.<br />
<br />
We now describe in general terms the particular operators that are instrumental to this form of analysis. The main series of operators all have the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathsf{W}<br />
& : &<br />
( U^\bullet \to X^\bullet )<br />
& \to &<br />
( \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet )<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
If we assume that the source universe <math>U^\bullet</math> and the target universe <math>X^\bullet</math> have finite dimensions <math>n\!</math> and <math>k,\!</math> respectively, then each operator <math>\mathsf{W}\!</math> is encompassed by the same abstract type:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathsf{W}<br />
& : &<br />
( [\mathbb{B}^n] \to [\mathbb{B}^k] )<br />
& \to &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k \times \mathbb{D}^k] )<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Since the range features of the operator result <math>\mathsf{W}F : [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k \times \mathbb{D}^k]</math> can be sorted by their ordinary versus differential qualities and the component maps can be examined independently, the complete operator <math>\mathsf{W}\!</math> can be separated accordingly into two components, in the form <math>\mathsf{W} = (\boldsymbol\varepsilon, \mathrm{W}).\!</math> Given a fixed context of source and target universes, <math>\boldsymbol\varepsilon\!</math> is always the same type of operator, a multiple component version of the tacit extension operators that were described earlier. In this context <math>\boldsymbol\varepsilon\!</math> has the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{array}{lccccc}<br />
\text{Concrete type}<br />
& \boldsymbol\varepsilon<br />
& : &<br />
( U^\bullet \to X^\bullet )<br />
& \to &<br />
( \mathrm{E}U^\bullet \to X^\bullet )<br />
\\[10pt]<br />
\text{Abstract type}<br />
& \boldsymbol\varepsilon<br />
& : &<br />
( [\mathbb{B}^n] \to [\mathbb{B}^k] )<br />
& \to &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k] )<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
On the other hand, the operator <math>\mathrm{W}\!</math> is specific to each <math>\mathsf{W}.\!</math> In this context <math>\mathrm{W}\!</math> always has the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{array}{lccccc}<br />
\text{Concrete type}<br />
& W<br />
& : &<br />
( U^\bullet \to X^\bullet )<br />
& \to &<br />
( \mathrm{E}U^\bullet \to \mathrm{d}X^\bullet )<br />
\\[10pt]<br />
\text{Abstract type}<br />
& W<br />
& : &<br />
( [\mathbb{B}^n] \to [\mathbb{B}^k] )<br />
& \to &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{D}^k] )<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
In the types just assigned to <math>\boldsymbol\varepsilon\!</math> and <math>\mathrm{W}\!</math> and by implication to their results <math>\boldsymbol\varepsilon F\!</math> and <math>\mathrm{W}F,\!</math> we have listed the most restrictive ranges defined for them rather than the more expansive target spaces that subsume these ranges. When there is need to recognize both, we may use type indications like the following:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\boldsymbol\varepsilon F<br />
& : &<br />
( \mathrm{E}U^\bullet \to X^\bullet \subseteq \mathrm{E}X^\bullet )<br />
& \cong &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k] \subseteq [\mathbb{B}^k \times \mathbb{D}^k] )<br />
\\[10pt]<br />
WF<br />
& : &<br />
( \mathrm{E}U^\bullet \to \mathrm{d}X^\bullet \subseteq \mathrm{E}X^\bullet )<br />
& \cong &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{D}^k] \subseteq [\mathbb{B}^k \times \mathbb{D}^k] )<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Hopefully, though, a general appreciation of these subsumptions will prevent us from having to make such declarations more often than absolutely necessary.<br />
<br />
In giving names to these operators we try to preserve as much of the traditional nomenclature and as many of the classical associations as possible. The chief difficulty in doing this is occasioned by the distinction between the &ldquo;sans&nbsp;serif&rdquo; operators <math>\mathsf{W}\!</math> and their &ldquo;serified&rdquo; components <math>\mathrm{W},\!</math> which forces us to find two distinct but parallel sets of terminology. Here is a plan to that purpose. First, the component operators <math>\mathrm{W}\!</math> are named by analogy with the corresponding operators in the classical difference calculus. Next, the complete operators <math>\mathsf{W} = (\boldsymbol\varepsilon, \mathrm{W})</math> are assigned titles according to their roles in a geometric or trigonometric allegory, if only to ensure that the tangent functor, that belongs to this family and whose exposition we are still working toward, comes out fit with its customary name. Finally, the operator results <math>\mathsf{W}F\!</math> and <math>\mathrm{W}F\!</math> can be fixed in our frame of reference by tethering the operative adjective for <math>\mathsf{W}\!</math> or <math>\mathrm{W}\!</math> to the anchoring epithet &ldquo;map&rdquo;, in conformity with an already standard practice.<br />
<br />
=====The Secant Operator : '''E'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Mr. Peirce, after pointing out that our beliefs are really rules for action, said that, to develop a thought's meaning, we need only determine what conduct it is fitted to produce: that conduct is for us its sole significance.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 46]<br />
|}<br />
<br />
Figures&nbsp;33-i and 33-ii depict two stages in the form of analysis that will be applied to transformations throughout the remainder of this study. From now on our interest is staked on an operator denoted <math>{}^{\backprime\backprime} \mathsf{E} {}^{\prime\prime},\!</math> which receives the principal investment of analytic attention, and on the constituent parts of <math>\mathsf{E},\!</math> which derive their shares of significance as developed by the analysis. In the sequel, we refer to <math>\mathsf{E}\!</math> as the ''secant operator'', taking it for granted that a context has been chosen that defines its type. The secant operator has the component description <math>\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}),\!</math> and its active ingredient <math>\mathrm{E}\!</math> is known as the ''enlargement operator''. (Here, we name <math>\mathrm{E}\!</math> after the literal ancestor of the shift operator in the calculus of finite differences, defined so that <math>\mathrm{E}f(x) = f(x+1)\!</math> for any suitable function <math>f,\!</math> though of course the logical analogue that we take up here must have a rather different definition.)<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
U% $E$ $E$U% $E$U% $E$U%<br />
o------------------>o============o============o<br />
| | | |<br />
| | | |<br />
| | | |<br />
| | | |<br />
F | | $E$F = | $d$^0.F + | $r$^0.F<br />
| | | |<br />
| | | |<br />
| | | |<br />
v v v v<br />
o------------------>o============o============o<br />
X% $E$ $E$X% $E$X% $E$X%<br />
<br />
Figure 33-i. Analytic Diagram (1)<br />
</pre><br />
|}<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
U% $E$ $E$U% $E$U% $E$U% $E$U%<br />
o------------------>o============o============o============o<br />
| | | | |<br />
| | | | |<br />
| | | | |<br />
| | | | |<br />
F | | $E$F = | $d$^0.F + | $d$^1.F + | $r$^1.F<br />
| | | | |<br />
| | | | |<br />
| | | | |<br />
v v v v v<br />
o------------------>o============o============o============o<br />
X% $E$ $E$X% $E$X% $E$X% $E$X%<br />
<br />
Figure 33-ii. Analytic Diagram (2)<br />
</pre><br />
|}<br />
<br />
In its action on universes <math>\mathsf{E}\!</math> yields the same result as <math>\mathrm{E},\!</math> a fact that can be expressed in equational form by writing <math>\mathsf{E}U^\bullet = \mathrm{E}U^\bullet\!</math> for any universe <math>U^\bullet.\!</math> Notice that the extended universes across the top and bottom of the diagram are indicated to be strictly identical, rather than requiring a corresponding decomposition for them. In a certain sense, the functional parts of <math>\mathsf{E}F\!</math> are partitioned into separate contexts that have to be re-integrated again, but the best image to use is that of making transparent copies of each universe and then overlapping their functional contents once more at the conclusion of the analysis, as suggested by the graphic conventions that are used at the top of Figure&nbsp;30.<br />
<br />
Acting on a transformation <math>F\!</math> from universe <math>U^\bullet\!</math> to universe <math>X^\bullet,\!</math> the operator <math>\mathsf{E}\!</math> determines a transformation <math>\mathsf{E}F\!</math> from <math>\mathsf{E}U^\bullet\!</math> to <math>\mathsf{E}X^\bullet.\!</math> The map <math>\mathsf{E}F\!</math> forms the main body of evidence to be investigated in performing a differential analysis of <math>F.\!</math> Because we shall frequently be focusing on small pieces of this map for considerable lengths of time, and consequently lose sight of the &ldquo;big picture&rdquo;, it is critically important to emphasize that the map <math>\mathsf{E}F\!</math> is a transformation that determines a relation from one extended universe into another. This means that we should not be satisfied with our understanding of a transformation <math>F\!</math> until we can lay out the full &ldquo;parts diagram&rdquo; of <math>\mathsf{E}F\!</math> along the lines of the generic frame in Figure&nbsp;30.<br />
<br />
Working within the confines of propositional calculus, it is possible to give an elementary definition of <math>\mathsf{E}F\!</math> by means of a system of propositional equations, as we now describe.<br />
<br />
Given a transformation<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>F = (F_1, \ldots, F_k) : \mathbb{B}^n \to \mathbb{B}^k\!</math><br />
|}<br />
<br />
of concrete type<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>F : [u_1, \ldots, u_n] \to [x_1, \ldots, x_k],\!</math><br />
|}<br />
<br />
the transformation<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\mathsf{E}F = (F_1, \ldots, F_k, \mathrm{E}F_1, \ldots, \mathrm{E}F_k) : \mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B}^k \times \mathbb{D}^k\!</math><br />
|}<br />
<br />
of concrete type<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\mathsf{E}F : [u_1, \dots, u_n, \mathrm{d}u_1, \dots, \mathrm{d}u_n] \to [x_1, \ldots, x_k, \mathrm{d}x_1, \ldots, \mathrm{d}x_k]\!</math><br />
|}<br />
<br />
is defined by means of the following system of logical equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\\[16pt]<br />
\mathrm{d}x_1<br />
& = & \mathrm{E}F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1 + \mathrm{d}u_1, \ldots, u_n + \mathrm{d}u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
\mathrm{d}x_k<br />
& = & \mathrm{E}F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1 + \mathrm{d}u_1, \ldots, u_n + \mathrm{d}u_n)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
It is important to note that this system of equations can be read as a conjunction of equational propositions, in effect, as a single proposition in the universe of discourse generated by all the named variables. Specifically, this is the universe of discourse over <math>2(n+k)\!</math> variables denoted by:<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
\mathrm{E}[\mathcal{U} \cup \mathcal{X}]<br />
& = &<br />
[u_1, \ldots, u_n, ~ x_1, \ldots, x_k, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n, ~ \mathrm{d}x_1, \ldots, \mathrm{d}x_k].<br />
\end{matrix}</math><br />
|}<br />
<br />
In this light, it should be clear that the system of equations defining <math>\mathsf{E}F\!</math> embodies, in a higher rank and differentially extended version, an analogy with the process of thematization that we treated earlier for propositions of type <math>F : \mathbb{B}^n \to \mathbb{B}.\!</math><br />
<br />
The entire collection of constraints that is represented in the above system of equations may be abbreviated by writing <math>\mathsf{E}F = (\boldsymbol\varepsilon F, \mathrm{E}F),\!</math> for any map <math>F.\!</math> This is tantamount to regarding <math>\mathsf{E}\!</math> as a complex operator, <math>\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}),\!</math> with a form of application that distributes each component of the operator to work on each component of the operand, as follows:<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
\mathsf{E}F<br />
& = &<br />
(\boldsymbol\varepsilon, \mathrm{E})F<br />
& = &<br />
(\boldsymbol\varepsilon F, \mathrm{E}F)<br />
& = &<br />
(\boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k, ~ \mathrm{E}F_1, \ldots, \mathrm{E}F_k).<br />
\end{matrix}</math><br />
|}<br />
<br />
Quite a lot of &ldquo;thematic infrastructure&rdquo; or interpretive information is being swept under the rug in the use of such abbreviations. When confusion arises about the meaning of such constructions, one always has recourse to the defining system of equations, in its totality a purely propositional expression. This means that the parenthesized argument lists, that were used in this context to build an image of multi-component transformations, should not be expected to determine a well-defined product in themselves but only to serve as reminders of the prior thematic decisions (choices of variable names, etc.) that have to be made in order to determine one. Accordingly, the argument list notation can be regarded as a kind of ''thematic frame'', an interpretive storage device that preserves the proper associations of concrete logical features between the extended universes at the source and target of <math>\mathsf{E}F.\!</math><br />
<br />
The generic notations <math>\mathsf{d}^0\!F, \mathsf{d}^1\!F, \ldots, \mathsf{d}^m\!F\!</math> in Figure&nbsp;33 refer to the increasing orders of differentials that are extracted in the course of analyzing <math>F.\!</math> When the analysis is halted at a partial stage of development, notations like <math>\mathsf{r}^0\!F, \mathsf{r}^1\!F, \ldots, \mathsf{r}^m\!F\!</math> may be used to summarize the contributions to <math>\mathsf{E}F\!</math> that remain to be analyzed. The Figure illustrates a convention that makes <math>\mathsf{r}^m\!F,\!</math> in effect, the sum of all differentials of order strictly greater than <math>m.\!</math><br />
<br />
We next discuss the operators that figure into this form of analysis, describing their effects on transformations. In simplified or specialized contexts these operators tend to take on a variety of different names and notations, some of whose number we introduce along the way.<br />
<br />
=====The Radius Operator : '''e'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
And the tangible fact at the root of all our thought-distinctions, however subtle, is that there is no one of them so fine as to consist in anything but a possible difference of practice.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 46]<br />
|}<br />
<br />
The operator identified as <math>\mathrm{d}^0\!</math> in the analytic diagram (Figure&nbsp;33) has the sole purpose of creating a proxy for <math>F\!</math> in the appropriately extended context. Construed in terms of its broadest components, <math>\mathrm{d}^0\!</math> is equivalent to the doubly tacit extension operator <math>(\boldsymbol\varepsilon, \boldsymbol\varepsilon),\!</math> in recognition of which let us redub it as <math>{}^{\backprime\backprime} \mathsf{e} {}^{\prime\prime}.\!</math> Pursuing a geometric analogy, we may refer to <math>\mathsf{e} =(\boldsymbol\varepsilon, \boldsymbol\varepsilon) = \mathrm{d}^0\!</math> as the ''radius operator''. The operation intended by all of these forms is defined by the following equation:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathsf{e}F<br />
& = &<br />
(\boldsymbol\varepsilon, \boldsymbol\varepsilon)F<br />
\\[4pt]<br />
& = &<br />
(\boldsymbol\varepsilon F, ~ \boldsymbol\varepsilon F)<br />
\\[4pt]<br />
& = &<br />
(\boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k, ~ \boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k).<br />
\end{array}</math><br />
|}<br />
<br />
which is tantamount to the system of equations below.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\\[16pt]<br />
\mathrm{d}x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
\mathrm{d}x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
=====The Phantom of the Operators : '''&eta;'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
I was wondering what the reason could be, when I myself raised my head and everything within me seemed drawn towards the Unseen, ''which was playing the most perfect music''!<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 81]<br />
|}<br />
<br />
We now describe an operator whose persistent but elusive action behind the scenes, whose slightly twisted and ambivalent character, and whose fugitive disposition, caught somewhere in flight between the arrantly negative and the positive but errant intent, has cost us some painstaking trouble to detect. In the end we shall place it among the other extensions and projections, as a shade among shadows, of muted tones and motley hue, that adumbrates its own thematic frame and paradoxically lights the way toward a whole new spectrum of values.<br />
<br />
Given a transformation <math>F : [u_1, \ldots, u_n] \to [x_1, \dots, x_k],\!</math> we often have call to consider a family of related transformations, all having the form:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>F^\dagger : [u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n] \to [\mathrm{d}x_1, \dots, \mathrm{d}x_k].\!</math><br />
|}<br />
<br />
The operator <math>\eta</math> is introduced to deal with the simplest one of these maps:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\eta F : [u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n] \to [\mathrm{d}x_1, \ldots \mathrm{d}x_k],\!</math><br />
|}<br />
<br />
which is defined by the following equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
\mathrm{d}x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
In effect, the operator <math>\eta\!</math> is nothing but the stand-alone version of a procedure that is otherwise invoked subordinate to the work of the radius operator <math>\mathsf{e}.\!</math> Operating independently, <math>\eta\!</math> achieves precisely the same results that the second <math>\boldsymbol\varepsilon\!</math> in <math>(\boldsymbol\varepsilon, \boldsymbol\varepsilon)\!</math> accomplishes by working within the context of its ordered pair thematic frame. From this point on, because the use of <math>\boldsymbol\varepsilon\!</math> and <math>\eta\!</math> in this setting combines the aims of both the tacit and the thematic extensions, and because <math>\eta\!</math> reflects in regard to <math>\boldsymbol\varepsilon\!</math> little more than the application of a differential twist, a mere turn of phrase, we refer to <math>\eta\!</math> as the ''trope extension'' operator.<br />
<br />
=====The Chord Operator : '''D'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
What difference would it practically make to any one if this notion rather than that notion were true? If no practical difference whatever can be traced, then the alternatives mean practically the same thing, and all dispute is idle.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 45]<br />
|}<br />
<br />
Next we discuss an operator that is always immanent in this form of analysis, and remains implicitly present in the entire proceeding. It may appear once as a record: a relic or revenant that reprises the reminders of an earlier stage of development. Or it may appear always as a resource: a reserve or redoubt that caches in advance an echo of what remains to be played out, cleared up, and requited in full at a future stage. And all of this remains true whether or not we recall the key at any time, and whether or not the subtending theme is recited explicitly at any stage of play.<br />
<br />
This is the operator that is referred to as <math>\mathsf{r}^0\!</math> in the initial stage of analysis (Figure&nbsp;33-i) and that is expanded as <math>\mathsf{d}^1 + \mathsf{r}^1\!</math> in the subsequent step (Figure&nbsp;33-ii). In congruence, but not quite harmony with our allusions of analogy that are not quite geometry, we call this the ''chord operator'' and denote it <math>\mathsf{D}.\!</math> In the more casual terms that are here introduced, <math>\mathsf{D}</math> is defined as the remainder of <math>\mathsf{E}\!</math> and <math>\mathsf{e}\!</math> and it assigns a due measure to each undertone of accord or discord that is struck between the note of enterprise <math>\mathsf{E}\!</math> and the bar of exigency <math>\mathsf{e}.\!</math><br />
<br />
The tension between these counterposed notions, in balance transient but regular in stridence, may be refracted along familiar lines, though never by any such fraction resolved. In this style we write <math>\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}),\!</math> calling <math>\mathrm{D}\!</math> the ''difference operator'' and noting that it plays a role in this realm of mutable and diverse discourse that is analogous to the part taken by the discrete difference operator in the ordinary difference calculus. Finally, we should note that the chord <math>\mathsf{D}\!</math> is not one that need be lost at any stage of development. At the <math>m^\text{th}\!</math> stage of play it can always be reconstituted in the following form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathsf{D}<br />
& = & \mathsf{E} - \mathsf{e}<br />
\\[6pt]<br />
& = & \mathsf{r}^0<br />
\\[6pt]<br />
& = & \mathsf{d}^1 + \mathsf{r}^1<br />
\\[6pt]<br />
& = & \mathsf{d}^1 + \ldots + \mathsf{d}^m + \mathsf{r}^m<br />
\\[6pt]<br />
& = & \displaystyle \sum_{i=1}^m \mathsf{d}^i + \mathsf{r}^m<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====The Tangent Operator : '''T'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
They take part in scenes of whose significance they have no inkling. They are merely tangent to curves of history the beginnings and ends and forms of which pass wholly beyond their ken. So we are tangent to the wider life of things.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 300]<br />
|}<br />
<br />
The operator tagged as <math>\mathsf{d}^1\!</math> in the analytic diagram (Figure&nbsp;33) is called the ''tangent operator'' and is usually denoted in this text as <math>\mathsf{d}\!</math> or <math>\mathsf{T}.\!</math> Because it has the properties required to qualify as a functor, namely, preserving the identity element of the composition operation and the articulated form of every composition of transformations, it also earns the title of a ''tangent functor''. According to the custom adopted here, we dissect it as <math>\mathsf{T} = \mathsf{d} = (\boldsymbol\varepsilon, \mathrm{d}),\!</math> where <math>\mathrm{d}\!</math> is the operator that yields the first order differential <math>\mathrm{d}F\!</math> when applied to a transformation <math>F,\!</math> and whose name is legion.<br />
<br />
Figure&nbsp;34 illustrates a stage of analysis where we ignore everything but the tangent functor <math>\mathsf{T}\!</math> and attend to it chiefly as it bears on the first order differential <math>\mathrm{d}F\!</math> in the analytic expansion of <math>F.\!</math> In this situation we often refer to the extended universes <math>\mathrm{E}U^\bullet\!</math> and <math>\mathrm{E}X^\bullet\!</math> under the equivalent designations <math>\mathsf{T}U^\bullet\!</math> and <math>\mathsf{T}X^\bullet,\!</math> respectively. The purpose of the tangent functor <math>\mathsf{T}\!</math> is to extract the tangent map <math>\mathsf{T}F\!</math> at each point of <math>U^\bullet,\!</math> and the tangent map <math>\mathsf{T}F = (\boldsymbol\varepsilon, \mathrm{d})F\!</math> tells us not only what the transformation <math>F\!</math> is doing at each point of the universe <math>U^\bullet\!</math> but also what <math>F\!</math> is doing to states in the neighborhood of that point, approximately, linearly, and relatively speaking.<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
U% $T$ $T$U% $T$U%<br />
o------------------>o============o<br />
| | |<br />
| | |<br />
| | |<br />
| | |<br />
F | | $T$F = | <!e!, d> F<br />
| | |<br />
| | |<br />
| | |<br />
v v v<br />
o------------------>o============o<br />
X% $T$ $T$X% $T$X%<br />
<br />
Figure 34. Tangent Functor Diagram<br />
</pre><br />
|}<br />
<br />
* '''NB.''' There is one aspect of the preceding construction that remains especially problematic. Why did we define the operators <math>\mathrm{W}\!</math> in <math>\{ \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}\!</math> so that the ranges of their resulting maps all fall within the realms of differential quality, even fabricating a variant of the tacit extension operator to have that character? Clearly, not all of the operator maps <math>\mathrm{W}F\!</math> have equally good reasons for placing their values in differential stocks. The reason for it appears to be that, without doing this, we cannot justify the comparison and combination of their functional values in the various analytic steps. By default, only those values in the same functional component can be brought into algebraic modes of interaction. Up till now the only mechanism provided for their broader association has been a purely logical one, their common placement in a target universe of discourse, but the task of converting this logical circumstance into algebraic forms of application has not yet been taken up.<br />
<br />
===Transformations of Type '''B'''<sup>2</sup> &rarr; '''B'''<sup>1</sup>===<br />
<br />
To study the effects of these analytic operators in the simplest possible setting, let us revert to a still more primitive case. Consider the singular proposition <math>J(u, v)= u\!\cdot\!v,\!</math> regarded either as the functional product of the maps <math>u\!</math> and <math>v\!</math> or as the logical conjunction of the features <math>u\!</math> and <math>v,\!</math> a map whose fiber of truth <math>J^{-1}(1)\!</math> picks out the single cell of that logical description in the universe of discourse <math>U^\bullet.\!</math> Thus <math>J,\!</math> or <math>u\!\cdot\!v,\!</math> may be treated as another name for the point whose coordinates are <math>(1, 1)\!</math> in <math>U^\bullet.\!</math><br />
<br />
====Analytic Expansion of Conjunction====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
<p>In her sufferings she read a great deal and discovered that she had lost something, the possession of which she had previously not been much aware of: a&nbsp;soul.</p><br />
<br />
<p>What is that? It is easily defined negatively: it is simply what curls up and hides when there is any mention of algebraic series.</p><br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 118]<br />
|}<br />
<br />
Figure&nbsp;35 pictures the form of conjunction <math>J : \mathbb{B}^2 \to \mathbb{B}\!</math> as a transformation from the <math>2\!</math>-dimensional universe <math>[u, v]\!</math> to the <math>1\!</math>-dimensional universe <math>[x].\!</math> This is a subtle but significant change of viewpoint on the proposition, attaching an arbitrary but concrete quality to its functional value. Using the language introduced earlier, we can express this change by saying that the proposition <math>J : \langle u, v \rangle \to \mathbb{B}\!</math> is being recast into the thematized role of a transformation <math>J : [u, v] \to [x],\!</math> where the new variable <math>x\!</math> takes the part of a thematic variable <math>\check{J}.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 35 -- A Conjunction Viewed as a Transformation.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 35.} ~~ \text{Conjunction as Transformation}\!</math><br />
|}<br />
<br />
=====Tacit Extension of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
I teach straying from me, yet who can stray from me?<br><br />
I follow you whoever you are from the present hour;<br><br />
My words itch at your ears till you understand them.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 83]<br />
|}<br />
<br />
Earlier we defined the tacit extension operators <math>\boldsymbol\varepsilon : X^\bullet \to Y^\bullet\!</math> as maps embedding each proposition of a given universe <math>X^\bullet~\!</math> in a more generously given universe <math>Y^\bullet \supset X^\bullet.\!</math> Of immediate interest are the tacit extensions <math>\boldsymbol\varepsilon : U^\bullet \to \mathrm{E}U^\bullet,\!</math> that locate each proposition of <math>U^\bullet\!</math> in the enlarged context of <math>\mathrm{E}U^\bullet.\!</math> In its application to the propositional conjunction <math>J = u\!\cdot\!v</math> in <math>[u, v],\!</math> the tacit extension operator <math>\boldsymbol\varepsilon\!</math> yields the proposition <math>\boldsymbol\varepsilon J\!</math> in <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v].\!</math> The extended proposition <math>\boldsymbol\varepsilon J\!</math> may be computed according to the scheme in Table&nbsp;36, in effect doing nothing more that conjoining a tautology of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> to <math>J\!</math> in <math>U^\bullet.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 36.} ~~ \text{Computation of}~ \boldsymbol\varepsilon J\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\boldsymbol\varepsilon J & = & J {}_{^\langle} u, v {}_{^\rangle}<br />
\\[4pt]<br />
& = & u \cdot v<br />
\\[4pt]<br />
& = & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{ } \mathrm{d}v \texttt{ }<br />
& + & u \cdot v \cdot \texttt{ } \mathrm{d}u \texttt{ } \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \cdot v \cdot \texttt{ } \mathrm{d}u \texttt{ } \cdot \texttt{ } \mathrm{d}v \texttt{ }<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{array}{*{4}{l}}<br />
\boldsymbol\varepsilon J<br />
& = && u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & u \cdot v \cdot \texttt{~} \mathrm{d}u \texttt{~} \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & u \cdot v \cdot \texttt{~} \mathrm{d}u \texttt{~} \cdot \texttt{~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
The lower portion of the Table contains the dispositional features of <math>\boldsymbol\varepsilon J\!</math> arranged in such a way that the variety of ordinary features spreads across the rows and the variety of differential features runs through the columns. This organization serves to facilitate pattern matching in the remainder of our computations. Again, the tacit extension is usually so trivial a concern that we do not always bother to make an explicit note of it, taking it for granted that any function <math>F\!</math> being employed in a differential context is equivalent to <math>\boldsymbol\varepsilon F\!</math> for a suitable <math>\boldsymbol\varepsilon.\!</math><br />
<br />
Figures&nbsp;37-a through 37-d present several pictures of the proposition <math>J\!</math> and its tacit extension <math>\boldsymbol\varepsilon J.\!</math> Notice in these Figures how <math>\boldsymbol\varepsilon J\!</math> in <math>\mathrm{E}U^\bullet\!</math> visibly extends <math>J\!</math> in <math>U^\bullet\!</math> by annexing to the indicated cells of <math>J\!</math> all the arcs that exit from or flow out of them. In effect, this extension attaches to these cells all the dispositions that spring from them, in other words, it attributes to these cells all the conceivable changes that are their issue.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-a -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-a.} ~~ \text{Tacit Extension of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-b -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-b.} ~~ \text{Tacit Extension of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-c -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-c.} ~~ \text{Tacit Extension of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-d -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-d.} ~~ \text{Tacit Extension of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
The computational scheme shown in Table&nbsp;36 treated <math>J\!</math> as a proposition in <math>U^\bullet\!</math> and formed <math>\boldsymbol\varepsilon J\!</math> as a proposition in <math>\mathrm{E}U^\bullet.\!</math> When <math>J\!</math> is regarded as a mapping <math>J : U^\bullet \to X^\bullet\!</math> then <math>\boldsymbol\varepsilon J\!</math> must be obtained as a mapping <math>\boldsymbol\varepsilon J : \mathrm{E}U^\bullet \to X^\bullet.\!</math> By default, the tacit extension of the map <math>J : [u, v] \to [x]\!</math> is naturally taken to be a particular map,<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\boldsymbol\varepsilon J : [u, v, \mathrm{d}u, \mathrm{d}v] \to [x] \subseteq [x, \mathrm{d}x],\!</math><br />
|}<br />
<br />
namely, the one that looks like <math>J\!</math> when painted in the frame of the extended source universe and that takes the same thematic variable in the extended target universe as the one that <math>J\!</math> already takes.<br />
<br />
But the choice of a particular thematic variable, for example <math>x\!</math> for <math>\check{J},\!</math> is a shade more arbitrary than the choice of original variable names <math>\{ u, v \},\!</math> so the map we are calling the ''trope extension'',<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\eta J : [u, v, \mathrm{d}u, \mathrm{d}v] \to [\mathrm{d}x] \subseteq [x, \mathrm{d}x],\!</math><br />
|}<br />
<br />
since it looks just the same as <math>\boldsymbol\varepsilon J\!</math> in the way its fibers paint the source domain, belongs just as fully to the family of tacit extensions, generically considered.<br />
<br />
These considerations have the practical consequence that all of our computations and illustrations of <math>\boldsymbol\varepsilon J\!</math> perform the double duty of capturing <math>\eta J\!</math> as well. In other words, we are saved the work of carrying out calculations and drawing figures for the trope extension <math>\eta J,\!</math> because it would be identical to the work already done for <math>\boldsymbol\varepsilon J.\!</math> Since the computations given for <math>\boldsymbol\varepsilon J\!</math> are expressed solely in terms of the variables <math>\{ u, v, \mathrm{d}u, \mathrm{d}v \},\!</math> they work equally well for finding <math>\eta J.\!</math> Further, since each of the above Figures shows only how the level sets of <math>\boldsymbol\varepsilon J\!</math> partition the extended source universe <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v],\!</math> all of them serve equally well as portraits of <math>\eta J.\!</math><br />
<br />
=====Enlargement Map of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
No one could have established the existence of any details that might not just as well have existed in earlier times too; but all the relations between things had shifted slightly. Ideas that had once been of lean account grew fat.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 62]<br />
|}<br />
<br />
The enlargement map <math>\mathrm{E}J\!</math> is computed from the proposition <math>J\!</math> by making a particular class of formal substitutions for its variables, in this case <math>u + \mathrm{d}u\!</math> for <math>u\!</math> and <math>v + \mathrm{d}v\!</math> for <math>v,\!</math> and afterwards expanding the result in whatever way is found convenient.<br />
<br />
Table&nbsp;38 shows a typical scheme of computation, following a systematic method of exploiting boolean expansions over selected variables and ultimately developing <math>\mathrm{E}J\!</math> over the cells of <math>[u, v].\!</math> The critical step of this procedure uses the facts that <math>\texttt{(} 0, x \texttt{)} = 0 + x = x\!</math> and <math>\texttt{(} 1, x \texttt{)} = 1 + x = \texttt{(} x \texttt{)}\!</math> for any boolean variable <math>x.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 38.} ~~ \text{Computation of}~ \mathrm{E}J ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{E}J & = & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}<br />
\\[4pt]<br />
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
& = & \texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot J_{(1 + \mathrm{d}u, 1 + \mathrm{d}v)}<br />
& + & \texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot J_{(1 + \mathrm{d}u, \mathrm{d}v)}<br />
& + & \texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot J_{(\mathrm{d}u, 1 + \mathrm{d}v)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot J_{(\mathrm{d}u, \mathrm{d}v)}<br />
\\[4pt]<br />
& = &<br />
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot J_{(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})}<br />
& + &<br />
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot J_{(\texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v)}<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot J_{(\mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)})}<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot J_{(\mathrm{d}u, \mathrm{d}v)}<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{E}J<br />
& = &<br />
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&& + &<br />
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{ } \mathrm{d}v \texttt{ }<br />
\\[4pt]<br />
&&&&& + &<br />
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot \texttt{ } \mathrm{d}u \texttt{ } \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&&&&&& + &<br />
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \texttt{ } \mathrm{d}u \texttt{ }~\texttt{ } \mathrm{d}v \texttt{ }<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;39 exhibits another method that happens to work quickly in this particular case, using distributive laws to multiply things out in an algebraic manner, arranging the notations of feature and fluxion according to a scale of simple character and degree. Proceeding this way leads through an intermediate step which, in chiming the changes of ordinary calculus, should take on a familiar ring. Consequential properties of exclusive disjunction then carry us on to the concluding line.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 39.} ~~ \text{Computation of}~ \mathrm{E}J ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{c}}<br />
\mathrm{E}J<br />
& = & (u + \mathrm{d}u) \cdot (v + \mathrm{d}v)<br />
\\[6pt]<br />
& = & u \cdot v<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = &<br />
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{ } \mathrm{d}v \texttt{ }<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot \texttt{ } \mathrm{d}u \texttt{ } \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \texttt{ } \mathrm{d}u \texttt{ }~\texttt{ } \mathrm{d}v \texttt{ }<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;40-a through 40-d present several views of the enlarged proposition <math>\mathrm{E}J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-a -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-a.} ~~ \text{Enlargement of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-b -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-b.} ~~ \text{Enlargement of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-c -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-c.} ~~ \text{Enlargement of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-d -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-d.} ~~ \text{Enlargement of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
An intuitive reading of the proposition <math>\mathrm{E}J\!</math> becomes available at this point. Recall that propositions in the extended universe <math>\mathrm{E}U^\bullet\!</math> express the ''dispositions'' of a system and the constraints that are placed on them. In other words, a differential proposition in <math>\mathrm{E}U^\bullet\!</math> can be read as referring to various changes that a system might undergo in and from its various states. In particular, we can understand <math>\mathrm{E}J\!</math> as a statement that tells us what changes need to be made with regard to each state in the universe of discourse in order to reach the ''truth'' of <math>J,\!</math> that is, the region of the universe where <math>J\!</math> is true. This interpretation is visibly clear in the Figures above and appeals to the imagination in a satisfying way but it has the added benefit of giving fresh meaning to the original name of the shift operator <math>\mathrm{E}.\!</math> Namely, <math>\mathrm{E}J\!</math> can be read as a proposition that ''enlarges'' on the meaning of <math>J,\!</math> in the sense of explaining its practical bearings and clarifying what it means in terms of actions and effects &mdash; the available options for differential action and the consequential effects that result from each choice.<br />
<br />
Read this way, the enlargement <math>\mathrm{E}J\!</math> has strong ties to the normal use of <math>J,\!</math> no matter whether it is understood as a proposition or a function, namely, to act as a figurative device for indicating the models of <math>J,\!</math> in effect, pointing to the interpretive elements in its fiber of truth <math>J^{-1}(1).\!</math> It is this kind of &ldquo;use&rdquo; that is often contrasted with the &ldquo;mention&rdquo; of a proposition, and thereby hangs a tale.<br />
<br />
=====Digression : Reflection on Use and Mention=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Reflection is turning a topic over in various aspects and in various lights so that nothing significant about it shall be overlooked &mdash; almost as one might turn a stone over to see what its hidden side is like or what is covered by it.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 57]<br />
|}<br />
<br />
The contrast drawn in logic between the ''use'' and the ''mention'' of a proposition corresponds to the difference that we observe in functional terms between using <math>{}^{\backprime\backprime} J \, {}^{\prime\prime}\!</math> to indicate the region <math>J^{-1}(1)\!</math> and using <math>{}^{\backprime\backprime} J \, {}^{\prime\prime}\!</math> to indicate the function <math>J.\!</math> You may think that one of these uses ought to be proscribed, and logicians are quick to prescribe against their confusion. But there seems to be no likelihood in practice that their interactions can be avoided. If the name <math>{}^{\backprime\backprime} J \, {}^{\prime\prime}\!</math> is used as a sign of the function <math>J,\!</math> and if the function <math>J\!</math> has its use in signifying something else, as would constantly be the case when some future theory of signs has given a functional meaning to every sign whatsoever, then is not <math>J,\!</math> by transitivity a sign of the thing itself? There are, of course, two answers to this question. Not every act of signifying or referring need be transitive. Not every warrant or guarantee or certificate is automatically transferable, indeed, not many. Not every feature of a feature is a feature of the featuree. Otherwise, if a buffalo is white, and white is a color, then a buffalo would ''be'' a color.<br />
<br />
The logical or pragmatic distinction between use and mention is cogent and necessary, and so is the analogous functional distinction between determining a value and determining what determines that value, but so are the normal techniques that we use to make these distinctions apply flexibly in practice. The way that the hue and cry about use and mention is raised in logical discussions, you might be led to think that this single dimension of choices embraces the only kinds of use worth mentioning and the only kinds of mention worth using. It will constitute the expeditionary and taxonomic tasks of that future theory of signs to explore and to classify the many other constellations and dimensions of use and mention that are yet to be opened up by the generative potential of full-fledged sign relations.<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
The well-known capacity that thoughts have &mdash; as doctors have discovered &mdash; for dissolving and dispersing those hard lumps of deep, ingrowing, morbidly entangled conflict that arise out of gloomy regions of the self probably rests on nothing other than their social and worldly nature, which links the individual being with other people and things; but unfortunately what gives them their power of healing seems to be the same as what diminishes the quality of personal experience in them.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 130]<br />
|}<br />
<br />
=====Difference Map of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
&ldquo;It doesn't matter what one does,&rdquo; the Man Without Qualities said to himself, shrugging his shoulders. &ldquo;In a tangle of forces like this it doesn't make a scrap of difference.&rdquo; He turned away like a man who has learned renunciation, almost indeed like a sick man who shrinks from any intensity of contact. And then, striding through his adjacent dressing-room, he passed a punching-ball that hung there; he gave it a blow far swifter and harder than is usual in moods of resignation or states of weakness.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 8]<br />
|}<br />
<br />
With the tacit extension map <math>\boldsymbol\varepsilon J\!</math> and the enlargement map <math>\mathrm{E}J\!</math> well in place, the difference map <math>\mathrm{D}J\!</math> can be computed along the lines displayed in Table&nbsp;41, ending up with an expansion of <math>\mathrm{D}J\!</math> over the cells of <math>[u, v].\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 41.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & \mathrm{E}J<br />
& + & \boldsymbol\varepsilon J<br />
\\[6pt]<br />
& = & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}<br />
& + & J_{(u, v)}<br />
\\[6pt]<br />
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \cdot v<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = &<br />
u \cdot v \cdot \qquad 0<br />
\\[6pt]<br />
& + &<br />
u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v<br />
& + &<br />
u \cdot \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v<br />
\\[6pt]<br />
& + &<br />
u \cdot v \cdot \texttt{~} \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
&&& + &<br />
\texttt{(} u \texttt{)} \cdot v \cdot \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
& + &<br />
u \cdot v \cdot \texttt{~} \mathrm{d}u \;\cdot\; \mathrm{d}v \texttt{~}<br />
&&&&& + &<br />
\texttt{(} u \texttt{)} \cdot \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = &<br />
u \cdot v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
u \cdot \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v<br />
& + &<br />
\texttt{(} u \texttt{)} \cdot v \cdot \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)} \cdot \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Alternatively, the difference map <math>\mathrm{D}J\!</math> can be expanded over the cells of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> to arrive at the formulation shown in Table&nbsp;42. The same development would be obtained from the previous Table by collecting terms in an alternate manner, along the rows rather than the columns in the middle portion of the Table.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 42.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & \boldsymbol\varepsilon J<br />
& + & \mathrm{E}J<br />
\\[6pt]<br />
& = & J_{(u, v)}<br />
& + & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}<br />
\\[6pt]<br />
& = & u \cdot v<br />
& + & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
& = & 0<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}J<br />
& = & 0<br />
& + & u \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Even more simply, the same result is reached by matching up the propositional coefficients of <math>\boldsymbol\varepsilon J</math> and <math>\mathrm{E}J\!</math> along the cells of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> and adding the pairs under boolean addition, that is, &ldquo;mod 2&rdquo;, where 1&nbsp;+&nbsp;1&nbsp;=&nbsp;0, as shown in Table&nbsp;43.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 43.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 3)}\!</math><br />
|<br />
<math>\begin{array}{*{5}{l}}<br />
\mathrm{D}J & = & \boldsymbol\varepsilon J & + & \mathrm{E}J<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\boldsymbol\varepsilon J<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ u \,\cdot\, v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~ u \;\cdot\; v \;\cdot\; \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u ~ \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} ~ v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot\, \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & ~~ 0 ~~ \,\cdot\, ~ \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~~ u ~ \,\cdot\, ~~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ ~ v ~~ \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
The difference map <math>\mathrm{D}J\!</math> can also be given a ''dispositional'' interpretation. First, recall that <math>\boldsymbol\varepsilon J\!</math> exhibits the dispositions to change from anywhere in <math>J\!</math> to anywhere at all in the universe of discourse and <math>\mathrm{E}J\!</math> exhibits the dispositions to change from anywhere in the universe to anywhere in <math>J.\!</math> Next, observe that each of these classes of dispositions may be divided in accordance with the case of <math>J\!</math> versus <math>\texttt{(} J \texttt{)}\!</math> that applies to their points of departure and destination, as shown below. Then, since the dispositions corresponding to <math>\boldsymbol\varepsilon J</math> and <math>\mathrm{E}J\!</math> have in common the dispositions to preserve <math>J,\!</math> their symmetric difference <math>\texttt{(} \boldsymbol\varepsilon J, \mathrm{E}J \texttt{)}\!</math> is made up of all the remaining dispositions, which are in fact disposed to cross the boundary of <math>J\!</math> in one direction or the other. In other words, we may conclude that <math>\mathrm{D}J\!</math> expresses the collective disposition to make a definite change with respect to <math>J,\!</math> no matter what value it holds in the current state of affairs.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
\boldsymbol\varepsilon J<br />
& = & \{ \text{Dispositions from}~ J ~\text{to}~ J \}<br />
& + & \{ \text{Dispositions from}~ J ~\text{to}~ \texttt{(} J \texttt{)} \}<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = & \{ \text{Dispositions from}~ J ~\text{to}~ J \}<br />
& + & \{ \text{Dispositions from}~ \texttt{(} J \texttt{)} ~\text{to}~ J \}<br />
\\[6pt]<br />
\mathrm{D}J<br />
& = & \{ \text{Dispositions from}~ J ~\text{to}~ \texttt{(} J \texttt{)} \}<br />
& + & \{ \text{Dispositions from}~ \texttt{(} J \texttt{)} ~\text{to}~ J \}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;44-a through 44-d illustrate the difference proposition <math>\mathrm{D}J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-a -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-a.} ~~ \text{Difference Map of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-b -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-b.} ~~ \text{Difference Map of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-c -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-c.} ~~ \text{Difference Map of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-d -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-d.} ~~ \text{Difference Map of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
=====Differential of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
By deploying discourse throughout a calendar, and by giving a date to each of its elements, one does not obtain a definitive hierarchy of precessions and originalities; this hierarchy is never more than relative to the systems of discourse that it sets out to evaluate.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Archaeology of Knowledge'', [Fou, 143]<br />
|}<br />
<br />
Finally, at long last, the differential proposition <math>\mathrm{d}J\!</math> can be gleaned from the difference proposition <math>\mathrm{D}J\!</math> by ranging over the cells of <math>[u, v]\!</math> and picking out the linear proposition of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> that is &ldquo;closest&rdquo; to the portion of <math>\mathrm{D}J\!</math> that touches on each point. The idea of distance that would give this definition unequivocal sense has been referred to in cautionary quotes, the kind we use to distance ourselves from taking a final position. There are obvious notions of approximation that suggest themselves, but finding one that can be justified as ultimately correct is not as straightforward as it seems.<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
He had drifted into the very heart of the world. From him to the distant beloved was as far as to the next tree.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 144]<br />
|}<br />
<br />
Let us venture a guess as to where these developments might be heading. From the present vantage point it appears that the ultimate answer to the quandary of distances and the question of a fitting measure may be that, rather than having the constitution of an analytic series depend on our familiar notions of approach, proximity, and approximation, it will be found preferable, and perhaps unavoidable, to turn the tables and let the orders of approximation be defined in terms of our favored and operative notions of formal analysis. Only the aftermath of this conversion, if it does converge, could be hoped to prove whether this hortatory form of analysis and the cohort idea of an analytic form &mdash; the limitary concept of a self-corrective process and the coefficient concept of a completable product &mdash; are truly (in practical reality) the more inceptive and persistent of principles and really (for all practical purposes) the more effective and regulative of ideas.<br />
<br />
Awaiting that determination, I proceed with what seems like the obvious course, and compute <math>\mathrm{d}J\!</math> according to the pattern in Table&nbsp;45.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 45.} ~~ \text{Computation of}~ \mathrm{d}J\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}J<br />
& = &<br />
u\!\cdot\!v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
u \, \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \, \mathrm{d}v<br />
& + &<br />
\texttt{(} u \texttt{)} \, v \cdot \mathrm{d}u \, \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \!\cdot\! \mathrm{d}v \texttt{~}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}J<br />
& = &<br />
u\!\cdot\!v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + &<br />
u \, \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + &<br />
\texttt{(} u \texttt{)} \, v \cdot \mathrm{d}u<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;46-a through 46-d illustrate the proposition <math>{\mathrm{d}J},\!</math> rounded out in our usual array of prospects. This proposition of <math>\mathrm{E}U^\bullet\!</math> is what we refer to as the (first order) differential of <math>J,\!</math> and normally regard as ''the'' differential proposition corresponding to <math>J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-a -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-a.} ~~ \text{Differential of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-b -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-b.} ~~ \text{Differential of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-c -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-c.} ~~ \text{Differential of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-d -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-d.} ~~ \text{Differential of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
=====Remainder of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
<p>I bequeath myself to the dirt to grow from the grass I love,<br><br />
If you want me again look for me under your bootsoles.</p><br />
<br />
<p>You will hardly know who I am or what I mean,<br><br />
But I shall be good health to you nevertheless,<br><br />
And filter and fibre your blood.</p><br />
<br />
<p>Failing to fetch me at first keep encouraged,<br><br />
Missing me one place search another,<br><br />
I stop some where waiting for you</p><br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]<br />
|}<br />
<br />
<br><br />
<br />
Let us recapitulate the story so far. We have in effect been carrying out a decomposition of the enlarged proposition <math>\mathrm{E}J\!</math> in a series of stages. First, we considered the equation <math>\mathrm{E}J = \boldsymbol\varepsilon J + \mathrm{D}J,\!</math> which was involved in the definition of <math>\mathrm{D}J\!</math> as the difference <math>\mathrm{E}J - \boldsymbol\varepsilon J.\!</math> Next, we contemplated the equation <math>\mathrm{D}J = \mathrm{d}J + \mathrm{r}J,\!</math> which expresses <math>\mathrm{D}J\!</math> in terms of two components, the differential <math>\mathrm{d}J\!</math> that was just extracted and the residual component <math>\mathrm{r}J = \mathrm{D}J - \mathrm{d}J.~\!</math> This remaining proposition <math>\mathrm{r}J\!</math> can be computed as shown in Table&nbsp;47.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 47.} ~~ \text{Computation of}~ \mathrm{r}J\!</math><br />
|<br />
<math>\begin{array}{*{5}{l}}<br />
\mathrm{r}J & = & \mathrm{D}J & + & \mathrm{d}J<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}J<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{r}J ~<br />
& = & u \!\cdot\! v \cdot ~ \mathrm{d}u \cdot \mathrm{d}v ~ ~ ~ ~ ~<br />
& + & u \texttt{(} v \texttt{)} \cdot \, \mathrm{d}u \cdot \mathrm{d}v \,<br />
& + & \texttt{(} u \texttt{)} v \cdot \, \mathrm{d}u \cdot \mathrm{d}v \,<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \, \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
As it happens, the remainder <math>\mathrm{r}J\!</math> falls under the description of a second order differential <math>\mathrm{r}J = \mathrm{d}^2 J.\!</math> This means that the expansion of <math>\mathrm{E}J\!</math> in the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|<br />
<math>\begin{array}{*{7}{l}}<br />
\mathrm{E}J<br />
& = & \boldsymbol\varepsilon J<br />
& + & \mathrm{D}J<br />
\\[6pt]<br />
& = & \boldsymbol\varepsilon J<br />
& + & \mathrm{d}J<br />
& + & \mathrm{r}J<br />
\\[6pt]<br />
& = & \mathrm{d}^0 J<br />
& + & \mathrm{d}^1 J<br />
& + & \mathrm{d}^2 J<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
which is nothing other than the propositional analogue of a Taylor series, is a decomposition that terminates in a finite number of steps.<br />
<br />
Figures&nbsp;48-a through 48-d illustrate the proposition <math>\mathrm{r}J = \mathrm{d}^2 J,\!</math> which forms the remainder map of <math>J\!</math> and also, in this instance, the second order differential of <math>J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-a -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-a.} ~~ \text{Remainder of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-b -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-b.} ~~ \text{Remainder of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-c -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-c.} ~~ \text{Remainder of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-d -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-d.} ~~ \text{Remainder of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
=====Summary of Conjunction=====<br />
<br />
To establish a convenient reference point for further discussion, Table&nbsp;49 summarizes the operator actions that have been computed for the form of conjunction, as exemplified by the proposition <math>J.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 49.} ~~ \text{Computation Summary for}~ J~\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon J<br />
& = & u \!\cdot\! v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}J<br />
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}J<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{r}J<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Analytic Series : Coordinate Method====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
And if he is told that something ''is'' the way it is, then he thinks: Well, it could probably just as easily be some other way. So the sense of possibility might be defined outright as the capacity to think how everything could &ldquo;just as easily&rdquo; be, and to attach no more importance to what is than to what is not.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 12]<br />
|}<br />
<br />
Table&nbsp;50 exhibits a truth table method for computing the analytic series (or the differential expansion) of a proposition in terms of coordinates.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:70%"<br />
|+ style="height:30px" | <math>\text{Table 50.} ~~ \text{Computation of an Analytic Series in Terms of Coordinates}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:8%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:8%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:8%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}u\!</math><br />
| style="width:8%; border-bottom:1px solid black" | <math>\mathrm{d}v\!</math><br />
| style="width:8%; border-bottom:1px solid black; border-left:1px solid black" | <math>u'\!</math><br />
| style="width:8%; border-bottom:1px solid black" | <math>v'\!</math><br />
| style="width:10%; border-bottom:1px solid black; border-left:4px double black" | <math>\boldsymbol\varepsilon J\!</math><br />
| style="width:10%; border-bottom:1px solid black" | <math>\mathrm{E}J\!</math><br />
| style="width:12%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{D}J\!</math><br />
| style="width:10%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}J\!</math><br />
| style="width:10%; border-bottom:1px solid black" | <math>\mathrm{d}^2\!J\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
The first six columns of the Table, taken as a whole, represent the variables of a construct called the ''contingent universe'' <math>[u, v, \mathrm{d}u, \mathrm{d}v, u', v'],\!</math> or the bundle of ''contingency spaces'' <math>[\mathrm{d}u, \mathrm{d}v, u', v']\!</math> over the universe <math>[u, v].\!</math> Their placement to the left of the double bar indicates that all of them amount to independent variables, but there is a co-dependency among them, as described by the following equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
u' & = & u + \mathrm{d}u & = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\\[8pt]<br />
v' & = & v + \mathrm{d}v & = & \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
These relations correspond to the formal substitutions that are made in defining <math>\mathrm{E}J\!</math> and <math>\mathrm{D}J.\!</math> For now, the whole rigamarole of contingency spaces can be regarded as a technical device for achieving the effect of these substitutions, adapted to a setting where functional compositions and other symbolic manipulations are difficult to contemplate and execute.<br />
<br />
The five columns to the right of the double bar in Table&nbsp;50 contain the values of the dependent variables <math>\{ \boldsymbol\varepsilon J, ~\mathrm{E}J, ~\mathrm{D}J, ~\mathrm{d}J, ~\mathrm{d}^2\!J \}.\!</math> These are normally interpreted as values of functions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B}\!</math> or as values of propositions in the extended universe <math>[u, v, \mathrm{d}u, \mathrm{d}v]\!</math> but the dependencies prevailing in the contingent universe make it possible to regard these same final values as arising via functions on alternative lists of arguments, for example, the set <math>\{ u, v, u', v' \}.\!</math><br />
<br />
The column for <math>\boldsymbol\varepsilon J\!</math> is computed as <math>J(u, v) = uv\!</math> and together with the columns for <math>u\!</math> and <math>v\!</math> illustrates how we &ldquo;share structure&rdquo; in the Table by listing only the first entries of each constant block.<br />
<br />
The column for <math>\mathrm{E}J\!</math> is computed by means of the following chain of identities, where the contingent variables <math>u'\!</math> and <math>v'\!</math> are defined as <math>u' = u + \mathrm{d}u\!</math> and <math>v' = v + \mathrm{d}v.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathrm{E}J(u, v, \mathrm{d}u, \mathrm{d}v)<br />
& = &<br />
J(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& = &<br />
J(u', v')<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
This makes it easy to determine <math>\mathrm{E}J\!</math> by inspection, computing the conjunction <math>J(u', v') = u'v'\!</math> from the columns headed <math>u'\!</math> and <math>v'.\!</math> Since each of these forms expresses the same proposition <math>\mathrm{E}J\!</math> in <math>\mathrm{E}U^\bullet,\!</math> the dependence on <math>\mathrm{d}u\!</math> and <math>\mathrm{d}v\!</math> is still present but merely left implicit in the final variant <math>J(u', v').\!</math><br />
<br />
* '''Note.''' On occasion, it is tempting to use the further notation <math>J'(u, v) = J(u', v'),\!</math> especially to suggest a transformation that acts on whole propositions, for example, taking the proposition <math>J\!</math> into the proposition <math>J' = \mathrm{E}J.\!</math> The prime <math>( {}^{\prime} )\!</math> then signifies an action that is mediated by a field of choices, namely, the values that are picked out for the contingent variables in sweeping through the initial universe. But this heaps an unwieldy lot of construed intentions on a rather slight character and puts too high a premium on the constant correctness of its interpretation. In practice, therefore, it is best to avoid this usage.<br />
<br />
Given the values of <math>\boldsymbol\varepsilon J\!</math> and <math>\mathrm{E}J,\!</math> the columns for the remaining functions can be filled in quickly. The difference map is computed according to the relation <math>\mathrm{D}J = \boldsymbol\varepsilon J + \mathrm{E}J.\!</math> The first order differential <math>\mathrm{d}J\!</math> is found by looking in each block of constant argument pairs <math>u, v\!</math> and choosing the linear function of <math>\mathrm{d}u, \mathrm{d}v\!</math> that best approximates <math>\mathrm{D}J\!</math> in that block. Finally, the remainder is computed as <math>\mathrm{r}J = \mathrm{D}J + \mathrm{d}J,\!</math> in this case yielding the second order differential <math>\mathrm{d}^2\!J.\!</math><br />
<br />
====Analytic Series : Recap====<br />
<br />
Let us now summarize the results of Table&nbsp;50 by writing down for each column and for each block of constant argument pairs <math>u, v\!</math> a reasonably canonical symbolic expression for the function of <math>\mathrm{d}u, \mathrm{d}v\!</math> that appears there. The synopsis formed in this way is presented in Table&nbsp;51. As one has a right to expect, it confirms the results that were obtained previously by operating solely in terms of the formal calculus.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:70%"<br />
|+ style="height:30px" | <math>\text{Table 51.} ~~ \text{Computation of an Analytic Series in Symbolic Terms}\!</math><br />
|- style="height:35px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black" | <math>J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{E}J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{D}J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{d}J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{d}^2\!J\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}u \!\;\cdot\;\! \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}u \!\;\cdot\;\! \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
\mathrm{d}u<br />
\\[4pt]<br />
\mathrm{d}v<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;52 and 53 provide a quick overview of the analysis performed so far, giving the successive decompositions of <math>\mathrm{E}J = J + \mathrm{D}J\!</math> and <math>\mathrm{D}J = \mathrm{d}J + \mathrm{r}J\!</math> in two different styles of diagram.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 52 -- Decomposition of EJ.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 52.} ~~ \text{Decomposition of}~ \mathrm{E}J\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 53 -- Decomposition of DJ.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 53.} ~~ \text{Decomposition of}~ \mathrm{D}J\!</math><br />
|}<br />
<br />
====Terminological Interlude====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Lastly, my attention was especially attracted, not so much to the scene, as to the mirrors that produced it. These mirrors were broken in parts. Yes, they were marked and scratched; they had been &ldquo;starred&rdquo;, in spite of their solidity &hellip;<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 230]<br />
|}<br />
<br />
At this point several issues of terminology have accrued enough substance to intrude on our discussion. The remarks of this Subsection are intended to accomplish two goals. First, we call attention to significant aspects of the previous series of Figures, translating into literal terms what they depict in iconic forms, and we re-stress the most important structural elements they indicate. Next, we prepare the way for taking on more complex examples of transformations, those whose target universes have more than one dimension.<br />
<br />
In talking about the actions of operators it is important to keep in mind the distinctions between the operators per&nbsp;se, their operands, and their results. Furthermore, in working with composite forms of operators <math>\mathrm{W} = (\mathrm{W}_1, \ldots, \mathrm{W}_n),\!</math> transformations <math>\mathrm{F} = (\mathrm{F}_1, \ldots, \mathrm{F}_n),\!</math> and target domains <math>X^\bullet = [x_1, \ldots, x_n],\!</math> we need to preserve a clear distinction between the compound entity of each given type and any one of its separate components. It is curious, given the usefulness of the concepts ''operator'' and ''operand'', that we seem to lack a generic term, formed on the same root, for the corresponding result of an operation. Following the obvious paradigm would lead to words like ''opus'', ''opera'', and ''operant'', but these words are too affected with clang associations to work well at present, though they might be adapted in time. One current usage gets around this problem by using the substantive ''map'' as a systematic epithet to express the result of each operator's action. We will follow this practice as far as possible, for example, using the phrase ''tangent map'' to denote the end product of the tangent functor acting on its operand map.<br />
<br />
* '''Scholium.''' See [JGH, 6-9] for a good account of tangent functors and tangent maps in ordinary analysis and for examples of their use in mechanics. This work as a whole is a model of clarity in applying functorial principles to problems in physical dynamics.<br />
<br />
Whenever we focus on isolated propositions, on single components of composite operators, or on the portions of transformations that have <math>1\!</math>-dimensional ranges, we are free to shift between the native form of a proposition <math>J : U \to \mathbb{B}\!</math> and the thematized form of a mapping <math>J : U^\bullet \to [x]\!</math> without much trouble. In these cases we are able to tolerate a higher degree of ambiguity about the precise nature of an operator's input and output domains than we otherwise might. For example, in the preceding treatment of the example <math>J,\!</math> and for each operator <math>\mathrm{W}\!</math> in the set <math>\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \},\!</math> both the operand <math>J\!</math> and the result <math>\mathrm{W}J\!</math> could be viewed in either one of two ways. On one hand we may treat them as propositions <math>J : U \to \mathbb{B}\!</math> and <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B},\!</math> ignoring the distinction between the range <math>[x] \cong \mathbb{B}\!</math> of <math>\boldsymbol\varepsilon J\!</math> and the range <math>[\mathrm{d}x] \cong \mathbb{D}\!</math> of the other types of <math>\mathrm{W}J.\!</math> This is what we usually do when we content ourselves with simply coloring in regions of venn diagrams. On the other hand we may view these entities as maps <math>J : U^\bullet \to [x] = X^\bullet\!</math> and <math>\boldsymbol\varepsilon J : \mathrm{E}U^\bullet \to [x] \subseteq \mathrm{E}X^\bullet\!</math> or <math>\mathrm{W}J : \mathrm{E}U^\bullet \to [\mathrm{d}x] \subseteq \mathrm{E}X^\bullet,\!</math> in which case the qualitative characters of the output features are not ignored.<br />
<br />
At the beginning of this Section we recast the natural form of a proposition <math>J : U \to \mathbb{B}\!</math> into the thematic role of a transformation <math>J : U^\bullet \to [x],\!</math> where <math>x\!</math> was a variable recruited to express the newly independent <math>\check{J}.\!</math> However, in our computations and representations of operator actions we immediately lapsed back to viewing the results as native elements of the extended universe <math>\mathrm{E}U^\bullet,\!</math> in other words, as propositions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B},\!</math> where <math>\mathrm{W}\!</math> ranged over the set <math>\{ \boldsymbol\varepsilon, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}.\!</math> That is as it should be. We have worked hard to devise a language that gives us these advantages &mdash; the flexibility to exchange terms and types of equal information value and the capacity to reflect as quickly and as wittingly as a controlled reflex on the fibers of our propositions, independently of whether they express amusements, beliefs, or conjectures.<br />
<br />
As we take on target spaces of increasing dimension, however, these types of confusions (and confusions of types) become less and less permissible. For this reason, Tables&nbsp;54 and 55 present a rather detailed summary of the notation and the terminology we are using, as applied to the case <math>J = uv.\!</math> The rationale of these Tables is not so much to train more elephant guns on this poor drosophila of a concrete example but to invest our paradigm with enough solidity to bear the weight of abstraction to come.<br />
<br />
Table&nbsp;54 provides basic notation and descriptive information for the objects and operators used in this Example, giving the generic type (or broadest defined type) for each entity. Here, the sans&nbsp;serif operators <math>\mathsf{W} \in \{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{d}, \mathsf{r} \}\!</math> and their components <math>\mathrm{W} \in \{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}\!</math> both have the same broad type <math>\mathsf{W}, \mathrm{W} : (U^\bullet \to X^\bullet) \to (\mathrm{E}U^\bullet \to \mathrm{E}X^\bullet),\!</math> as appropriate to operators that map transformations <math>J : U^\bullet \to X^\bullet\!</math> to extended transformations <math>\mathsf{W}J, \mathrm{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 54.} ~~ \text{Cast of Characters : Expansive Subtypes of Objects and Operators}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| align="center" | <math>\text{Symbol}\!</math><br />
| align="center" | <math>\text{Notation}\!</math><br />
| align="center" | <math>\text{Description}\!</math><br />
| align="center" | <math>\text{Type}\!</math><br />
|-<br />
| align="center" | <math>U^\bullet\!</math><br />
| <math>= [u, v]\!</math><br />
| <math>\text{Source universe}\!</math><br />
| <math>[\mathbb{B}^2]\!</math><br />
|-<br />
| align="center" | <math>X^\bullet~\!</math><br />
| <math>= [x]\!</math><br />
| <math>\text{Target universe}\!</math><br />
| <math>[\mathbb{B}^1]~\!</math><br />
|-<br />
| align="center" | <math>\mathrm{E}U^\bullet\!</math><br />
| <math>= [u, v, \mathrm{d}u, \mathrm{d}v]\!</math><br />
| <math>\text{Extended source universe}\!</math><br />
| <math>[\mathbb{B}^2 \!\times\! \mathbb{D}^2]</math><br />
|-<br />
| align="center" | <math>\mathrm{E}X^\bullet\!</math><br />
| <math>= [x, \mathrm{d}x]~\!</math><br />
| <math>\text{Extended target universe}\!</math><br />
| <math>[\mathbb{B}^1 \!\times\! \mathbb{D}^1]</math><br />
|-<br />
| align="center" | <math>J\!</math><br />
| <math>J : U \!\to\! \mathbb{B}\!</math><br />
| <math>\text{Proposition}\!</math><br />
| <math>(\mathbb{B}^2 \!\to\! \mathbb{B}) \in [\mathbb{B}^2]\!</math><br />
|-<br />
| align="center" | <math>J\!</math><br />
| <math>J : U^\bullet \!\to\! X^\bullet\!</math><br />
| <math>\text{Transformation or Map}\!</math><br />
| <math>[\mathbb{B}^2] \!\to\! [\mathbb{B}^1]\!</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\boldsymbol\varepsilon<br />
\\<br />
\eta<br />
\\<br />
\mathrm{E}<br />
\\<br />
\mathrm{D}<br />
\\<br />
\mathrm{d}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{W} : U^\bullet \!\to\! \mathrm{E}U^\bullet,<br />
\\<br />
\mathrm{W} : X^\bullet \!\to\! \mathrm{E}X^\bullet,<br />
\\<br />
\mathrm{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\\<br />
\text{for each}~ \mathrm{W} ~\text{in the set:}<br />
\\<br />
\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{ll}<br />
\text{Tacit extension operator} & \boldsymbol\varepsilon<br />
\\<br />
\text{Trope extension operator} & \eta<br />
\\<br />
\text{Enlargement operator} & \mathrm{E}<br />
\\<br />
\text{Difference operator} & \mathrm{D}<br />
\\<br />
\text{Differential operator} & \mathrm{d}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^2] \!\to\! [\mathbb{B}^2 \!\times\! \mathbb{D}^2]},<br />
\\<br />
{[\mathbb{B}^1] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]},<br />
\\\\<br />
([\mathbb{B}^2] \!\to\! [\mathbb{B}^1]) \!\to\!<br />
\\<br />
([\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1])<br />
\end{array}</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\mathsf{e}<br />
\\<br />
\mathsf{E}<br />
\\<br />
\mathsf{D}<br />
\\<br />
\mathsf{T}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{W} : U^\bullet \!\to\! \mathsf{T}U^\bullet = \mathrm{E}U^\bullet,<br />
\\<br />
\mathsf{W} : X^\bullet \!\to\! \mathsf{T}X^\bullet = \mathrm{E}X^\bullet,<br />
\\<br />
\mathsf{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathsf{T}U^\bullet \!\to\! \mathsf{T}X^\bullet)<br />
\\<br />
\text{for each}~ \mathsf{W} ~\text{in the set:}<br />
\\<br />
\{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{T} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{lll}<br />
\text{Radius operator} & \mathsf{e} & = (\boldsymbol\varepsilon, \eta)<br />
\\<br />
\text{Secant operator} & \mathsf{E} & = (\boldsymbol\varepsilon, \mathrm{E})<br />
\\<br />
\text{Chord operator} & \mathsf{D} & = (\boldsymbol\varepsilon, \mathrm{D})<br />
\\<br />
\text{Tangent functor} & \mathsf{T} & = (\boldsymbol\varepsilon, \mathrm{d})<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^2] \!\to\! [\mathbb{B}^2 \!\times\! \mathbb{D}^2]},<br />
\\<br />
{[\mathbb{B}^1] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]},<br />
\\\\<br />
([\mathbb{B}^2] \!\to\! [\mathbb{B}^1]) \!\to\!<br />
\\<br />
([\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1])<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;55 supplies a more detailed outline of terminology for operators and their results. Here, we list the restrictive subtype (or narrowest defined subtype) that applies to each entity and we indicate across the span of the Table the whole spectrum of alternative types that color the interpretation of each symbol. For example, all the component operator maps <math>\mathrm{W}J\!</math> have <math>1\!</math>-dimensional ranges, either <math>\mathbb{B}^1\!</math> or <math>\mathbb{D}^1,\!</math> and so they can be viewed either as propositions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B}\!</math> or as logical transformations <math>\mathrm{W}J : \mathrm{E}U^\bullet \to X^\bullet.\!</math> As a rule, the plan of the Table allows us to name each entry by detaching the underlined adjective at the left of its row and prefixing it to the generic noun at the top of its column. In one case, however, it is customary to depart from this scheme. Because the phrase ''differential proposition'', applied to the result <math>\mathrm{d}J : \mathrm{E}U \to \mathbb{D},\!</math> does not distinguish it from the general run of differential propositions <math>\mathrm{G}: \mathrm{E}U \to \mathbb{B},\!</math> it is usual to single out <math>\mathrm{d}J\!</math> as the ''tangent proposition'' of <math>J.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 55.} ~~ \text{Synopsis of Terminology : Restrictive and Alternative Subtypes}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| &nbsp;<br />
| align="center" | <math>\text{Operator}\!</math><br />
| align="center" | <math>\text{Proposition}\!</math><br />
| align="center" | <math>\text{Map}\!</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tacit}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\boldsymbol\varepsilon : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\boldsymbol\varepsilon : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{B} \\<br />
\boldsymbol\varepsilon J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{B}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x] \\<br />
\boldsymbol\varepsilon J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Trope}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\eta : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\eta : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\eta J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\eta J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Enlargement}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{E} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{E}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{E}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Difference}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{D} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{D}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{D}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Differential}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{d} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{d} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{d}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}\!</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Remainder}}\\\text{operator}\end{matrix}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{r} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{r} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{r}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{r}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Radius}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e} = (\boldsymbol\varepsilon, \eta) \\<br />
\mathsf{e} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{e}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Secant}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}) \\<br />
\mathsf{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{E}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Chord}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}) \\<br />
\mathsf{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{D}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tangent}}\\\text{functor}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T} = (\boldsymbol\varepsilon, \mathrm{d}) \\<br />
\mathsf{T} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{T}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====End of Perfunctory Chatter : Time to Roll the Clip!====<br />
<br />
Two steps remain to finish the analysis of <math>J\!</math> that we began so long ago. First, we need to paste our accumulated heap of flat pictures into the frames of transformations, filling out the shapes of the operator maps <math>\mathsf{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet.~\!</math> This scheme is executed in two styles, using the ''areal views'' in Figures&nbsp;56-a and the ''box views'' in Figures&nbsp;56-b. Finally, in Figures&nbsp;57-1 to 57-4 we put all the pieces together to construct the full operator diagrams for <math>\mathsf{W} : J \to \mathsf{W}J.\!</math> There is a considerable amount of redundancy among the following three series of Figures but that will hopefully provide a fuller picture of the operations under review, enabling these snapshots to serve as successive frames in the animation of logic they are meant to become.<br />
<br />
=====Operator Maps : Areal Views=====<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a1 -- Radius Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a1.} ~~ \text{Radius Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a2 -- Secant Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a2.} ~~ \text{Secant Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a3 -- Chord Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a3.} ~~ \text{Chord Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a4 -- Tangent Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a4.} ~~ \text{Tangent Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
=====Operator Maps : Box Views=====<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b1 -- Radius Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b1.} ~~ \text{Radius Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b2 -- Secant Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b2.} ~~ \text{Secant Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b3 -- Chord Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b3.} ~~ \text{Chord Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b4 -- Tangent Map of J ISW.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b4.} ~~ \text{Tangent Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
=====Operator Diagrams for the Conjunction J = uv=====<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-1 -- Radius Operator Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-1.} ~~ \text{Radius Operator Diagram for the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-2 -- Secant Operator Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-2.} ~~ \text{Secant Operator Diagram for the Conjunction}~ J = uv~\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-3 -- Chord Operator Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-3.} ~~ \text{Chord Operator Diagram for the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-4 -- Tangent Functor Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-4.} ~~ \text{Tangent Functor Diagram for the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
===Taking Aim at Higher Dimensional Targets===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
The past and present wilt . . . . I have filled them and<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;emptied them,<br><br />
And proceed to fill my next fold of the future.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 87]<br />
|}<br />
<br />
In the next Section we consider a transformation <math>F\!</math> of concrete type <math>F : [u, v] \to [x, y]\!</math> and abstract type <math>F : [\mathbb{B}^2] \to [\mathbb{B}^2].\!</math> From the standpoint of propositional calculus we naturally approach the task of understanding such a transformation by parsing it into component maps with <math>1\!</math>-dimensional ranges, as follows:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{ccccccl}<br />
F & = & (F_1, F_2) & = & (f, g) & : & [u, v] \to [x, y],<br />
\\[6pt]<br />
&& F_1 & = & f & : & [u, v] \to [x],<br />
\\[6pt]<br />
&& F_2 & = & g & : & [u, v] \to [y].<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Then we tackle the separate components, now viewed as propositions <math>F_i : U \to \mathbb{B},\!</math> one at a time. At the completion of this analytic phase, we return to the task of synthesizing these partial and transient impressions into an agile form of integrity, a solidly coordinated and deeply integrated comprehension of the ongoing transformation. (Very often, of course, in tangling with refractory cases, we never get as far as the beginning again.)<br />
<br />
Let us now refer to the dimension of the target space or codomain as the ''toll'' (or ''tole'') of a transformation, as distinguished from the dimension of the range or image that is customarily called the ''rank''. When we keep to transformations with a toll of <math>1,\!</math> as <math>J : [u, v] \to [x],\!</math> we tend to get lazy about distinguishing a logical transformation from its component propositions. However, if we deal with transformations of a higher toll, this form of indolence can no longer be tolerated.<br />
<br />
Well, perhaps we can carry it a little further. After all, the operator result <math>\mathrm{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet\!</math> is a map of toll <math>2,\!</math> and cannot be unfolded in one piece as a proposition. But when a map has rank <math>1,\!</math> like <math>\boldsymbol\varepsilon J : \mathrm{E}U \to X \subseteq \mathrm{E}X\!</math> or <math>\mathrm{d}J : \mathrm{E}U \to \mathrm{d}X \subseteq \mathrm{E}X,\!</math> we naturally choose to concentrate on the <math>1\!</math>-dimensional range of the operator result <math>\mathrm{W}J,\!</math> ignoring the final difference in quality between the spaces <math>X\!</math> and <math>\mathrm{d}X,\!</math> and view <math>\mathrm{W}J\!</math> as a proposition about <math>\mathrm{E}U.\!</math><br />
<br />
In this way, an initial ambivalence about the role of the operand <math>J\!</math> conveys a double duty to the result <math>\mathrm{W}J.\!</math> The pivot that is formed by our focus of attention is essential to the linkage that transfers this double moment, as the whole process takes its bearing and wheels around the precise measure of a narrow bead that we can draw on the range of <math>\mathrm{W}J.\!</math> This is the escapement that it takes to get away with what may otherwise seem to be a simple duplicity, and this is the tolerance that is needed to counterbalance a certain arrogance of equivocation, by all of which machinations we make ourselves free to indicate the operator results <math>\mathrm{W}J\!</math> as propositions or as transformations, indifferently.<br />
<br />
But that's it, and no further. Neglect of these distinctions in range and target universes of higher dimensions is bound to cause a hopeless confusion. To guard against these adverse prospects, Tables&nbsp;58 and 59 lay the groundwork for discussing a typical map <math>F : [\mathbb{B}^2] \to [\mathbb{B}^2],\!</math> and begin to pave the way to some extent for discussing any transformation of the form <math>F : [\mathbb{B}^n] \to [\mathbb{B}^k].\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 58.} ~~ \text{Cast of Characters : Expansive Subtypes of Objects and Operators}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| align="center" | <math>\text{Symbol}\!</math><br />
| align="center" | <math>\text{Notation}\!</math><br />
| align="center" | <math>\text{Description}\!</math><br />
| align="center" | <math>\text{Type}\!</math><br />
|-<br />
| align="center" | <math>U^\bullet\!</math><br />
| <math>= [u, v]\!</math><br />
| <math>\text{Source universe}\!</math><br />
| <math>[\mathbb{B}^n]\!</math><br />
|-<br />
| align="center" | <math>X^\bullet~\!</math><br />
| <math>\begin{array}{l}<br />
= [x, y] \\<br />
= [f, g]<br />
\end{array}</math><br />
| <math>\text{Target universe}\!</math><br />
| <math>[\mathbb{B}^k]\!</math><br />
|-<br />
| align="center" | <math>\mathrm{E}U^\bullet\!</math><br />
| <math>= [u, v, \mathrm{d}u, \mathrm{d}v]\!</math><br />
| <math>\text{Extended source universe}\!</math><br />
| <math>[\mathbb{B}^n \!\times\! \mathbb{D}^n]\!</math><br />
|-<br />
| align="center" | <math>\mathrm{E}X^\bullet\!</math><br />
| <math>\begin{array}{l}<br />
= [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
= [f, g, \mathrm{d}f, \mathrm{d}g]<br />
\end{array}</math><br />
| <math>\text{Extended target universe}\!</math><br />
| <math>[\mathbb{B}^k \!\times\! \mathbb{D}^k]\!</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
f \\ g<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{ll}<br />
f : U \!\to\! [x] \cong \mathbb{B} \\<br />
g : U \!\to\! [y] \cong \mathbb{B}<br />
\end{array}</math><br />
| <math>\text{Proposition}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathbb{B}^n \!\to\! \mathbb{B} \\<br />
\in (\mathbb{B}^n, \mathbb{B}^n \!\to\! \mathbb{B}) = [\mathbb{B}^n]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>F\!</math><br />
| <math>F = (f, g) : U^\bullet \!\to\! X^\bullet\!</math><br />
| <math>\text{Transformation of Map}\!</math><br />
| <math>[\mathbb{B}^n] \!\to\! [\mathbb{B}^k]</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\boldsymbol\varepsilon<br />
\\<br />
\eta<br />
\\<br />
\mathrm{E}<br />
\\<br />
\mathrm{D}<br />
\\<br />
\mathrm{d}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{W} : U^\bullet \!\to\! \mathrm{E}U^\bullet,<br />
\\<br />
\mathrm{W} : X^\bullet \!\to\! \mathrm{E}X^\bullet,<br />
\\<br />
\mathrm{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\\<br />
\text{for each}~ \mathrm{W} ~\text{in the set:}<br />
\\<br />
\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{ll}<br />
\text{Tacit extension operator} & \boldsymbol\varepsilon<br />
\\<br />
\text{Trope extension operator} & \eta<br />
\\<br />
\text{Enlargement operator} & \mathrm{E}<br />
\\<br />
\text{Difference operator} & \mathrm{D}<br />
\\<br />
\text{Differential operator} & \mathrm{d}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^n] \!\to\! [\mathbb{B}^n \!\times\! \mathbb{D}^n]},<br />
\\<br />
{[\mathbb{B}^k] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]},<br />
\\\\<br />
([\mathbb{B}^n] \!\to\! [\mathbb{B}^k]) \!\to\!<br />
\\<br />
([\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k])<br />
\end{array}</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\mathsf{e}<br />
\\<br />
\mathsf{E}<br />
\\<br />
\mathsf{D}<br />
\\<br />
\mathsf{T}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{W} : U^\bullet \!\to\! \mathsf{T}U^\bullet = \mathrm{E}U^\bullet,<br />
\\<br />
\mathsf{W} : X^\bullet \!\to\! \mathsf{T}X^\bullet = \mathrm{E}X^\bullet,<br />
\\<br />
\mathsf{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathsf{T}U^\bullet \!\to\! \mathsf{T}X^\bullet)<br />
\\<br />
\text{for each}~ \mathsf{W} ~\text{in the set:}<br />
\\<br />
\{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{T} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{lll}<br />
\text{Radius operator} & \mathsf{e} & = (\boldsymbol\varepsilon, \eta)<br />
\\<br />
\text{Secant operator} & \mathsf{E} & = (\boldsymbol\varepsilon, \mathrm{E})<br />
\\<br />
\text{Chord operator} & \mathsf{D} & = (\boldsymbol\varepsilon, \mathrm{D})<br />
\\<br />
\text{Tangent functor} & \mathsf{T} & = (\boldsymbol\varepsilon, \mathrm{d})<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^n] \!\to\! [\mathbb{B}^n \!\times\! \mathbb{D}^n]},<br />
\\<br />
{[\mathbb{B}^k] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]},<br />
\\\\<br />
([\mathbb{B}^n] \!\to\! [\mathbb{B}^k]) \!\to\!<br />
\\<br />
([\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k])<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 59.} ~~ \text{Synopsis of Terminology : Restrictive and Alternative Subtypes}~\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| &nbsp;<br />
| align="center" | <math>\begin{matrix}\text{Operator}\\\text{or}\\\text{Operand}\end{matrix}</math><br />
| align="center" | <math>\begin{matrix}\text{Proposition}\\\text{or}\\\text{Component}\end{matrix}</math><br />
| align="center" | <math>\begin{matrix}\text{Transformation}\\\text{or}\\\text{Map}\end{matrix}</math><br />
|-<br />
| align="center" | <math>\underline{\text{Operand}}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
F = (F_1, F_2) \\<br />
F = (f, g) : U \!\to\! X<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
F_i : \langle u, v \rangle \!\to\! \mathbb{B} \\<br />
F_i : \mathbb{B}^n \!\to\! \mathbb{B}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
F : [u, v] \!\to\! [x, y] \\<br />
F : [\mathbb{B}^n] \!\to\! [\mathbb{B}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tacit}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\boldsymbol\varepsilon : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\boldsymbol\varepsilon : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{B} \\<br />
\boldsymbol\varepsilon F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{B}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y] \\<br />
\boldsymbol\varepsilon F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Trope}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\eta : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\eta : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\eta F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\eta F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Enlargement}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{E} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{E}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{E}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Difference}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{D} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{D}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{D}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Differential}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{d} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{d} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{d}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}\!</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Remainder}}\\\text{operator}\end{matrix}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{r} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{r} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{r}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{r}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Radius}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e} = (\boldsymbol\varepsilon, \eta) \\<br />
\mathsf{e} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{e}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Secant}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}) \\<br />
\mathsf{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{E}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Chord}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}) \\<br />
\mathsf{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{D}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tangent}}\\\text{functor}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T} = (\boldsymbol\varepsilon, \mathrm{d}) \\<br />
\mathsf{T} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{T}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
===Transformations of Type '''B'''<sup>2</sup> &rarr; '''B'''<sup>2</sup>===<br />
<br />
To take up a slightly more complex example, but one that remains simple enough to pursue through a complete series of developments, consider the transformation from <math>U^\bullet = [u, v]\!</math> to <math>X^\bullet = [x, y]\!</math> that is defined by the following system of equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
x<br />
& = & f(u, v)<br />
& = & \texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[8pt]<br />
y<br />
& = & g(u, v)<br />
& = & \texttt{((} u \texttt{,} v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
The component notation <math>F = (F_1, F_2) = (f, g) : U^\bullet \to X^\bullet\!</math> allows us to give a name and a type to this transformation and permits defining it by the compact description that follows:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
(x, y)<br />
& = & F(u, v)<br />
& = & (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Logical Transformations====<br />
<br />
The information that defines the logical transformation <math>F\!</math> can be represented in the form of a truth table, as shown in Table&nbsp;60. To cut down on subscripts in this example we continue to use plain letter equivalents for all components of spaces and maps.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:60%"<br />
|+ style="height:30px" | <math>\text{Table 60.} ~~ \text{A Propositional Transformation}\!</math><br />
|- style="height:40px; background:ghostwhite; width:100%"<br />
| style="width:25%" | <math>u\!</math><br />
| style="width:25%" | <math>v\!</math><br />
| style="width:25%; border-left:1px solid black" | <math>f\!</math><br />
| style="width:25%" | <math>g\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="border-top:1px solid black" | <math>u\!</math><br />
| style="border-top:1px solid black" | <math>v\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;61 shows how we might paint a picture of the transformation <math>F\!</math> in the manner of Figure&nbsp;30.<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 61 -- Propositional Transformation.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 61.} ~~ \text{A Propositional Transformation}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;62 extracts the gist of Figure&nbsp;61, exhibiting a style of diagram that is adequate for most purposes.<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 62 -- Propositional Transformation (Short Form).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 62.} ~~ \text{A Propositional Transformation (Short Form)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Local Transformations====<br />
<br />
Figure&nbsp;63 gives a more complete picture of the transformation <math>F,\!</math> showing how the points of <math>U^\bullet\!</math> are transformed into points of <math>X^\bullet.\!</math> The bold lines crossing from one universe to the other trace the action that <math>F\!</math> induces on points, in other words, they show how the transformation acts as a mapping from points to points and chart its effects on the elements that are variously called cells, points, positions, or singular propositions.<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 63 -- Transformation of Positions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 63.} ~~ \text{A Transformation of Positions}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;64 shows how the action of <math>F\!</math> on cells or points can be computed in terms of coordinates.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 64.} ~~ \text{A Transformation of Positions}\!</math><br />
|- style="height:40px; background:ghostwhite; width:100%"<br />
| style="width:8%" | <math>u\!</math><br />
| style="width:8%" | <math>v\!</math><br />
| style="width:12%; border-left:1px solid black" | <math>x\!</math><br />
| style="width:12%" | <math>y\!</math><br />
| style="width:10%; border-left:1px solid black" | <math>x~y\!</math><br />
| style="width:10%" | <math>x \texttt{(} y \texttt{)}\!</math><br />
| style="width:10%" | <math>\texttt{(} x \texttt{)} y\!</math><br />
| style="width:10%" | <math>\texttt{(} x \texttt{)(} y \texttt{)}\!</math><br />
| style="width:20%; border-left:1px solid black" | <math>X^\bullet = [x, y]\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\uparrow<br />
\\[4pt]<br />
F =<br />
\\[4pt]<br />
(f, g)<br />
\\[4pt]<br />
\uparrow<br />
\end{matrix}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="border-top:1px solid black" | <math>u\!</math><br />
| style="border-top:1px solid black" | <math>v\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
| style="border-top:1px solid black" | <math>\texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>u~v\!</math><br />
| style="border-top:1px solid black" | <math>\texttt{(} u \texttt{,} v \texttt{)}\!</math><br />
| style="border-top:1px solid black" | <math>\texttt{(} u \texttt{)(} v \texttt{)}\!</math><br />
| style="border-top:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>U^\bullet = [u, v]\!</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;65 extends this scheme from single cells to arbitrary regions, showing how we might compute the action of a logical transformation on arbitrary propositions in the universe of discourse. The effect of a point-transformation on arbitrary propositions, or any other structures erected on points, is referred to as the ''induced action'' of the transformation on the structures in question.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 65-a.} ~~ \text{An Induced Transformation on Propositions}\!</math><br />
|- style="height:50px; background:ghostwhite"<br />
| style="width:20%" | <math>X^\bullet~\!</math><br />
| style="width:20%; border-right:none" | <math>\longleftarrow\!</math><br />
| style="width:20%; border-left:none; border-right:none" | <math>F = (f, g)\!</math><br />
| style="width:20%; border-left:none" | <math>\longleftarrow\!</math><br />
| style="width:20%" | <math>U^\bullet~\!</math><br />
|- style="background:ghostwhite"<br />
| rowspan="2" | <math>f_i (x, y)\!</math><br />
| align="right" | <math>\begin{matrix}u = \\ v =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~0~0\\1~0~1~0\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= u \\ = v\end{matrix}</math><br />
| rowspan="2" style="width:20%" | <math>f_j (u, v)\!</math><br />
|- style="background:ghostwhite"<br />
| align="right" | <math>\begin{matrix}x = \\ y =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~1~0\\1~0~0~1\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= f(u, v) \\ = g(u, v)\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{2}<br />
\\[2pt]<br />
f_{3}<br />
\\[2pt]<br />
f_{4}<br />
\\[2pt]<br />
f_{5}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{7}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)~ ~}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{~ ~(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{,~} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{~~} y \texttt{)}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~0<br />
\\[2pt]<br />
0~0~0~0<br />
\\[2pt]<br />
0~0~0~1<br />
\\[2pt]<br />
0~0~0~1<br />
\\[2pt]<br />
0~1~1~0<br />
\\[2pt]<br />
0~1~1~0<br />
\\[2pt]<br />
0~1~1~1<br />
\\[2pt]<br />
0~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{,~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{,~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{~~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{~~} v \texttt{)}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[2pt]<br />
f_{0}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{7}<br />
\\[2pt]<br />
f_{7}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{8}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{10}<br />
\\[2pt]<br />
f_{11}<br />
\\[2pt]<br />
f_{12}<br />
\\[2pt]<br />
f_{13}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{15}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[2pt]<br />
\texttt{~ ~ ~ ~} y \texttt{~~}<br />
\\[2pt]<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[2pt]<br />
\texttt{~~} x \texttt{~ ~ ~ ~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
1~0~0~0<br />
\\[2pt]<br />
1~0~0~0<br />
\\[2pt]<br />
1~0~0~1<br />
\\[2pt]<br />
1~0~0~1<br />
\\[2pt]<br />
1~1~1~0<br />
\\[2pt]<br />
1~1~1~0<br />
\\[2pt]<br />
1~1~1~1<br />
\\[2pt]<br />
1~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~~} u \texttt{~~} v \texttt{~~}<br />
\\[2pt]<br />
\texttt{~~} u \texttt{~~} v \texttt{~~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{8}<br />
\\[2pt]<br />
f_{8}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{15}<br />
\\[2pt]<br />
f_{15}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 65-b.} ~~ \text{An Induced Transformation on Propositions}\!</math><br />
|- style="height:50px; background:ghostwhite"<br />
| style="width:20%" | <math>X^\bullet~\!</math><br />
| style="width:20%; border-right:none" | <math>\longleftarrow\!</math><br />
| style="width:20%; border-left:none; border-right:none" | <math>F = (f, g)\!</math><br />
| style="width:20%; border-left:none" | <math>\longleftarrow\!</math><br />
| style="width:20%" | <math>U^\bullet~\!</math><br />
|- style="background:ghostwhite"<br />
| rowspan="2" | <math>f_i (x, y)\!</math><br />
| align="right" | <math>\begin{matrix}u = \\ v =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~0~0\\1~0~1~0\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= u \\ = v\end{matrix}</math><br />
| rowspan="2" style="width:20%" | <math>f_j (u, v)\!</math><br />
|- style="background:ghostwhite"<br />
| align="right" | <math>\begin{matrix}x = \\ y =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~1~0\\1~0~0~1\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= f(u, v) \\ = g(u, v)\end{matrix}</math><br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>f_{0}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[2pt]<br />
f_{2}<br />
\\[2pt]<br />
f_{4}<br />
\\[2pt]<br />
f_{8}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~0<br />
\\[2pt]<br />
0~0~0~1<br />
\\[2pt]<br />
0~1~1~0<br />
\\[2pt]<br />
1~0~0~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{,~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{~} u \texttt{~~} v \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{8}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{3}<br />
\\[2pt]<br />
f_{12}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[2pt]<br />
1~1~1~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} u \texttt{)(} v \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[2pt]<br />
f_{14}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[2pt]<br />
f_{9}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[2pt]<br />
1~0~0~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{~~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{~} u \texttt{~~} v \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[2pt]<br />
f_{8}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{5}<br />
\\[2pt]<br />
f_{10}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[2pt]<br />
1~0~0~1<br />
\end{matrix}~\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} u \texttt{,~} v \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[2pt]<br />
f_{9}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[2pt]<br />
f_{11}<br />
\\[2pt]<br />
f_{13}<br />
\\[2pt]<br />
f_{14}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[2pt]<br />
1~0~0~1<br />
\\[2pt]<br />
1~1~1~0<br />
\\[2pt]<br />
1~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} u \texttt{~~} v \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{15}<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>f_{15}\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Difference Operators and Tangent Functors====<br />
<br />
Given the alphabets <math>\mathcal{U} = \{ u, v \}\!</math> and <math>\mathcal{X} = \{ x, y \},\!</math> along with the corresponding universes of discourse <math>U^\bullet, X^\bullet \cong [\mathbb{B}^2],\!</math> how many logical transformations of the general form <math>G = (G_1, G_2) : U^\bullet \to X^\bullet\!</math> are there? Since <math>G_1\!</math> and <math>G_2\!</math> can be any propositions of the type <math>\mathbb{B}^2 \to \mathbb{B},\!</math> there are <math>2^4 = 16\!</math> choices for each of the maps <math>G_1\!</math> and <math>G_2\!</math> and thus there are <math>2^4 \cdot 2^4 = 2^8 = 256\!</math> different mappings altogether of the form <math>G : U^\bullet \to X^\bullet.\!</math> The set of functions of a given type is denoted by placing its type indicator in parentheses, in the present instance writing <math>(U^\bullet \to X^\bullet) = \{ G : U^\bullet \to X^\bullet \},\!</math> and so the cardinality of the ''function space'' <math>(U^\bullet \to X^\bullet)\!</math> is summed up by writing <math>|(U^\bullet \to X^\bullet)| = |(\mathbb{B}^2 \to \mathbb{B}^2)| = 4^4 = 256.\!</math><br />
<br />
Given a transformation <math>G = (G_1, G_2) : U^\bullet \to X^\bullet\!</math> of this type, we proceed to define a pair of further transformations, related to <math>G,\!</math> that operate between the extended universes, <math>\mathrm{E}U^\bullet\!</math> and <math>\mathrm{E}X^\bullet,\!</math> of its source and target domains.<br />
<br />
First, the ''enlargement map'' (or ''secant transformation'') <math>\mathrm{E}G = (\mathrm{E}G_1, \mathrm{E}G_2) : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet\!</math> is defined by the following set of component equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}G_i<br />
& = & G_i (u + \mathrm{d}u, v + \mathrm{d}v)<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Next, the ''difference map'' (or ''chordal transformation'') <math>\mathrm{D}G = (\mathrm{D}G_1, \mathrm{D}G_2) : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet~\!</math> is defined in component-wise fashion as the boolean sum of the initial proposition <math>G_i\!</math> and the enlarged proposition <math>\mathrm{E}G_i,\!</math> for <math>i = 1, 2,\!</math> according to the following set of equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
\mathrm{D}G_i<br />
& = & G_i (u, v)<br />
& + & \mathrm{E}G_i (u, v, \mathrm{d}u, \mathrm{d}v)<br />
\\[8pt]<br />
& = & G_i (u, v)<br />
& + & G_i (u + \mathrm{d}u, v + \mathrm{d}v)<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Maintaining a strict analogy with ordinary difference calculus would perhaps have us write <math>\mathrm{D}G_i = \mathrm{E}G_i - G_i,\!</math> but the sum and difference operations are the same thing in boolean arithmetic. It is more often natural in the logical context to consider an initial proposition <math>q,\!</math> then to compute the enlargement <math>\mathrm{E}q,\!</math> and finally to determine the difference <math>\mathrm{D}q = q + \mathrm{E}q,\!</math> so we let the variant order of terms reflect this sequence of considerations.<br />
<br />
Viewed in this light the difference operator <math>\mathrm{D}\!</math> is imagined to be a function of very wide scope and polymorphic application, one that is able to realize the association between each transformation <math>G\!</math> and its difference map <math>\mathrm{D}G,\!</math> for example, taking the function space <math>(U^\bullet \to X^\bullet)\!</math> into <math>(\mathrm{E}U^\bullet \to \mathrm{E}X^\bullet).\!</math> When we consider the variety of interpretations permitted to propositions over the contexts in which we put them to use, it should be clear that an operator of this scope is not at all a trivial matter to define in general and that it may take some trouble to work out. For the moment we content ourselves with returning to particular cases.<br />
<br />
Acting on the logical transformation <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;),\!</math> the operators <math>\mathrm{E}\!</math> and <math>\mathrm{D}\!</math> yield the enlarged map <math>\mathrm{E}F = (\mathrm{E}f, \mathrm{E}g)\!</math> and the difference map <math>\mathrm{D}F = (\mathrm{D}f, \mathrm{D}g),\!</math> respectively, whose components are given as follows.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}f<br />
& = & \texttt{((} u + \mathrm{d}u \texttt{)(} v + \mathrm{d}v \texttt{))}<br />
\\[8pt]<br />
\mathrm{E}g<br />
& = & \texttt{((} u + \mathrm{d}u \texttt{,~} v + \mathrm{d}v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
\mathrm{D}f<br />
& = & \texttt{((} u \texttt{)(} v \texttt{))}<br />
& + & \texttt{((} u + \mathrm{d}u \texttt{)(} v + \mathrm{d}v \texttt{))}<br />
\\[8pt]<br />
\mathrm{D}g<br />
& = & \texttt{((} u \texttt{,~} v \texttt{))}<br />
& + & \texttt{((} u + \mathrm{d}u \texttt{,~} v + \mathrm{d}v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
But these initial formulas are purely definitional, and help us little in understanding either the purpose of the operators or the meaning of their results. Working symbolically, let us apply the same method to the separate components <math>f\!</math> and <math>g\!</math> that we earlier used on <math>J.\!</math> This work is recorded in Appendix&nbsp;3 and a summary of the results is presented in Tables&nbsp;66-i and 66-ii.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 66-i.} ~~ \text{Computation Summary for}~ f(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f<br />
& = & u \!\cdot\! v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 1<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}f<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \cdot \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{d}f<br />
& = & u \!\cdot\! v \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}f<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 66-ii.} ~~ \text{Computation Summary for}~ g(u, v) = \texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon g<br />
& = & u \!\cdot\! v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}g<br />
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}g<br />
& = & u \!\cdot\! v \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;67 shows how to compute the analytic series for <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math> in terms of coordinates, and Table&nbsp;68 recaps these results in symbolic terms, agreeing with earlier derivations.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 67.} ~~ \text{Computation of an Analytic Series in Terms of Coordinates}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:6%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}u\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>\mathrm{d}v\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>u'\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>v'\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:4px double black" | <math>f\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>g\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{E}f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{E}g}\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{D}f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{D}g}\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{d}f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{d}g}\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{d}^2\!f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{d}^2\!g}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 68.} ~~ \text{Computation of an Analytic Series in Symbolic Terms}\!</math><br />
|- style="height:40px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black" | <math>f\!</math><br />
| <math>g\!</math><br />
| style="border-left:1px solid black" | <math>{\mathrm{D}f}\!</math><br />
| <math>{\mathrm{D}g}\!</math><br />
| style="border-left:1px solid black" | <math>{\mathrm{d}f}\!</math><br />
| <math>{\mathrm{d}g}\!</math><br />
| style="border-left:1px solid black" | <math>{\mathrm{d}^2\!f}\!</math><br />
| <math>{\mathrm{d}^2\!g}\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[4pt]<br />
\texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
\\[4pt]<br />
\texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
\\[4pt]<br />
\texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;69 gives a graphical picture of the difference map <math>\mathrm{D}F = (\mathrm{D}f, \mathrm{D}g)\!</math> for the transformation <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math> This represents the same information about <math>\mathrm{D}f~\!</math> and <math>\mathrm{D}g~\!</math> that was given in the corresponding rows of Tables&nbsp;66-i and 66-ii, for ease of reference repeated below.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[8pt]<br />
\mathrm{D}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 69 -- Difference Map (Short Form).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 69.} ~~ \text{Difference Map of}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;70-a shows a way of visualizing the tangent functor map <math>\mathrm{d}F = (\mathrm{d}f, \mathrm{d}g)\!</math> for the transformation <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math> This amounts to the same information about <math>\mathrm{d}f~\!</math> and <math>\mathrm{d}g~\!</math> that was given in Tables&nbsp;66-i and 66-ii, the corresponding rows of which are repeated below.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{d}f<br />
& = & u \!\cdot\! v \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[8pt]<br />
\mathrm{d}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 70-a -- Tangent Functor Diagram.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 70-a.} ~~ \text{Tangent Functor Diagram for}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math><br />
|}<br />
<br />
<br><br />
<br />
Figure 70-b shows another way to picture the action of the tangent functor on the logical transformation <math>F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 70-b -- Tangent Functor Diagram.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 70-b.} ~~ \text{Tangent Functor Ferris Wheel for}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math><br />
|}<br />
<br />
<br><br />
<br />
* '''Note.''' The original Figure&nbsp;70-b lost some of its labeling in a succession of platform metamorphoses over the years, so we have included an ASCII version below to indicate where the missing labels go.<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| /////\ /////\ | | /XXXX\ /XXXX\ | | /\\\\\ /\\\\\ |<br />
| ///////o//////\ | | /XXXXXXoXXXXXX\ | | /\\\\\\o\\\\\\\ |<br />
| //////// \//////\ | | /XXXXXX/ \XXXXXX\ | | /\\\\\\/ \\\\\\\\ |<br />
| o/////// \//////o | | oXXXXXX/ \XXXXXXo | | o\\\\\\/ \\\\\\\o |<br />
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |<br />
| |/du//| |//dv/| | | |XXXXX| |XXXXX| | | |\du\\| |\\dv\| |<br />
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |<br />
| o//////\ ///////o | | oXXXXXX\ /XXXXXXo | | o\\\\\\\ /\\\\\\o |<br />
| \//////\ //////// | | \XXXXXX\ /XXXXXX/ | | \\\\\\\\ /\\\\\\/ |<br />
| \//////o/////// | | \XXXXXXoXXXXXX/ | | \\\\\\\o\\\\\\/ |<br />
| \///// \///// | | \XXXX/ \XXXX/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ (u)(v) o-----------------------o dv' @ (u)(v) =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| / \ /////\ | | /\\\\\ /XXXX\ | | /\\\\\ /\\\\\ |<br />
| / o//////\ | | /\\\\\\oXXXXXX\ | | /\\\\\\o\\\\\\\ |<br />
| / //\//////\ | | /\\\\\\//\XXXXXX\ | | /\\\\\\/ \\\\\\\\ |<br />
| o ////\//////o | | o\\\\\\////\XXXXXXo | | o\\\\\\/ \\\\\\\o |<br />
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |<br />
| | du |/////|//dv/| | | |\\\\\|/////|XXXXX| | | |\du\\| |\\dv\| |<br />
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |<br />
| o \//////////o | | o\\\\\\\////XXXXXXo | | o\\\\\\\ /\\\\\\o |<br />
| \ \///////// | | \\\\\\\\//XXXXXX/ | | \\\\\\\\ /\\\\\\/ |<br />
| \ o/////// | | \\\\\\\oXXXXXX/ | | \\\\\\\o\\\\\\/ |<br />
| \ / \///// | | \\\\\/ \XXXX/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ (u) v o-----------------------o dv' @ (u) v =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| /////\ / \ | | /XXXX\ /\\\\\ | | /\\\\\ /\\\\\ |<br />
| ///////o \ | | /XXXXXXo\\\\\\\ | | /\\\\\\o\\\\\\\ |<br />
| /////////\ \ | | /XXXXXX//\\\\\\\\ | | /\\\\\\/ \\\\\\\\ |<br />
| o//////////\ o | | oXXXXXX////\\\\\\\o | | o\\\\\\/ \\\\\\\o |<br />
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |<br />
| |/du//|/////| dv | | | |XXXXX|/////|\\\\\| | | |\du\\| |\\dv\| |<br />
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |<br />
| o//////\//// o | | oXXXXXX\////\\\\\\o | | o\\\\\\\ /\\\\\\o |<br />
| \//////\// / | | \XXXXXX\//\\\\\\/ | | \\\\\\\\ /\\\\\\/ |<br />
| \//////o / | | \XXXXXXo\\\\\\/ | | \\\\\\\o\\\\\\/ |<br />
| \///// \ / | | \XXXX/ \\\\\/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ u (v) o-----------------------o dv' @ u (v) =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| / \ / \ | | /\\\\\ /\\\\\ | | /\\\\\ /\\\\\ |<br />
| / o \ | | /\\\\\\o\\\\\\\ | | /\\\\\\o\\\\\\\ |<br />
| / / \ \ | | /\\\\\\/ \\\\\\\\ | | /\\\\\\/ \\\\\\\\ |<br />
| o / \ o | | o\\\\\\/ \\\\\\\o | | o\\\\\\/ \\\\\\\o |<br />
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |<br />
| | du | | dv | | | |\\\\\| |\\\\\| | | |\du\\| |\\dv\| |<br />
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |<br />
| o \ / o | | o\\\\\\\ /\\\\\\o | | o\\\\\\\ /\\\\\\o |<br />
| \ \ / / | | \\\\\\\\ /\\\\\\/ | | \\\\\\\\ /\\\\\\/ |<br />
| \ o / | | \\\\\\\o\\\\\\/ | | \\\\\\\o\\\\\\/ |<br />
| \ / \ / | | \\\\\/ \\\\\/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ u v o-----------------------o dv' @ u v =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| U | |\U\\\\\\\\\\\\\\\\\\\\\| |\U\\\\\\\\\\\\\\\\\\\\\|<br />
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|<br />
| /////\ /////\ | |\\\\\/////\\/////\\\\\\| |\\\\\/ \\/ \\\\\\|<br />
| ///////o//////\ | |\\\\///////o//////\\\\\| |\\\\/ o \\\\\|<br />
| /////////\//////\ | |\\\////////X\//////\\\\| |\\\/ /\\ \\\\|<br />
| o//////////\//////o | |\\o///////XXX\//////o\\| |\\o /\\\\ o\\|<br />
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|<br />
| |//u//|/////|//v//| | |\\|//u//|XXXXX|//v//|\\| |\\| u |\\\\\| v |\\|<br />
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|<br />
| o//////\//////////o | |\\o//////\XXX///////o\\| |\\o \\\\/ o\\|<br />
| \//////\///////// | |\\\\//////\X////////\\\| |\\\\ \\/ /\\\|<br />
| \//////o/////// | |\\\\\//////o///////\\\\| |\\\\\ o /\\\\|<br />
| \///// \///// | |\\\\\\/////\\/////\\\\\| |\\\\\\ /\\ /\\\\\|<br />
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|<br />
| | |\\\\\\\\\\\\\\\\\\\\\\\| |\\\\\\\\\\\\\\\\\\\\\\\|<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= u' o-----------------------o v' =<br />
= | U' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/u'//|XXXXX|\\v'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
Figure 70-b. Tangent Functor Ferris Wheel for F<u, v> = <((u)(v)), ((u, v))><br />
</pre><br />
|}<br />
<br />
==Epilogue, Enchoiry, Exodus==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
It is time to explain myself . . . . let us stand up.<br />
| width="4%" | &nbsp;<br />
|- <br />
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 79]<br />
|}<br />
<br />
==Appendices==<br />
<br />
===Appendix 1. Propositional Forms and Differential Expansions===<br />
<br />
====Table A1. Propositional Forms on Two Variables====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A1.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="background:ghostwhite"<br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}</math><br />
| width="25%" | <math>\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{0}\\f_{1}\\f_{2}\\f_{3}\\f_{4}\\f_{5}\\f_{6}\\f_{7}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0000}\\f_{0001}\\f_{0010}\\f_{0011}\\f_{0100}\\f_{0101}\\f_{0110}\\f_{0111}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~0\\0~0~0~1\\0~0~1~0\\0~0~1~1\\0~1~0~0\\0~1~0~1\\0~1~1~0\\0~1~1~1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)~ ~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~ ~(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{,~} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{~~} y \texttt{)}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{false}<br />
\\<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\<br />
y ~\text{without}~ x<br />
\\<br />
\text{not}~ x<br />
\\<br />
x ~\text{without}~ y<br />
\\<br />
\text{not}~ y<br />
\\<br />
x ~\text{not equal to}~ y<br />
\\<br />
\text{not both}~ x ~\text{and}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\<br />
\lnot x \land \lnot y<br />
\\<br />
\lnot x \land y<br />
\\<br />
\lnot x<br />
\\<br />
x \land \lnot y<br />
\\<br />
\lnot y<br />
\\<br />
x \ne y<br />
\\<br />
\lnot x \lor \lnot y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{8}\\f_{9}\\f_{10}\\f_{11}\\f_{12}\\f_{13}\\f_{14}\\f_{15}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{1000}\\f_{1001}\\f_{1010}\\f_{1011}\\f_{1100}\\f_{1101}\\f_{1110}\\f_{1111}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1~0~0~0\\1~0~0~1\\1~0~1~0\\1~0~1~1\\1~1~0~0\\1~1~0~1\\1~1~1~0\\1~1~1~1<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~ ~ ~ ~} y \texttt{~~}<br />
\\<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{~~} x \texttt{~ ~ ~ ~}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{((~))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{and}~ y<br />
\\<br />
x ~\text{equal to}~ y<br />
\\<br />
y<br />
\\<br />
\text{not}~ x ~\text{without}~ y<br />
\\<br />
x<br />
\\<br />
\text{not}~ y ~\text{without}~ x<br />
\\<br />
x ~\text{or}~ y<br />
\\<br />
\text{true}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x \land y<br />
\\<br />
x = y<br />
\\<br />
y<br />
\\<br />
x \Rightarrow y<br />
\\<br />
x<br />
\\<br />
x \Leftarrow y<br />
\\<br />
x \lor y<br />
\\<br />
1<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A2. Propositional Forms on Two Variables====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A2.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="background:ghostwhite"<br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}</math><br />
| width="25%" | <math>\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>f_{0000}\!</math><br />
| <math>0~0~0~0</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0001}\\f_{0010}\\f_{0100}\\f_{1000}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~1\\0~0~1~0\\0~1~0~0\\1~0~0~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\<br />
y ~\text{without}~ x<br />
\\<br />
x ~\text{without}~ y<br />
\\<br />
x ~\text{and}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot x \land \lnot y<br />
\\<br />
\lnot x \land y<br />
\\<br />
x \land \lnot y<br />
\\<br />
x \land y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{3}\\f_{12}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0011}\\f_{1100}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~1~1\\1~1~0~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ x<br />
\\<br />
x<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot x<br />
\\<br />
x<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{6}\\f_{9}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0110}\\f_{1001}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~0\\1~0~0~1<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{not equal to}~ y<br />
\\<br />
x ~\text{equal to}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x \ne y<br />
\\<br />
x = y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{5}\\f_{10}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0101}\\f_{1010}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~0~1\\1~0~1~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ y<br />
\\<br />
y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot y<br />
\\<br />
y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{7}\\f_{11}\\f_{13}\\f_{14}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0111}\\f_{1011}\\f_{1101}\\f_{1110}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~1\\1~0~1~1\\1~1~0~1\\1~1~1~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not both}~ x ~\text{and}~ y<br />
\\<br />
\text{not}~ x ~\text{without}~ y<br />
\\<br />
\text{not}~ y ~\text{without}~ x<br />
\\<br />
x ~\text{or}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot x \lor \lnot y<br />
\\<br />
x \Rightarrow y<br />
\\<br />
x \Leftarrow y<br />
\\<br />
x \lor y<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>f_{1111}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A3. E''f'' Expanded Over Differential Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A3.} ~~ \mathrm{E}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{T}_{11}f\\\mathrm{E}f|_{\mathrm{d}x ~ \mathrm{d}y}\end{matrix}</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{T}_{10}f\\\mathrm{E}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}\end{matrix}</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{T}_{01}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}\end{matrix}</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{T}_{00}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{3}\\f_{12}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{6}\\f_{9}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{5}\\f_{10}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{7}\\f_{11}\\f_{13}\\f_{14}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
|- style="background:ghostwhite"<br />
| style="border-top:1px solid black" colspan="2" | <math>\text{Fixed Point Total}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>4\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>4\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>4\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>16\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A4. D''f'' Expanded Over Differential Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A4.} ~~ \mathrm{D}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\mathrm{D}f|_{\mathrm{d}x ~ \mathrm{d}y}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}~\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
y<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
y<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
x<br />
\\<br />
x<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
x<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
y<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
y<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <br />
<math>\begin{matrix}<br />
x<br />
\\<br />
x<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A5. E''f'' Expanded Over Ordinary Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A5.} ~~ \mathrm{E}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\mathrm{E}f|_{xy}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{E}f|_{x \texttt{(} y \texttt{)}}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{E}f|_{\texttt{(} x \texttt{)} y}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{E}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{3}\\f_{12}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{6}\\f_{9}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{5}\\f_{10}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{7}\\f_{11}\\f_{13}\\f_{14}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A6. D''f'' Expanded Over Ordinary Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A6.} ~~ \mathrm{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\mathrm{D}f|_{xy}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{x \texttt{(} y \texttt{)}}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} x \texttt{)} y}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\texttt{(} x \texttt{)}\\\texttt{~} x \texttt{~}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Appendix 2. Differential Forms===<br />
<br />
The actions of the difference operator <math>\mathrm{D}\!</math> and the tangent operator <math>\mathrm{d}\!</math> on the 16 bivariate propositions are shown in Tables&nbsp;A7 and A8.<br />
<br />
Table A7 expands the differential forms that result over a ''logical basis'':<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
|<br />
<math>\{~ \texttt{(}\mathrm{d}x\texttt{)(}\mathrm{d}y\texttt{)}, ~\mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}, ~\texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.\!</math><br />
|}<br />
<br />
This set consists of the singular propositions in the first order differential variables, indicating mutually exclusive and exhaustive ''cells'' of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the cell-basis, point-basis, or singular differential basis. In this setting it is frequently convenient to use the following abbreviations:<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
|<br />
<math>\partial x ~=~ \mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}\!</math> &nbsp; &nbsp; and &nbsp; &nbsp; <math>\partial y ~=~ \texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y.\!</math><br />
|}<br />
<br />
Table A8 expands the differential forms that result over an ''algebraic basis'':<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
| <math>\{~ 1, ~\mathrm{d}x, ~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.\!</math><br />
|}<br />
<br />
This set consists of the ''positive propositions'' in the first order differential variables, indicating overlapping positive regions of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the ''positive differential basis''.<br />
<br />
====Table A7. Differential Forms Expanded on a Logical Basis====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A7.} ~~ \text{Differential Forms Expanded on a Logical Basis}\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| &nbsp;<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" | <math>\mathrm{D}f~\!</math><br />
| <math>\mathrm{d}f~\!</math><br />
|-<br />
| <math>f_{0}\!</math><br />
| style="border-right:none" | <math>\texttt{(~)}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x) ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\\<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\\<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\\<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\partial x<br />
\\<br />
\partial x<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
\\<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\partial x & + & \partial y<br />
\\<br />
\partial x & + & \partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\partial y<br />
\\<br />
\partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x) ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\\<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\\<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\\<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| style="border-right:none" | <math>\texttt{((~))}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A8. Differential Forms Expanded on an Algebraic Basis====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A8.} ~~ \text{Differential Forms Expanded on an Algebraic Basis}\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| &nbsp;<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" | <math>\mathrm{D}f~\!</math><br />
| <math>\mathrm{d}f~\!</math><br />
|-<br />
| <math>f_{0}\!</math><br />
| style="border-right:none" | <math>\texttt{(~)}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x<br />
\\<br />
\mathrm{d}x<br />
\end{matrix}\!</math><br />
| <math>\begin{matrix}<br />
\mathrm{d}x<br />
\\<br />
\mathrm{d}x<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\\<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\\<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}y<br />
\\<br />
\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y<br />
\\<br />
\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| style="border-right:none" | <math>\texttt{((~))}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A9. Tangent Proposition as Pointwise Linear Approximation====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A9.} ~~ \text{Tangent Proposition}~ \mathrm{d}f = \text{Pointwise Linear Approximation to the Difference Map}~ \mathrm{D}f\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}f =<br />
\\[2pt]<br />
\partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}^2\!f =<br />
\\[2pt]<br />
\partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\mathrm{d}f|_{x \, y}</math><br />
| <math>\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math><br />
|-<br />
| style="border-right:none" | <math>f_0\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{5}\\f_{10}\end{matrix}\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\end{matrix}\!</math><br />
| <math>\begin{matrix}<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>f_{15}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A10. Taylor Series Expansion Df = d''f'' + d<sup>2</sup>''f''====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" |<br />
<math>\text{Table A10.} ~~ \text{Taylor Series Expansion}~ {\mathrm{D}f = \mathrm{d}f + \mathrm{d}^2\!f}\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{D}f<br />
\\<br />
= & \mathrm{d}f & + & \mathrm{d}^2\!f<br />
\\<br />
= & \partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y & + & \partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\mathrm{d}f|_{x \, y}</math><br />
| <math>\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math><br />
|-<br />
| style="border-right:none" | <math>f_0\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>f_{15}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A11. Partial Differentials and Relative Differentials====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A11.} ~~ \text{Partial Differentials and Relative Differentials}\!</math><br />
|- style="background:ghostwhite; height:50px"<br />
| &nbsp;<br />
| <math>f\!</math><br />
| <math>\frac{\partial f}{\partial x}\!</math><br />
| <math>\frac{\partial f}{\partial y}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}f =<br />
\\[2pt]<br />
\partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\left. \frac{\partial x}{\partial y} \right| f\!</math><br />
| <math>\left. \frac{\partial y}{\partial x} \right| f\!</math><br />
|-<br />
| <math>f_0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A12. Detail of Calculation for the Difference Map====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:4px double black; border-left:4px double black; border-right:4px double black; border-top:4px double black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A12.} ~~ \text{Detail of Calculation for}~ {\mathrm{E}f + f = \mathrm{D}f}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:6%" | &nbsp;<br />
| style="width:14%; border-left:1px solid black" | <math>f\!</math><br />
| style="width:20%; border-left:4px double black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\mathrm{d}x ~ \mathrm{d}y}<br />
\\[4pt]<br />
+ & f|_{\mathrm{d}x ~ \mathrm{d}y}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\mathrm{d}x ~ \mathrm{d}y}<br />
\end{array}</math><br />
| style="width:20%; border-left:1px solid black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}<br />
\\[4pt]<br />
+ & f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}<br />
\end{array}</math><br />
| style="width:20%; border-left:1px solid black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
+ & f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}<br />
\end{array}</math><br />
| style="width:20%; border-left:1px solid black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
+ & f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}<br />
\end{array}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{0}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:4px double black; border-left:4px double black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{1}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{)(} y \texttt{)~}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{2}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{)~} y \texttt{~~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{4}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~~} x \texttt{~(} y \texttt{)~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{8}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~~} x \texttt{~~} y \texttt{~~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{3}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{(} x \texttt{)}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{12}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>x\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{6}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{,~} y \texttt{)~}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{9}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} x \texttt{,~} y \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{5}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{(} y \texttt{)}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{10}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>y\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{7}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{~~} y \texttt{)~}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{11}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{~(} y \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{13}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} x \texttt{)~} y \texttt{)~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{14}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} x \texttt{)(} y \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{15}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:4px double black; border-left:4px double black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Appendix 3. Computational Details===<br />
<br />
====Operator Maps for the Logical Conjunction ''f''<sub>8</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{8}~\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{8}<br />
& = && f_{8}(u, v)<br />
\\[4pt]<br />
& = && uv<br />
\\[4pt]<br />
& = && uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & uv \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & uv \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{8}<br />
& = && uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & uv \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & uv \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & uv \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.2-i} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{E}f_{8}<br />
& = & f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{8}(\mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{8}(\mathrm{d}u, \mathrm{d}v)<br />
\\[4pt]<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[20pt]<br />
\mathrm{E}f_{8}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
\\[4pt]<br />
&&&&& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&&&&&& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.2-ii} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{c}}<br />
\mathrm{E}f_{8}<br />
& = & (u + \mathrm{d}u) \cdot (v + \mathrm{d}v)<br />
\\[6pt]<br />
& = & u \cdot v<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}f_{8}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{8}<br />
& = && \mathrm{E}f_{8}<br />
& + & \boldsymbol\varepsilon f_{8}<br />
\\[4pt]<br />
& = && f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{8}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & uv<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{~}<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}f_{8}<br />
& = & \boldsymbol\varepsilon f_{8}<br />
& + & \mathrm{E}f_{8}<br />
\\[6pt]<br />
& = & f_{8}(u, v)<br />
& + & f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[6pt]<br />
& = & uv<br />
& + & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
& = & 0<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}f_{8}<br />
& = & 0<br />
& + & u \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.3-iii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 3)}\!</math><br />
|<br />
<math>\begin{array}{c*{9}{l}}<br />
\mathrm{D}f_{8}<br />
& = & \boldsymbol\varepsilon f_{8} ~+~ \mathrm{E}f_{8}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{8}<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ u \,\cdot\, v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~ u \;\cdot\; v \;\cdot\; \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}f_{8}<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u ~ \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} ~ v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot\, \mathrm{d}u ~ \mathrm{d}v<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = & ~ ~ 0 ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~ ~ u ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ ~ ~ v ~~ \cdot ~ \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
=====Computation of d''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.4} ~~ \text{Computation of}~ \mathrm{d}f_{8}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{8}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.5} ~~ \text{Computation of}~ \mathrm{r}f_{8}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{8} & = & \mathrm{D}f_{8} ~+~ \mathrm{d}f_{8}<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[20pt]<br />
\mathrm{r}f_{8}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Conjunction=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.6} ~~ \text{Computation Summary for}~ f_{8}(u, v) = uv\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{8}<br />
& = & uv \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}f_{8}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{r}f_{8}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Operator Maps for the Logical Equality ''f''<sub>9</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{9}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{9}<br />
& = && f_{9}(u, v)<br />
\\[4pt]<br />
& = && \texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{9}(1, 1)<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{9}(1, 0)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{9}(0, 1)<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{9}(0, 0)<br />
\\[4pt]<br />
& = && u v & + & 0 & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{9}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.2} ~~ \text{Computation of}~ \mathrm{E}f_{9}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{E}f_{9}<br />
& = && f_{9}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! f_{9}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! f_{9}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{ } \mathrm{d}v \texttt{ })<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! f_{9}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! f_{9}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{ } \mathrm{d}v \texttt{ })<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{ (} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{) }<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{ (} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{) }<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{E}f_{9}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{9}<br />
& = && \mathrm{E}f_{9}<br />
& + & \boldsymbol\varepsilon f_{9}<br />
\\[4pt]<br />
& = && f_{9}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{9}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{9}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[20pt]<br />
\mathrm{D}f_{9}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}f_{9}<br />
& = & 0 \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & 1 \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & 1 \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & 0 \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of d''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.4} ~~ \text{Computation of}~ \mathrm{d}f_{9}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.5} ~~ \text{Computation of}~ \mathrm{r}f_{9}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{9} & = & \mathrm{D}f_{9} ~+~ \mathrm{d}f_{9}<br />
\\[20pt]<br />
\mathrm{D}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[20pt]<br />
\mathrm{r}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Equality=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.6} ~~ \text{Computation Summary for}~ f_{9}(u, v) = \texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{9}<br />
& = & uv \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}f_{9}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f_{9}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{9}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}f_{9}<br />
& = & uv \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Operator Maps for the Logical Implication ''f''<sub>11</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{11}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{11}<br />
& = && f_{11}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(} u \texttt{(} v \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{11}(1, 1)<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{11}(1, 0)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{11}(0, 1)<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{11}(0, 0)<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ }<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ }<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{11}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.2} ~~ \text{Computation of}~ \mathrm{E}f_{11}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{E}f_{11}<br />
& = && f_{11}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = &&<br />
\texttt{(}<br />
\\<br />
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\\<br />
&&& \texttt{(}<br />
\\<br />
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\<br />
&&& \texttt{))}<br />
\\[4pt]<br />
& = &&<br />
u v<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)((} \mathrm{d}v \texttt{)))}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u \texttt{((} \mathrm{d}v \texttt{)))}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[4pt]<br />
& = &&<br />
u v<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{E}f_{11}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{11}<br />
& = && \mathrm{E}f_{11}<br />
& + & \boldsymbol\varepsilon f_{11}<br />
\\[4pt]<br />
& = && f_{11}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{11}(u, v)<br />
\\[4pt]<br />
& = &&<br />
\texttt{(} \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\texttt{(} \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{(} v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & u v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & 0<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & 0<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = && u v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{c*{9}{l}}<br />
\mathrm{D}f_{11}<br />
& = & \boldsymbol\varepsilon f_{11} ~+~ \mathrm{E}f_{11}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{11}<br />
& = & u v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}f_{11}<br />
& = &<br />
u v<br />
\cdot<br />
\texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\cdot<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\cdot<br />
\texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\cdot<br />
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = &<br />
u v<br />
\cdot<br />
\texttt{~(} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{~}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\cdot<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\cdot<br />
\texttt{~} \mathrm{d}u ~ \mathrm{d}v \texttt{~}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\cdot<br />
\texttt{~} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of d''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.4} ~~ \text{Computation of}~ \mathrm{d}f_{11}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{11}<br />
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{11}<br />
& = & u v \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.5} ~~ \text{Computation of}~ \mathrm{r}f_{11}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{11} & = & \mathrm{D}f_{11} ~+~ \mathrm{d}f_{11}<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{11}<br />
& = & u v \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u<br />
\\[20pt]<br />
\mathrm{r}f_{11}<br />
& = & u v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Implication=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.6} ~~ \text{Computation Summary for}~ f_{11}(u, v) = \texttt{(} u \texttt{(} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{11}<br />
& = & u v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}f_{11}<br />
& = & u v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f_{11}<br />
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{11}<br />
& = & u v \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u<br />
\\[6pt]<br />
\mathrm{r}f_{11}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Operator Maps for the Logical Disjunction ''f''<sub>14</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{14}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{14}<br />
& = && f_{14}(u, v)<br />
\\[4pt]<br />
& = && \texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{14}(1, 1)<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{14}(1, 0)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{14}(0, 1)<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{14}(0, 0)<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ }<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ }<br />
& + & 0<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{14}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.2} ~~ \text{Computation of}~ \mathrm{E}f_{14}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{E}f_{14}<br />
& = && f_{14}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = &&<br />
\texttt{((}<br />
\\<br />
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\\<br />
&&& \texttt{)(}<br />
\\<br />
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\<br />
&&& \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! f_{14}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! f_{14}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{ } \mathrm{d}v \texttt{ })<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! f_{14}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! f_{14}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{ } \mathrm{d}v \texttt{ })<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{E}f_{14}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{14}<br />
& = && \mathrm{E}f_{14}<br />
& + & \boldsymbol\varepsilon f_{14}<br />
\\[4pt]<br />
& = && f_{14}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{14}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{))((} v \texttt{,} \mathrm{d}v \texttt{)))}<br />
& + & \texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{14}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
\\[4pt]<br />
&& + & 0<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
\\[4pt]<br />
&& + & uv \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
\\[20pt]<br />
\mathrm{D}f_{14}<br />
& = && uv \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}f_{14}<br />
& = & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & 0 \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of d''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.4} ~~ \text{Computation of}~ \mathrm{d}f_{14}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.5} ~~ \text{Computation of}~ \mathrm{r}f_{14}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{14} & = & \mathrm{D}f_{14} ~+~ \mathrm{d}f_{14}<br />
\\[20pt]<br />
\mathrm{D}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{d}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[20pt]<br />
\mathrm{r}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Disjunction=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.6} ~~ \text{Computation Summary for}~ f_{14}(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{14}<br />
& = & uv \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 1<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}f_{14}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f_{14}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{d}f_{14}<br />
& = & uv \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}f_{14}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
===Appendix 4. Source Materials===<br />
<br />
===Appendix 5. Various Definitions of the Tangent Vector===<br />
<br />
==References==<br />
<br />
===Works Cited===<br />
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{| cellpadding=3<br />
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|}<br />
<br />
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| Shakespeare, William, ''A Midsummer Night's Dream'', Washington Square Press, New York, NY, 1958.<br />
|-<br />
| valign=top | [Sh2]<br />
| Shakespeare, William, ''The Tragedy of Hamlet, Prince of Denmark'', In [Sha], pp. 654&ndash;690.<br />
|-<br />
| valign=top | [Sh3]<br />
| Shakespeare, William, ''Measure for Measure'', Washington Square Press, New York, NY, 1965.<br />
|-<br />
| valign=top | [Web]<br />
| ''Webster's Ninth New Collegiate Dictionary'', Merriam-Webster, Springfield, MA, 1983.<br />
|-<br />
| valign=top | [Whi]<br />
| Whitman, Walt, ''Leaves of Grass'', Vintage Books / The Library of America, New York, NY, 1992. Originally published in numerous editions, 1855&ndash;1892.<br />
|-<br />
| valign=top | [Wil]<br />
| Wilhelm, R., and Baynes, C.F. (trans.), ''The I Ching, or Book of Changes'', foreword by C.G. Jung, preface by H. Wilhelm, 3rd edition, Bollingen Series XIX, Princeton University Press, Princeton, NJ, 1967.<br />
|}<br />
<br />
==Document History==<br />
<br />
<p align="center"><math>\begin{array}{lcr}<br />
& \text{Differential Logic and Dynamic Systems} &<br />
\\<br />
\text{Author:} & \text{Jon Awbrey} & \text{October 20, 1994}<br />
\\<br />
\text{Course:} & \text{Engineering 690, Graduate Project} & \text{Winter Term 1994}<br />
\\<br />
\text{Supervisor:} & \text{M.A. Zohdy} & \text{Oakland University}<br />
\\<br />
\text{Created:} && \text{16 Dec 1993}<br />
\\<br />
\text{Relayed:} && \text{31 Oct 1994}<br />
\\<br />
\text{Revised:} && \text{03 Jun 2003}<br />
\\<br />
\text{Recoded:} && \text{03 Jun 2007}<br />
\end{array}</math></p><br />
<br />
[[Category:Adaptive Systems]]<br />
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[[Category:Formal Sciences]]<br />
[[Category:Formal Systems]]<br />
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[[Category:Visualization]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems&diff=480649Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems2023-03-03T14:30:10Z<p>Jon Awbrey: /* Document History */</p>
<hr />
<div>{{DISPLAYTITLE:Differential Logic and Dynamic Systems}}<br />
{| align="center" cellpadding="10" width="100%"<br />
| '''''NOTE.''' The current version of this document is '''[[Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0|Differential Logic and Dynamic Systems 2.0]].'''''<br />
|}<br />
<br />
{| align="center" cellpadding="10"<br />
| [[Image:Tangent_Functor_Ferris_Wheel.gif]]<br />
|}<br />
<br />
{| style="height:36px; width:100%"<br />
| align="left" | ''Stand and unfold yourself.''<br />
| align="right" | Hamlet: Francsico&mdash;1.1.2<br />
|}<br />
<br />
This article develops a differential extension of [[propositional calculus]] and applies it to a context of problems arising in dynamic systems. The work pursued here is coordinated with a parallel application that focuses on neural network systems, but the dependencies are arranged to make the present article the main and the more self-contained work, to serve as a conceptual frame and a technical background for the network project.<br />
<br />
==Review and Transition==<br />
<br />
This note continues a previous discussion on the problem of dealing with change and diversity in logic-based intelligent systems. For ease of reference, I begin by summarizing essential material from previous reports.<br />
<br />
Table 1 outlines the notation that I use for propositional calculus. Explained as briefly as possible, I am using only two basic kinds of truth-functional connectives, both of variable ''k''-ary scope.<br />
<br />
# For the first, I use concatenation as a connective to indicate the logical conjunction of ''k'' arguments.<br />
# For the other, I use a bracket of the form (&nbsp;,&nbsp;,&nbsp;,&nbsp;) as a connective which says that exactly one of its ''k'' arguments is false.<br />
<br />
All other truth-functional connectives can be obtained in a very efficient style of representation through combinations of these two forms.<br />
<br />
This treatment of propositional logic is derived from the work of [[C.S. Peirce]] [P1, P2], who gave this approach an extensive development in his graphical systems of predicate, relational, and modal logic [Rob]. More recently, these ideas were revived and supplemented in an alternative interpretation by G. Spencer-Brown [SpB]. Both of these authors used other forms of enclosure where I use parentheses, but the structural topologies of expression and the functional varieties of interpretation are fundamentally the same.<br />
<br />
While working with expressions solely in propositional calculus, the use of plain parentheses to represent logical connectives is simplest to write and easiest to read for both human and machine parsers. In the present text I preserve this form of expression in tables and set-off displays, but in contexts where parentheses are needed for functional notation I will use a distinctive font for logical operators.<br />
<br />
The briefest expression for logical truth is the empty word, usually denoted by &epsilon; or &lambda; in formal languages, where it forms the identity element for concatenation. To make it visible in this text, I denote it by the equivalent expression "(())", or, especially if operating in an algebraic context, by a simple "1". Also when working in an algebraic mode, I use the plus sign "+" for exclusive disjunction. Thus, we may express the following paraphrases of algebraic forms:<br />
<br />
:{| cellpadding="4"<br />
| ''A'' + ''B''<br />
| =<br />
| (''A'', ''B'')<br />
|-<br />
| ''A'' + ''B'' + ''C''<br />
| =<br />
| ((''A'', ''B''), ''C'')<br />
| =<br />
| (''A'', (''B'', ''C''))<br />
|}<br />
<br />
One should be careful to observe that these last two expressions are not equivalent to the form (''A'',&nbsp;''B'',&nbsp;''C'').<br />
<br />
<font face="courier new"><br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"<br />
|+ '''Table 1. Syntax and Semantics of a Calculus for Propositional Logic'''<br />
|- style="background:paleturquoise"<br />
! Expression<br />
! Interpretation<br />
! Other Notations<br />
|-<br />
| "&nbsp;"<br />
| True.<br />
| 1<br />
|-<br />
| (&nbsp;)<br />
| False.<br />
| 0<br />
|-<br />
| A<br />
| A.<br />
| A<br />
|-<br />
| (A)<br />
| Not A.<br />
| &nbsp;A’ <br> ~A <br> &not;A<br />
|-<br />
| A B C<br />
| A and B and C.<br />
| A &and; B &and; C<br />
|-<br />
| ((A)(B)(C))<br />
| A or B or C.<br />
| A &or; B &or; C<br />
|-<br />
| (A (B))<br />
| A implies B. <br> If A then B.<br />
| A &rArr; B<br />
|-<br />
| (A, B)<br />
| A not equal to B. <br> A exclusive-or B.<br />
| A &ne; B <br> A + B<br />
|-<br />
| ((A, B))<br />
| A is equal to B. <br> A if & only if B.<br />
| A = B <br> A &hArr; B<br />
|-<br />
| (A, B, C)<br />
| Just one of <br> A, B, C <br> is false.<br />
|<br />
A’B C &or;<br><br />
A B’C &or;<br><br />
A B C’<br />
|-<br />
| ((A),(B),(C))<br />
| Just one of <br> A, B, C <br> is true. <br><br><br />
Partition all <br> into A, B, C.<br />
|<br />
A B’C’ &or;<br><br />
A’B C’ &or;<br><br />
A’B’C<br />
|-<br />
| ((A, B), C) <br> &nbsp; <br> (A, (B, C))<br />
| Oddly many of <br> A, B, C <br> are true.<br />
|<br />
A + B + C<br>&nbsp;<br><br />
A B C &nbsp;&or;<br><br />
A B’C’ &or;<br><br />
A’B C’ &or;<br><br />
A’B’C<br />
|-<br />
| (Q, (A),(B),(C))<br />
| Partition Q <br> into A, B, C.<br><br />
Genus Q comprises <br> species A, B, C.<br />
|<br />
Q’A’B’C’ &or;<br><br />
Q A B’C’ &or;<br><br />
Q A’B C’ &or;<br><br />
Q A’B’C<br />
|}<br />
</font><br><br />
<br />
NB. The usage that one often sees, of a plus sign "+" to represent inclusive disjunction, and the reference to this operation as ''boolean addition'', is a misnomer on at least two counts. Boole used the plus sign to represent exclusive disjunction (at any rate, an operation of aggregation restricted in its logical interpretation to cases where the represented sets are disjoint (Boole, 32)), as any mathematician with a sensitivity to the ring and field properties of algebra would do:<br />
<br />
<blockquote><br />
The expression ''x'' + ''y'' seems indeed uninterpretable, unless it be assumed that the things represented by ''x'' and the things represented by ''y'' are entirely separate; that they embrace no individuals in common. (Boole, 66).<br />
</blockquote><br />
<br />
It was only later that Peirce and Jevons treated inclusive disjunction as a fundamental operation, but these authors, with a respect for the algebraic properties that were already associated with the plus sign, used a variety of other symbols for inclusive disjunction (Sty, 177, 189). It seems to have been Schroeder who later reassigned the plus sign to inclusive disjunction (Sty, 208). Additional information, discussion, and references can be found in (Boole) and (Sty, 177-263). Aside from these historical points, which never really count against a current practice that has gained a life of its own, this usage does have a further disadvantage of cutting or confounding the lines of communication between algebra and logic. For this reason, I am forced to avoid it here.<br />
<br />
==A Functional Conception of Propositional Calculus==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Out of the dimness opposite equals advance . . . .<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Always substance and increase,<br><br />
Always a knit of identity . . . . always distinction . . . .<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;always a breed of life.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]<br />
|}<br />
<br />
In the general case, we start with a set of logical features {''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>} that represent properties of objects or propositions about the world. In concrete examples the features {''a''<sub>''i''</sub>} commonly appear as capital letters from an ''alphabet'' like {''A'', ''B'', ''C'', &hellip;} or as meaningful words from a linguistic ''vocabulary'' of codes. This language can be drawn from any sources, whether natural, technical, or artificial in character and interpretation. In the application to dynamic systems we tend to use the letters {''x''<sub>1</sub>, &hellip;, ''x''<sub>''n''</sub>} as our coordinate propositions, and to interpret them as denoting properties of a system's ''state'', that is, as propositions about its location in configuration space. Because I have to consider non-deterministic systems from the outset, I often use the word ''state'' in a loose sense, to denote the position or configuration component of a contemplated state vector, whether or not it ever gets a deterministic completion.<br />
<br />
The set of logical features {''a''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''a''<sub>''n''</sub>} provides a basis for generating an ''n''-dimensional ''universe of discourse'' that I denote as [''a''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''a''<sub>''n''</sub>]. It is useful to consider each universe of discourse as a unified categorical object that incorporates both the set of points 〈''a''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''a''<sub>''n''</sub>〉 and the set of propositions ''f''&nbsp;:&nbsp;〈''a''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''a''<sub>''n''</sub>〉&nbsp;&rarr;&nbsp;'''B''' that are implicit with the ordinary picture of a venn diagram on ''n'' features. Thus, we may regard the universe of discourse [''a''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''a''<sub>''n''</sub>] as an ordered pair having the type ('''B'''<sup>''n''</sup>,&nbsp;('''B'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''B'''), and we may abbreviate this last type designation as '''B'''<sup>''n''</sup>&nbsp;+&rarr;&nbsp;'''B''', or even more succinctly as ['''B'''<sup>''n''</sup>]. (Used this way, the angle brackets 〈&hellip;〉 are referred to as ''generator brackets''.)<br />
<br />
Table 2 exhibits the scheme of notation I use to formalize the domain of propositional calculus, corresponding to the logical content of truth tables and venn diagrams. Although it overworks the square brackets a bit, I also use either one of the equivalent notations [''n''] or '''''n''''' to denote the data type of a finite set on n elements.<br />
<br />
<font face="courier new"><br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"<br />
|+ '''Table 2. Fundamental Notations for Propositional Calculus'''<br />
|- style="background:paleturquoise"<br />
! Symbol<br />
! Notation<br />
! Description<br />
! Type<br />
|-<br />
| <font face="lucida calligraphy">A<font><br />
| {''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>}<br />
| Alphabet<br />
| [''n''] = '''n'''<br />
|-<br />
| ''A''<sub>''i''</sub><br />
| {(''a''<sub>''i''</sub>), ''a''<sub>''i''</sub>}<br />
| Dimension ''i''<br />
| '''B'''<br />
|-<br />
| ''A''<br />
|<br />
〈<font face="lucida calligraphy">A</font>〉<br><br />
〈''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>〉<br><br />
{‹''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>›}<br><br />
''A''<sub>1</sub> &times; &hellip; &times; ''A''<sub>''n''</sub><br><br />
&prod;<sub>''i''</sub> ''A''<sub>''i''</sub><br />
|<br />
Set of cells,<br><br />
coordinate tuples,<br><br />
points, or vectors<br><br />
in the universe<br><br />
of discourse<br />
| '''B'''<sup>''n''</sup><br />
|-<br />
| ''A''*<br />
| (hom : ''A'' &rarr; '''B''')<br />
| Linear functions<br />
| ('''B'''<sup>''n''</sup>)* = '''B'''<sup>''n''</sup><br />
|-<br />
| ''A''^<br />
| (''A'' &rarr; '''B''')<br />
| Boolean functions<br />
| '''B'''<sup>''n''</sup> &rarr; '''B'''<br />
|-<br />
| ''A''<sup>&bull;</sup><br />
|<br />
[<font face="lucida calligraphy">A</font>]<br><br />
(''A'', ''A''^)<br><br />
(''A'' +&rarr; '''B''')<br><br />
(''A'', (''A'' &rarr; '''B'''))<br><br />
[''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>]<br />
|<br />
Universe of discourse<br><br />
based on the features<br><br />
{''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>}<br />
|<br />
('''B'''<sup>''n''</sup>, ('''B'''<sup>''n''</sup> &rarr; '''B'''))<br><br />
('''B'''<sup>''n''</sup> +&rarr; '''B''')<br><br />
['''B'''<sup>''n''</sup>]<br />
|}<br />
</font><br><br />
<br />
===Qualitative Logic and Quantitative Analogy===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
''Logical'', however, is used in a third sense, which is at once more vital and more practical; to denote, namely, the systematic care, negative and positive, taken to safeguard reflection so that it may yield the best results under the given conditions.<br />
| width="4%" | &nbsp;<br />
|- <br />
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 56]<br />
|}<br />
<br />
These concepts and notations can now be explained in greater detail. In order to begin as simply as possible, I distinguish two levels of analysis and set out initially on the easier path. On the first level of analysis, I take spaces like '''B''', '''B'''<sup>''n''</sup>, and ('''B'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''B''') at face value and treat them as the primary objects of interest. On the second level of analysis, I use these spaces as coordinate charts for talking about points and functions in more fundamental spaces.<br />
<br />
A pair of spaces, of types '''B'''<sup>''n''</sup> and ('''B'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''B'''), give typical expression to everything that we commonly associate with the ordinary picture of a venn diagram. The dimension, ''n'', counts the number of "circles" or simple closed curves that are inscribed in the universe of discourse, corresponding to its relevant logical features or basic propositions. Elements of type '''B'''<sup>''n''</sup> correspond to what are often called propositional<br />
''interpretations'' in logic, that is, the different assignments of truth values to sentence letters. Relative to a given universe of discourse, these interpretations are visualized as its ''cells'', in other words, the smallest enclosed areas or undivided regions of the venn diagram. The functions ''f''&nbsp;:&nbsp;'''B'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''B''' correspond to the different ways of shading<br />
the venn diagram to indicate arbitrary propositions, regions, or sets. Regions included under a shading indicate the ''models'', and regions excluded represent the ''non-models'' of a proposition. To recognize and formalize the natural cohesion of these two layers of concepts into a single universe of discourse, I introduce the type notations ['''B'''<sup>''n''</sup>] = '''B'''<sup>''n''</sup>&nbsp;+&rarr;&nbsp;'''B''' to stand for the pair of types ('''B'''<sup>''n''</sup>, ('''B'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''B''')). The resulting "stereotype" serves to frame the universe of discourse as a unified categorical object, and makes it subject to prescribed sets of evaluations and transformations (categorical morphisms or ''arrows'') that affect the universe of discourse as an integrated whole.<br />
<br />
Most of the time we can serve the algebraic, geometric, and logical interests of our study without worrying about their occasional conflicts and incidental divergences. The conventions and definitions already set down will continue to cover most of the algebraic and functional aspects of our discussion, but to handle the logical and qualitative aspects we will need to add a few more. In general, abstract sets may be denoted by gothic, greek, or script capital variants of ''A'', ''B'', ''C'', and so on, with elements denoted by a corresponding set of subscripted letters in plain lower case, for example, <font face="lucida calligraphy">A</font> = {''a''<sub>''i''</sub>}. Most<br />
of the time, a set such as <font face="lucida calligraphy">A</font> = {''a''<sub>''i''</sub>} will be employed as the ''alphabet'' of<br />
a [[formal language]]. These alphabet letters serve to name the logical features (properties or propositions) that generate a particular universe of discourse. When we want to discuss the particular features of a universe of discourse,<br />
beyond the abstract designation of a type like ('''B'''<sup>''n''</sup>&nbsp;+&rarr;&nbsp;'''B'''), then we may use the following notations. If <font face="lucida calligraphy">A</font> = {''a''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''a''<sub>''n''</sub>} is an alphabet of logical<br />
features, then ''A'' = 〈<font face="lucida calligraphy">A</font>〉 = 〈''a''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''a''<sub>''n''</sub>〉 is the set of interpretations, ''A''^ = (''A'' &rarr; '''B''') is the set of propositions, and ''A''<sup>&nbsp;&bull;</sup> = [<font face="lucida calligraphy">A</font>] = [''a''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''a''<sub>''n''</sub>] is<br />
the combination of these interpretations and propositions into the universe of discourse that is based on the features {''a''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''a''<sub>''n''</sub>}.<br />
<br />
As always, especially in concrete examples, these rules may be dropped whenever necessary, reverting to a free assortment of feature labels. However, when we need to talk about the logical aspects of a space that is already named as a vector space, it will be necessary to make special provisions. At any rate, these elaborations can be deferred until actually needed.<br />
<br />
===Philosophy of Notation : Formal Terms and Flexible Types===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Where number is irrelevant, regimented mathematical technique has hitherto tended to be lacking. Thus it is that the progress of natural science has depended so largely upon the discernment of measurable quantity of one sort or another.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]<br />
|}<br />
<br />
For much of our discussion propositions and boolean functions are treated as the same formal objects, or as different interpretations of the same formal calculus. This rule of interpretation has exceptions, though. There is a distinctively logical interest in the use of propositional calculus that is not exhausted by its functional interpretation. It is part of our task in this study to deal with these uniquely logical characteristics as they present themselves both in our subject matter and in our formal calculus. Just to provide a hint of what's at stake: In logic, as opposed to the more imaginative realms of mathematics, we consider it a good thing to always know what we are talking about. Where mathematics encourages tolerance for uninterpreted symbols as intermediate terms, logic exerts a keener effort to interpret directly each oblique carrier of meaning, no matter how slight, and to unfold the complicities of every indirection in the flow of information. Translated into functional terms, this means that we want to maintain a continual, immediate, and persistent sense of both the converse relation ''f''<sup>&ndash;1</sup> &sube; '''B'''&nbsp;&times;&nbsp;'''B'''<sup>''n''</sup>, or what is the same thing, ''f''<sup>&ndash;1</sup> : '''B''' &rarr; ''Pow''('''B'''<sup>''n''</sup>), and the ''fibers'' or inverse images ''f''<sup>&ndash;1</sup>(0) and ''f''<sup>&ndash;1</sup>(1), associated with each boolean function ''f'' : '''B'''<sup>''n''</sup> &rarr; '''B''' that we use. In practical terms, the desired implementation of a propositional interpreter should incorporate our intuitive recognition that the induced partition of the functional domain into level sets ''f''<sup>&ndash;1</sup>(''b''), for ''b'' &isin; '''B''', is part and parcel of understanding the denotative uses of each propositional function ''f''.<br />
<br />
===Special Classes of Propositions===<br />
<br />
It is important to remember that the coordinate propositions {''a''<sub>''i''</sub>}, besides being projection maps ''a''<sub>''i''</sub>&nbsp;:&nbsp;'''B'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''B''', are propositions on an equal footing with all others, even though employed as a basis in a particular moment. This set of ''n'' propositions may sometimes be referred to as the ''basic'' or ''simple'' propositions that found the universe of discourse. As typical and collective notations, we may use the forms {''a''<sub>''i''</sub> : '''B'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''B'''} = ('''B'''<sup>''n''</sup>&nbsp;<font face=symbol>'''¸>'''</font>&nbsp;'''B''') to indicate the adoption of a set of ''a''<sub>''i''</sub> as a basis for discourse.<br />
<br />
Among the <math>2^{2^n}</math> propositions or functions in ('''B'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''B''') are several fundamental sets of 2<sup>''n''</sup> propositions each that take on special forms with respect to a given basis <font face="lucida calligraphy">A</font>&nbsp;=&nbsp;{''a''<sub>''i''</sub>}. Three of these forms are especially common, the ''linear'', the ''positive'', and the ''singular''<br />
propositions. Each set is naturally parameterized by the coordinate vectors in '''B'''<sup>''n''</sup> and falls into ''n''+1 ranks, with a binomial coefficient <math>\tbinom{n}{k}</math> giving the number of propositions that have rank or weight ''k''.<br />
<br />
The ''linear propositions'', {hom&nbsp;:&nbsp;'''B'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''B'''} = ('''B'''<sup>''n''</sup>&nbsp;<font face=symbol>'''+>'''</font>&nbsp;'''B'''), may be expressed as sums of the following form:<br />
<br />
: <math>\textstyle \sum_{i=1}^n e_i = e_1 + \ldots + e_n \ \mbox{where} \ \forall_{i=1}^n \ e_i = a_i \ \mbox{or} \ e_i = 0.</math><br />
<br />
The ''positive propositions'', {pos&nbsp;:&nbsp;'''B'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''B'''} = ('''B'''<sup>''n''</sup>&nbsp;<font face=symbol>'''¥>'''</font>&nbsp;'''B'''), may be expressed as products of the following form:<br />
<br />
: <math>\textstyle \prod_{i=1}^n e_i = e_1 \cdot \ldots \cdot e_n \ \mbox{where} \ \forall_{i=1}^n \ e_i = a_i \ \mbox{or} \ e_i = 1.</math><br />
<br />
The ''singular propositions'', {''x''&nbsp;:&nbsp;'''B'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''B'''} = ('''B'''<sup>''n''</sup>&nbsp;<font face=symbol>'''××>'''</font>&nbsp;'''B'''), may be expressed as products of the following form:<br />
<br />
: <math>\textstyle \prod_{i=1}^n e_i = e_1 \cdot \ldots \cdot e_n \ \mbox{where} \ \forall_{i=1}^n \ e_i = a_i \ \mbox{or} \ e_i = (a_i) = \lnot a_i.</math><br />
<br />
In each case the rank ''k'' ranges from 0 to ''n'' and counts the number of positive appearances of coordinate propositions ''a''<sub>''i''</sub> in the resulting expression.<br />
<br />
For example, for ''n'' = 3, the linear proposition of rank 0 is 0, the positive proposition of rank 0 is 1, and the singular proposition of rank 0 is (''a''<sub>1</sub>)(''a''<sub>2</sub>)(''a''<sub>3</sub>).<br />
<br />
The coordinate projections or simple propositions ''a''<sub>''i''</sub>&nbsp;:&nbsp;'''B'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''B''' are both linear and positive. So these two kinds of propositions, the linear or the positive, may be viewed as two different ways of generalizing the class of simple projections. The linear and the positive propositions are generated by taking boolean sums and products, respectively, over selected subsets of the basic propositions in {''a''<sub>''i''</sub>}. Therefore, each set of functions can be parameterized by the subsets ''J'' of the basic index set <font face="lucida calligraphy">I</font>&nbsp;=&nbsp;{1,&nbsp;&hellip;,&nbsp;''n''}.<br />
<br />
Let us define <font face="lucida calligraphy">A</font><sub>''J''</sub> as the subset of <font face="lucida calligraphy">A</font> that is given by {''a''<sub>''i''</sub> : ''i'' &isin; ''J''}. Then we may comprehend the action of the linear and the positive propositions in the following terms:<br />
<br />
* The linear proposition <font face="mt extra">l</font><sub>''J''</sub> : '''B'''<sup>''n''</sup> &rarr; '''B''' evaluates each cell ''x'' of '''B'''<sup>''n''</sup> by looking at the coefficients of ''x'' with respect to the features that <font face="mt extra">l</font><sub>''J''</sub> "likes", namely those in <font face="lucida calligraphy">A</font><sub>''J''</sub>, and then adds them up in '''B'''. Thus, <font face="mt extra">l</font><sub>''J''</sub>(''x'') computes the parity of the number of features that ''x'' has in <font face="lucida calligraphy">A</font><sub>''J''</sub>, yielding one for odd and zero for even. Expressed in this idiom, <font face="mt extra">l</font><sub>''J''</sub>(''x'') = 1 says that ''x'' seems ''odd'' (or ''oddly true'') to <font face="lucida calligraphy">A</font><sub>''J''</sub>, whereas <font face="mt extra">l</font><sub>''J''</sub>(''x'') = 0 says that ''x'' seems ''even'' (or ''evenly true'') to <font face="lucida calligraphy">A</font><sub>''J''</sub>, so long as we recall that ''zero times'' is evenly often, too.<br />
<br />
* The positive proposition ''p''<sub>''J''</sub> : '''B'''<sup>''n''</sup> &rarr; '''B''' evaluates each cell ''x'' of '''B'''<sup>''n''</sup> by looking at the coefficients of ''x'' with regard to the features that ''p''<sub>''J''</sub> "likes", namely those in <font face="lucida calligraphy">A</font><sub>''J''</sub>, and then takes their product in ''B''. Thus, ''p''<sub>''J''</sub>(''x'') assesses the unanimity of the multitude of features that ''x'' has in <font face="lucida calligraphy">A</font><sub>''J''</sub>, yielding one for all and aught for else. In these consensual or contractual terms, ''p''<sub>''J''</sub>(''x'') = 1 means that x is ''AOK'' or congruent with all of the conditions of <font face="lucida calligraphy">A</font><sub>''J''</sub>, while ''p''<sub>''J''</sub>(''x'') = 0 means that ''x'' defaults or dissents from some condition of <font face="lucida calligraphy">A</font><sub>''J''</sub>.<br />
<br />
===Basis Relativity and Type Ambiguity===<br />
<br />
Finally, two things are important to keep in mind with regard to the simplicity, linearity, positivity, and singularity of propositions.<br />
<br />
First, all of these properties are relative to a particular basis. For example, a singular proposition with respect to a basis <font face="lucida calligraphy">A</font> will not remain singular if <font face="lucida calligraphy">A</font> is extended by a number of new and independent features. Even if we stick to the original set of pairwise options {''a''<sub>''i''</sub>} &cup; {(''a''<sub>''i''</sub>)} to select a new basis, the sets of linear and positive propositions are determined by the choice of simple propositions, and this determination is tantamount to the conventional choice of a cell as origin.<br />
<br />
Second, the singular propositions '''B'''<sup>''n''</sup>&nbsp;<font face=symbol>'''××>'''</font>&nbsp;'''B''', picking out as they do a single cell or a coordinate tuple of '''B'''<sup>''n''</sup>, become the carriers or the vehicles of a certain type-ambiguity that vacillates between the dual forms '''B'''<sup>''n''</sup> and ('''B'''<sup>''n''</sup>&nbsp;<font face=symbol>'''××>'''</font>&nbsp;'''B''') and infects the whole hierarchy of types built on them. In plainer language, the terms that signify the interpretations ''x''&nbsp;:&nbsp;'''B'''<sup>''n''</sup> and the singular propositions ''x''&nbsp;:&nbsp;'''B'''<sup>''n''</sup>&nbsp;<font face=symbol>'''××>'''</font>&nbsp;'''B''' are fully equivalent in information, and this means that every<br />
token of the type '''B'''<sup>''n''</sup> can be reinterpreted as an appearance of the subtype '''B'''<sup>''n''</sup>&nbsp;<font face=symbol>'''××>'''</font>&nbsp;'''B'''. And vice versa, the two types can be exchanged with each other everywhere that they turn up. In practical terms, this allows the use of singular propositions as a way of denoting points, forming an alternative to coordinate tuples.<br />
<br />
For example, relative to the universe of discourse [''a''<sub>1</sub>,&nbsp;''a''<sub>2</sub>,&nbsp;''a''<sub>3</sub>] the singular proposition ''a''<sub>1</sub>&nbsp;''a''<sub>2</sub>&nbsp;''a''<sub>3</sub>&nbsp;:&nbsp;'''B'''<sup>3</sup>&nbsp;<font face=symbol>'''××>'''</font>&nbsp;'''B''' could be explicitly retyped as ''a''<sub>1</sub>&nbsp;''a''<sub>2</sub>&nbsp;''a''<sub>3</sub>&nbsp;:&nbsp;'''B'''<sup>3</sup> to indicate the point <font face=system>‹1,&nbsp;1,&nbsp;1›</font>, but in most cases the proper interpretation could be gathered from context. Both notations remain dependent on a particular basis, but the code that is generated under the singular option has the advantage in its self-commenting features, in other words, it constantly reminds us of its basis in the process of denoting points. When the time comes to put a multiplicity of different bases into play, and to search for objects and properties that remain invariant under the transformations between them, this infinitesimal potential advantage may well evolve into an overwhelming practical necessity.<br />
<br />
===The Analogy Between Real and Boolean Types===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Measurement consists in correlating our subject matter with the series of real numbers; and such correlations are desirable because, once they are set up, all the well-worked theory of numerical mathematics lies ready at hand as a tool for our further reasoning.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]<br />
|}<br />
<br />
There are two further reasons why I am spending so much time on a careful treatment of types, and they both have to do with our being able to take full computational advantage of certain dimensions of flexibility in the types that apply to terms. First, the domains of differential geometry and logic programming are connected by analogies between real and boolean types of the same pattern. Second, the types involved in these patterns have important isomorphisms connecting them that apply on both the real and the boolean sides of the picture.<br />
<br />
Amazingly enough, these isomorphisms are themselves schematized by the axioms and theorems of propositional logic. This fact is known as the ''propositions as types'' analogy or the Curry-Howard isomorphism [How]. In another formulation it says that terms are to types as proofs are to propositions. This principle seems to have more implications for our subject than I can fully comprehend at present, though I sense that they must be crucial. (Cf. [LaS, 42-46] and [SeH] for a good discussion and further references.) To anticipate the bearing of these issues on our immediate topic, Table 3 sketches a partial overview of the Real to Boolean analogy that may serve to illustrate the paradigm that I have in mind.<br />
<br />
<font face="courier new"><br />
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"<br />
|+ '''Table 3. Analogy of Real and Boolean Types'''<br />
|- style="background:paleturquoise"<br />
! Real Domain '''R'''<br />
! &larr;&rarr;<br />
! Boolean Domain '''B'''<br />
|-<br />
| '''R'''<sup>''n''</sup><br />
| Basic Space<br />
| '''B'''<sup>''n''</sup><br />
|-<br />
| '''R'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''R'''<br />
| Function Space<br />
| '''B'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''B'''<br />
|-<br />
| ('''R'''<sup>''n''</sup>&rarr;'''R''')&nbsp;&rarr;&nbsp;'''R'''<br />
| Tangent Vector<br />
| ('''B'''<sup>''n''</sup>&rarr;'''B''')&nbsp;&rarr;&nbsp;'''B'''<br />
|-<br />
| '''R'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;(('''R'''<sup>''n''</sup>&rarr;'''R''')&rarr;'''R''')<br />
| Vector Field<br />
| '''B'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;(('''B'''<sup>''n''</sup>&rarr;'''B''')&rarr;'''B''')<br />
|-<br />
| ('''R'''<sup>''n''</sup>&nbsp;&times;&nbsp;('''R'''<sup>''n''</sup>&rarr; '''R'''))&nbsp;&rarr;&nbsp;'''R'''<br />
| ditto<br />
| ('''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;('''B'''<sup>''n''</sup>&rarr; '''B'''))&nbsp;&rarr;&nbsp;'''B'''<br />
|-<br />
| (('''R'''<sup>''n''</sup>&rarr;'''R''')&nbsp;&times;&nbsp;'''R'''<sup>''n''</sup>)&nbsp;&rarr;&nbsp;'''R'''<br />
| ditto<br />
| (('''B'''<sup>''n''</sup>&rarr;'''B''')&nbsp;&times;&nbsp;'''B'''<sup>''n''</sup>)&nbsp;&rarr;&nbsp;'''B'''<br />
|-<br />
| ('''R'''<sup>''n''</sup>&rarr;'''R''')&nbsp;&rarr;&nbsp;('''R'''<sup>''n''</sup>&rarr;'''R''')<br />
| Derivation<br />
| ('''B'''<sup>''n''</sup>&rarr;'''B''')&nbsp;&rarr;&nbsp;('''B'''<sup>''n''</sup>&rarr;'''B''')<br />
|-<br />
| '''R'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''R'''<sup>''m''</sup><br />
| Basic Transformation<br />
| '''B'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''B'''<sup>''m''</sup><br />
|-<br />
| ('''R'''<sup>''n''</sup>&rarr;'''R''')&nbsp;&rarr;&nbsp;('''R'''<sup>''m''</sup>&rarr;'''R''')<br />
| Function Transformation<br />
| ('''B'''<sup>''n''</sup>&rarr;'''B''')&nbsp;&rarr;&nbsp;('''B'''<sup>''m''</sup>&rarr;'''B''')<br />
|-<br />
| ...<br />
| ...<br />
| ...<br />
|}<br />
</font><br><br />
<br />
The Table exhibits a sample of likely parallels between the real and boolean domains. The central column gives a selection of terminology that I borrow from typical usage in differential geometry and extend in its meaning to the logical side of the Table. These are the varieties of spaces that come up when we turn to analyzing the dynamics of processes that followe their courses through the states of an arbitrary space ''X''. Moreover, when it becomes necessary to approach situations of overwhelming dynamic complexity in a succession of qualitative reaches, then the methods of logic that are afforded by the boolean domains, with their declarative means of synthesis and deductive modes of analysis, supply a natural battery of tools for the task.<br />
<br />
It is usually expedient to take these spaces two at a time, in dual pairs of the form ''X'' and (''X''&nbsp;&rarr;&nbsp;'''K'''). In general, one creates pairs of type schemas by replacing any space ''X'' with its dual (''X''&nbsp;&rarr;&nbsp;'''K'''), for example, pairing the type ''X''&nbsp;&rarr;&nbsp;''Y'' with the type (''X''&nbsp;&rarr;&nbsp;'''K''')&nbsp;&rarr;&nbsp;(''Y''&nbsp;&rarr;&nbsp;'''K'''), and ''X''&nbsp;&times;&nbsp;''Y'' with (''X''&nbsp;&rarr;&nbsp;'''K''')&nbsp;&times;&nbsp;(''Y''&nbsp;&rarr;&nbsp;'''K'''). Here, I am using<br />
the word ''dual'' in its larger sense to mean all of the functionals, not just the linear ones. Given any function ''f''&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;'''K''', the ''converse'' or inverse relation corresponding to ''f'' is denoted as ''f''<sup>&ndash;1</sup>, and the subsets of ''X'' that are defined by ''f''<sup>&ndash;1</sup>(''k''), taken over ''k'' in '''K''', are called the ''fibers'' or the ''level sets'' of the function ''f''.<br />
<br />
===Theory of Control and Control of Theory===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
You will hardly know who I am or what I mean,<br><br />
But I shall be good health to you nevertheless,<br><br />
And filter and fibre your blood.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]<br />
|}<br />
<br />
In the boolean context, a function ''f''&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;'''B''' is tantamount to a ''proposition'' about elements of ''X'', and the elements of ''X'' constitute the ''interpretations'' of that proposition. The fiber ''f''<sup>&ndash;1</sup>(1) comprises the set of ''models'' of ''f'', or examples of elements in ''X'' satisfying the proposition ''f''. The fiber ''f''<sup>&ndash;1</sup>(0) collects the complementary set of ''anti-models'', or the exceptions to the proposition ''f'' that exist in ''X''. Of course, the space of functions (''X''&nbsp;&rarr;&nbsp;'''B''') is isomorphic to the set of all subsets of X, called the ''power set'' of ''X'' and often denoted as <font face="lucida calligraphy">Pow</font>(''X'') or 2<sup>''X''</sup>.<br />
<br />
The operation of replacing ''X'' by (''X''&nbsp;&rarr;&nbsp;'''B''') in a type schema corresponds to a certain shift of attitude towards the space ''X'', in which one passes from a focus on the ostensibly individual elements of ''X'' to a concern with the states of information and uncertainty that one possesses about objects and situations in ''X''. The conceptual obstacles in the path of this transition can be smoothed over by using singular functions (''X''&nbsp;<font face=symbol>'''××>'''</font>&nbsp;'''B''') as stepping stones. First of all, it's an easy step from an element ''x'' of type '''B'''<sup>''n''</sup> to the equivalent information of a singular proposition ''x''&nbsp;:&nbsp;''X''&nbsp;<font face=symbol>'''××>'''</font>&nbsp;'''B''', and then only a small jump of generalization remains to reach the type of an arbitrary proposition ''f''&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;'''B''', perhaps understood to indicate a relaxed constraint on the singularity of points or a neighborhood circumscribing the original ''x''. I have frequently discovered this to be a useful transformation, communicating between the ''objective'' and the ''intentional'' perspectives, in spite perhaps of the open objection that this distinction is transient in the mean time and ultimately superficial.<br />
<br />
It is hoped that this measure of flexibility, allowing us to stretch a point into a proposition, can be useful in the examination of inquiry driven systems, where the differences between empirical, intentional, and theoretical propositions constitute the discrepancies and the distributions that drive experimental activity. I can give this model of inquiry a cybernetic cast by realizing that theory change and theory evolution, as well as the choice and the evaluation of experiments, are actions that are taken by a system or its agent in response to the differences that are detected between observational contents and theoretical coverage.<br />
<br />
All of the above notwithstanding, there are several points that distinguish these two tasks, namely, the ''theory of control'' and the ''control of theory'', features that are often obscured by too much precipitation in the quickness with which we understand their similarities. In the control of uncertainty through inquiry, some of the actuators that we need to be concerned with are axiom changers and theory modifiers, operators with the power to compile and to revise the theories that generate expectations and predictions, effectors that form and edit our grammars for the languages of observational data, and agencies that rework the proposed model to fit the actual sequences of events and the realized relationships of values that are observed in the environment. Moreover, when steps must be taken to carry out an experimental action, there must be something about the particular shape of our uncertainty that guides us in choosing what directions to explore, and this impression is more than likely influenced by previous accumulations of experience. Thus it must be anticipated that much of what goes into scientific progress, or any sustainable effort toward a goal of knowledge, is necessarily predicated on long term observation and modal expectations, not only on the more local or short term prediction and correction.<br />
<br />
===Propositions as Types and Higher Order Types===<br />
<br />
The arrangement of types collected in Table&nbsp;3 (repeated below) can serve as a good introduction to several ideas about ''higher order propositional expressions'' (HOPE's) and also about the ''propositions as types'' (PAT) isomorphism.<br />
<br />
<font face="courier new"><br />
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"<br />
|+ '''Table 3. Analogy of Real and Boolean Types'''<br />
|- style="background:paleturquoise"<br />
! Real Domain '''R'''<br />
! &larr;&rarr;<br />
! Boolean Domain '''B'''<br />
|-<br />
| '''R'''<sup>''n''</sup><br />
| Basic Space<br />
| '''B'''<sup>''n''</sup><br />
|-<br />
| '''R'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''R'''<br />
| Function Space<br />
| '''B'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''B'''<br />
|-<br />
| ('''R'''<sup>''n''</sup>&rarr;'''R''')&nbsp;&rarr;&nbsp;'''R'''<br />
| Tangent Vector<br />
| ('''B'''<sup>''n''</sup>&rarr;'''B''')&nbsp;&rarr;&nbsp;'''B'''<br />
|-<br />
| '''R'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;(('''R'''<sup>''n''</sup>&rarr;'''R''')&rarr;'''R''')<br />
| Vector Field<br />
| '''B'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;(('''B'''<sup>''n''</sup>&rarr;'''B''')&rarr;'''B''')<br />
|-<br />
| ('''R'''<sup>''n''</sup>&nbsp;&times;&nbsp;('''R'''<sup>''n''</sup>&rarr; '''R'''))&nbsp;&rarr;&nbsp;'''R'''<br />
| ditto<br />
| ('''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;('''B'''<sup>''n''</sup>&rarr; '''B'''))&nbsp;&rarr;&nbsp;'''B'''<br />
|-<br />
| (('''R'''<sup>''n''</sup>&rarr;'''R''')&nbsp;&times;&nbsp;'''R'''<sup>''n''</sup>)&nbsp;&rarr;&nbsp;'''R'''<br />
| ditto<br />
| (('''B'''<sup>''n''</sup>&rarr;'''B''')&nbsp;&times;&nbsp;'''B'''<sup>''n''</sup>)&nbsp;&rarr;&nbsp;'''B'''<br />
|-<br />
| ('''R'''<sup>''n''</sup>&rarr;'''R''')&nbsp;&rarr;&nbsp;('''R'''<sup>''n''</sup>&rarr;'''R''')<br />
| Derivation<br />
| ('''B'''<sup>''n''</sup>&rarr;'''B''')&nbsp;&rarr;&nbsp;('''B'''<sup>''n''</sup>&rarr;'''B''')<br />
|-<br />
| '''R'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''R'''<sup>''m''</sup><br />
| Basic Transformation<br />
| '''B'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''B'''<sup>''m''</sup><br />
|-<br />
| ('''R'''<sup>''n''</sup>&rarr;'''R''')&nbsp;&rarr;&nbsp;('''R'''<sup>''m''</sup>&rarr;'''R''')<br />
| Function Transformation<br />
| ('''B'''<sup>''n''</sup>&rarr;'''B''')&nbsp;&rarr;&nbsp;('''B'''<sup>''m''</sup>&rarr;'''B''')<br />
|-<br />
| ...<br />
| ...<br />
| ...<br />
|}<br />
</font><br><br />
<br />
First, observe that the type of a ''tangent vector at a point'', also known as a ''directional derivative'' at that point, has the form ('''K'''<sup>''n''</sup>&nbsp;&rarr; '''K''')&nbsp;&rarr;&nbsp;'''K''', where '''K''' is the chosen ground field, in the present case either '''R''' or '''B'''. At a point in a space of type '''K'''<sup>''n''</sup>, a directional derivative operator <math>\vartheta\!</math> takes a function on that space, an ''f'' of type ('''K'''<sup>''n''</sup>&nbsp;&rarr; '''K'''), and maps it to a ground field value of type '''K'''. This value is known as the ''derivative'' of ''f'' in the direction <math>\vartheta\!</math> [Che46, 76-77]. In the boolean case, <math>\vartheta\!</math>&nbsp;:&nbsp;('''B'''<sup>''n''</sup>&nbsp;&rarr; '''B''')&nbsp;&rarr;&nbsp;'''B''' has the form of a proposition about propositions, in other words, a proposition of the next higher type.<br />
<br />
Next, by way of illustrating the propositions as types theme, consider a proposition of the form ''X''&nbsp;&rArr;&nbsp;(''Y''&nbsp;&rArr;&nbsp;''Z''). One knows from propositional calculus that this is logically equivalent to a proposition of the form (''X''&nbsp;&and;&nbsp;''Y'')&nbsp;&rArr;&nbsp;''Z''. But this equivalence should remind us of the functional isomorphism that exists between a construction of the type ''X''&nbsp;&rarr;&nbsp;(''Y''&nbsp;&rarr;&nbsp;''Z'') and a construction of the type (''X''&nbsp;&times;&nbsp;''Y'')&nbsp;&rarr;&nbsp;''Z''. The propositions as types analogy permits us to take a functional type like this and, under the right conditions, replace the functional arrows "&rarr;" and products "&times;" with the respective logical arrows "&rArr;" and products "&and;". Accordingly, viewing the result as a proposition, we can employ axioms and theorems of propositional calculus to suggest appropriate isomorphisms among the categorical and functional constructions.<br />
<br />
Finally, examine the middle four rows of Table 3. These display a series of isomorphic types that stretch from the categories that are labeled ''Vector Field'' to those that are labeled ''Derivation''. A ''vector field'', also known as an ''infinitesimal transformation'', associates a tangent vector at a point with each point of a space. In symbols, a vector field is a function of the form <math>\chi\!</math>&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;<math>\bigcup_x \ \chi_x\!</math> that assigns to each point ''x'' of the space ''X'' a tangent vector to ''X'' at that point, namely, the tangent vector <math>\chi_x\!</math> [Che46, 82-83]. If ''X'' is of type '''K'''<sup>''n''</sup>, then <math>\chi\!</math> is of type '''K'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;(('''K'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''K''')&nbsp;&rarr;&nbsp;'''K'''). This has the pattern ''X''&nbsp;&rarr;&nbsp;(''Y''&nbsp;&rarr;&nbsp;''Z''), with ''X''&nbsp;=&nbsp;'''K'''<sup>''n''</sup>, ''Y''&nbsp;=&nbsp;('''K'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''K'''), and ''Z''&nbsp;=&nbsp;'''K'''.<br />
<br />
Applying the propositions as types analogy, one can follow this pattern through a series of metamorphoses from the type of a vector field to the type of a derivation, as traced out in Table 4. Observe how the function ''f''&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;'''K''', associated with the place of ''Y'' in the pattern, moves through its paces from the second to the first position. In this way, the vector field <math>\chi\!</math>, initially viewed as attaching each tangent vector <math>\chi_x\!</math> to the site ''x'' where it acts in ''X'', now comes to be seen as acting on each scalar potential ''f''&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;'''K''' like a generalized species of differentiation, producing another function <math>\chi\!</math>''f''&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;'''K''' of the same type.<br />
<br />
<font face="courier new"><br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"<br />
|+ '''Table 4. An Equivalence Based on the Propositions as Types Analogy<br />
'''<br />
|- style="background:paleturquoise"<br />
! Pattern<br />
! Construction<br />
! Instance<br />
|-<br />
| ''X''&nbsp;&rarr;&nbsp;(''Y''&nbsp;&rarr;&nbsp;''Z'')<br />
| Vector Field<br />
| '''K'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;(('''K'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''K''')&nbsp;&rarr;&nbsp;'''K''')<br />
|-<br />
|(''X''&nbsp;&times;&nbsp;''Y'')&nbsp;&rarr;&nbsp;''Z''<br />
| &nbsp;<br />
| ('''K'''<sup>''n''</sup>&nbsp;&times;&nbsp;('''K'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''K'''))&nbsp;&rarr;&nbsp;'''K'''<br />
|-<br />
| (''Y''&nbsp;&times;&nbsp;''X'')&nbsp;&rarr;&nbsp;''Z''<br />
| &nbsp;<br />
| (('''K'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''K''')&nbsp;&times;&nbsp;'''K'''<sup>''n''</sup>)&nbsp;&rarr;&nbsp;'''K'''<br />
|-<br />
| ''Y''&nbsp;&rarr;&nbsp;(''X''&nbsp;&rarr;&nbsp;''Z'')<br />
| Derivation<br />
| ('''K'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''K''')&nbsp;&rarr;&nbsp;('''K'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''K''')<br />
|}<br />
</font><br><br />
<br />
===Reality at the Threshold of Logic===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
But no science can rest entirely on measurement, and many scientific investigations are quite out of reach of that device. To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]<br />
|}<br />
<br />
Table 5 accumulates an array of notation that I hope will not be too distracting. Some of it is rarely needed, but has been filled in for the sake of completeness. Its purpose is simple, to give literal expression to the visual intuitions that come with venn diagrams, and to help build a bridge between our qualitative and quantitative outlooks on dynamic systems.<br />
<br />
<font face="courier new"><br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"<br />
|+ '''Table 5. A Bridge Over Troubled Waters'''<br />
|- style="background:paleturquoise"<br />
! Linear Space<br />
! Liminal Space<br />
! Logical Space<br />
|-<br />
|<br />
<font face="lucida calligraphy">X</font><br><br />
{''x''<sub>1</sub>, &hellip;, ''x''<sub>''n''</sub>}<br><br />
cardinality ''n''<br />
|<br />
<font face="lucida calligraphy"><u>X</u></font><br><br />
{<u>''x''</u><sub>1</sub>, &hellip;, <u>''x''</u><sub>''n''</sub>}<br><br />
cardinality ''n''<br />
|<br />
<font face="lucida calligraphy">A</font><br><br />
{''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>}<br><br />
cardinality ''n''<br />
|-<br />
|<br />
''X''<sub>''i''</sub><br><br />
〈''x''<sub>''i''</sub>〉<br><br />
isomorphic to '''K'''<br />
|<br />
<u>''X''</u><sub>''i''</sub><br><br />
{(<u>''x''</u><sub>''i''</sub>), <u>''x''</u><sub>''i''</sub>}<br><br />
isomorphic to '''B'''<br />
|<br />
''A''<sub>''i''</sub><br><br />
{(''a''<sub>''i''</sub>), ''a''<sub>''i''</sub>}<br><br />
isomorphic to '''B'''<br />
|-<br />
|<br />
''X''<br><br />
〈<font face="lucida calligraphy">X</font>〉<br><br />
〈''x''<sub>1</sub>, &hellip;, ''x''<sub>''n''</sub>〉<br><br />
{‹''x''<sub>1</sub>, &hellip;, ''x''<sub>''n''</sub>›}<br><br />
''X''<sub>1</sub> &times; &hellip; &times; ''X''<sub>''n''</sub><br><br />
&prod;<sub>''i''</sub> ''X''<sub>''i''</sub><br><br />
isomorphic to '''K'''<sup>''n''</sup><br />
|<br />
<u>''X''</u><br><br />
〈<font face="lucida calligraphy"><u>X</u></font>〉<br><br />
〈<u>''x''</u><sub>1</sub>, &hellip;, <u>''x''</u><sub>''n''</sub>〉<br><br />
{‹<u>''x''</u><sub>1</sub>, &hellip;, <u>''x''</u><sub>''n''</sub>›}<br><br />
<u>''X''</u><sub>1</sub> &times; &hellip; &times; <u>''X''</u><sub>''n''</sub><br><br />
&prod;<sub>''i''</sub> <u>''X''</u><sub>''i''</sub><br><br />
isomorphic to '''B'''<sup>''n''</sup><br />
|<br />
''A''<br><br />
〈<font face="lucida calligraphy">A</font>〉<br><br />
〈''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>〉<br><br />
{‹''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>›}<br><br />
''A''<sub>1</sub> &times; &hellip; &times; ''A''<sub>''n''</sub><br><br />
&prod;<sub>''i''</sub> ''A''<sub>''i''</sub><br><br />
isomorphic to '''B'''<sup>''n''</sup><br />
|-<br />
|<br />
''X''*<br><br />
(hom : ''X'' &rarr; '''K''')<br><br />
isomorphic to '''K'''<sup>''n''</sup><br />
|<br />
<u>''X''</u>*<br><br />
(hom : <u>''X''</u> &rarr; '''B''')<br><br />
isomorphic to '''B'''<sup>''n''</sup><br />
|<br />
''A''*<br><br />
(hom : ''A'' &rarr; '''B''')<br><br />
isomorphic to '''B'''<sup>''n''</sup><br />
|-<br />
|<br />
''X''^<br><br />
(''X'' &rarr; '''K''')<br><br />
isomorphic to:<br><br />
('''K'''<sup>''n''</sup> &rarr; '''K''')<br />
|<br />
<u>''X''</u>^<br><br />
(<u>''X''</u> &rarr; '''B''')<br><br />
isomorphic to:<br><br />
('''B'''<sup>''n''</sup> &rarr; '''B''')<br />
|<br />
''A''^<br><br />
(''A'' &rarr; '''B''')<br><br />
isomorphic to:<br><br />
('''B'''<sup>''n''</sup> &rarr; '''B''')<br />
|-<br />
|<br />
''X''<sup>&bull;</sup><br><br />
[<font face="lucida calligraphy">X</font>]<br><br />
[''x''<sub>1</sub>, &hellip;, ''x''<sub>''n''</sub>]<br><br />
(''X'', ''X''^)<br><br />
(''X'' +&rarr; '''K''')<br><br />
(''X'', (''X'' &rarr; '''K'''))<br><br />
isomorphic to:<br><br />
('''K'''<sup>''n''</sup>, ('''K'''<sup>''n''</sup> &rarr; '''K'''))<br><br />
('''K'''<sup>''n''</sup> +&rarr; '''K''')<br><br />
['''K'''<sup>''n''</sup>]<br />
|<br />
<u>''X''</u><sup>&bull;</sup><br><br />
[<font face="lucida calligraphy"><u>X</u></font>]<br><br />
[<u>''x''</u><sub>1</sub>, &hellip;, <u>''x''</u><sub>''n''</sub>]<br><br />
(<u>''X''</u>, <u>''X''</u>^)<br><br />
(<u>''X''</u> +&rarr; '''B''')<br><br />
(<u>''X''</u>, (<u>''X''</u> &rarr; '''B'''))<br><br />
isomorphic to:<br><br />
('''B'''<sup>''n''</sup>, ('''B'''<sup>''n''</sup> &rarr; '''B'''))<br><br />
('''B'''<sup>''n''</sup> +&rarr; '''B''')<br><br />
['''B'''<sup>''n''</sup>]<br />
|<br />
''A''<sup>&bull;</sup><br><br />
[<font face="lucida calligraphy">A</font>]<br><br />
[''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>]<br><br />
(''A'', ''A''^)<br><br />
(''A'' +&rarr; '''B''')<br><br />
(''A'', (''A'' &rarr; '''B'''))<br><br />
isomorphic to:<br><br />
('''B'''<sup>''n''</sup>, ('''B'''<sup>''n''</sup> &rarr; '''B'''))<br><br />
('''B'''<sup>''n''</sup> +&rarr; '''B''')<br><br />
['''B'''<sup>''n''</sup>]<br />
|}<br />
</font><br><br />
<br />
The left side of the Table collects mostly standard notation for an ''n''-dimensional vector space over a field '''K'''. The right side of the table repeats the first elements of a notation that I sketched above, to be used in further developments of propositional calculus. (I plan to use this notation in the logical analysis of neural network systems.) The middle column of the table is designed as a transitional step from the case of an arbitrary field '''K''', with a special interest in the continuous line '''R''', to the qualitative and discrete situations that are instanced and typified by '''B'''.<br />
<br />
I now proceed to explain these concepts in more detail. The two most important ideas developed in the table are:<br />
<br />
* The idea of a universe of discourse, which includes both a space of ''points'' and a space of ''maps'' on those points.<br />
<br />
* The idea of passing from a more complex universe to a simpler universe by a process of ''thresholding'' each dimension of variation down to a single bit of information.<br />
<br />
For the sake of concreteness, let us suppose that we start with a continuous ''n''-dimensional vector space like ''X''&nbsp;=&nbsp;〈''x''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''x''<sub>''n''</sub>〉 <math>\cong</math> '''R'''<sup>''n''</sup>. The coordinate<br />
system <font face=lucida calligraphy">X</font> = {''x''<sub>''i''</sub>} is a set of maps ''x''<sub>''i''</sub>&nbsp;:&nbsp;'''R'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''R''', also known as the coordinate projections. Given a "dataset" of points ''x'' in '''R'''<sup>''n''</sup>, there are numerous ways of sensibly reducing the data down to one bit for each dimension. One strategy that is general enough for our present purposes is as follows. For each ''i'' we choose an ''n''-ary relation ''L''<sub>''i''</sub> on '''R''', that is, a subset of '''R'''<sup>''n''</sup>, and then we define the ''i''<sup>th</sup> threshold map, or ''limen'' <u>''x''</u><sub>''i''</sub> as follows:<br />
<br />
: <u>''x''</u><sub>''i''</sub> : '''R'''<sup>''n''</sup> &rarr; '''B''' such that:<br />
<br />
: <u>''x''</u><sub>''i''</sub>(''x'') = 1 if ''x'' &isin; ''L''<sub>''i''</sub>,<br />
<br />
: <u>''x''</u><sub>''i''</sub>(''x'') = 0 if otherwise.<br />
<br />
In other notations that are sometimes used, the operator <math>\chi (\ )</math> or the corner brackets <math>\lceil \ldots \rceil</math> can be used to denote a ''characteristic function'', that is, a mapping from statements to their truth values, given as elements of '''B'''. Finally, it is not uncommon to use the name of the relation itself as a predicate that maps ''n''-tuples into truth values. In each of these notations, the above definition could be expressed as follows:<br />
<br />
: <u>''x''</u><sub>''i''</sub>(''x'') = <math>\chi (x \in L_i)</math> = <math>\lceil x \in L_i \rceil</math> = ''L''<sub>''i''</sub>(''x'').<br />
<br />
Notice that, as defined here, there need be no actual relation between the ''n''-dimensional subsets {''L''<sub>''i''</sub>} and the coordinate axes corresponding to {''x''<sub>''i''</sub>}, aside from the circumstance that the two sets have the same cardinality. In concrete cases, though, one usually has some reason for associating these "volumes" with these "lines", for instance, ''L''<sub>''i''</sub> is bounded by some hyperplane that intersects the ''i''<sup>th</sup> axis at a unique threshold value ''r''<sub>''i''</sub> &isin; '''R'''. Often, the hyperplane is chosen normal to the axis. In recognition of this motive, let us make the following convention. When the set ''L''<sub>''i''</sub> has points on the ''i''<sup>th</sup> axis, that is, points of the form ‹0,&nbsp;&hellip;,&nbsp;0,&nbsp;''r''<sub>''i''</sub>,&nbsp;0,&nbsp;&hellip;,&nbsp;0› where only the ''x''<sub>''i''</sub> coordinate is possibly non-zero, we may pick any one of these coordinate values as a parametric index of the relation. In this case we say that the indexing is ''real'', otherwise the indexing is ''imaginary''. For a knowledge based system ''X'', this should serve once again to mark the distinction between ''acquaintance'' and ''opinion''.<br />
<br />
States of knowledge about the location of a system or about the distribution of a population of systems in a state space ''X'' = '''R'''<sup>''n''</sup> can now be expressed by taking the set <font face="lucida calligraphy"><u>X</u></font>&nbsp;=&nbsp;{<u>''x''</u><sub>''i''</sub>} as a basis of logical features. In picturesque terms, one may think of the underscore and the subscript as combining to form a subtextual spelling for the ''i''<sup>th</sup> threshold map. This can help to remind us that the ''threshold operator'' (<u>&nbsp;</u>)<sub>''i''</sub> acts on ''x'' by setting up a kind of a "hurdle" for it. In this interpretation, the coordinate proposition <u>''x''</u><sub>''i''</sub> asserts that the representative point ''x'' resides ''above'' the ''i''<sup>th</sup> threshold.<br />
<br />
Primitive assertions of the form <u>''x''</u><sub>''i''</sub>(''x'') can then be negated and joined by means of propositional connectives in the usual ways to provide information about the state ''x'' of a contemplated system or a statistical ensemble of systems. Parentheses "(&nbsp;)" may be used to indicate negation. Eventually one discovers the usefulness of the ''k''-ary ''just one false'' operators of the form "(&nbsp;,&nbsp;,&nbsp;,&nbsp;)", as treated in earlier reports. This much tackle generates a space of points (cells, interpretations), <u>''X''</u>&nbsp;=&nbsp;〈<font face="lucida calligraphy"><u>X</u></font>〉&nbsp;<math>\cong</math>&nbsp;'''B'''<sup>''n''</sup>, and<br />
a space of functions (regions, propositions), <u>''X''</u>^&nbsp;<math>\cong</math>&nbsp;('''B'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''B'''). Together these form a new universe of discourse <u>''X''</u><sup>&nbsp;&bull;</sup> = [<font face="lucida calligraphy"><u>X</u></font>] of the type ('''B'''<sup>''n''</sup>,&nbsp;('''B'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''B''')), which we may abbreviate as '''B'''<sup>''n''</sup>&nbsp;+&rarr;&nbsp;'''B''', or most succinctly as ['''B'''<sup>''n''</sup>].<br />
<br />
The square brackets have been chosen to recall the rectangular frame of a venn diagram. In thinking about a universe of discourse it is a good idea to keep this picture in mind, where we constantly think of the elementary cells <u>''x''</u>, the defining features <u>''x''</u><sub>''i''</sub>, and the potential shadings ''f''&nbsp;:&nbsp;<u>''X''</u>&nbsp;&rarr;&nbsp;'''B''', all at the same time, remaining aware of the arbitrariness of the way that we choose to inscribe our distinctions in the medium of a continuous space.<br />
<br />
Finally, let ''X''* denote the space of linear functions, (hom&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;'''K'''), which in the finite case has the same dimensionality as ''X'', and let the same notation be extended across the table.<br />
<br />
We have just gone through a lot of work, apparently doing nothing more substantial than spinning a complex spell of notational devices through a labyrinth of baffled spaces and baffling maps. The reason for doing this was to bind together and to constitute the intuitive concept of a universe of discourse into a coherent categorical object, the kind of thing, once grasped, which can be turned over in the mind and considered in all its manifold changes and facets. The effort invested in these preliminary measures is intended to pay off later, when we need to consider the state transformations and the time evolution of neural network systems.<br />
<br />
===Tables of Propositional Forms===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope. It provides explicit techniques for manipulating the most basic ingredients of discourse.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7&ndash;8]<br />
|}<br />
<br />
To prepare for the next phase of discussion, Tables 6 and 7 collect and summarize all of the propositional forms on one and two variables. These propositional forms are represented over bases of boolean variables as complete sets of boolean-valued functions. Adjacent to their names and specifications are listed what are roughly the simplest expressions in the ''[[cactus language]]'', the particular syntax for propositional calculus that I use in formal and computational contexts. For the sake of orientation, the English paraphrases and the more common notations are listed in the last two columns. As simple and circumscribed as these low-dimensional universes may appear to be, a careful exploration of their differential extensions will involve us in complexities sufficient to demand our attention for some time to come.<br />
<br />
Propositional forms on one variable correspond to boolean functions ''f''&nbsp;:&nbsp;'''B'''<sup>1</sup>&nbsp;&rarr;&nbsp;'''B'''. In Table 6 these functions are listed in a variant form of [[truth table]], one which rotates the axes of the usual arrangement. Each function ''f''<sub>''i''</sub> is indexed by the string of values that it takes on the points of the universe ''X''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;[''x'']&nbsp;<math>\cong</math>&nbsp;'''B'''<sup>1</sup>. The binary index generated in this way is converted to its decimal equivalent, and these are used as conventional names for the ''f''<sub>''i''</sub>&nbsp;, as shown in the first column of the Table. In their own right the 2<sup>1</sup> points of the universe ''X''<sup>&nbsp;&bull;</sup> are coordinated as a space of type '''B'''<sup>1</sup>, this in light of the universe ''X''<sup>&nbsp;&bull;</sup> being a functional domain where the coordinate projection ''x'' takes on its values in '''B'''.<br />
<br />
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"<br />
|+ '''Table 6. Propositional Forms on One Variable'''<br />
|- style="background:paleturquoise"<br />
! style="width:16%" | L<sub>1</sub><br>Decimal<br />
! style="width:16%" | L<sub>2</sub><br>Binary<br />
! style="width:16%" | L<sub>3</sub><br>Vector<br />
! style="width:16%" | L<sub>4</sub><br>Cactus<br />
! style="width:16%" | L<sub>5</sub><br>English<br />
! style="width:16%" | L<sub>6</sub><br>Ordinary<br />
|- style="background:paleturquoise"<br />
| &nbsp;<br />
| align="right" | x :<br />
| 1 0 <br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| f<sub>0</sub><br />
| f<sub>00</sub><br />
| 0 0<br />
| ( )<br />
| false<br />
| 0<br />
|-<br />
| f<sub>1</sub><br />
| f<sub>01</sub><br />
| 0 1<br />
| (x)<br />
| not x<br />
| ~x<br />
|-<br />
| f<sub>2</sub><br />
| f<sub>10</sub><br />
| 1 0<br />
| x<br />
| x<br />
| x<br />
|-<br />
| f<sub>3</sub><br />
| f<sub>11</sub><br />
| 1 1<br />
| (( ))<br />
| true<br />
| 1<br />
|}<br />
<br><br />
<br />
Propositional forms on two variables correspond to boolean functions ''f''&nbsp;:&nbsp;'''B'''<sup>2</sup>&nbsp;&rarr;&nbsp;'''B'''. In Table 7 each function ''f''<sub>''i''</sub> is indexed by the values that it takes on the points of the universe ''X''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;[''x'',&nbsp;''y'']&nbsp;<math>\cong</math>&nbsp;'''B'''<sup>2</sup>. Converting the binary index thus generated to a decimal equivalent, we obtain the functional nicknames that are listed in the first column. The 2<sup>2</sup> points of the universe ''X''<sup>&nbsp;&bull;</sup> are coordinated as a space of type '''B'''<sup>2</sup>, as indicated under the heading of the Table, where the coordinate projections ''x'' and ''y'' run through the various combinations of their values in '''B'''.<br />
<br />
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"<br />
|+ '''Table 7. Propositional Forms on Two Variables'''<br />
|- style="background:paleturquoise"<br />
! style="width:16%" | L<sub>1</sub><br>Decimal<br />
! style="width:16%" | L<sub>2</sub><br>Binary<br />
! style="width:16%" | L<sub>3</sub><br>Vector<br />
! style="width:16%" | L<sub>4</sub><br>Cactus<br />
! style="width:16%" | L<sub>5</sub><br>English<br />
! style="width:16%" | L<sub>6</sub><br>Ordinary<br />
|- style="background:paleturquoise"<br />
| &nbsp;<br />
| align="right" | x :<br />
| 1 1 0 0 <br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:paleturquoise"<br />
| &nbsp;<br />
| align="right" | y :<br />
| 1 0 1 0<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || (&nbsp;) || false || 0<br />
|-<br />
| f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || &not;x &and; &not;y<br />
|-<br />
| f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || &not;x &and; y<br />
|-<br />
| f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || &not;x<br />
|-<br />
| f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x &and; &not;y<br />
|-<br />
| f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || &not;y<br />
|-<br />
| f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x &ne; y<br />
|-<br />
| f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x&nbsp;y) || not both x and y || &not;x &or; &not;y<br />
|-<br />
| f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x&nbsp;y || x and y || x &and; y<br />
|-<br />
| f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y<br />
|-<br />
| f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y<br />
|-<br />
| f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x &rarr; y<br />
|-<br />
| f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x<br />
|-<br />
| f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x &larr; y<br />
|-<br />
| f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y || x &or; y<br />
|-<br />
| f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || ((&nbsp;)) || true || 1<br />
|}<br />
<br><br />
<br />
==A Differential Extension of Propositional Calculus==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Fire over water:<br><br />
The image of the condition before transition.<br><br />
Thus the superior man is careful<br><br />
In the differentiation of things,<br><br />
So that each finds its place.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; ''I Ching'', Hexagram 64, [Wil, 249]<br />
|}<br />
<br />
This much preparation is enough to begin introducing my subject, if I excuse myself from giving full arguments for my definitional choices until some later stage. I am trying to develop a ''differential theory of qualitative equations'' that parallels the application of differential geometry to dynamic systems. The idea of a tangent vector is key to this work and a major goal is to find the right logical analogues of tangent spaces, bundles, and functors. The strategy is taken of looking for the simplest versions of these constructions that can be discovered within the realm of propositional calculus, so long as they serve to fill out the general theme.<br />
<br />
===Differential Propositions : The Qualitative Analogues of Differential Equations===<br />
<br />
In order to define the differential extension of a universe of discourse [<font face="lucida calligraphy">A</font>], the initial alphabet <font face="lucida calligraphy">A</font> must be extended to include a collection of symbols for ''differential features'', or basic ''changes'' that are capable of occurring in [<font face="lucida calligraphy">A</font>]. Intuitively, these symbols may be construed as denoting primitive features of change, qualitative attributes of motion, or propositions about how things or points in [<font face="lucida calligraphy">A</font>] may change or move with respect to the features that are noted in the initial alphabet.<br />
<br />
Hence, let us define the corresponding ''differential alphabet'' or ''tangent alphabet'' as d<font face="lucida calligraphy">A</font>&nbsp;=&nbsp;{d''a''<sub>1</sub>,&nbsp;&hellip;,&nbsp;d''a''<sub>''n''</sub>}, in principle just an arbitrary alphabet of symbols, disjoint from the initial alphabet <font face="lucida calligraphy">A</font>&nbsp;=&nbsp;{''a''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''a''<sub>''n''</sub>}, that is intended to be interpreted in the way just indicated. It only remains to be understood that the precise interpretation of the symbols in d<font face="lucida calligraphy">A</font> is often conceived to be changeable from point to point of the underlying space ''A''. (For all we know, the state space ''A'' might well be the state space of a language interpreter, one that is concerned, among other things, with the idiomatic meanings of the dialect generated by <font face="lucida calligraphy">A</font> and d<font face="lucida calligraphy">A</font>.)<br />
<br />
In the above terms, a typical tangent space of ''A'' at a point ''x'', frequently denoted as T<sub>''x''</sub>(''A''), can be characterized as having the generic construction d''A''&nbsp;=&nbsp;〈d<font face="lucida calligraphy">A</font>〉&nbsp;=&nbsp;〈d''a''<sub>1</sub>,&nbsp;&hellip;,&nbsp;d''a''<sub>''n''</sub>〉. Strictly speaking, the name ''cotangent space'' is probably more correct for this construction, but the fact that we take up spaces and their duals in pairs to form our universes of discourse allows our language to be pliable here.<br />
<br />
Proceeding as we did before with the base space ''A'', we can analyze the individual tangent space at a point of ''A'' as a product of distinct and independent factors:<br />
<br />
: d''A'' = &prod;<sub>''i''</sub> d''A''<sub>''i''</sub> = d''A''<sub>1</sub> &times; &hellip; &times; d''A''<sub>''n''</sub>.<br />
<br />
Here, d<font face="lucida calligraphy">A</font><sub>''i''</sub> is an alphabet of two symbols, d<font face="lucida calligraphy">A</font><sub>''i''</sub>&nbsp;=&nbsp;{(d''a''<sub>''i''</sub>),&nbsp;d''a''<sub>''i''</sub>}, where (d''a''<sub>''i''</sub>) is a symbol with the logical value of "not d''a''<sub>''i''</sub>". Each component d''A''<sub>''i''</sub> has the type '''B''', under the correspondence {(d''a''<sub>''i''</sub>),&nbsp;d''a''<sub>''i''</sub>} <math>\cong</math> {0,&nbsp;1}. However, clarity is often served by acknowledging this differential usage with a superficially distinct type '''D''', whose intension may be indicated as follows:<br />
<br />
: '''D''' = {(d''a''<sub>''i''</sub>),&nbsp;d''a''<sub>''i''</sub>} = {same,&nbsp;different} = {stay,&nbsp;change} = {stop,&nbsp;step}.<br />
<br />
Viewed within a coordinate representation, spaces of type '''B'''<sup>''n''</sup> and '''D'''<sup>''n''</sup> may appear to be identical sets of binary vectors, but taking a view at this level of abstraction would be like ignoring the qualitative units and the diverse dimensions that distinguish position and momentum, or the different roles of quantity and impulse.<br />
<br />
===An Interlude on the Path===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
There would have been no beginnings: instead, speech would proceed from me, while I stood in its path &ndash; a slender gap &ndash; the point of its possible disappearance.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
It may help to get a sense of the relation between '''B''' and '''D''' by considering the ''path classifier'' (or equivalence class of curves) approach to tangent vectors. As if by reflex, the thought of physical motion makes us cross over to a universe marked by the nominal character [<font face="lucida calligraphy">X</font>]. Given the boolean value system, a path in the space ''X'' = 〈<font face="lucida calligraphy">X</font>〉 is a map ''q'' : '''B''' &rarr; ''X''. In this case, the set of paths ('''B''' &rarr; ''X'') is isomorphic to the cartesian square ''X''<sup>2</sup> = ''X'' &times; ''X'', or the set of ordered pairs from ''X''.<br />
<br />
We may analyze ''X''<sup>2</sup> = {‹''u'', ''v''› : ''u'', ''v'' &isin; ''X''} into two parts, specifically, the pairs that lie on and off the diagonal:<br />
<br />
: ''X''<sup>2</sup> = {‹''u'', ''v''› : ''u'' = ''v''} &cup; {‹''u'', ''v''› : ''u'' &ne; ''v''}<br />
<br />
In symbolic terms, this partition may be expressed as:<br />
<br />
: ''X''<sup>2</sup> <math>\cong</math> Diag(''X'') + 2 * Comb(''X'', 2),<br />
<br />
where:<br />
<br />
: Diag(''X'') = {‹''x'', ''x''› : ''x'' &isin; ''X''},<br />
<br />
and where:<br />
<br />
: Comb(''X'', ''k'') = "''X'' choose ''k''" = {''k''-sets from ''X''},<br />
<br />
so that:<br />
<br />
: Comb(''X'', 2) = {{''u'', ''v''} : ''u'', ''v'' &isin; ''X''}.<br />
<br />
We can now use the features in d<font face="lucida calligraphy">X</font> = {d''x''<sub>''i''</sub>} = {d''x''<sub>1</sub>,&nbsp;&hellip;,&nbsp;d''x''<sub>''n''</sub>} to classify the paths of ('''B''' &rarr; ''X'') by way of the pairs in ''X''<sup>2</sup>. If ''X'' <math>\cong</math> '''B'''<sup>''n''</sup> then a path in ''X'' has the form ''q'' : ('''B''' &rarr; '''B'''<sup>''n''</sup>) <math>\cong</math> '''B'''<sup>''n''</sup> &times; '''B'''<sup>''n''</sup> <math>\cong</math> '''B'''<sup>2''n''</sup> <math>\cong</math> ('''B'''<sup>2</sup>)<sup>''n''</sup>. Intuitively, we want to map this ('''B'''<sup>2</sup>)<sup>''n''</sup> onto ''D''<sup>''n''</sup> by mapping each component '''B'''<sup>2</sup> onto a copy of '''D'''. But in our current situation "'''D'''" is just a name we give, or an accidental quality we attribute, to coefficient values in '''B''' when they are attached to features in d<font face="lucida calligraphy">X</font>.<br />
<br />
Therefore, define d''x''<sub>''i''</sub> : ''X''<sup>2</sup> &rarr; '''B''' such that:<br />
<br />
:{| cellpadding=2<br />
| d''x''<sub>''i''</sub>(‹''u'', ''v''›)<br />
| =<br />
| <font face=system>(</font> ''x''<sub>''i''</sub>(''u'') , ''x''<sub>''i''</sub>(''v'') <font face=system>)</font><br />
|-<br />
| &nbsp;<br />
| =<br />
| ''x''<sub>''i''</sub>(''u'') + ''x''<sub>''i''</sub>(''v'')<br />
|-<br />
| &nbsp;<br />
| =<br />
| ''x''<sub>''i''</sub>(''v'') &ndash; ''x''<sub>''i''</sub>(''u'').<br />
|}<br />
<br />
In the above transcription, the operator bracket of the form "<font face=system>(&nbsp;&hellip;&nbsp;,&nbsp;&hellip;&nbsp;)</font>" is a ''cactus lobe'', signifying ''just one false'', in this case among two boolean variables, while "+" is boolean addition in the proper sense of addition in GF(2), and is thus equivalent to "&ndash;", in the sense of adding the additive inverse.<br />
<br />
The above definition is equivalent to defining d''x''<sub>''i''</sub>&nbsp;:&nbsp;(''B''&nbsp;&rarr;&nbsp;''X'')&nbsp;&rarr;&nbsp;'''B''' such that:<br />
<br />
:{| cellpadding=2<br />
| d''x''<sub>''i''</sub>(''q'')<br />
| =<br />
| <font face=system>(</font> ''x''<sub>''i''</sub>(''q''<sub>0</sub>) , ''x''<sub>''i''</sub>(''q''<sub>1</sub>) <font face=system>)</font><br />
|-<br />
| &nbsp;<br />
| =<br />
| ''x''<sub>''i''</sub>(''q''<sub>0</sub>) + ''x''<sub>''i''</sub>(''q''<sub>1</sub>)<br />
|-<br />
| &nbsp;<br />
| =<br />
| ''x''<sub>''i''</sub>(''q''<sub>1</sub>) &ndash; ''x''<sub>''i''</sub>(''q''<sub>0</sub>),<br />
|}<br />
<br />
where ''q''<sub>''b''</sub> = ''q''(''b''), for each ''b'' in '''B'''. Thus, the proposition d''x''<sub>''i''</sub> is true of the path ''q'' = ‹''u'',&nbsp;''v''› exactly if the terms of ''q'', the endpoints ''u'' and ''v'', lie on different sides of the question ''x''<sub>''i''</sub>.<br />
<br />
Now we can use the language of features in 〈d<font face="lucida calligraphy">X</font>〉, indeed the whole calculus of propositions in [d<font face="lucida calligraphy">X</font>], to classify paths and sets of paths. In other words, the paths can be taken as models of the propositions ''g''&nbsp;:&nbsp;d''X''&nbsp;&rarr;&nbsp;'''B'''. For example, the paths corresponding to ''Diag''(''X'') fall under the description <font face=system>(</font>d''x''<sub>1</sub><font face=system>)</font>&hellip;<font face=system>(</font>d''x''<sub>''n''</sub><font face=system>)</font>, which says that nothing changes among the set of features {''x''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''x''<sub>''n''</sub>}.<br />
<br />
Finally, a few words of explanation may be in order. If this concept of a path appears to be described in a roundabout fashion, it is because I am trying to avoid using any assumption of vector space properties for the space ''X'' which contains its range. In many ways the treatment is still unsatisfactory, but improvements will have to wait for the introduction of substitution operators acting on singular propositions.<br />
<br />
===The Extended Universe of Discourse===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
At the moment of speaking, I would like to have perceived a nameless voice, long preceding me, leaving me merely to enmesh myself in it, taking up its cadence, and to lodge myself, when no one was looking, in its interstices as if it had paused an instant, in suspense, to beckon to me.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
Next, we define the so-called ''extended alphabet'' or ''bundled alphabet'' E<font face="lucida calligraphy">A</font> as:<br />
<br />
: E<font face="lucida calligraphy">A</font> = <font face="lucida calligraphy">A</font> &cup; d<font face="lucida calligraphy">A</font> = {''a''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''a''<sub>''n''</sub>,&nbsp;d''a''<sub>1</sub>,&nbsp;&hellip;,&nbsp;d''a''<sub>''n''</sub>}<br />
<br />
This supplies enough material to construct the ''differential extension'' E''A'', or the ''tangent bundle'' over the initial space ''A'', in the following fashion:<br />
<br />
:{| cellpadding=2<br />
| E''A''<br />
| =<br />
| ''A'' &times; d''A''<br />
|-<br />
| &nbsp;<br />
| =<br />
| 〈E<font face="lucida calligraphy">A</font>〉<br />
|-<br />
| &nbsp;<br />
| =<br />
| 〈<font face="lucida calligraphy">A</font> &cup; d<font face="lucida calligraphy">A</font>〉<br />
|-<br />
| &nbsp;<br />
| =<br />
| 〈''a''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''a''<sub>''n''</sub>,&nbsp;d''a''<sub>1</sub>,&nbsp;&hellip;,&nbsp;d''a''<sub>''n''</sub>〉,<br />
|}<br />
<br />
thus giving E''A'' the type '''B'''<sup>''n''</sup> &times; '''D'''<sup>''n''</sup>.<br />
<br />
Finally, the tangent universe E''A''<sup>&nbsp;&bull;</sup> = [E<font face="lucida calligraphy">A</font>] is constituted from the totality of points and maps, or interpretations and propositions, that are based on the extended set of features E<font face="lucida calligraphy">A</font>:<br />
<br />
: E''A''<sup>&nbsp;&bull;</sup> = [E<font face="lucida calligraphy">A</font>] = [''a''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''a''<sub>''n''</sub>,&nbsp;d''a''<sub>1</sub>,&nbsp;&hellip;,&nbsp;d''a''<sub>''n''</sub>],<br />
<br />
thus giving the tangent universe E''A''<sup>&nbsp;&bull;</sup> the type<br />
('''B'''<sup>''n''</sup> &times; '''D'''<sup>''n''</sup> +&rarr; '''B''') = ('''B'''<sup>''n''</sup> &times; '''D'''<sup>''n''</sup>, ('''B'''<sup>''n''</sup> &times; '''D'''<sup>''n''</sup> &rarr; '''B''')).<br />
<br />
A proposition in the tangent universe [E<font face="lucida calligraphy">A</font>] is called a ''differential proposition'' and forms the analogue of a system of differential equations, constraints, or relations in ordinary calculus.<br />
<br />
With these constructions, to be specific, the differential extension E''A'' and the differential proposition ''h''&nbsp;:&nbsp;E''A''&nbsp;&rarr;&nbsp;'''B''', we have arrived, in concept at least, at one of the major subgoals of this study. At this juncture, I pause by way of summary to set another Table with the current crop of mathematical produce (Table 8).<br />
<br />
<font face="courier new"><br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"<br />
|+ '''Table 8. Notation for the Differential Extension of Propositional Calculus'''<br />
|- style="background:paleturquoise"<br />
! Symbol<br />
! Notation<br />
! Description<br />
! Type<br />
|-<br />
| d<font face="lucida calligraphy">A<font><br />
| {d''a''<sub>1</sub>, &hellip;, d''a''<sub>''n''</sub>}<br />
|<br />
Alphabet of<br><br />
differential<br><br />
features<br />
| [''n''] = '''n'''<br />
|-<br />
| d''A''<sub>''i''</sub><br />
| {(d''a''<sub>''i''</sub>), d''a''<sub>''i''</sub>}<br />
|<br />
Differential<br><br />
dimension ''i''<br />
| '''D'''<br />
|-<br />
| d''A''<br />
|<br />
〈d<font face="lucida calligraphy">A</font>〉<br><br />
〈d''a''<sub>1</sub>, &hellip;, d''a''<sub>''n''</sub>〉<br><br />
{‹d''a''<sub>1</sub>, &hellip;, d''a''<sub>''n''</sub>›}<br><br />
d''A''<sub>1</sub> &times; &hellip; &times; d''A''<sub>''n''</sub><br><br />
&prod;<sub>''i''</sub> d''A''<sub>''i''</sub><br />
|<br />
Tangent space<br><br />
at a point:<br><br />
Set of changes,<br><br />
motions, steps,<br><br />
tangent vectors<br><br />
at a point<br />
| '''D'''<sup>''n''</sup><br />
|-<br />
| d''A''*<br />
| (hom : d''A'' &rarr; '''B''')<br />
|<br />
Linear functions<br><br />
on d''A''<br />
| ('''D'''<sup>''n''</sup>)* = '''D'''<sup>''n''</sup><br />
|-<br />
| d''A''^<br />
| (d''A'' &rarr; '''B''')<br />
|<br />
Boolean functions<br><br />
on d''A''<br />
| '''D'''<sup>''n''</sup> &rarr; '''B'''<br />
|-<br />
| d''A''<sup>&bull;</sup><br />
|<br />
[d<font face="lucida calligraphy">A</font>]<br><br />
(d''A'', d''A''^)<br><br />
(d''A'' +&rarr; '''B''')<br><br />
(d''A'', (d''A'' &rarr; '''B'''))<br><br />
[d''a''<sub>1</sub>, &hellip;, d''a''<sub>''n''</sub>]<br />
|<br />
Tangent universe<br><br />
at a point of ''A''<sup>&bull;</sup>,<br><br />
based on the<br><br />
tangent features<br><br />
{d''a''<sub>1</sub>, &hellip;, d''a''<sub>''n''</sub>}<br />
|<br />
('''D'''<sup>''n''</sup>, ('''D'''<sup>''n''</sup> &rarr; '''B'''))<br><br />
('''D'''<sup>''n''</sup> +&rarr; '''B''')<br><br />
['''D'''<sup>''n''</sup>]<br />
|}<br />
</font><br><br />
<br />
The adjectives ''differential'' or ''tangent'' are systematically attached to every construct based on the differential alphabet d<font face="lucida calligraphy">A</font>, taken by itself. Strictly speaking, we probably ought to call d<font face="lucida calligraphy">A</font> the set of ''cotangent'' features derived from <font face="lucida calligraphy">A</font>, but the only time this distinction really seems to matter is when we need to distinguish the tangent vectors as maps of type ('''B'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''B''')&nbsp;&rarr;&nbsp;'''B''' from cotangent vectors as elements of type '''D'''<sup>''n''</sup>. In like fashion, having defined E<font face="lucida calligraphy">A</font> = <font face="lucida calligraphy">A</font>&nbsp;&cup;&nbsp;d<font face="lucida calligraphy">A</font>, we can systematically attach the adjective ''extended'' or the substantive ''bundle'' to all of the constructs associated with this full complement of 2''n'' features.<br />
<br />
Eventually we may want to extend our basic alphabet even further, to allow for discussion of higher order differential expressions. For those who want to run ahead, and would like to play through, I submit the following gamut of notation (Table 9).<br />
<br />
<font face="courier new"><br />
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"<br />
|+ '''Table 9. Higher Order Differential Features'''<br />
| width=50% |<br />
{| cellpadding="4" style="background:lightcyan"<br />
| <font face="lucida calligraphy">A</font><br />
| =<br />
| d<sup>0</sup><font face="lucida calligraphy">A</font><br />
| =<br />
| {''a''<sub>1</sub>,<br />
| &hellip;,<br />
| ''a''<sub>''n''</sub>}<br />
|-<br />
| d<font face="lucida calligraphy">A</font><br />
| =<br />
| d<sup>1</sup><font face="lucida calligraphy">A</font><br />
| =<br />
| {d''a''<sub>1</sub>,<br />
| &hellip;,<br />
| d''a''<sub>''n''</sub>}<br />
|-<br />
| &nbsp;<br />
| &nbsp;<br />
| d<sup>''k''</sup><font face="lucida calligraphy">A</font><br />
| =<br />
| {d<sup>''k''</sup>''a''<sub>''1''</sub>,<br />
| &hellip;,<br />
| d<sup>''k''</sup>''a''<sub>''n''</sub>}<br />
|-<br />
| d<sup>*</sup><font face="lucida calligraphy">A</font><br />
| =<br />
| {d<sup>0</sup><font face="lucida calligraphy">A</font>,<br />
| &hellip;,<br />
| d<sup>''k''</sup><font face="lucida calligraphy">A</font>,<br />
| &hellip;}<br />
|}<br />
| width=50% |<br />
{| cellpadding="4" style="background:lightcyan"<br />
| E<sup>0</sup><font face="lucida calligraphy">A</font><br />
| =<br />
| d<sup>0</sup><font face="lucida calligraphy">A</font><br />
|-<br />
| E<sup>1</sup><font face="lucida calligraphy">A</font><br />
| =<br />
| d<sup>0</sup><font face="lucida calligraphy">A</font> &cup; d<sup>1</sup><font face="lucida calligraphy">A</font><br />
|-<br />
| E<sup>''k''</sup><font face="lucida calligraphy">A</font><br />
| =<br />
| d<sup>0</sup><font face="lucida calligraphy">A</font> &cup; &hellip; &cup; d<sup>''k''</sup><font face="lucida calligraphy">A</font><br />
|-<br />
| E<sup>&infin;</sup><font face="lucida calligraphy">A</font><br />
| =<br />
| &cup; d<sup>*</sup><font face="lucida calligraphy">A</font><br />
|}<br />
|}<br />
</font><br><br />
<br />
===Intentional Propositions===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Do you guess I have some intricate purpose?<br><br />
Well I have . . . . for the April rain has, and the mica on<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the side of a rock has.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 45]<br />
|}<br />
<br />
In order to analyze the behavior of a system at successive moments in time, while staying within the limitations of propositional logic, it is necessary to create independent alphabets of logical features for each moment of time that we contemplate using in our discussion. These moments have reference to typical instances and relative intervals, not actual or absolute times. For example, to discuss ''velocities'' (first order rates of change) we need<br />
to consider points of time in pairs. There are a number of natural ways of doing this. Given an initial alphabet, we could use its symbols as a lexical basis to generate successive alphabets of compound symbols, say, with temporal markers appended as suffixes.<br />
<br />
As a standard way of dealing with these situations, I produce the following scheme of notation, which extends any alphabet of logical features through as many temporal moments as a particular order of analysis may demand. The lexical operators p<sup>''k''</sup> and Q<sup>''k''</sup> are convenient in many contexts where the accumulation of prime symbols and union symbols would otherwise be cumbersome.<br />
<br />
<font face="courier new"><br />
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"<br />
|+ '''Table 10. A Realm of Intentional Features'''<br />
| width=50% |<br />
{| cellpadding="4" style="background:lightcyan"<br />
| p<sup>0</sup><font face="lucida calligraphy">A</font><br />
| =<br />
| <font face="lucida calligraphy">A</font><br />
| =<br />
| {''a''<sub>1</sub>&nbsp;,<br />
| &hellip;,<br />
| ''a''<sub>''n''</sub>&nbsp;}<br />
|-<br />
| p<sup>1</sup><font face="lucida calligraphy">A</font><br />
| =<br />
| <font face="lucida calligraphy">A</font>&prime;<br />
| =<br />
| {''a''<sub>1</sub>&prime;,<br />
| &hellip;,<br />
| ''a''<sub>''n''</sub>&prime;}<br />
|-<br />
| p<sup>2</sup><font face="lucida calligraphy">A</font><br />
| =<br />
| <font face="lucida calligraphy">A</font>&Prime;<br />
| =<br />
| {''a''<sub>1</sub>&Prime;,<br />
| &hellip;,<br />
| ''a''<sub>''n''</sub>&Prime;}<br />
|-<br />
| ...<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| ...<br />
|-<br />
| p<sup>''k''</sup><font face="lucida calligraphy">A</font><br />
| =<br />
| &nbsp;<br />
| &nbsp;<br />
| {p<sup>''k''</sup>''a''<sub>1</sub>,<br />
| &hellip;,<br />
| p<sup>''k''</sup>''a''<sub>''n''</sub>}<br />
|}<br />
| width=50% |<br />
{| cellpadding="4" style="background:lightcyan"<br />
| Q<sup>0</sup><font face="lucida calligraphy">A</font><br />
| =<br />
| <font face="lucida calligraphy">A</font><br />
|-<br />
| Q<sup>1</sup><font face="lucida calligraphy">A</font><br />
| =<br />
| <font face="lucida calligraphy">A</font> &cup; <font face="lucida calligraphy">A</font>&prime;<br />
|-<br />
| Q<sup>2</sup><font face="lucida calligraphy">A</font><br />
| =<br />
| <font face="lucida calligraphy">A</font> &cup; <font face="lucida calligraphy">A</font>&prime; &cup; <font face="lucida calligraphy">A</font>&Prime;<br />
|-<br />
| ...<br />
| &nbsp;<br />
| ...<br />
|-<br />
| Q<sup>''k''</sup><font face="lucida calligraphy">A</font><br />
| =<br />
| <font face="lucida calligraphy">A</font> &cup; <font face="lucida calligraphy">A</font>&prime; &cup; &hellip; &cup; p<sup>''k''</sup><font face="lucida calligraphy">A</font><br />
|}<br />
|}<br />
</font><br><br />
<br />
The resulting augmentations of our logical basis found a series of discursive universes that may be called the ''intentional extension'' of propositional calculus. The pattern of this extension is analogous to that of the differential extension, which was developed in terms of the operators d<sup>''k''</sup> and E<sup>''k''</sup>, and there is an obvious and natural relation between these two extensions that falls within our purview to explore. In contexts displaying this regular pattern, where a series of domains stretches up from an anchoring domain ''X'' through an indefinite number of higher reaches, I refer to a particular collection of domains based on ''X'' as a ''realm'' of ''X'', and when the succession exhibits a temporal aspect, as a ''reign'' of ''X''.<br />
<br />
For the purposes of this discussion, let us define an ''intentional proposition'' as a proposition in the universe of discourse Q''X''<sup>&nbsp;&bull;</sup> = [Q<font face="lucida calligraphy">X</font>], in other words, a map ''q''&nbsp;:&nbsp;Q''X''&nbsp;&rarr;&nbsp;'''B'''. The sense of this definition may be seen if we consider the following facts. First, the equivalence Q''X''&nbsp;=&nbsp;''X''&nbsp;&times;&nbsp;''X''&prime; motivates the following chain of isomorphisms between spaces:<br />
<br />
:{| cellpadding=2<br />
| (Q''X'' &rarr; '''B''')<br />
| <math>\cong</math><br />
| (''X'' &times; ''X''&prime; &rarr; '''B''')<br />
|-<br />
| &nbsp;<br />
| <math>\cong</math><br />
| (''X'' &rarr; (''X''&prime; &rarr; '''B'''))<br />
|-<br />
| &nbsp;<br />
| <math>\cong</math><br />
| (''X''&prime; &rarr; (''X'' &rarr; '''B''')).<br />
|}<br />
<br />
Viewed in this light, an intentional proposition ''q'' may be rephrased as a map ''q''&nbsp;:&nbsp;''X''&nbsp;&times;&nbsp;''X''&prime;&nbsp;&rarr;&nbsp;'''B''', which judges the juxtaposition of states in ''X'' from one moment to the next. Alternatively, ''q'' may be parsed in two stages in two different ways, as ''q''&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;(''X''&prime;&nbsp;&rarr;&nbsp;'''B''') and as ''q''&nbsp;:&nbsp;''X''&prime;&nbsp;&rarr;&nbsp;(''X''&nbsp;&rarr;&nbsp;'''B'''), which associate to each point of ''X'' or ''X''&prime; a proposition about states in ''X''&prime; or ''X'', respectively. In this way, an intentional proposition embodies a type of value system, in effect, a proposal that places a value on a collection of ends-in-view, or a project that evaluates a set of goals as regarded from each point of view in the state space of a system.<br />
<br />
In sum, the intentional proposition q indicates a method for the systematic selection of local goals. As a general form of description, we may refer to a map of the type ''q''&nbsp;:&nbsp;Q<sup>''i''</sup>''X''&nbsp;&rarr;&nbsp;'''B''' as an "''i''<sup>th</sup>&nbsp;order intentional proposition". Naturally, when we speak of intentional propositions without qualification, we usually mean first order intentions.<br />
<br />
Many different realms of discourse have the same structure as the extensions that have been indicated here. From a strictly logical point of view, each new layer of terms is composed of independent logical variables that are no different in kind from those that go before, and each further course of logical atoms is treated like so many individual, but otherwise indifferent bricks by the prototype computer program that I use as a propositional interpreter. Thus, the names that I use to single out the differential and the intentional extensions, and the lexical paradigms that I follow to construct them, are meant to suggest the interpretations that I have in mind, but they can only hint at the extra meanings that human communicators may pack into their terms and inflections.<br />
<br />
As applied here, the word ''intentional'' is drawn from common use and may have little bearing on its technical use in other, more properly philosophical, contexts. I am merely using the complex of intentional concepts - aims, ends, goals, objectives, purposes, and so on - metaphorically to flesh out and vividly to represent any situation where one needs to contemplate a system in multiple aspects of state and destination, that is, its being in certain states and at the same time acting as if headed through certain states. If confusion arises, more neutral words like ''conative'', ''contingent'', ''discretionary'', ''experimental'', ''kinetic'', ''progressive'', ''tentative'', or ''trial'' would probably serve as well.<br />
<br />
===Life on Easy Street===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Failing to fetch me at first keep encouraged,<br><br />
Missing me one place search another,<br><br />
I stop some where waiting for you<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]<br />
|}<br />
<br />
The finite character of the extended universe [E<font face="lucida calligraphy">A</font>] makes the problem of solving differential propositions relatively straightforward, at least,<br />
in principle. The solution set of the differential proposition ''q''&nbsp;:&nbsp;E''A''&nbsp;&rarr;&nbsp;'''B''' is the set of models ''q''<sup>&ndash;1</sup>(1) in E''A''. Finding all of the models of ''q'', the extended interpretations in E''A'' that satisfy ''q'', can be carried out by a finite search. Being in possession of complete algorithms for propositional calculus theorem proving makes the analytic task fairly simple in principle, though the question of efficiency in the face of arbitrary complexity may always remain another matter entirely. While the fact that propositional satisfiability is NP-complete may be discouraging for the prospects of a single efficient algorithm that covers the whole space of [E<font face="lucida calligraphy">A</font>] with equal facility, there appears to be much room for improvement in classifying special forms and in developing algorithms that are tailored to their practical processing.<br />
<br />
In view of these constraints and contingencies, my focus shifts to the tasks of approximation and interpretation that support intuition, especially in dealing with the natural kinds of differential propositions that arise in applications, and in the effort to understand, in succinct and adaptive forms, their dynamic implications. In the absence of direct insights, these tasks are partially carried out by forging analogies with the familiar situations and customary routines of ordinary calculus. But the indirect approach, going by way of specious analogy and intuitive habit, forces us to remain on guard against the circumstance that occurs when the word ''forging'' takes on its shadier nuance, indicting the constant risk of a counterfeit in the proportion.<br />
<br />
==Back to the Beginning : Exemplary Universes==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
I would have preferred to be enveloped in words, borne way beyond all possible beginnings.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
To anchor our understanding of differential logic, let us look at how the various concepts apply in the simplest possible concrete cases, where the initial dimension is only 1 or 2. In spite of the obvious simplicity of these cases, it is possible to observe how central difficulties of the subject begin to arise already at this stage.<br />
<br />
===A One-Dimensional Universe===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
There was never any more inception than there is now,<br><br />
Nor any more youth or age than there is now;<br><br />
And will never be any more perfection than there is now,<br><br />
Nor any more heaven or hell than there is now.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, Leaves of Grass, [Whi, 28]<br />
|}<br />
<br />
Let <font face="lucida calligraphy">X</font> = {''x''<sub>1</sub>} = {''A''} be an alphabet that represents one boolean variable or a single logical feature. In this example I am using the capital letter "''A''" in a more usual informal way, to name a feature and not a space, at variance with my formerly stated formal conventions. At any rate, the basis element<br />
''A'' = ''x''<sub>1</sub> may be interpreted as a simple proposition or a coordinate projection ''A'' = ''x''<sub>1</sub> : '''B'''<sub>1</sub> <font face=symbol>'''¸>'''</font> '''B'''. The space ''X'' = 〈''A'' 〉 = {(''A''), ''A''} of points (cells, vectors, interpretations) has cardinality 2<sup>''n''</sup> = 2<sup>1</sup> = 2 and is isomorphic to '''B''' = {0,&nbsp;1}. Moreover, ''X'' may be identified with the set of singular propositions {''x'' : '''B''' <font face=symbol>'''××>'''</font> '''B'''}. The space of linear propositions ''X''* = {hom : '''B''' <font face=symbol>'''+>'''</font> '''B'''} = {0,&nbsp;''A''} is algebraically dual to ''X'' and also has cardinality 2. Here, "0" is interpreted as denoting the constant function 0 : '''B''' &rarr; '''B''', amounting to the linear proposition of rank 0, while ''A'' is the linear proposition of rank 1. Last but not least we have the positive propositions {pos : '''B''' <font face=symbol>'''¥>'''</font> '''B'''} = {''A'',&nbsp;1}, of rank 1 and 0, respectively, where "1" is understood as denoting the constant function 1 : '''B''' &rarr; '''B'''. In sum, there are <math>2^{2^n} = 2^{2^1} = 4</math> propositions altogether in the universe of discourse, comprising the set ''X''^ = {f : ''X'' &rarr; '''B'''} = {0, (''A''), ''A'', 1} <math>\cong</math> ('''B''' &rarr; '''B''').<br />
<br />
The first order differential extension of <font face="lucida calligraphy">X</font> is E<font face="lucida calligraphy">X</font> = {''x''<sub>1</sub>, d''x''<sub>1</sub>} = {''A'', d''A''}. If the feature "''A''" is understood as applying to some object or state, then the feature "d''A''" may be interpreted as an attribute of the same object or state that says that it is changing ''significantly'' with respect to the property ''A'', or that it has an ''escape velocity'' with respect to the state ''A''. In practice, differential features acquire their logical meaning through a class of ''temporal inference rules''.<br />
<br />
For example, relative to a frame of observation that is left implicit for now, one is permitted to make the following sorts of inference: From the fact that ''A'' and d''A'' are true at a given moment one may infer that (''A'') will be true in the next moment of observation. Altogether in the present instance, there is the fourfold scheme of inference that is shown below:<br />
<br />
<br><font face="courier new"><br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"<br />
|<br />
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"<br />
| &nbsp; || From || (''A'') || and || (d''A'') || infer || (''A'') || next. || &nbsp;<br />
|-<br />
| &nbsp; || From || (''A'') || and || d''A'' || infer || ''A'' || next. || &nbsp;<br />
|-<br />
| &nbsp; || From || ''A'' || and || (d''A'') || infer || ''A'' || next. || &nbsp;<br />
|-<br />
| &nbsp; || From || ''A'' || and || d''A'' || infer || (''A'') || next. || &nbsp;<br />
|}<br />
|}<br />
</font><br><br />
<br />
It might be thought that we need to bring in an independent time variable at this point, but an insight of fundamental importance appears to be that the idea of process is more basic than the notion of time. A time variable is actually a reference to a ''clock'', that is, a canonical or a convenient process that is established or accepted as a standard of measurement, but in essence no different than any other process. This raises the question of how different subsystems in a more global process can be brought into comparison, and what it means for one process to serve the function of a local standard for others. But these inquiries only wrap up puzzles in further riddles, and are obviously too involved to be handled at our current level of approximation.<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
The clock indicates the moment . . . . but what does<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;eternity indicate?<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, 'Leaves of Grass', [Whi, 79]<br />
|}<br />
<br />
Observe that the secular inference rules, used by themselves, involve a loss of information, since nothing in them can tell us whether the momenta {(d''A''),&nbsp;d''A''} are preserved or changed in the next instance. In order to know this, we would have to determine d<sup>2</sup>''A'', and so on, pursuing an infinite regress. Ultimately, in order to rest with a finitely determinate system, it is necessary to make an infinite assumption, for example, that d<sup>''k''</sup>''A'' = 0 for all ''k'' greater than some fixed value ''M''. Another way to escape the regress is through the provision of a dynamic law, in typical form making higher order differentials dependent on lower degrees and estates.<br />
<br />
===Example 1. A Square Rigging===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Urge and urge and urge,<br><br />
Always the procreant urge of the world.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]<br />
|}<br />
<br />
By way of example, suppose that we are given the initial condition ''A''&nbsp;=&nbsp;d''A'' and the law d<sup>2</sup>''A''&nbsp;=&nbsp;(''A''). Then, since "''A''&nbsp;=&nbsp;d''A''" &hArr; "''A''&nbsp;d''A'' or (''A'')(d''A'')", we may infer two possible trajectories, as displayed in Table 11. In either of these cases, the state ''A''(d''A'')(d<sup>2</sup>''A'') is a stable attractor or a terminal condition for both starting points.<br />
<br />
<font face="courier new"><br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"<br />
|+ '''Table 11. A Pair of Commodious Trajectories'''<br />
|- style="background:paleturquoise"<br />
! Time<br />
! Trajectory 1<br />
! Trajectory 2<br />
|-<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; text-align:center"<br />
| 0<br />
|-<br />
| 1<br />
|-<br />
| 2<br />
|-<br />
| 3<br />
|-<br />
| 4<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; text-align:center"<br />
| ''A'' || d''A'' || (d<sup>2</sup>''A'')<br />
|-<br />
| (''A'') || d''A'' || d<sup>2</sup>''A''<br />
|-<br />
| ''A'' || (d''A'') || (d<sup>2</sup>''A'')<br />
|-<br />
| ''A'' || (d''A'') || (d<sup>2</sup>''A'')<br />
|-<br />
| " || " || "<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; text-align:center"<br />
| (''A'') || (d''A'') || d<sup>2</sup>''A''<br />
|-<br />
| (''A'') || d''A'' || d<sup>2</sup>''A''<br />
|-<br />
| ''A'' || (d''A'') || (d<sup>2</sup>''A'')<br />
|-<br />
| ''A'' || (d''A'') || (d<sup>2</sup>''A'')<br />
|-<br />
| " || " || "<br />
|}<br />
|}<br />
</font><br><br />
<br />
Because the initial space ''X''&nbsp;=&nbsp;〈''A''〉 is one-dimensional, we can easily fit the second order extension E<sup>2</sup>''X''&nbsp;=&nbsp;〈''A'',&nbsp;d''A'',&nbsp;d<sup>2</sup>''A''〉 within the compass of a single venn diagram, charting the couple of converging trajectories as shown in Figure 12.<br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 12 -- The Anchor.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 12. The Anchor'''</font></center></p><br />
<br />
If we eliminate from view the regions of E<sup>2</sup>''X'' that are ruled out by the dynamic law d<sup>2</sup>''A''&nbsp;=&nbsp;(''A''), then what remains is the quotient structure that is shown in Figure 13. This picture makes it easy to see that the dynamically allowable portion of the universe is partitioned between the properties ''A'' and d<sup>2</sup>''A''. As it happens, this fact might have been expressed "right off the bat" by an equivalent formulation of the differential law, one that uses the exclusive disjunction to state the law as (''A'',&nbsp;d<sup>2</sup>''A'').<br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 13 -- The Tiller.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 13. The Tiller'''</font></center></p><br />
<br />
What we have achieved in this example is to give a differential description of a simple dynamic process. In effect, we did this by embedding a directed graph, which can be taken to represent the state transitions of a finite automaton, in a dynamically allotted quotient structure that is created from a boolean lattice or an ''n''-cube by nullifying all of the regions that the dynamics outlaws. With growth in the dimensions of our contemplated universes, it becomes essential, both for human comprehension and for computer implementation, that the dynamic structures of interest to us be represented not actually, by acquaintance, but virtually, by description. In our present study, we are using the language of propositional calculus to express the relevant descriptions, and to comprehend the structure that is implicit in the subsets of a ''n''-cube without necessarily being forced to actualize all of its points.<br />
<br />
One of the reasons for engaging in this kind of extremely reduced, but explicitly controlled case study is to throw light on the general study of languages, formal and natural, in their full array of syntactic, semantic, and pragmatic aspects. Propositional calculus is one of the last points of departure where we can view these three aspects interacting in a non-trivial way without being immediately and totally overwhelmed by the complexity they generate. Often this complexity causes investigators of formal and natural languages to adopt the strategy of focusing on a single aspect and to abandon all hope of understanding the whole, whether it's the still living natural language or the dynamics of inquiry that lies crystallized in formal logic.<br />
<br />
From the perspective that I find most useful here, a language is a syntactic system that is designed or evolved in part to express a set of descriptions. When the explicit symbols of a language have extensions in its object world that are actually infinite, or when the implicit categories and generative devices of a linguistic theory have extensions in its subject matter that are potentially infinite, then the finite characters of terms, statements, arguments, grammars, logics, and rhetorics force an excess of intension to reside in all these symbols and functions, across the spectrum from the object language to the metalinguistic uses. In the aphorism from W. von Humboldt that Chomsky often cites, for example, in [Cho86, 30] and [Cho93, 49], language requires "the infinite use of finite means". This is necessarily true when the extensions are infinite, when the referential symbols and grammatical categories of a language possess infinite sets of models and instances. But it also voices a practical truth when the extensions, though finite at every stage, tend to grow at exponential rates.<br />
<br />
This consequence of dealing with extensions that are "practically infinite" becomes crucial when one tries to build neural network systems that learn, since the learning competence of any intelligent system is limited to the objects and domains that it is able to represent. If we want to design systems that operate intelligently with the full deck of propositions dealt by intact universes of discourse, then we must supply them with succinct representations and efficient transformations in this domain. Furthermore, in the project of constructing inquiry driven systems, we find ourselves forced to contemplate the level of generality that is embodied in propositions, because the dynamic evolution of these systems is driven by the measurable discrepancies that occur among their expectations, intentions, and observations, and because each of these subsystems or components of knowledge constitutes a propositional modality that can take on the fully generic character of an empirical summary or an axiomatic theory.<br />
<br />
A compression scheme by any other name is a symbolic representation, and this is what the differential extension of propositional calculus, through all of its many universes of discourse, is intended to supply. Why is this particular program of mental calisthenics worth carrying out in general? By providing a uniform logical medium for describing dynamic systems we can make the task of understanding complex systems much easier, both in looking for invariant representations of individual cases and in finding points of comparison among diverse structures that would otherwise appear as isolated systems. All of this goes to facilitate the search for compact knowledge and to adapt what is learned from individual cases to the general realm.<br />
<br />
===Back to the Feature===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
I guess it must be the flag of my disposition, out of hopeful<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;green stuff woven.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 31]<br />
|}<br />
<br />
Let us assume that the sense intended for differential features is well enough established in the intuition, for now, that I may continue with outlining the structure of the differential extension [E<font face="lucida calligraphy">X</font>]&nbsp;=&nbsp;[''A'',&nbsp;d''A'']. Over the extended alphabet E<font face="lucida calligraphy">X</font> = {''x''<sub>1</sub>,&nbsp;d''x''<sub>1</sub>} = {''A'',&nbsp;d''A''}, of cardinality 2<sup>''n''</sup> = 2, we generate the set of points, E''X'', of cardinality 2<sup>2''n''</sup> = 4, that bears the following chain of equivalent descriptions:<br />
<br />
:{| cellpadding=2<br />
| E''X''<br />
| =<br />
| 〈''A'', d''A''〉<br />
|-<br />
| &nbsp;<br />
| =<br />
| {(''A''), ''A''} &times; {(d''A''), d''A''}<br />
|-<br />
| &nbsp;<br />
| =<br />
| {(''A'')(d''A''), (''A'') d''A'', ''A'' (d''A''), ''A'' d''A''}.<br />
|}<br />
<br />
The space E''X'' may be assigned the mnemonic type '''B'''&nbsp;&times;&nbsp;'''D''', which is really no different than '''B'''&nbsp;&times;&nbsp;'''B'''&nbsp;=&nbsp;'''B'''<sup>2</sup>. An individual element of E''X'' may be regarded as a ''disposition at a point'' or a ''situated direction'', in effect, a singular mode of change occurring at a single point in the universe of discourse. In applications, the modality of this change can be interpreted in various ways, for example, as an expectation, an intention, or an observation with respect to the behavior of a system.<br />
<br />
To complete the construction of the extended universe of discourse E''X''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;[''x''<sub>1</sub>,&nbsp;d''x''<sub>1</sub>]&nbsp;=&nbsp;[''A'',&nbsp;d''A''], one must add the set of differential propositions E''X''^&nbsp;=&nbsp;{''g''&nbsp;:&nbsp;E''X''&nbsp;&rarr;&nbsp;'''B'''}&nbsp;<math>\cong</math>&nbsp;('''B'''&nbsp;&times;&nbsp;'''D'''&nbsp;&rarr;&nbsp;'''B''') to the set of dispositions in E''X''. There are <math>2^{2^{2n}}</math>&nbsp;=&nbsp;16 propositions in E''X''^, as detailed in Table 14.<br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"<br />
|+ '''Table 14. Differential Propositions'''<br />
|- style="background:paleturquoise"<br />
| &nbsp;<br />
| align="right" | A :<br />
| 1 1 0 0 <br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:paleturquoise"<br />
| &nbsp;<br />
| align="right" | dA :<br />
| 1 0 1 0<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| f<sub>0</sub><br />
| g<sub>0</sub><br />
| 0 0 0 0<br />
| (&nbsp;)<br />
| False<br />
| 0<br />
|-<br />
|<br />
{| style="background:lightcyan"<br />
|<br />
&nbsp;<br><br />
&nbsp;<br><br />
&nbsp;<br><br />
&nbsp;<br />
|}<br />
|<br />
{| style="background:lightcyan"<br />
|<br />
g<sub>1</sub><br><br />
g<sub>2</sub><br><br />
g<sub>4</sub><br><br />
g<sub>8</sub><br />
|}<br />
|<br />
{| style="background:lightcyan"<br />
|<br />
0 0 0 1<br><br />
0 0 1 0<br><br />
0 1 0 0<br><br />
1 0 0 0<br />
|}<br />
|<br />
{| style="background:lightcyan"<br />
|<br />
(A)(dA)<br><br />
(A) dA <br><br />
A (dA)<br><br />
A dA<br />
|}<br />
|<br />
{| style="background:lightcyan"<br />
|<br />
Neither A nor dA<br><br />
Not A but dA<br><br />
A but not dA<br><br />
A and dA<br />
|}<br />
|<br />
{| style="background:lightcyan"<br />
|<br />
&not;A &and; &not;dA<br><br />
&not;A &and; dA<br><br />
A &and; &not;dA<br><br />
A &and; dA<br />
|}<br />
|-<br />
|<br />
{| style="background:lightcyan"<br />
|<br />
f<sub>1</sub><br><br />
f<sub>2</sub><br />
|}<br />
|<br />
{| style="background:lightcyan"<br />
|<br />
g<sub>3</sub><br><br />
g<sub>12</sub><br />
|}<br />
|<br />
{| style="background:lightcyan"<br />
|<br />
0 0 1 1<br><br />
1 1 0 0<br />
|}<br />
|<br />
{| style="background:lightcyan"<br />
|<br />
(A)<br><br />
A<br />
|}<br />
|<br />
{| style="background:lightcyan"<br />
|<br />
Not A<br><br />
A<br />
|}<br />
|<br />
{| style="background:lightcyan"<br />
|<br />
&not;A<br><br />
A<br />
|}<br />
|-<br />
|<br />
{| style="background:lightcyan"<br />
|<br />
&nbsp;<br><br />
&nbsp;<br />
|}<br />
|<br />
{| style="background:lightcyan"<br />
|<br />
g<sub>6</sub><br><br />
g<sub>9</sub><br />
|}<br />
|<br />
{| style="background:lightcyan"<br />
|<br />
0 1 1 0<br><br />
1 0 0 1<br />
|}<br />
|<br />
{| style="background:lightcyan"<br />
|<br />
(A, dA)<br><br />
((A, dA))<br />
|}<br />
|<br />
{| style="background:lightcyan"<br />
|<br />
A not equal to dA<br><br />
A equal to dA<br />
|}<br />
|<br />
{| style="background:lightcyan"<br />
|<br />
A &ne; dA<br><br />
A = dA<br />
|}<br />
|-<br />
|<br />
{| style="background:lightcyan"<br />
|<br />
&nbsp;<br><br />
&nbsp;<br />
|}<br />
|<br />
{| style="background:lightcyan"<br />
|<br />
g<sub>5</sub><br><br />
g<sub>10</sub><br />
|}<br />
|<br />
{| style="background:lightcyan"<br />
|<br />
0 1 0 1<br><br />
1 0 1 0<br />
|}<br />
|<br />
{| style="background:lightcyan"<br />
|<br />
(dA)<br><br />
dA<br />
|}<br />
|<br />
{| style="background:lightcyan"<br />
|<br />
Not dA<br><br />
dA<br />
|}<br />
|<br />
{| style="background:lightcyan"<br />
|<br />
&not;dA<br><br />
dA<br />
|}<br />
|-<br />
|<br />
{| style="background:lightcyan"<br />
|<br />
&nbsp;<br><br />
&nbsp;<br><br />
&nbsp;<br><br />
&nbsp;<br />
|}<br />
|<br />
{| style="background:lightcyan"<br />
|<br />
g<sub>7</sub><br><br />
g<sub>11</sub><br><br />
g<sub>13</sub><br><br />
g<sub>14</sub><br />
|}<br />
|<br />
{| style="background:lightcyan"<br />
|<br />
0 1 1 1<br><br />
1 0 1 1<br><br />
1 1 0 1<br><br />
1 1 1 0<br />
|}<br />
|<br />
{| style="background:lightcyan"<br />
|<br />
(A dA)<br><br />
(A (dA))<br><br />
((A) dA)<br><br />
((A)(dA))<br />
|}<br />
|<br />
{| style="background:lightcyan"<br />
|<br />
Not both A and dA<br><br />
Not A without dA<br><br />
Not dA without A<br><br />
A or dA<br />
|}<br />
|<br />
{| style="background:lightcyan"<br />
|<br />
&not;A &or; &not;dA<br><br />
A &rarr; dA<br><br />
A &larr; dA<br><br />
A &or; dA<br />
|}<br />
|-<br />
| f<sub>3</sub><br />
| g<sub>15</sub><br />
| 1 1 1 1<br />
| ((&nbsp;))<br />
| True<br />
| 1<br />
|}<br />
<br><br />
<br />
Aside from changing the names of variables and shuffling the order of rows, this Table follows the format that was used previously for boolean functions of two variables. The rows are grouped to reflect natural similarity classes among the propositions. In a future discussion, these classes will be given additional explanation and motivation as the orbits of a certain transformation group acting on the set of 16 propositions. Notice that four of the propositions, in their logical expressions, resemble those given in the table for ''X''^. Thus the first set of propositions {''f''<sub>''i''</sub>} is automatically embedded in the present set {''g''<sub>''j''</sub>}, and the corresponding inclusions are indicated at the far left margin of the table.<br />
<br />
===Tacit Extensions===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
I would really like to have slipped imperceptibly into this lecture, as into all the others I shall be delivering, perhaps over the years ahead.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
Strictly speaking, however, there is a subtle distinction in type between the function ''f''<sub>''i''</sub>&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;'''B''' and the corresponding function ''g''<sub>''j''</sub>&nbsp;:&nbsp;E''X''&nbsp;&rarr;&nbsp;'''B''', even though they share the same logical expression. Being human, we insist on preserving all the aesthetic delights afforded by the abstractly unified form of the "cake" while giving up none of the diverse contents that its substantive consummation can provide. In short, we want to maintain the logical equivalence of expressions that represent the same proposition, while appreciating the full diversity of that proposition's functional and typical representatives. Both perspectives, and all the levels of abstraction extending through them, have their reasons, as will develop in time.<br />
<br />
Because this special circumstance points up an important general theme, it is a good idea to discuss it more carefully. Whenever there arises a situation like this, where one alphabet <font face="lucida calligraphy">X</font> is a subset of another alphabet <font face="lucida calligraphy">Y</font>, then we say that any proposition ''f''&nbsp;:&nbsp;〈<font face="lucida calligraphy">X</font>〉&nbsp;&rarr;&nbsp;''B'' has a ''tacit extension'' to a proposition <math>\epsilon</math>''f''&nbsp;:&nbsp;〈<font face="lucida calligraphy">Y</font>〉&nbsp;&rarr;&nbsp;'''B''', and that the space (〈<font face="lucida calligraphy">X</font>〉&nbsp;&rarr;&nbsp;'''B''') has an ''automatic embedding'' within the space (〈<font face="lucida calligraphy">Y</font>〉&nbsp;&rarr;&nbsp;'''B'''). The extension is defined in such a way that <math>\epsilon</math>''f'' puts the same constraint on the variables of <font face="lucida calligraphy">X</font> that are contained in <font face="lucida calligraphy">Y</font> as the proposition ''f'' initially did, while it puts no constraint on the variables of <font face="lucida calligraphy">Y</font> outside of <font face="lucida calligraphy">X</font>, in effect, conjoining the two constraints.<br />
<br />
If the variables in question are indexed as <font face="lucida calligraphy">X</font>&nbsp;=&nbsp;{''x''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''x''<sub>''n''</sub>} and <font face="lucida calligraphy">Y</font>&nbsp;=&nbsp;{''x''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''x''<sub>''n''</sub>,&nbsp;&hellip;,&nbsp;''x''<sub>''n''+''k''</sub>}, then the definition of the tacit extension from <font face="lucida calligraphy">X</font> to <font face="lucida calligraphy">Y</font> may be expressed in the form of an equation:<br />
<br />
: <math>\epsilon</math>''f''(''x''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''x''<sub>''n''</sub>,&nbsp;&hellip;,&nbsp;''x''<sub>''n''+''k''</sub>) = ''f''(''x''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''x''<sub>''n''</sub>).<br />
<br />
On formal occasions, such as the present context of definition, the tacit extension from <font face="lucida calligraphy">X</font> to <font face="lucida calligraphy">Y</font> is explicitly symbolized by the operator <math>\epsilon</math>&nbsp;:&nbsp;(〈<font face="lucida calligraphy">X</font>〉&nbsp;&rarr;&nbsp;'''B''')&nbsp;&rarr;&nbsp;(〈<font face="lucida calligraphy">Y</font>〉&nbsp;&rarr;&nbsp;'''B'''), where the appropriate alphabets <font face="lucida calligraphy">X</font> and <font face="lucida calligraphy">Y</font> are understood from context, but normally one may leave the "<math>\epsilon</math>" silent.<br />
<br />
Let's explore what this means for the present Example. Here, <font face="lucida calligraphy">X</font> = {''A''} and <font face="lucida calligraphy">Y</font> = E<font face="lucida calligraphy">X</font> = {''A'',&nbsp;d''A''}. For each of the propositions ''f''<sub>''i''</sub> over ''X'', specifically, those whose expression ''e''<sub>''i''</sub> lies in the collection {0,&nbsp;(''A''),&nbsp;''A'',&nbsp;1}, the tacit extension <math>\epsilon</math>''f'' of ''f'' to E''X'' can be phrased as a logical conjunction of two factors, ''f''<sub>''i''</sub> = ''e''<sub>''i''</sub>&nbsp;'''·'''&nbsp;<math>\tau</math>&nbsp;, where <math>\tau</math> is a logical tautology that uses all the variables of <font face="lucida calligraphy">Y</font>&nbsp;&ndash;&nbsp;<font face="lucida calligraphy">X</font>. Working in these terms, the tacit extensions <math>\epsilon</math>''f'' of ''f'' to E''X'' may be explicated as shown in Table&nbsp;15.<br />
<br />
<font face="courier new"><br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"<br />
|+ '''Table 15. Tacit Extension of [''A''] to [''A'', d''A'']'''<br />
|<br />
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"<br />
| &nbsp;<br />
| 0<br />
| =<br />
| 0<br />
| &middot;<br />
| ((d''A''),&nbsp;d''A'')<br />
| =<br />
| 0<br />
| &nbsp;<br />
|-<br />
| &nbsp;<br />
| (''A'')<br />
| =<br />
| (''A'')<br />
| &middot;<br />
| ((d''A''),&nbsp;d''A'')<br />
| =<br />
| (''A'')(d''A'') + (''A'') d''A''&nbsp;<br />
| &nbsp;<br />
|-<br />
| &nbsp;<br />
| ''A''<br />
| =<br />
| ''A''<br />
| &middot;<br />
| ((d''A''),&nbsp;d''A'')<br />
| =<br />
| &nbsp;''A'' (d''A'') + &nbsp;''A''&nbsp;&nbsp;d''A''&nbsp;<br />
| &nbsp;<br />
|-<br />
| &nbsp;<br />
| 1<br />
| =<br />
| 1<br />
| &middot;<br />
| ((d''A''),&nbsp;d''A'')<br />
| =<br />
| 1<br />
|}<br />
|}<br />
</font><br><br />
<br />
In its effect on the singular propositions over ''X'', this analysis has an interesting interpretation. The tacit extension takes us from thinking about a particular state, like ''A'' or (''A''), to considering the collection of outcomes, the outgoing changes or the singular dispositions, that spring from that state.<br />
<br />
===Example 2. Drives and Their Vicissitudes===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
I open my scuttle at night and see the far-sprinkled systems,<br><br />
And all I see, multiplied as high as I can cipher, edge but<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the rim of the farther systems.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 81]<br />
|}<br />
<br />
Before we leave the one-feature case let's look at a more substantial example, one that illustrates a general class of curves that can be charted through the extended feature spaces and that provides an opportunity to discuss a number of important themes concerning their structure and dynamics.<br />
<br />
Again, let <font face="lucida calligraphy">X</font>&nbsp;=&nbsp;{''x''<sub>1</sub>}&nbsp;=&nbsp;{''A''}. In the discussion that follows I will consider a class of trajectories having the property that d<sup>''k''</sup>''A''&nbsp;=&nbsp;0 for all ''k'' greater than some fixed ''m'', and I indulge in the use of some picturesque terms that describe salient classes of such curves. Given the finite order condition, there is a highest order non-zero difference d<sup>''m''</sup>''A'' exhibited at each point in the course of any determinate trajectory that one may wish to consider. With respect to any point of the corresponding orbit or curve let us call this highest order differential feature d<sup>''m''</sup>''A'' the ''drive'' at that point. Curves of constant drive d<sup>''m''</sup>''A'' are then referred to as "''m''<sup>th</sup> gear curves".<br />
<br />
* '''Scholium.''' The fact that a difference calculus can be developed for boolean functions is well known [Fuji], [Koh, &sect; 8-4] and was probably familiar to Boole, who was an expert in difference equations before he turned to logic. And of course there is the strange but true story of how the Turin machines of the 1840's prefigured the Turing machines of the 1940's [Men, 225-297]. At the very outset of general purpose, mechanized computing we find that the motive power driving the Analytical Engine of Babbage, the kernel of an idea behind all of his wheels, was exactly his notion that difference operations, suitably trained, can serve as universal joints for any conceivable computation [M&M], [Mel, ch. 4].<br />
<br />
Given this language, the particular Example that I take up here can be described as the family of 4<sup>th</sup> gear curves through E<sup>4</sup>''X'' = 〈''A'',&nbsp;d''A'',&nbsp;d<sup>2</sup>''A'',&nbsp;d<sup>3</sup>''A'',&nbsp;d<sup>4</sup>''A''〉. These are the trajectories generated subject to the dynamic law d<sup>4</sup>''A''&nbsp;=&nbsp;1, where it is understood in such a statement that all higher order differences are equal to 0. Since d<sup>4</sup>''A'' and all higher d<sup>''k''</sup>''A'' are fixed, the temporal or transitional conditions (initial, mediate, terminal - transient or stable states) vary only with respect to their projections as points of E<sup>3</sup>''X'' = 〈''A'',&nbsp;d''A'',&nbsp;d<sup>2</sup>''A'',&nbsp;d<sup>3</sup>''A''〉. Thus, there is just enough space in a planar venn diagram to plot all of these orbits and to show how they partition the points of E<sup>3</sup>''X''. It turns out that there are exactly two possible orbits, of eight points each, as illustrated in Figure&nbsp;16.<br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 16 -- A Couple of Fourth Gear Orbits.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 16. A Couple of Fourth Gear Orbits'''</font></center></p><br />
<br />
With a little thought it is possible to devise an indexing scheme for the general run of dynamic states that allows for comparing universes of discourse that weigh in on different scales of observation. With this end in sight, let us index the states ''q'' in E<sup>''m''</sup>''X'' with the dyadic rationals (or the binary fractions) in the half-open interval [0,&nbsp;2). Formally and canonically, a state ''q''<sub>''r''</sub> is indexed by a fraction ''r''&nbsp;=&nbsp;''s''/''t'' whose denominator is the power of two ''t''&nbsp;=&nbsp;2<sup>''m''</sup> and whose numerator is a binary numeral that is formed from the coefficients of state in a manner to be described next. The ''differential coefficients'' of the state ''q'' are just the values d<sup>''k''</sup>''A''(''q''), for ''k''&nbsp;=&nbsp;0&nbsp;to&nbsp;''m'', where d<sup>0</sup>''A'' is defined as being identical to ''A''. To form the binary index d<sub>0</sub>'''.'''d<sub>1</sub>&hellip;d<sub>''m''</sub> of the state ''q'' the coefficient d<sup>''k''</sup>''A''(''q'') is read off as the binary digit ''d''<sub>''k''</sub> associated with the place value 2<sup>&ndash;''k''</sup>. Expressed by way of algebraic formulas, the rational index ''r'' of the state ''q'' can be given by the following equivalent formulations:<br />
<br />
<br><font face="courier new"><br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"<br />
|<br />
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"<br />
| <math>r(q)\!</math><br />
| <math>=</math><br />
| <math>\sum_k d_k \cdot 2^{-k}</math><br />
| <math>=</math><br />
| <math>\sum_k \mbox{d}^k A(q) \cdot 2^{-k}</math><br />
|-<br />
| <math>=</math><br />
|-<br />
| <math>\frac{s(q)}{t}</math><br />
| <math>=</math><br />
| <math>\frac{\sum_k d_k \cdot 2^{(m-k)}}{2^m}</math><br />
| <math>=</math><br />
| <math>\frac{\sum_k \mbox{d}^k A(q) \cdot 2^{(m-k)}}{2^m}</math><br />
|}<br />
|}<br />
</font><br><br />
<br />
Applied to the example of fourth gear curves, this scheme results in the data of Tables&nbsp;17-a and 17-b, which exhibit one period for each orbit. The states in each orbit are listed as ordered pairs ‹''p''<sub>''i''</sub>,&nbsp;''q''<sub>''j''</sub>›, where ''p''<sub>''i''</sub> may be read as a temporal parameter that indicates the present time of the state, and where ''j'' is the decimal equivalent of the binary numeral ''s''. Informally and more casually, the Tables exhibit the states ''q''<sub>''s''</sub> as subscripted with the numerators of their rational indices, taking for granted the constant denominators of 2<sup>''m''</sup>&nbsp;=&nbsp;2<sup>4</sup>&nbsp;=&nbsp;16. Within this set-up, the temporal successions of states can be reckoned as given by a kind of ''parallel round-up rule''. That is, if ‹''d''<sub>''k''</sub>,&nbsp;''d''<sub>''k''+1</sub>› is any pair of adjacent digits in the state index ''r'', then the value of ''d''<sub>''k''</sub> in the next state is ''d''<sub>''k''</sub>&prime;&nbsp;=&nbsp;''d''<sub>''k''</sub>&nbsp;+&nbsp;''d''<sub>''k''+1</sub>.<br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"<br />
|+ '''Table 17-a. A Couple of Orbits in Fourth Gear: Orbit 1'''<br />
|- style="background:paleturquoise"<br />
| Time<br />
| State<br />
| ''A''<br />
| d''A''<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:paleturquoise"<br />
| ''p''<sub>''i''</sub><br />
| ''q''<sub>''j''</sub><br />
| d<sup>0</sup>''A''<br />
| d<sup>1</sup>''A''<br />
| d<sup>2</sup>''A''<br />
| d<sup>3</sup>''A''<br />
| d<sup>4</sup>''A''<br />
|-<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center"<br />
| ''p''<sub>0</sub><br />
|-<br />
| ''p''<sub>1</sub><br />
|-<br />
| ''p''<sub>2</sub><br />
|-<br />
| ''p''<sub>3</sub><br />
|-<br />
| ''p''<sub>4</sub><br />
|-<br />
| ''p''<sub>5</sub><br />
|-<br />
| ''p''<sub>6</sub><br />
|-<br />
| ''p''<sub>7</sub><br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center"<br />
| ''q''<sub>01</sub><br />
|-<br />
| ''q''<sub>03</sub><br />
|-<br />
| ''q''<sub>05</sub><br />
|-<br />
| ''q''<sub>15</sub><br />
|-<br />
| ''q''<sub>17</sub><br />
|-<br />
| ''q''<sub>19</sub><br />
|-<br />
| ''q''<sub>21</sub><br />
|-<br />
| ''q''<sub>31</sub><br />
|}<br />
| colspan="5" |<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0. || 0 || 0 || 0 || 1<br />
|-<br />
| 0. || 0 || 0 || 1 || 1<br />
|-<br />
| 0. || 0 || 1 || 0 || 1<br />
|-<br />
| 0. || 1 || 1 || 1 || 1<br />
|-<br />
| 1. || 0 || 0 || 0 || 1<br />
|-<br />
| 1. || 0 || 0 || 1 || 1<br />
|-<br />
| 1. || 0 || 1 || 0 || 1<br />
|-<br />
| 1. || 1 || 1 || 1 || 1<br />
|}<br />
|}<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"<br />
|+ '''Table 17-b. A Couple of Orbits in Fourth Gear: Orbit 2'''<br />
|- style="background:paleturquoise"<br />
| Time<br />
| State<br />
| ''A''<br />
| d''A''<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:paleturquoise"<br />
| ''p''<sub>''i''</sub><br />
| ''q''<sub>''j''</sub><br />
| d<sup>0</sup>''A''<br />
| d<sup>1</sup>''A''<br />
| d<sup>2</sup>''A''<br />
| d<sup>3</sup>''A''<br />
| d<sup>4</sup>''A''<br />
|-<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center"<br />
| ''p''<sub>0</sub><br />
|-<br />
| ''p''<sub>1</sub><br />
|-<br />
| ''p''<sub>2</sub><br />
|-<br />
| ''p''<sub>3</sub><br />
|-<br />
| ''p''<sub>4</sub><br />
|-<br />
| ''p''<sub>5</sub><br />
|-<br />
| ''p''<sub>6</sub><br />
|-<br />
| ''p''<sub>7</sub><br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center"<br />
| ''q''<sub>25</sub><br />
|-<br />
| ''q''<sub>11</sub><br />
|-<br />
| ''q''<sub>29</sub><br />
|-<br />
| ''q''<sub>07</sub><br />
|-<br />
| ''q''<sub>09</sub><br />
|-<br />
| ''q''<sub>27</sub><br />
|-<br />
| ''q''<sub>13</sub><br />
|-<br />
| ''q''<sub>23</sub><br />
|}<br />
| colspan="5" |<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 1. || 1 || 0 || 0 || 1<br />
|-<br />
| 0. || 1 || 0 || 1 || 1<br />
|-<br />
| 1. || 1 || 1 || 0 || 1<br />
|-<br />
| 0. || 0 || 1 || 1 || 1<br />
|-<br />
| 0. || 1 || 0 || 0 || 1<br />
|-<br />
| 1. || 1 || 0 || 1 || 1<br />
|-<br />
| 0. || 1 || 1 || 0 || 1<br />
|-<br />
| 1. || 0 || 1 || 1 || 1<br />
|}<br />
|}<br />
<br><br />
<br />
==Transformations of Discourse==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
It is understandable that an engineer should be completely absorbed in his speciality, instead of pouring himself out into the freedom and vastness of the world of thought, even though his machines are being sent off to the ends of the earth; for he no more needs to be capable of applying to his own personal soul what is daring and new in the soul of his subject than a machine is in fact capable of applying to itself the differential calculus on which it is based. The same thing cannot, however, be said about mathematics; for here we have the new method of thought, pure intellect, the very well-spring of the times, the ''fons et origo'' of an unfathomable transformation.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 39]<br />
|}<br />
<br />
In this section we take up the general study of logical transformations, or maps that relate one universe of discourse to another. In many ways, and especially as applied to the subject of intelligent dynamic systems, my argument develops the antithesis of the statement just quoted. Along the way, if incidental to my ends, I hope this essay can pose a fittingly irenic epitaph to the frankly ironic epigraph inscribed at its head.<br />
<br />
My goal in this section is to answer a single question: What is a propositional tangent functor? In other words, my aim is to develop a clear conception of what manner of thing would pass in the logical realm for a genuine analogue of the tangent functor, an object conceived to generalize as far as possible in the abstract terms of category theory the ordinary notions of functional differentiation and the all too familiar operations of taking derivatives.<br />
<br />
As a first step I discuss the kinds of transformations that we already know as ''extensions'' and ''projections'', and I use these special cases to illustrate several different styles of logical and visual representation that will figure heavily in the sequel.<br />
<br />
===Foreshadowing Transformations : Extensions and Projections of Discourse===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
And, despite the care which she took to look behind her at every moment, she failed to see a shadow which followed her like her own shadow, which stopped when she stopped, which started again when she did, and which made no more noise than a well-conducted shadow should.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 126]<br />
|}<br />
<br />
Many times in our discussion we have occasion to place one universe of discourse in the context of a larger universe of discourse. An embedding of the general type [<font face="lucida calligraphy">X</font>]&nbsp;&rarr;&nbsp;[<font face="lucida calligraphy">Y</font>] is implied any time that we make use of one alphabet <font face="lucida calligraphy">X</font> that happens to be included in another alphabet <font face="lucida calligraphy">Y</font>. When we are discussing differential issues we usually have in mind that the extended alphabet <font face="lucida calligraphy">Y</font> has a special construction or a specific lexical relation with respect to the initial alphabet <font face="lucida calligraphy">X</font>, one that is marked by characteristic types of accents, indices, or inflected forms.<br />
<br />
====Extension from 1 to 2 Dimensions====<br />
<br />
Figure 18-a lays out the ''angular form'' of venn diagram for universes of 1 and 2 dimensions, indicating the embedding map of type '''B'''<sup>1</sup>&nbsp;&rarr;&nbsp;'''B'''<sup>2</sup> and detailing the coordinates that are associated with individual cells. Because all points, cells, or logical interpretations are represented as connected geometric areas, we can say that these pictures provide us with an ''areal view'' of each universe of discourse.<br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 18-a -- Extension from 1 to 2 Dimensions.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 18-a. Extension from 1 to 2 Dimensions: Areal'''</font></center></p><br />
<br />
Figure 18-b shows the differential extension from ''X''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;[''x''] to E''X''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;[''x'',&nbsp;d''x''] in a ''bundle of boxes'' form of venn diagram. As awkward as it may seem at first, this type of picture is often the most natural and the most easily available representation when we want to conceptualize the localized information or momentary knowledge of an intelligent dynamic system. It gives a ready picture of a ''proposition at a point'', in the present instance, of a proposition about changing states which is itself associated with a particular dynamic state of a system. It is easy to see how this application might be extended to conceive of more general types of instantaneous knowledge that are possessed by a system.<br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 18-b -- Extension from 1 to 2 Dimensions.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 18-b. Extension from 1 to 2 Dimensions: Bundle'''</font></center></p><br />
<br />
Figure 18-c shows the same extension in a ''compact'' style of venn diagram, where the differential features at each position are represented by arrows extending from that position that cross or do not cross, as the case may be, the corresponding feature boundaries.<br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 18-c -- Extension from 1 to 2 Dimensions.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 18-c. Extension from 1 to 2 Dimensions: Compact'''</font></center></p><br />
<br />
Figure 18-d compresses the picture of the differential extension even further, yielding a directed graph or ''digraph'' form of representation. (Notice that my definition of a digraph allows for loops or ''slings'' at individual points, in addition to arcs or ''arrows'' between the points.)<br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 18-d -- Extension from 1 to 2 Dimensions.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 18-d. Extension from 1 to 2 Dimensions: Digraph'''</font></center></p><br />
<br />
====Extension from 2 to 4 Dimensions====<br />
<br />
Figure 19-a lays out the ''areal view'' or the ''angular form'' of venn diagram for universes of 2 and 4 dimensions, indicating the embedding map of type '''B'''<sup>2</sup>&nbsp;&rarr;&nbsp;'''B'''<sup>4</sup>. In many ways these pictures are the best kind there is, giving full canvass to an ideal vista. Their style allows the clearest, the fairest, and the plainest view that we can form of a universe of discourse, affording equal representation to all dispositions and maintaining a balance with respect to ordinary and differential features. If only we could extend this view! Unluckily, an obvious difficulty beclouds this prospect, and that is how precipitately we run into the limits of our plane and visual intuitions. Even within the scope of the spare few dimensions that we have scanned up to this point subtle discrepancies have crept in already. The circumstances that bind us and the frameworks that block us, the flat distortion of the planar projection and the inevitable ineffability that precludes us from wrapping its rhomb figure into rings around a torus, all of these factors disguise the underlying but true connectivity of the universe of discourse.<br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 19-a -- Extension from 2 to 4 Dimensions.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 19-a. Extension from 2 to 4 Dimensions: Areal'''</font></center></p><br />
<br />
Figure 19-b shows the differential extension from ''U''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;[''u'',&nbsp;''v''] to E''U''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;[''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''] in the ''bundle of boxes'' form of venn diagram.<br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 19-b -- Extension from 2 to 4 Dimensions.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 19-b. Extension from 2 to 4 Dimensions: Bundle'''</font></center></p><br />
<br />
As dimensions increase, this factorization of the extended universe along the lines that are marked out by the bundle picture begins to look more and more like a practical necessity. But whenever we use a propositional model to address a real situation in the context of nature we need to remain aware that this articulation into factors, affecting our description, may be wholly artificial in nature and cleave to nothing, no joint in nature, nor any juncture in time to be in or out of joint.<br />
<br />
Figure 19-c illustrates the extension from 2 to 4 dimensions in the ''compact'' style of venn diagram. Here, just the changes with respect to the center cell are shown.<br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 19-c -- Extension from 2 to 4 Dimensions.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 19-c. Extension from 2 to 4 Dimensions: Compact'''</font></center></p><br />
<br />
Figure 19-d gives the ''digraph'' form of representation for the differential extension ''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''U''<sup>&nbsp;&bull;</sup>, where the 4 nodes marked "@" are the cells ''uv'', ''u''(''v''), (''u'')''v'', (''u'')(''v''), respectively, and where a 2-headed arc counts as two arcs of the differential digraph.<br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 19-d -- Extension from 2 to 4 Dimensions.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 19-d. Extension from 2 to 4 Dimensions: Digraph'''</font></center></p><br />
<br />
===Thematization of Functions : And a Declaration of Independence for Variables===<br />
<br />
{| width="100%"<br />
| align="left" |<br />
''And as imagination bodies forth''<br><br />
''The forms of things unknown, the poet's pen''<br><br />
''Turns them to shapes, and gives to airy nothing''<br><br />
''A local habitation and a name.''<br />
| align="right" valign="bottom" | A Midsummer Night's Dream, 5.1.18<br />
|}<br />
<br />
In the representation of propositions as functions it is possible to notice different degrees of explicitness in the way their functional character is symbolized. To indicate what I mean by this, the next series of Figures illustrates a set of graphic conventions that will be put to frequent use in the remainder of this discussion, both to mark the relevant distinctions and to help us convert between related expressions at different levels of explicitness in their functionality.<br />
<br />
====Thematization : Venn Diagrams====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
The known universe has one complete lover and that is the greatest poet. He consumes an eternal passion and is indifferent which chance happens and which possible contingency of fortune or misfortune and persuades daily and hourly his delicious pay.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 11&ndash;12]<br />
|}<br />
<br />
Figure 20-i traces the first couple of steps in this order of ''thematic'' progression, that will gradually run the gamut through a complete series of degrees of functional explicitness in the expression of logical propositions. The first venn diagram represents a situation where the function is indicated by a shaded figure and a logical expression. At this stage one may be thinking of the proposition only as expressed by a formula in a particular language and its content only as a subset of the universe of discourse, as when one considers the proposition ''u''<b>·</b>''v'' in [''u'',&nbsp;''v''].<br />
<br />
The second venn diagram depicts a situation in which two significant steps have been taken. First, one has taken the trouble to give the proposition ''u''<b>·</b>''v'' a distinctive functional name "''J''&nbsp;". Second, one has come to think explicitly about the target domain that contains the functional values of ''J'', as when one writes ''J''&nbsp;:&nbsp;〈''u'',&nbsp;''v''〉&nbsp;&rarr;&nbsp;'''B'''.<br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 20-i -- Thematization of Conjunction (Stage 1).gif|center]]</p><br />
<p><center><font size="+1">'''Figure 20-i. Thematization of Conjunction (Stage 1)'''</font></center></p><br />
<br />
In Figure 20-ii the proposition ''J'' is viewed explicitly as a transformation from one universe of discourse to another.<br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 20-ii -- Thematization of Conjunction (Stage 2).gif|center]]</p><br />
<p><center><font size="+1">'''Figure 20-ii. Thematization of Conjunction (Stage 2)'''</font></center></p><br />
<br />
<pre><br />
o-------------------------------o o-------------------------------o<br />
| | | |<br />
| o-----o o-----o | | o-----o o-----o |<br />
| / \ / \ | | / \ / \ |<br />
| / o \ | | / o \ |<br />
| / /`\ \ | | / /`\ \ |<br />
| o o```o o | | o o```o o |<br />
| | u |```| v | | | | u |```| v | |<br />
| o o```o o | | o o```o o |<br />
| \ \`/ / | | \ \`/ / |<br />
| \ o / | | \ o / |<br />
| \ / \ / | | \ / \ / |<br />
| o-----o o-----o | | o-----o o-----o |<br />
| | | |<br />
o-------------------------------o o-------------------------------o<br />
\ / \ /<br />
\ / \ /<br />
\ / \ J /<br />
\ / \ /<br />
\ / \ /<br />
o----------\---------/----------o o----------\---------/----------o<br />
| \ / | | \ / |<br />
| \ / | | \ / |<br />
| o-----@-----o | | o-----@-----o |<br />
| /`````````````\ | | /`````````````\ |<br />
| /```````````````\ | | /```````````````\ |<br />
| /`````````````````\ | | /`````````````````\ |<br />
| o```````````````````o | | o```````````````````o |<br />
| |```````````````````| | | |```````````````````| |<br />
| |```````` J ````````| | | |```````` x ````````| |<br />
| |```````````````````| | | |```````````````````| |<br />
| o```````````````````o | | o```````````````````o |<br />
| \`````````````````/ | | \`````````````````/ |<br />
| \```````````````/ | | \```````````````/ |<br />
| \`````````````/ | | \`````````````/ |<br />
| o-----------o | | o-----------o |<br />
| | | |<br />
| | | |<br />
o-------------------------------o o-------------------------------o<br />
J = u v x = J<u, v><br />
<br />
Figure 20-ii. Thematization of Conjunction (Stage 2)<br />
</pre><br />
<br />
In the first venn diagram the name that is assigned to a composite proposition, function, or region in the source universe is delegated to a simple feature in the target universe. This can result in a single character or term exceeding the responsibilities it can carry off well. Allowing the name of a function ''J''&nbsp;:&nbsp;〈''u'',&nbsp;''v''〉&nbsp;&rarr;&nbsp;'''B''' to serve as the name of its dependent variable ''J''&nbsp;:&nbsp;'''B''' does not mean that one has to confuse a function with any of its values, but it does put one at risk for a number of obvious problems, and we should not be surprised, on numerous and limiting occasions, when quibbling arises from the attempts of a too original syntax to serve these two masters.<br />
<br />
The second venn diagram circumvents these difficulties by introducing a new variable name for each basic feature of the target universe, as when one writes ''J''&nbsp;:&nbsp;〈''u'',&nbsp;''v''〉&nbsp;&rarr;&nbsp;〈''x''〉 and thereby assigns a concrete type 〈''x''〉 to the abstract codomain '''B'''. To make this induction of variables more formal one can append subscripts, as in ''x''<sub>''J''</sub>, to indicate the origin or the derivation of these parvenu characters. However, it is not always convenient to keep inventing new variable names in this way. For use at these times, I introduce a lexical operator "¢", read ''cents'' or ''obelus'', that converts a function name into a variable name. For example, one may think of ''x'' = ''x''<sub>''J''</sub> = ¢(''J'') = ''J''&nbsp;¢ = ''J''<sup>&nbsp;¢</sup> as "the cache variable of ''J''&nbsp;", "''J'' circumscript", "''J'' made circumstantial", or "''J'' considered as a contingent variable".<br />
<br />
In Figure 20-iii we arrive at a stage where the functional equations, ''J''&nbsp;=&nbsp;''u''<b>·</b>''v'' and ''x''&nbsp;=&nbsp;''u''<b>·</b>''v'', are regarded as propositions in their own right, reigning in and ruling over 3-feature universes of discourse, [''u'',&nbsp;''v'',&nbsp;''J''] and [''u'',&nbsp;''v'',&nbsp;''x''], respectively. Subject to the cautions already noted, the function name "''J''&nbsp;" can be reinterpreted as the name of a feature ''J''<sup>&nbsp;¢</sup>, and the equation ''J''&nbsp;=&nbsp;''u''<b>·</b>''v'' can be read as the logical equivalence ((''J'',&nbsp;''u''&nbsp;''v'')). To give it a generic name let us call this newly expressed, collateral proposition the ''thematization'' or the ''thematic extension'' of the original proposition ''J''.<br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 20-iii -- Thematization of Conjunction (Stage 3).gif|center]]</p><br />
<p><center><font size="+1">'''Figure 20-iii. Thematization of Conjunction (Stage 3)'''</font></center></p><br />
<br />
The first venn diagram represents the thematization of the conjunction ''J'' with shading in the appropriate regions of the universe [''u'',&nbsp;''v'',&nbsp;''J'']. Also, it illustrates a quick way of constructing a thematic extension. First, draw a line, in practice or the imagination, that bisects every cell of the original universe, placing half of each cell under the aegis of the thematized proposition and the other half under its antithesis. Next, on the scene where the theme applies leave the shade wherever it lies, and off the stage, where it plays otherwise, stagger the pattern in a harlequin guise.<br />
<br />
In the final venn diagram of this sequence the thematic progression comes full circle and completes one round of its development. The ambiguities that were occasioned by the changing role of the name "''J''&nbsp;" are resolved by introducing a new variable name "''x''&nbsp;" to take the place of ''J''<sup>&nbsp;¢</sup>, and the region that represents this fresh featured ''x'' is circumscribed in a more conventional symmetry of form and placement. Just as we once gave the name "''J''&nbsp;" to the proposition ''u''<b>·</b>''v'', we now give the name "&iota;" to its thematization ((''x'',&nbsp;''u''&nbsp;''v'')). Already, again, at this culminating stage of reflection, we begin to think of the newly named proposition as a distinctive individual, a particular function &iota;&nbsp;:&nbsp;〈''u'',&nbsp;''v'',&nbsp;''x''〉&nbsp;&rarr;&nbsp;'''B'''.<br />
<br />
From now on, the terms ''thematic extension'' and ''thematization'' will be used to describe both the process and degree of explication that progresses through this series of pictures, both the operation of increasingly explicit symbolization and the dimension of variation that is swept out by it. To speak of this change in general, that takes us in our current example from ''J'' to &iota;, I introduce a class of operators symbolized by the Greek letter &theta;, writing &iota; = &theta;''J'' in the present instance. The operator &theta;, in the present situation bearing the type &theta;&nbsp;:&nbsp;[''u'',&nbsp;''v'']&nbsp;&rarr;>&nbsp;[''u'',&nbsp;''v'',&nbsp;''x''], provides us with a convenient way of recapitulating and summarizing the complete cycle of thematic developments.<br />
<br />
Figure 21 shows how the thematic extension operator &theta; acts on two further examples, the disjunction ((''u'')(''v'')) and the equality ((''u'',&nbsp;''v'')). Referring to the disjunction as ''f''‹''u'',&nbsp;''v''› and the equality as ''g''‹''u'',&nbsp;''v''›, I write the thematic extensions as &phi; = &theta;''f'' and &gamma; = &theta;''g''.<br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 21 -- Thematization of Disjunction and Equality.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 21. Thematization of Disjunction and Equality'''</font></center></p><br />
<br />
====Thematization : Truth Tables====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
That which distorts honest shapes or which creates unearthly beings or places or contingencies is a nuisance and a revolt.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 19]<br />
|}<br />
<br />
Tables 22 through 25 outline a method for computing the thematic extensions of propositions in terms of their coordinate values.<br />
<br />
A preliminary step, as illustrated in Table 22, is to write out the truth table representations of the propositional forms whose thematic extensions one wants to compute, in the present instance, the functions ''f''‹''u'',&nbsp;''v''›&nbsp;=&nbsp;((''u'')(''v'')) and ''g''‹''u'',&nbsp;''v''›&nbsp;=&nbsp;((''u'',&nbsp;''v'')).<br />
<br />
<br><font face="courier new"><br />
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"<br />
|+ '''Table 22. Disjunction ''f'' and Equality ''g'' '''<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| ''u'' || ''v''<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| ''f'' || ''g''<br />
|}<br />
|-<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|-<br />
| 0 || 1<br />
|-<br />
| 1 || 0<br />
|-<br />
| 1 || 1<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 1<br />
|-<br />
| 1 || 0<br />
|-<br />
| 1 || 0<br />
|-<br />
| 1 || 1<br />
|}<br />
|}<br />
</font><br><br />
<br />
Next, each propositional form is individually represented in the fashion shown in Tables&nbsp;23-i and 23-ii, using "''f''&nbsp;" and "''g''&nbsp;" as function names and creating new variables ''x'' and ''y'' to hold the associated functional values. This pair of Tables outlines the first stage in the transition from the 2-dimensional universes of ''f'' and ''g'' to the 3-dimensional universes of &theta;''f'' and &theta;''g''. The top halves of the Tables replicate the truth table patterns for ''f'' and ''g'' in the form ''f''&nbsp;:&nbsp;[''u'',&nbsp;''v'']&nbsp;&rarr;&nbsp;[''x''] and ''g''&nbsp;:&nbsp;[''u'',&nbsp;''v'']&nbsp;&rarr;&nbsp;[''y'']. The bottom halves of the tables print the negatives of these pictures, as it were, and paste the truth tables for (''f'') and (''g'') under the copies for ''f'' and ''g''. At this stage, the columns for &theta;''f'' and &theta;''g'' are appended almost as afterthoughts, amounting to indicator functions for the sets of ordered triples that make up the functions ''f'' and ''g''.<br />
<br />
<br><br />
{| align="center" style="width:96%"<br />
|+ '''Tables 23-i and 23-ii. Thematics of Disjunction and Equality (1)'''<br />
|<br />
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"<br />
|+ '''Table 23-i. Disjunction ''f'' '''<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| ''u'' || ''v'' || ''f''<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| ''x'' || &phi;<br />
|}<br />
|-<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0 || &rarr;<br />
|-<br />
| 0 || 1 || &rarr;<br />
|-<br />
| 1 || 0 || &rarr;<br />
|-<br />
| 1 || 1 || &rarr;<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 1<br />
|-<br />
| 1 || 1<br />
|-<br />
| 1 || 1<br />
|-<br />
| 1 || 1<br />
|}<br />
|-<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0 || &nbsp;&nbsp;<br />
|-<br />
| 0 || 1 || &nbsp;&nbsp;<br />
|-<br />
| 1 || 0 || &nbsp;&nbsp;<br />
|-<br />
| 1 || 1 || &nbsp;&nbsp;<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 1 || 0<br />
|-<br />
| 0 || 0<br />
|-<br />
| 0 || 0<br />
|-<br />
| 0 || 0<br />
|}<br />
|}<br />
|<br />
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"<br />
|+ '''Table 23-ii. Equality ''g'' '''<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| ''u'' || ''v'' || ''g''<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| ''y'' || &gamma;<br />
|}<br />
|-<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0 || &rarr;<br />
|-<br />
| 0 || 1 || &rarr;<br />
|-<br />
| 1 || 0 || &rarr;<br />
|-<br />
| 1 || 1 || &rarr;<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 1 || 1<br />
|-<br />
| 0 || 1<br />
|-<br />
| 0 || 1<br />
|-<br />
| 1 || 1<br />
|}<br />
|-<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0 || &nbsp;&nbsp;<br />
|-<br />
| 0 || 1 || &nbsp;&nbsp;<br />
|-<br />
| 1 || 0 || &nbsp;&nbsp;<br />
|-<br />
| 1 || 1 || &nbsp;&nbsp;<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|-<br />
| 1 || 0<br />
|-<br />
| 1 || 0<br />
|-<br />
| 0 || 0<br />
|}<br />
|}<br />
|}<br />
<br><br />
<br />
All the data are now in place to give the truth tables for &theta;''f'' and &theta;''g''. In the remaining steps all we do is to permute the rows and change the roles of ''x'' and ''y'' from dependent to independent variables. In Tables&nbsp;24-i and 24-ii the rows are arranged in such a way as to put the 3-tuples ‹''u'',&nbsp;''v'',&nbsp;''x''› and ‹''u'',&nbsp;''v'',&nbsp;''y''› in binary numerical order, suitable for viewing as the arguments of the maps &theta;''f''&nbsp;=&nbsp;&phi;&nbsp;:&nbsp;[''u'',&nbsp;''v'',&nbsp;''x'']&nbsp;&rarr;&nbsp;'''B''' and &theta;''g''&nbsp;=&nbsp;&gamma;&nbsp;:&nbsp;[''u'',&nbsp;''v'',&nbsp;''y'']&nbsp;&rarr;&nbsp;'''B'''. Moreover, the structure of the tables is altered slightly, allowing the now vestigial functions ''f'' and ''g'' to be passed over without further attention and shifting the heavy vertical bars a notch to the right. In effect, this clinches the fact that the thematic variables ''x''&nbsp;:=&nbsp;''f''<sup>&nbsp;¢</sup> and ''y''&nbsp;:=&nbsp;''g''<sup>&nbsp;¢</sup> are now to be regarded as independent variables.<br />
<br />
<br><br />
{| align="center" style="width:96%"<br />
|+ '''Tables 24-i and 24-ii. Thematics of Disjunction and Equality (2)'''<br />
|<br />
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"<br />
|+ '''Table 24-i. Disjunction ''f'' '''<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| ''u'' || ''v'' || ''f'' || ''x''<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| &phi;<br />
|}<br />
|-<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0 || &rarr; || 0<br />
|-<br />
| 0 || 0 || &nbsp;&nbsp; || 1<br />
|-<br />
| 0 || 1 || &nbsp;&nbsp; || 0<br />
|-<br />
| 0 || 1 || &rarr; || 1<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 1<br />
|-<br />
| 0<br />
|-<br />
| 0<br />
|-<br />
| 1<br />
|}<br />
|-<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 1 || 0 || &nbsp;&nbsp; || 0<br />
|-<br />
| 1 || 0 || &rarr; || 1<br />
|-<br />
| 1 || 1 || &nbsp;&nbsp; || 0<br />
|-<br />
| 1 || 1 || &rarr; || 1<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0<br />
|-<br />
| 1<br />
|-<br />
| 0<br />
|-<br />
| 1<br />
|}<br />
|}<br />
|<br />
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"<br />
|+ '''Table 24-ii. Equality ''g'' '''<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| ''u'' || ''v'' || ''g'' || ''y''<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| &gamma;<br />
|}<br />
|-<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0 || &nbsp;&nbsp; || 0<br />
|-<br />
| 0 || 0 || &rarr; || 1<br />
|-<br />
| 0 || 1 || &rarr; || 0<br />
|-<br />
| 0 || 1 || &nbsp;&nbsp; || 1<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0<br />
|-<br />
| 1<br />
|-<br />
| 1<br />
|-<br />
| 0<br />
|}<br />
|-<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 1 || 0 || &rarr; || 0<br />
|-<br />
| 1 || 0 || &nbsp;&nbsp; || 1<br />
|-<br />
| 1 || 1 || &nbsp;&nbsp; || 0<br />
|-<br />
| 1 || 1 || &rarr; || 1<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 1<br />
|-<br />
| 0<br />
|-<br />
| 0<br />
|-<br />
| 1<br />
|}<br />
|}<br />
|}<br />
<br><br />
<br />
An optional reshuffling of the rows brings additional features of the thematic extensions to light. Leaving the columns in place for the sake of comparison, Tables&nbsp;25-i and 25-ii sort the rows in a different order, in effect treating ''x'' and ''y'' as the primary variables in their respective 3-tuples. Regarding the thematic extensions in the form &phi;&nbsp;:&nbsp;[''x'',&nbsp;''u'',&nbsp;''v'']&nbsp;&rarr;&nbsp;'''B''' and &gamma;&nbsp;:&nbsp;[''y'',&nbsp;''u'',&nbsp;''v'']&nbsp;&rarr;&nbsp;'''B''' makes it easier to see in this tabular setting a property that was graphically obvious in the venn diagrams above. Specifically, when the thematic variable ''F''<sup>&nbsp;¢</sup> is true then &theta;''F'' exhibits the pattern of the original ''F'', and when ''F''<sup>&nbsp;¢</sup> is false then &theta;''F'' exhibits the pattern of its negation (''F'').<br />
<br />
<br><br />
{| align="center" style="width:96%"<br />
|+ '''Tables 25-i and 25-ii. Thematics of Disjunction and Equality (3)'''<br />
|<br />
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"<br />
|+ '''Table 25-i. Disjunction ''f'' '''<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| ''u'' || ''v'' || ''f'' || ''x''<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| &phi;<br />
|}<br />
|-<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0 || &rarr; || 0<br />
|-<br />
| 0 || 1 || &nbsp;&nbsp; || 0<br />
|-<br />
| 1 || 0 || &nbsp;&nbsp; || 0<br />
|-<br />
| 1 || 1 || &nbsp;&nbsp; || 0<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 1<br />
|-<br />
| 0<br />
|-<br />
| 0<br />
|-<br />
| 0<br />
|}<br />
|-<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0 || &nbsp;&nbsp; || 1<br />
|-<br />
| 0 || 1 || &rarr; || 1<br />
|-<br />
| 1 || 0 || &rarr; || 1<br />
|-<br />
| 1 || 1 || &rarr; || 1<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0<br />
|-<br />
| 1<br />
|-<br />
| 1<br />
|-<br />
| 1<br />
|}<br />
|}<br />
|<br />
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"<br />
|+ '''Table 25-ii. Equality ''g'' '''<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| ''u'' || ''v'' || ''g'' || ''y''<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| &gamma;<br />
|}<br />
|-<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0 || &nbsp;&nbsp; || 0<br />
|-<br />
| 0 || 1 || &rarr; || 0<br />
|-<br />
| 1 || 0 || &rarr; || 0<br />
|-<br />
| 1 || 1 || &nbsp;&nbsp; || 0<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0<br />
|-<br />
| 1<br />
|-<br />
| 1<br />
|-<br />
| 0<br />
|}<br />
|-<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0 || &rarr; || 1<br />
|-<br />
| 0 || 1 || &nbsp;&nbsp; || 1<br />
|-<br />
| 1 || 0 || &nbsp;&nbsp; || 1<br />
|-<br />
| 1 || 1 || &rarr; || 1<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 1<br />
|-<br />
| 0<br />
|-<br />
| 0<br />
|-<br />
| 1<br />
|}<br />
|}<br />
|}<br />
<br><br />
<br />
Finally, Tables&nbsp;26-i and 26-ii compare the tacit extensions <math>\epsilon</math>&nbsp;:&nbsp;[''u'',&nbsp;''v'']&nbsp;&rarr;&nbsp;[''u'',&nbsp;''v'',&nbsp;''x''] and <math>\epsilon</math>&nbsp;:&nbsp;[''u'',&nbsp;''v'']&nbsp;&rarr;&nbsp;[''u'',&nbsp;''v'',&nbsp;''y''] with the thematic extensions of the same types, as applied to the propositions ''f'' and ''g'', respectively.<br />
<br />
<br><br />
{| align="center" style="width:96%"<br />
|+ '''Tables 26-i and 26-ii. Tacit Extension and Thematization'''<br />
|<br />
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"<br />
|+ '''Table 26-i. Disjunction ''f'' '''<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| ''u'' || ''v'' || ''x''<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| &epsilon;''f'' || &theta;''f''<br />
|}<br />
|-<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0 || 0<br />
|-<br />
| 0 || 0 || 1<br />
|-<br />
| 0 || 1 || 0<br />
|-<br />
| 0 || 1 || 1<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 1<br />
|-<br />
| 0 || 0<br />
|-<br />
| 1 || 0<br />
|-<br />
| 1 || 1<br />
|}<br />
|-<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 1 || 0 || 0<br />
|-<br />
| 1 || 0 || 1<br />
|-<br />
| 1 || 1 || 0<br />
|-<br />
| 1 || 1 || 1<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 1 || 0<br />
|-<br />
| 1 || 1<br />
|-<br />
| 1 || 0<br />
|-<br />
| 1 || 1<br />
|}<br />
|}<br />
|<br />
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"<br />
|+ '''Table 26-ii. Equality ''g'' '''<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| ''u'' || ''v'' || ''y''<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| &epsilon;''g'' || &theta;''g''<br />
|}<br />
|-<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0 || 0<br />
|-<br />
| 0 || 0 || 1<br />
|-<br />
| 0 || 1 || 0<br />
|-<br />
| 0 || 1 || 1<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 1 || 0<br />
|-<br />
| 1 || 1<br />
|-<br />
| 0 || 1<br />
|-<br />
| 0 || 0<br />
|}<br />
|-<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 1 || 0 || 0<br />
|-<br />
| 1 || 0 || 1<br />
|-<br />
| 1 || 1 || 0<br />
|-<br />
| 1 || 1 || 1<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 1<br />
|-<br />
| 0 || 0<br />
|-<br />
| 1 || 0<br />
|-<br />
| 1 || 1<br />
|}<br />
|}<br />
|}<br />
<br><br />
<br />
Table 27 summarizes the thematic extensions of all propositions on two variables. Column 4 lists the equations of form ((&nbsp;''f''<sup>&nbsp;¢</sup>&nbsp;,&nbsp;''f''<sup>&nbsp;¢</sup>‹''u'',&nbsp;''v''›&nbsp;)) and Column 5 simplifies these equations into the form of algebraic expressions. (As always, "+" refers to exclusive disjunction, and "''f''&nbsp;" should be read as "''f''<sub>''i''</sub><sup>¢</sup>" in the body of the Table.)<br />
<br />
<br><font face="courier new"><br />
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"<br />
|+ Table 27. Thematization of Bivariate Propositions<br />
|- style="background:paleturquoise"<br />
|<br />
{| align="right" style="background:paleturquoise; text-align:right"<br />
| u :<br />
|-<br />
| v :<br />
|}<br />
|<br />
{| style="background:paleturquoise"<br />
| 1100<br />
|-<br />
| 1010<br />
|}<br />
|<br />
{| style="background:paleturquoise"<br />
| f<br />
|-<br />
| &nbsp;<br />
|}<br />
|<br />
{| style="background:paleturquoise"<br />
| &theta;f<br />
|-<br />
| &nbsp;<br />
|}<br />
|<br />
{| style="background:paleturquoise"<br />
| &theta;f<br />
|-<br />
| &nbsp;<br />
|}<br />
|-<br />
|<br />
{| cellpadding="2" style="background:lightcyan"<br />
| f<sub>0</sub><br />
|-<br />
| f<sub>1</sub><br />
|-<br />
| f<sub>2</sub><br />
|-<br />
| f<sub>3</sub><br />
|-<br />
| f<sub>4</sub><br />
|-<br />
| f<sub>5</sub><br />
|-<br />
| f<sub>6</sub><br />
|-<br />
| f<sub>7</sub><br />
|}<br />
|<br />
{| cellpadding="2" style="background:lightcyan"<br />
| 0000<br />
|-<br />
| 0001<br />
|-<br />
| 0010<br />
|-<br />
| 0011<br />
|-<br />
| 0100<br />
|-<br />
| 0101<br />
|-<br />
| 0110<br />
|-<br />
| 0111<br />
|}<br />
|<br />
{| cellpadding="2" style="background:lightcyan"<br />
| ()<br />
|-<br />
| &nbsp;(u)(v)&nbsp;<br />
|-<br />
| &nbsp;(u)&nbsp;v&nbsp;&nbsp;<br />
|-<br />
| &nbsp;(u)&nbsp;&nbsp;&nbsp;&nbsp;<br />
|-<br />
| &nbsp;&nbsp;u&nbsp;(v)&nbsp;<br />
|-<br />
| &nbsp;&nbsp;&nbsp;&nbsp;(v)&nbsp;<br />
|-<br />
| &nbsp;(u,&nbsp;v)&nbsp;<br />
|-<br />
| &nbsp;(u&nbsp;&nbsp;v)&nbsp;<br />
|}<br />
|<br />
{| cellpadding="2" style="background:lightcyan"<br />
| ((&nbsp;f&nbsp;,&nbsp;&nbsp;&nbsp;&nbsp;()&nbsp;&nbsp;&nbsp;&nbsp;))<br />
|-<br />
| ((&nbsp;f&nbsp;,&nbsp;&nbsp;(u)(v)&nbsp;&nbsp;))<br />
|-<br />
| ((&nbsp;f&nbsp;,&nbsp;&nbsp;(u)&nbsp;v&nbsp;&nbsp;&nbsp;))<br />
|-<br />
| ((&nbsp;f&nbsp;,&nbsp;&nbsp;(u)&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;))<br />
|-<br />
| ((&nbsp;f&nbsp;,&nbsp;&nbsp;&nbsp;u&nbsp;(v)&nbsp;&nbsp;))<br />
|-<br />
| ((&nbsp;f&nbsp;,&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(v)&nbsp;&nbsp;))<br />
|-<br />
| ((&nbsp;f&nbsp;,&nbsp;&nbsp;(u,&nbsp;v)&nbsp;&nbsp;))<br />
|-<br />
| ((&nbsp;f&nbsp;,&nbsp;&nbsp;(u&nbsp;&nbsp;v)&nbsp;&nbsp;))<br />
|}<br />
|<br />
{| align="left" cellpadding="2" style="background:lightcyan; text-align:left"<br />
| &nbsp;f + 1<br />
|-<br />
| &nbsp;f + u + v + uv<br />
|-<br />
| &nbsp;f + v + uv + 1<br />
|-<br />
| &nbsp;f + u<br />
|-<br />
| &nbsp;f + u + uv + 1<br />
|-<br />
| &nbsp;f + v<br />
|-<br />
| &nbsp;f + u + v + 1<br />
|-<br />
| &nbsp;f + uv<br />
|}<br />
|-<br />
|<br />
{| cellpadding="2" style="background:lightcyan"<br />
| f<sub>8</sub><br />
|-<br />
| f<sub>9</sub><br />
|-<br />
| f<sub>10</sub><br />
|-<br />
| f<sub>11</sub><br />
|-<br />
| f<sub>12</sub><br />
|-<br />
| f<sub>13</sub><br />
|-<br />
| f<sub>14</sub><br />
|-<br />
| f<sub>15</sub><br />
|}<br />
|<br />
{| cellpadding="2" style="background:lightcyan"<br />
| 1000<br />
|-<br />
| 1001<br />
|-<br />
| 1010<br />
|-<br />
| 1011<br />
|-<br />
| 1100<br />
|-<br />
| 1101<br />
|-<br />
| 1110<br />
|-<br />
| 1111<br />
|}<br />
|<br />
{| cellpadding="2" style="background:lightcyan"<br />
| &nbsp;&nbsp;u&nbsp;&nbsp;v&nbsp;&nbsp;<br />
|-<br />
| ((u,&nbsp;v))<br />
|-<br />
| &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;v&nbsp;&nbsp;<br />
|-<br />
| &nbsp;(u&nbsp;(v))<br />
|-<br />
| &nbsp;&nbsp;u&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br />
|-<br />
| ((u)&nbsp;v)&nbsp;<br />
|-<br />
| ((u)(v))<br />
|-<br />
| (())<br />
|}<br />
|<br />
{| cellpadding="2" style="background:lightcyan"<br />
| ((&nbsp;f&nbsp;,&nbsp;&nbsp;&nbsp;u&nbsp;&nbsp;v&nbsp;&nbsp;&nbsp;))<br />
|-<br />
| ((&nbsp;f&nbsp;,&nbsp;((u,&nbsp;v))&nbsp;))<br />
|-<br />
| ((&nbsp;f&nbsp;,&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;v&nbsp;&nbsp;&nbsp;))<br />
|-<br />
| ((&nbsp;f&nbsp;,&nbsp;&nbsp;(u&nbsp;(v))&nbsp;))<br />
|-<br />
| ((&nbsp;f&nbsp;,&nbsp;&nbsp;&nbsp;u&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;))<br />
|-<br />
| ((&nbsp;f&nbsp;,&nbsp;((u)&nbsp;v)&nbsp;&nbsp;))<br />
|-<br />
| ((&nbsp;f&nbsp;,&nbsp;((u)(v))&nbsp;))<br />
|-<br />
| ((&nbsp;f&nbsp;,&nbsp;&nbsp;&nbsp;(())&nbsp;&nbsp;&nbsp;))<br />
|}<br />
|<br />
{| align="left" cellpadding="2" style="background:lightcyan; text-align:left"<br />
| &nbsp;f + uv + 1<br />
|-<br />
| &nbsp;f + u + v<br />
|-<br />
| &nbsp;f + v + 1<br />
|-<br />
| &nbsp;f + u + uv<br />
|-<br />
| &nbsp;f + u + 1<br />
|-<br />
| &nbsp;f + v + uv<br />
|-<br />
| &nbsp;f + u + v + uv + 1<br />
|-<br />
| &nbsp;f<br />
|}<br />
|}<br />
</font><br><br />
<br />
In order to show what all of the thematic extensions from two dimensions to three dimensions look like in terms of coordinates, Tables&nbsp;28 and 29 present ordinary truth tables for the functions ''f''<sub>''i''</sub>&nbsp;:&nbsp;'''B'''<sup>2</sup>&nbsp;&rarr;&nbsp;'''B''' and for the corresponding thematizations &theta;''f''<sub>''i''</sub>&nbsp;=&nbsp;&phi;<sub>''i''</sub>&nbsp;:&nbsp;'''B'''<sup>3</sup>&nbsp;&rarr;&nbsp;'''B'''.<br />
<br />
<br><br />
{| align="center" border="1" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"<br />
|+ Table 28. Propositions on Two Variables<br />
|<br />
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
|- style="background:paleturquoise"<br />
| u || v || &nbsp;<br />
|f<sub>00</sub>||f<sub>01</sub>||f<sub>02</sub>||f<sub>03</sub><br />
|f<sub>04</sub>||f<sub>05</sub>||f<sub>06</sub>||f<sub>07</sub><br />
|f<sub>08</sub>||f<sub>09</sub>||f<sub>10</sub>||f<sub>11</sub><br />
|f<sub>12</sub>||f<sub>13</sub>||f<sub>14</sub>||f<sub>15</sub><br />
|-<br />
| 0 || 0 || &nbsp;<br />
|0||1||0||1||0||1||0||1||0||1||0||1||0||1||0||1<br />
|-<br />
| 0 || 1 || &nbsp;<br />
|0||0||1||1||0||0||1||1||0||0||1||1||0||0||1||1<br />
|-<br />
| 1 || 0 || &nbsp;<br />
|0||0||0||0||1||1||1||1||0||0||0||0||1||1||1||1<br />
|-<br />
| 1 || 1 || &nbsp;<br />
|0||0||0||0||0||0||0||0||1||1||1||1||1||1||1||1<br />
|}<br />
|}<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"<br />
|+ Table 29. Thematic Extensions of Bivariate Propositions<br />
|<br />
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
|- style="background:paleturquoise"<br />
| u || v || f<sup>&cent;</sup><br />
| &phi;<sub>00</sub> || &phi;<sub>01</sub><br />
| &phi;<sub>02</sub> || &phi;<sub>03</sub><br />
| &phi;<sub>04</sub> || &phi;<sub>05</sub><br />
| &phi;<sub>06</sub> || &phi;<sub>07</sub><br />
| &phi;<sub>08</sub> || &phi;<sub>09</sub><br />
| &phi;<sub>10</sub> || &phi;<sub>11</sub><br />
| &phi;<sub>12</sub> || &phi;<sub>13</sub><br />
| &phi;<sub>14</sub> || &phi;<sub>15</sub><br />
|-<br />
| 0 || 0 || 0 ||1||0||1||0||1||0||1||0||1||0||1||0||1||0||1||0<br />
|-<br />
| 0 || 0 || 1 ||0||1||0||1||0||1||0||1||0||1||0||1||0||1||0||1<br />
|-<br />
| 0 || 1 || 0 ||1||1||0||0||1||1||0||0||1||1||0||0||1||1||0||0<br />
|-<br />
| 0 || 1 || 1 ||0||0||1||1||0||0||1||1||0||0||1||1||0||0||1||1<br />
|-<br />
| 1 || 0 || 0 ||1||1||1||1||0||0||0||0||1||1||1||1||0||0||0||0<br />
|-<br />
| 1 || 0 || 1 ||0||0||0||0||1||1||1||1||0||0||0||0||1||1||1||1<br />
|-<br />
| 1 || 1 || 0 ||1||1||1||1||1||1||1||1||0||0||0||0||0||0||0||0<br />
|-<br />
| 1 || 1 || 1 ||0||0||0||0||0||0||0||0||1||1||1||1||1||1||1||1<br />
|}<br />
|}<br />
<br><br />
<br />
===Propositional Transformations===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
If only the word 'artificial' were associated with the idea of ''art'', or expert skill gained through voluntary apprenticeship (instead of suggesting the factitious and unreal), we might say that ''logical'' refers to artificial thought.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 56&ndash;57]<br />
|}<br />
<br />
In this Subdivision I develop a comprehensive set of concepts for dealing with transformations between universes of discourse. In this most general context the source and the target universes of a transformation are allowed to be distinct, but may also be one and the same. When these concepts are applied to dynamic systems one focuses on the important special cases of transformations that map a universe into itself, and transformations of this shape may be interpreted as the state transitions of a discrete dynamical process, as these take place among the myriad ways that a universe of discourse might change, and by that change turn into itself.<br />
<br />
====Alias and Alibi Transformations====<br />
<br />
There are customarily two modes of understanding a transformation, at least, when we try to interpret its relevance to something in reality. A transformation always refers to a changing prospect, to say it in a unified but equivocal way, but this can be taken to mean either a subjective change in the interpreting observer's point of view or an objective change in the systematic subject of discussion. In practice these variant uses of the transformation concept are distinguished in the following terms:<br />
<br />
# A ''perspectival'' or ''alias'' transformation refers to a shift in perspective or a change in language that takes place in the observer's frame of reference.<br />
# A ''transitional'' or ''alibi'' transformation refers to a change of position or an alteration of state that occurs in the object system as it falls under study.<br />
<br />
(For a recent discussion of the alias vs. alibi issue, as it relates to linear transformations in vector spaces and to other issues of an algebraic nature, see [MaB, 256, 582-4].)<br />
<br />
Naturally, when we are concerned with the dynamical properties of a system, the transitional aspect of transformation is the factor that comes to the fore, and this involves us in contemplating all of the ways of changing a universe into itself while remaining under the rule of established dynamical laws. In the prospective application to dynamic systems, and to neural networks viewed in this light, our interest lies chiefly with the transformations of a state space into itself that constitute the state transitions of a discrete dynamic process. Nevertheless, many important properties of these transformations, and some constructions that we need to see most clearly, are independent of the transitional interpretation and are likely to be confounded with irrelevant features if presented first and only in that association.<br />
<br />
In addition, and in partial contrast, intelligent systems are exactly that species of dynamic agents that have the capacity to have a point of view, and we cannot do justice to their peculiar properties without examining their ability to form and transform their own frames of reference in exposure to the elements of their own experience. In this setting, the perspectival aspect of transformation is the facet that shines most brightly, perhaps too often leaving us fascinated with mere glimmerings of its actual potential. It needs to be emphasized that nothing of the ordinary sort needs be moved in carrying out a transformation under the alias interpretation, that it may only involve a change in the forms of address, an amendment of the terms which are customed to approach and fashioned to describe the very same things in the very same world. But again, working within a discipline of realistic computation, we know how formidably complex and resource-consuming such transformations of perspective can be to implement in practice, much less to endow in the self-governed form of a nascently intelligent dynamical system.<br />
<br />
====Transformations of General Type====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
''Es ist passiert'', "it just sort of happened", people said there when other people in other places thought heaven knows what had occurred. It was a peculiar phrase, not known in this sense to the Germans and with no equivalent in other languages, the very breath of it transforming facts and the bludgeonings of fate into something light as eiderdown, as thought itself.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 34]<br />
|}<br />
<br />
Consider the situation illustrated in Figure 30, where the alphabets <font face="lucida calligraphy">U</font>&nbsp;=&nbsp;{''u'',&nbsp;''v''} and <font face="lucida calligraphy">X</font>&nbsp;=&nbsp;{''x'',&nbsp;''y'',&nbsp;''z''} are used to label basic features in two different logical universes, ''U''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;[''u'',&nbsp;''v''] and ''X''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;[''x'',&nbsp;''y'',&nbsp;''z''].<br />
<br />
<pre><br />
o-------------------------------------------------------o<br />
| U |<br />
| |<br />
| o-----------o o-----------o |<br />
| / \ / \ |<br />
| / o \ |<br />
| / / \ \ |<br />
| / / \ \ |<br />
| o o o o |<br />
| | | | | |<br />
| | u | | v | |<br />
| | | | | |<br />
| o o o o |<br />
| \ \ / / |<br />
| \ \ / / |<br />
| \ o / |<br />
| \ / \ / |<br />
| o-----------o o-----------o |<br />
| |<br />
| |<br />
o---------------------------o---------------------------o<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
o-------------------------o o-------------------------o o-------------------------o<br />
| U | | U | | U |<br />
| o---o o---o | | o---o o---o | | o---o o---o |<br />
| / \ / \ | | / \ / \ | | / \ / \ |<br />
| / o \ | | / o \ | | / o \ |<br />
| / / \ \ | | / / \ \ | | / / \ \ |<br />
| o o o o | | o o o o | | o o o o |<br />
| | u | | v | | | | u | | v | | | | u | | v | |<br />
| o o o o | | o o o o | | o o o o |<br />
| \ \ / / | | \ \ / / | | \ \ / / |<br />
| \ o / | | \ o / | | \ o / |<br />
| \ / \ / | | \ / \ / | | \ / \ / |<br />
| o---o o---o | | o---o o---o | | o---o o---o |<br />
| | | | | |<br />
o-------------------------o o-------------------------o o-------------------------o<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ g | \ f / | h /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ o----------|-----------\-----/-----------|----------o /<br />
\ | X | \ / | | /<br />
\ | | \ / | | /<br />
\ | | o-----o-----o | | /<br />
\| | / \ | |/<br />
\ | / \ | /<br />
|\ | / \ | /|<br />
| \ | / \ | / |<br />
| \ | / \ | / |<br />
| \ | o x o | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \| | | |/ |<br />
| o--o--------o o--------o--o |<br />
| / \ \ / / \ |<br />
| / \ \ / / \ |<br />
| / \ o / \ |<br />
| / \ / \ / \ |<br />
| / \ / \ / \ |<br />
| o o--o-----o--o o |<br />
| | | | | |<br />
| | | | | |<br />
| | | | | |<br />
| | y | | z | |<br />
| | | | | |<br />
| | | | | |<br />
| o o o o |<br />
| \ \ / / |<br />
| \ \ / / |<br />
| \ o / |<br />
| \ / \ / |<br />
| \ / \ / |<br />
| o-----------o o-----------o |<br />
| |<br />
| |<br />
o---------------------------------------------------o<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ p , q /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
o<br />
<br />
Figure 30. Generic Frame of a Logical Transformation<br />
</pre><br />
<br />
Enter the picture, as we usually do, in the middle of things, with features like ''x'',&nbsp;''y'',&nbsp;''z'' that present themselves to be simple enough in their own right and that form a satisfactory, if a temporary, foundation to provide a basis for discussion. In this universe and on these terms we find expression for various propositions and questions of principal interest to ourselves, as indicated by the maps ''p'',&nbsp;''q''&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;'''B'''. Then we discover that the simple features {''x'',&nbsp;''y'',&nbsp;''z''} are really more complex than we thought at first, and it becomes useful to regard them as functions {''f'',&nbsp;''g'',&nbsp;''h''} of other features {''u'',&nbsp;''v''}, that we place in a preface to our original discourse, or suppose as topics of a preliminary universe of discourse ''U''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;[''u'',&nbsp;''v'']. It may happen that these late-blooming but pre-ambling features are found to lie closer, in a sense that may be our job to determine, to the central nature of the situation of interest, in which case they earn our regard as being more fundamental, but these functions and features are only required to supply a critical stance on the universe of discourse or an alternate perspective on the nature of things in order to be preserved as useful.<br />
<br />
A particular transformation ''F''&nbsp;:&nbsp;[''u'',&nbsp;''v'']&nbsp;&rarr;&nbsp;[''x'',&nbsp;''y'',&nbsp;''z''] may be expressed by a system of equations, as shown below. Here, ''F'' is defined by its component maps ''F''&nbsp;=&nbsp;‹F<sub>1</sub>,&nbsp;F<sub>2</sub>,&nbsp;F<sub>3</sub>›&nbsp;=&nbsp;‹''f'',&nbsp;''g'',&nbsp;''h''›, where each component map in {''f'',&nbsp;''g'',&nbsp;''h''} is a proposition of type '''B'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''B'''<sup>1</sup>.<br />
<br />
<br><font face="courier new"><br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"<br />
|<br />
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"<br />
| width="20%" | &nbsp;<br />
| width="20%" | ''x''<br />
| width="20%" | =<br />
| width="20%" | ''f''‹''u'', ''v''›<br />
| width="20%" | &nbsp;<br />
|-<br />
| &nbsp; || ''y'' || = || ''g''‹''u'', ''v''› || &nbsp;<br />
|-<br />
| &nbsp; || ''z'' || = || ''h''‹''u'', ''v''› || &nbsp;<br />
|}<br />
|}<br />
</font><br><br />
<br />
Regarded as a logical statement, this system of equations expresses a relation between a collection of freely chosen propositions {''f'',&nbsp;''g'',&nbsp;''h''} in one universe of discourse and the special collection of simple propositions {''x'',&nbsp;''y'',&nbsp;''z''} on which are founded another universe of discourse. Growing familiarity with a particular transformation of discourse, and the desire to achieve a ready understanding of its implications, requires that we be able to convert this information about generals and simples into information about all the main subtypes of propositions, including the linear and singular propositions.<br />
<br />
===Analytic Expansions : Operators and Functors===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Consider what effects that might ''conceivably'' have practical bearings you ''conceive'' the objects of your ''conception'' to have. Then, your ''conception'' of those effects is the whole of your ''conception'' of the object.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; C.S. Peirce, "The Maxim of Pragmatism", CP 5.438<br />
|}<br />
<br />
Given the barest idea of a logical transformation, as suggested by the sketch in Figure&nbsp;30, and having conceptualized the universe of discourse, with all of its points and propositions, as a beginning object of discussion, we are ready to enter the next phase of our investigation.<br />
<br />
====Operators on Propositions and Transformations====<br />
<br />
The next step is naturally inclined toward objects of the next higher order, namely, with operators that take in argument lists of logical transformations and that give back specified types of logical transformations as their results. For our present aims, we do not need to consider the most general class of such operators, nor any one of them for its own sake. Rather, we are interested in the special sorts of operators that arise in the study and analysis of logical transformations. Figuratively speaking, these operators serve as instruments for the live tomography (and hopefully not the vivisection) of the forms of change under view. Beyond that, they open up ways to implement the changes of view that we need to grasp all the variations on a transformational theme, or to appreciate enough of its significant features to "get the drift" of the change occurring, to form a passing acquaintance or a synthetic comprehension of its general character and disposition.<br />
<br />
The simplest type of operator is one that takes a single transformation as an argument and returns a single transformation as a result, and most of the operators that I will explicitly consider here are of this kind. Figure&nbsp;31 illustrates the typical situation.<br />
<br />
<pre><br />
o---------------------------------------o<br />
| |<br />
| |<br />
| U% F X% |<br />
| o------------------>o |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| !W! | | !W! |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| v v |<br />
| o------------------>o |<br />
| !W!U% !W!F !W!X% |<br />
| |<br />
| |<br />
o---------------------------------------o<br />
Figure 31. Operator Diagram (1)<br />
</pre><br />
<br />
In this Figure "<font face=georgia>'''W'''</font>" serves as a generic name for an operator, in this case one that takes a logical transformation ''F'' of type (''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup>) into a logical transformation <font face=georgia>'''W'''</font>''F'' of type (<font face=georgia>'''W'''</font>''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;<font face=georgia>'''W'''</font>''X''<sup>&nbsp;&bull;</sup>). Thus, the operator <font face=georgia>'''W'''</font> must be viewed as making assignments for both families of objects that we have previously considered, both for universes of discourse like ''U''<sup>&nbsp;&bull;</sup> and ''X''<sup>&nbsp;&bull;</sup> and for logical transformations like ''F''.<br />
<br />
NB. Strictly speaking, an operator like <font face=georgia>'''W'''</font> works between two whole categories of universes and transformations, which we call the ''source'' and the ''target'' categories of <font face=georgia>'''W'''</font>. Given this setting, <font face=georgia>'''W'''</font> specifies for each universe ''U''<sup>&nbsp;&bull;</sup> in its source category a definite universe <font face=georgia>'''W'''</font>''U''<sup>&nbsp;&bull;</sup> in its target category, and to each transformation ''F'' in its source category it assigns a unique transformation <font face=georgia>'''W'''</font>''F'' in its target category. Naturally, this only works if <font face=georgia>'''W'''</font> takes the source ''U''<sup>&nbsp;&bull;</sup> and the target ''X''<sup>&nbsp;&bull;</sup> of the map F over to the source <font face=georgia>'''W'''</font>''U''<sup>&nbsp;&bull;</sup> and the target <font face=georgia>'''W'''</font>''X''<sup>&nbsp;&bull;</sup> of the map <font face=georgia>'''W'''</font>''F''. With luck or care enough, we can avoid ever having to put anything like that in words again, letting diagrams do the work. In the situations of present concern we are usually focused on a single transformation ''F'', and thus we can take it for granted that the assignment of universes under <font face=georgia>'''W'''</font> is defined appropriately at the source and the target ends of ''F''. It is not always the case, though, that we need to use the particular names (like "<font face=georgia>'''W'''</font>''U''<sup>&nbsp;&bull;</sup>" and "<font face=georgia>'''W'''</font>''X''<sup>&nbsp;&bull;</sup>") that <font face=georgia>'''W'''</font> assigns by default to its operative image universes. In most contexts we will usually have a prior acquaintance with these universes under other names, and it is only necessary that we can tell from the information associated with an operator <font face=georgia>'''W'''</font> what universes they are.<br />
<br />
In Figure&nbsp;31 the maps ''F'' and <font face=georgia>'''W'''</font>''F'' are displayed horizontally, the way that one normally orients functional arrows in a written text, and <font face=georgia>'''W'''</font> rolls the map ''F'' downward into the images that are associated with <font face=georgia>'''W'''</font>''F''. In Figure&nbsp;32 the same information is redrawn so that the maps ''F'' and <font face=georgia>'''W'''</font>''F'' flow down the page, and <font face=georgia>'''W'''</font> unfurls the map ''F'' rightward into domains that are the eminent purview of <font face=georgia>'''W'''</font>''F''.<br />
<br />
<pre><br />
o---------------------------------------o<br />
| |<br />
| |<br />
| U% !W! !W!U% |<br />
| o------------------>o |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| F | | !W!F |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| v v |<br />
| o------------------>o |<br />
| X% !W! !W!X% |<br />
| |<br />
| |<br />
o---------------------------------------o<br />
Figure 32. Operator Diagram (2)<br />
</pre><br />
<br />
The latter arrangement, as it appears in Figure&nbsp;32, is more congruent with the thinking about operators that we shall be doing in the rest of this discussion, since all logical transformations from here on out will be pictured vertically, after the fashion of Figure&nbsp;30.<br />
<br />
====Differential Analysis of Propositions and Transformations====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
he resultant metaphysical problem now is this: ''Does the man go round the squirrel or not?''<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 43]<br />
|}<br />
<br />
The approach to the differential analysis of logical propositions and transformations of discourse that will be pursued here is carried out in terms of particular operators <font face=georgia>'''W'''</font> that act on propositions ''F'' or on transformations ''F'' to yield the corresponding operator maps <font face=georgia>'''W'''</font>''F''. The operator results then become the subject of a series of further stages of analysis, which take them apart into their propositional components, rendering them as a set of purely logical constituents. After this is done, all the parts are then re-integrated to reconstruct the original object in the light of a more complete understanding, at least in ways that enable one to appreciate certain aspects of it with fresh insight.<br />
<br />
* '''Remark on Strategy.''' At this point I run into a set of conceptual difficulties that force me to make a strategic choice in how I proceed. Part of the problem can be remedied by extending my discussion of tacit extensions to the transformational context. But the troubles that remain are much more obstinate and lead me to try two different types of solution. The approach that I develop first makes use of a variant type of extension operator, the ''trope extension'', to be defined below. This method is more conservative and requires less preparation, but has features which make it seem unsatisfactory in the long run. A more radical approach, but one with a better hope of long term success, makes use of the notion of ''contingency spaces''. These are an even more generous type of extended universe than the kind I currently use, but are defined subject to certain internal constraints. The extra work needed to set up this method forces me to put it off to a later stage. However, as a compromise, and to prepare the ground for the next pass, I call attention to the various conceptual difficulties as they arise along the way and try to give an honest estimate of how well my first approach deals with them.<br />
<br />
I now describe in general terms the particular operators that are instrumental to this form of analysis. The main series of operators all have the form <font face=georgia>'''W'''</font> : (''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup>)&nbsp;&rarr;&nbsp;(E''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''X''<sup>&nbsp;&bull;</sup>). If we assume that the source universe ''U''<sup>&nbsp;&bull;</sup> and the target universe ''X''<sup>&nbsp;&bull;</sup> have finite dimensions ''n'' and ''k'', respectively, then each operator <font face=georgia>'''W'''</font> is encompassed by the same<br />
abstract type:<br />
<br />
:{| cellpadding=1 style="height:40px"<br />
| <font face=georgia>'''W'''</font><br />
| :<br />
| (<br />
| [<br />
| '''B'''<sup>''n''</sup><br />
| ]<br />
| &rarr;<br />
| [<br />
| '''B'''<sup>''k''</sup><br />
| ]<br />
| )<br />
| &nbsp;<br />
| &rarr;<br />
| &nbsp;<br />
| (<br />
| [<br />
| '''B'''<sup>''n''</sup><br />
| &times;<br />
| '''D'''<sup>''n''</sup><br />
| ]<br />
| &rarr;<br />
| [<br />
| '''B'''<sup>''k''</sup><br />
| &times;<br />
| '''D'''<sup>''k''</sup><br />
| ]<br />
| )<br />
| .<br />
|}<br />
<br />
Since the range features of the operator result <font face=georgia>'''W'''</font>''F'' : ['''B'''<sup>''n''</sup> &times; '''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''B'''<sup>''k''</sup> &times; '''D'''<sup>''k''</sup>] can be sorted out by their ordinary versus their differential qualities and the component maps can be examined independently, the complete operator <font face=georgia>'''W'''</font> can be separated accordingly into two components, in the form <font face=georgia>'''W'''</font> = ‹<math>\epsilon</math>,&nbsp;W›. Given a fixed context of source and target universes of discourse, <math>\epsilon</math> is always the same type of operator, a multiple component elaboration of the tacit extension operators that were articulated earlier. In this context <math>\epsilon</math> has the shape:<br />
<br />
:{| style="height:80px; text-align:center; width:90%"<br />
| align=left width=20%| Concrete type<br />
| width=8% | <math>\epsilon</math><br />
| :<br />
| (<br />
| ''U''<sup>&nbsp;&bull;</sup><br />
| &rarr;<br />
| ''X''<sup>&nbsp;&bull;</sup><br />
| )<br />
| width=16% | &rarr;<br />
| (<br />
| E''U''<sup>&nbsp;&bull;</sup><br />
| &rarr;<br />
| ''X''<sup>&nbsp;&bull;</sup><br />
| )<br />
|-<br />
| align=left width=20%| Abstract type<br />
| width=8% | <math>\epsilon</math><br />
| :<br />
| (<br />
| ['''B'''<sup>''n''</sup>]<br />
| &rarr;<br />
| ['''B'''<sup>''k''</sup>]<br />
| )<br />
| width=16% | &rarr;<br />
| (<br />
| ['''B'''<sup>''n''</sup> &times; '''D'''<sup>''n''</sup>]<br />
| &rarr;<br />
| ['''B'''<sup>''k''</sup>]<br />
| )<br />
|}<br />
<br />
On the other hand, the operator W is specific to each <font face=georgia>'''W'''</font>. In this context W always has the form:<br />
<br />
:{| style="height:80px; text-align:center; width:90%"<br />
| align=left width=20%| Concrete type<br />
| width=8% | W<br />
| :<br />
| (<br />
| ''U''<sup>&nbsp;&bull;</sup><br />
| &rarr;<br />
| ''X''<sup>&nbsp;&bull;</sup><br />
| )<br />
| width=16% | &rarr;<br />
| (<br />
| E''U''<sup>&nbsp;&bull;</sup><br />
| &rarr;<br />
| d''X''<sup>&nbsp;&bull;</sup><br />
| )<br />
|-<br />
| align=left width=20%| Abstract type<br />
| width=8% | W<br />
| :<br />
| (<br />
| ['''B'''<sup>''n''</sup>]<br />
| &rarr;<br />
| ['''B'''<sup>''k''</sup>]<br />
| )<br />
| width=16% | &rarr;<br />
| (<br />
| ['''B'''<sup>''n''</sup> &times; '''D'''<sup>''n''</sup>]<br />
| &rarr;<br />
| ['''D'''<sup>''k''</sup>]<br />
| )<br />
|}<br />
<br />
In the types just assigned to <math>\epsilon</math> and W, and implicitly to their results <math>\epsilon</math>''F'' and W''F'', I have listed the most restrictive ranges defined for them, rather than the more expansive target spaces that subsume these ranges. When there is need to recognize both, we may use type indications like the following:<br />
<br />
:{| style="height:80px; text-align:center; width:90%"<br />
| width=6% | <math>\epsilon</math>''F''<br />
| width=2% | :<br />
| width=2% | (<br />
| width=8% | E''U''<sup>&nbsp;&bull;</sup><br />
| width=4% | &rarr;<br />
| width=8% | ''X''<sup>&nbsp;&bull;</sup><br />
| width=4% | &sube;<br />
| width=8% | E''X''<sup>&nbsp;&bull;</sup><br />
| width=2% | )<br />
| width=4% | <math>\cong</math><br />
| width=2% | (<br />
| width=16% | ['''B'''<sup>''n''</sup> &times; '''D'''<sup>''n''</sup>]<br />
| width=4% | &rarr;<br />
| width=8% | ['''B'''<sup>''k''</sup>]<br />
| width=4% | &sube;<br />
| width=16% | ['''B'''<sup>''k''</sup> &times; '''D'''<sup>''k''</sup>]<br />
| width=2% | )<br />
|-<br />
| width=6% | W''F''<br />
| width=2% | :<br />
| width=2% | (<br />
| width=8% | E''U''<sup>&nbsp;&bull;</sup><br />
| width=4% | &rarr;<br />
| width=8% | d''X''<sup>&nbsp;&bull;</sup><br />
| width=4% | &sube;<br />
| width=8% | E''X''<sup>&nbsp;&bull;</sup><br />
| width=2% | )<br />
| width=4% | <math>\cong</math><br />
| width=2% | (<br />
| width=16% | ['''B'''<sup>''n''</sup> &times; '''D'''<sup>''n''</sup>]<br />
| width=4% | &rarr;<br />
| width=8% | ['''D'''<sup>''k''</sup>]<br />
| width=4% | &sube;<br />
| width=16% | ['''B'''<sup>''k''</sup> &times; '''D'''<sup>''k''</sup>]<br />
| width=2% | )<br />
|}<br />
<br />
Hopefully, though, a general appreciation of these subsumptions will prevent us from having to make such declarations more often than absolutely necessary.<br />
<br />
In giving names to these operators I am attempting to preserve as much of the traditional nomenclature and as many of the classical associations as possible. The chief difficulty in doing this is occasioned by the distinction between the operators <font face=georgia>'''W'''</font> and their components W, which forces me to find two distinct but parallel sets of terminology. Here is the plan that I have settled on. First, the component operators W are named by analogy with the corresponding operators in the classical difference calculus. Next, the complete operators <font face=georgia>'''W'''</font> = ‹<math>\epsilon</math>, W› are assigned their titles according to their roles in a geometric or trigonometric allegory, if only to ensure that the tangent functor, that belongs to this family and whose exposition I am still working toward, comes out fit with its customary name. Finally, the operator results <font face=georgia>'''W'''</font>''F'' and W''F'' can be fixed in this frame of reference by tethering the operative adjective for <font face=georgia>'''W'''</font> or W to the anchoring epithet ''map'', in conformity with an already standard practice.<br />
<br />
=====The Secant Operator : <font face=georgia>'''E'''</font>=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Mr. Peirce, after pointing out that our beliefs are really rules for action, said that, to develop a thought's meaning, we need only determine what conduct it is fitted to produce: that conduct is for us its sole significance.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 46]<br />
|}<br />
<br />
Figures&nbsp;33-i and 33-ii depict two stages in the form of analysis that will be applied to transformations throughout the remainder of this study. From now on our interest is staked on an operator denoted "<font face=georgia>'''E'''</font>", which receives the principal investment of analytic attention, and on the constituent parts of <font face=georgia>'''E'''</font>, which derive their shares of significance as developed by the analysis. In the sequel, I refer to <font face=georgia>'''E'''</font> as the ''secant operator'', taking it for granted that a context has been chosen that defines its type. The secant operator has the component description <font face=georgia>'''E'''</font>&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;E›, and its active ingredient E is known as the ''enlargement operator''. (Here, I have named E after the literal ancestor of the shift operator in the calculus of finite differences, defined so that E''f''(''x'')&nbsp;=&nbsp;''f''(''x''+1) for any suitable function ''f'', though of course the logical analogue that we take up here must have a rather different definition.)<br />
<br />
<pre><br />
U% $E$ $E$U% $E$U% $E$U%<br />
o------------------>o============o============o<br />
| | | |<br />
| | | |<br />
| | | |<br />
| | | |<br />
F | | $E$F = | $d$^0.F + | $r$^0.F<br />
| | | |<br />
| | | |<br />
| | | |<br />
v v v v<br />
o------------------>o============o============o<br />
X% $E$ $E$X% $E$X% $E$X%<br />
<br />
Figure 33-i. Analytic Diagram (1)<br />
</pre><br />
<br />
<pre><br />
U% $E$ $E$U% $E$U% $E$U% $E$U%<br />
o------------------>o============o============o============o<br />
| | | | |<br />
| | | | |<br />
| | | | |<br />
| | | | |<br />
F | | $E$F = | $d$^0.F + | $d$^1.F + | $r$^1.F<br />
| | | | |<br />
| | | | |<br />
| | | | |<br />
v v v v v<br />
o------------------>o============o============o============o<br />
X% $E$ $E$X% $E$X% $E$X% $E$X%<br />
<br />
Figure 33-ii. Analytic Diagram (2)<br />
</pre><br />
<br />
In its action on universes <font face=georgia>'''E'''</font> yields the same result as E, a fact that can be expressed in equational form by writing <font face=georgia>'''E'''</font>''U''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;E''U''<sup>&nbsp;&bull;</sup> for any universe ''U''<sup>&nbsp;&bull;</sup>. Notice that the extended universes across the top and bottom of the diagram are indicated to be strictly identical, rather than requiring a corresponding decomposition for them. In a certain sense, the functional parts of <font face=georgia>'''E'''</font>''F'' are partitioned into separate contexts that have to be re-integrated again, but the best image to use is that of making transparent copies of each universe and then overlapping their functional contents once more at the conclusion of the analysis, as suggested by the graphic conventions that are used at the top of Figure&nbsp;30.<br />
<br />
Acting on a transformation ''F'' from universe ''U''<sup>&nbsp;&bull;</sup> to universe ''X''<sup>&nbsp;&bull;</sup>, the operator <font face=georgia>'''E'''</font> determines a transformation <font face=georgia>'''E'''</font>''F'' from <font face=georgia>'''E'''</font>''U''<sup>&nbsp;&bull;</sup> to <font face=georgia>'''E'''</font>''X''<sup>&nbsp;&bull;</sup>. The map <font face=georgia>'''E'''</font>''F'' forms the main body of evidence to be investigated in performing a differential analysis of ''F''. Because we shall frequently be focusing on small pieces of this map for considerable lengths of time, and consequently lose sight of the "big picture", it is critically important to emphasize that the map <font face=georgia>'''E'''</font>''F'' is a transformation that determines a relation from one extended universe into another. This means that we should not be satisfied with our understanding of a transformation ''F'' until we can lay out the full "parts diagram" of <font face=georgia>'''E'''</font>''F'' along the lines of the generic frame in Figure&nbsp;30.<br />
<br />
If one is working within the confines of propositional calculus, it is possible to give an elementary definition of <font face=georgia>'''E'''</font>''F'' by means of a system of propositional equations, as will now be described.<br />
<br />
Given a transformation:<br />
<br />
: ''F'' = ‹''F''<sub>1</sub>, &hellip;, ''F''<sub>''k''</sub>› : '''B'''<sup>''n''</sup> &rarr; '''B'''<sup>''k''</sup><br />
<br />
of concrete type:<br />
<br />
: ''F'' : [''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>] &rarr; [''x''<sub>1</sub>, &hellip;, ''x''<sub>''k''</sub>]<br />
<br />
the transformation:<br />
<br />
: <font face=georgia>'''E'''</font>''F'' = ‹''F''<sub>1</sub>, &hellip;, ''F''<sub>''k''</sub>, E''F''<sub>1</sub>, &hellip;, E''F''<sub>''k''</sub>› : '''B'''<sup>''n''</sup> &times; '''D'''<sup>''n''</sup> &rarr; '''B'''<sup>''k''</sup> &times; '''D'''<sup>''k''</sup><br />
<br />
of concrete type:<br />
<br />
: <font face=georgia>'''E'''</font>''F'' : [''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, &hellip;, d''u''<sub>''n''</sub>] &rarr; [''x''<sub>1</sub>, &hellip;, ''x''<sub>''k''</sub>, d''x''<sub>1</sub>, &hellip;, d''x''<sub>''k''</sub>]<br />
<br />
is defined by means of the following system of logical equations:<br />
<br />
<br><font face="courier new"><br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"<br />
|<br />
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"<br />
| width="8%" | ''x''<sub>1</sub><br />
| width="4%" | =<br />
| width="44%" | <math>\epsilon</math>''F''<sub>1</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, &hellip;, d''u''<sub>''n''</sub>›<br />
| width="4%" | =<br />
| width="40%" | ''F''<sub>1</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>›<br />
|-<br />
| ...<br />
|-<br />
| width="8%" | ''x''<sub>''k''</sub><br />
| width="4%" | =<br />
| width="44%" | <math>\epsilon</math>''F''<sub>''k''</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, &hellip;, d''u''<sub>''n''</sub>›<br />
| width="4%" | =<br />
| width="40%" | ''F''<sub>''k''</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>›<br />
|}<br />
|-<br />
|<br />
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"<br />
| width="8%" | d''x''<sub>1</sub><br />
| width="4%" | =<br />
| width="44%" | E''F''<sub>1</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, &hellip;, d''u''<sub>''n''</sub>›<br />
| width="4%" | =<br />
| width="40%" | ''F''<sub>1</sub>‹''u''<sub>1</sub> + d''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub> + d''u''<sub>''n''</sub>›<br />
|-<br />
| ...<br />
|-<br />
| width="8%" | d''x''<sub>''k''</sub><br />
| width="4%" | =<br />
| width="44%" | E''F''<sub>''k''</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, &hellip;, d''u''<sub>''n''</sub>›<br />
| width="4%" | =<br />
| width="40%" | ''F''<sub>''k''</sub>‹''u''<sub>1</sub> + d''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub> + d''u''<sub>''n''</sub>›<br />
|}<br />
|}<br />
</font><br><br />
<br />
It is important to note that this system of equations can be read as a conjunction of equational propositions, in effect, as a single proposition in the universe of discourse that is generated by all of the named variables. Specifically, this is the universe of discourse over 2(''n''+''k'') variables that is denoted by:<br />
<br />
: E[<font face="lucida calligraphy">U</font> &cup; <font face="lucida calligraphy">X</font>] = [''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>, ''x''<sub>1</sub>, &hellip;, ''x''<sub>''k''</sub>, d''u''<sub>1</sub>, &hellip;, d''u''<sub>''n''</sub>, d''x''<sub>1</sub>, &hellip;, d''x''<sub>''k''</sub>].<br />
<br />
In this light, it should be clear that the system of equations defining <font face=georgia>'''E'''</font>''F'' embodies, in a higher rank and in a differentially extended version, an analogy with the process of thematization that was treated earlier for propositions of the type ''F'' : '''B'''<sup>''n''</sup> &rarr; '''B'''.<br />
<br />
The entire collection of constraints that is represented in the above system of equations may be abbreviated by writing <font face=georgia>'''E'''</font>''F'' = ‹<math>\epsilon</math>''F'',&nbsp;E''F''›, for any map ''F''. This is tantamount to regarding <font face=georgia>'''E'''</font> as a complex operator, <font face=georgia>'''E'''</font> = ‹<math>\epsilon</math>,&nbsp;E›, with a form of application that distributes each component of the operator to work on each component of the operand:<br />
<br />
: <font face=georgia>'''E'''</font>''F'' = ‹<math>\epsilon</math>, E›''F'' = ‹<math>\epsilon</math>''F'',&nbsp;E''F''› = ‹<math>\epsilon</math>''F''<sub>1</sub>, &hellip;, <math>\epsilon</math>''F''<sub>''k''</sub>, E''F''<sub>1</sub>, &hellip;, E''F''<sub>''k''</sub>›.<br />
<br />
Quite a lot of "thematic infrastructure" or interpretive information is being swept under the rug in the use of such abbreviations. When confusion arises about the meaning of such constructions, one always has recourse to the defining system of equations, in its totality a purely propositional expression. This means that the angle brackets, which were used in this context to build an image of multi-component transformations, should not be expected to determine a well-defined product in themselves, but only to serve as reminders of the prior thematic decisions (choices of variable names, etc.) that have to be made in order to determine one. Accordingly, the angle bracket notation ‹&nbsp;,&nbsp;› can be regarded as a kind of ''thematic frame'', an interpretive storage device that preserves the proper associations of concrete logical features between the extended universes at the source and target of <font face=georgia>'''E'''</font>F.<br />
<br />
The generic notations <font face=georgia>'''d'''</font><sup>0</sup>''F'', <font face=georgia>'''d'''</font><sup>1</sup>''F'', &hellip;, <font face=georgia>'''d'''</font><sup>''m''</sup>''F'' in Figure&nbsp;33 refer to the increasing orders of differentials that are extracted in the course of analyzing ''F''. When the analysis is halted at a partial stage of development, notations like <font face=georgia>'''r'''</font><sup>0</sup>''F'', <font face=georgia>'''r'''</font><sup>1</sup>''F'', &hellip;, <font face=georgia>'''r'''</font><sup>''m''</sup>''F'' may be used to summarize the contributions to <font face=georgia>'''E'''</font>''F'' that remain to be analyzed. The Figure illustrates a convention that renders the remainder term <font face=georgia>'''r'''</font><sup>''m''</sup>''F'', in effect, the sum of all differentials of order strictly greater than ''m''.<br />
<br />
I next discuss the set of operators that figure into this form of analysis, describing their effects on transformations. In simplified or specialized contexts these operators tend to take on a variety of different names and notations, some of whose number I will introduce along the way.<br />
<br />
=====The Radius Operator : <font face=georgia>'''e'''</font>=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
And the tangible fact at the root of all our thought-distinctions, however subtle, is that there is no one of them so fine as to consist in anything but a possible difference of practice.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 46]<br />
|}<br />
<br />
The operator identified as <font face=georgia>'''d'''</font><sup>0</sup> in the analytic diagram (Figure&nbsp;33) has the sole purpose of creating a proxy for ''F'' in the appropriately extended context. Construed in terms of its broadest components, <font face=georgia>'''d'''</font><sup>0</sup> is equivalent to the doubly tacit extension operator ‹<math>\epsilon</math>,&nbsp;<math>\epsilon</math>›, in recognition of which let us redub it as "<font face=georgia>'''e'''</font>". Pursuing a geometric analogy, we may refer to <font face=georgia>'''e'''</font>&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;<math>\epsilon</math>›&nbsp;=&nbsp;<font face=georgia>'''d'''</font><sup>0</sup> as the ''radius operator''. The operation that is intended by all of these forms is defined by the equation:<br />
<br />
:{| cellpadding=2<br />
| <font face=georgia>'''e'''</font>''F''<br />
| =<br />
| ‹<math>\epsilon</math>, <math>\epsilon</math>›''F''<br />
|-<br />
| &nbsp;<br />
| =<br />
| ‹<math>\epsilon</math>''F'', <math>\epsilon</math>''F''›<br />
|-<br />
| &nbsp;<br />
| =<br />
| ‹<math>\epsilon</math>''F''<sub>1</sub>, &hellip;, <math>\epsilon</math>F<sub>''k''</sub>, <math>\epsilon</math>F<sub>1</sub>, &hellip;, <math>\epsilon</math>F<sub>''k''</sub>›&nbsp;,<br />
|}<br />
<br />
which is tantamount to the system of equations given below.<br />
<br />
<br><font face="courier new"><br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"<br />
|<br />
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"<br />
| width="8%" | ''x''<sub>1</sub><br />
| width="4%" | =<br />
| width="44%" | <math>\epsilon</math>''F''<sub>1</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, &hellip;, d''u''<sub>''n''</sub>›<br />
| width="4%" | =<br />
| width="40%" | ''F''<sub>1</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>›<br />
|-<br />
| ...<br />
|-<br />
| width="8%" | ''x''<sub>''k''</sub><br />
| width="4%" | =<br />
| width="44%" | <math>\epsilon</math>''F''<sub>''k''</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, &hellip;, d''u''<sub>''n''</sub>›<br />
| width="4%" | =<br />
| width="40%" | ''F''<sub>''k''</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>›<br />
|}<br />
|-<br />
|<br />
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"<br />
| width="8%" | d''x''<sub>1</sub><br />
| width="4%" | =<br />
| width="44%" | <math>\epsilon</math>''F''<sub>1</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, &hellip;, d''u''<sub>''n''</sub>›<br />
| width="4%" | =<br />
| width="40%" | ''F''<sub>1</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>›<br />
|-<br />
| ...<br />
|-<br />
| width="8%" | d''x''<sub>''k''</sub><br />
| width="4%" | =<br />
| width="44%" | <math>\epsilon</math>''F''<sub>''k''</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, &hellip;, d''u''<sub>''n''</sub>›<br />
| width="4%" | =<br />
| width="40%" | ''F''<sub>''k''</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>›<br />
|}<br />
|}<br />
</font><br><br />
<br />
=====The Phantom of the Operators : '''&eta;'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
I was wondering what the reason could be, when I myself raised my head and everything within me seemed drawn towards the Unseen, ''which was playing the most perfect music''!<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 81]<br />
|}<br />
<br />
I now describe an operator whose persistent but elusive action behind the scenes, whose slightly twisted and ambivalent character, and whose fugitive disposition, caught somewhere in flight between the arrantly negative and the positive but errant intent, has cost me some painstaking trouble to detect. In the end I shall place it among the other extensions and projections, as a shade among shadows, of muted tones and motley hue, that adumbrates its own thematic frame and paradoxically lights the way toward a whole new spectrum of values.<br />
<br />
Given a transformation ''F'' : [''u''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''u''<sub>''n''</sub>]&nbsp;&rarr;&nbsp;[''x''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''x''<sub>''k''</sub>], we often need to make a separate treatment of a related family of transformations of the form ''F''*&nbsp;:&nbsp;[''u''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''u''<sub>''n''</sub>,&nbsp;d''u''<sub>1</sub>,&nbsp;&hellip;,&nbsp;d''u''<sub>''n''</sub>]&nbsp;&rarr;&nbsp;[d''x''<sub>1</sub>,&nbsp;&hellip;,&nbsp;d''x''<sub>''k''</sub>]. The operator <math>\eta</math> is introduced to deal with the simplest one of these maps:<br />
<br />
: <math>\eta</math>''F'' : [''u''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''u''<sub>''n''</sub>,&nbsp;d''u''<sub>1</sub>,&nbsp;&hellip;,&nbsp;d''u''<sub>''n''</sub>]&nbsp;&rarr;&nbsp;[d''x''<sub>1</sub>,&nbsp;&hellip;,&nbsp;d''x''<sub>''k''</sub>]<br />
<br />
which is defined by the equations:<br />
<br />
<br><font face="courier new"><br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"<br />
|<br />
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"<br />
| width="8%" | d''x''<sub>1</sub><br />
| width="4%" | =<br />
| width="44%" | <math>\epsilon</math>''F''<sub>1</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, &hellip;, d''u''<sub>''n''</sub>›<br />
| width="4%" | =<br />
| width="40%" | ''F''<sub>1</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>›<br />
|-<br />
| ...<br />
|-<br />
| width="8%" | d''x''<sub>''k''</sub><br />
| width="4%" | =<br />
| width="44%" | <math>\epsilon</math>''F''<sub>''k''</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, &hellip;, d''u''<sub>''n''</sub>›<br />
| width="4%" | =<br />
| width="40%" | ''F''<sub>''k''</sub>‹''u''<sub>1</sub>, &hellip;, ''u''<sub>''n''</sub>›<br />
|}<br />
|}<br />
</font><br><br />
<br />
In effect, the operator <math>\eta</math> is nothing but the stand-alone version of a procedure that is otherwise invoked subordinate to the work of the radius operator <font face=georgia>'''e'''</font>. Operating independently, <math>\eta</math> achieves precisely the same results that the second <math>\epsilon</math> in ‹<math>\epsilon</math>,&nbsp;<math>\epsilon</math>› accomplishes by working within the context of its adjuvant thematic frame, "‹&nbsp;,&nbsp;›". From this point on, because the use of <math>\epsilon</math> and <math>\eta</math> in this setting combines the aims of both the tacit and the thematic extensions, and because <math>\eta</math> reflects in regard to <math>\epsilon</math> little more than the application of a differential twist, a mere turn of phrase, I refer to <math>\eta</math> as the ''trope extension'' operator.<br />
<br />
=====The Chord Operator : <font face=georgia>'''D'''</font>=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
What difference would it practically make to any one if this notion rather than that notion were true? If no practical difference whatever can be traced, then the alternatives mean practically the same thing, and all dispute is idle.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 45]<br />
|}<br />
<br />
Next I discuss an operator that is always immanent in this form of analysis, and remains implicitly present in the entire proceeding. It may appear once as a record: a relic or revenant that reprises the reminders of an earlier stage of development. Or it may appear always as a resource: a reserve or redoubt that caches in advance an echo of what remains to be played out, cleared up, and requited in full at a future stage. And all of this remains true whether or not we recall the key at any time, and whether or not the subtending theme is recited explicitly at any stage of play.<br />
<br />
This is the operator that is referred to as <font face=georgia>'''r'''</font><sup>0</sup> in the initial stage of analysis (Figure&nbsp;33-i), and that is expanded as <font face=georgia>'''d'''</font><sup>1</sup>&nbsp;+&nbsp;<font face=georgia>'''r'''</font><sup>1</sup> in the subsequent step (Figure&nbsp;33-ii). In congruence, but not quite harmony, with my allusions of analogy that are not quite geometry, I call this the ''chord operator'' and denote it <font face=georgia>'''D'''</font>. In the more casual terms that are here introduced, <font face=georgia>'''D'''</font> is defined as the remainder of <font face=georgia>'''E'''</font> and <font face=georgia>'''e'''</font>, and it assigns a due measure to each undertone of accord or discord that is struck between the note of enterprise <font face=georgia>'''E'''</font> and the bar of exigency <font face=georgia>'''e'''</font>.<br />
<br />
The tension between these counterposed notions, in balance transient but regular in stridence, may be refracted along familiar lines, though never by any such fraction resolved. In this style we may write <font face=georgia>'''D'''</font>&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;D›, calling D the ''difference operator'' and noting that it plays a role in this realm of mutable and diverse discourse that is analogous to the part taken by the discrete difference operator in the ordinary difference calculus. Finally, we should note that the chord <font face=georgia>'''D'''</font> is not one that need be lost at any stage of development. At the ''m''<sup>th</sup> stage of play it can always be reconstituted in the following form:<br />
<br />
<br><font face="courier new"><br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"<br />
|<br />
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"<br />
| <font face=georgia>'''D'''</font><br />
| =<br />
| <font face=georgia>'''E'''</font> &ndash; <font face=georgia>'''e'''</font><br />
|-<br />
| &nbsp;<br />
| =<br />
| <font face=georgia>'''r'''</font><sup>0</sup><br />
|-<br />
| &nbsp;<br />
| =<br />
| <font face=georgia>'''d'''</font><sup>1</sup> + <font face=georgia>'''r'''</font><sup>1</sup><br />
|-<br />
| &nbsp;<br />
| =<br />
| <font face=georgia>'''d'''</font><sup>1</sup> + &hellip; + <font face=georgia>'''d'''</font><sup>''m''</sup> + <font face=georgia>'''r'''</font><sup>''m''</sup><br />
|-<br />
| &nbsp;<br />
| =<br />
| <font size="+2">&sum;</font><sub>(''i'' = 1 &hellip; ''m'')</sub> <font face=georgia>'''d'''</font><sup>''i''</sup> + <font face=georgia>'''r'''</font><sup>''m''</sup><br />
|}<br />
|}<br />
</font><br><br />
<br />
=====The Tangent Operator : <font face=georgia>'''T'''</font>=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
They take part in scenes of whose significance they have no inkling. They are merely tangent to curves of history the beginnings and ends and forms of which pass wholly beyond their ken. So we are tangent to the wider life of things.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 300]<br />
|}<br />
<br />
The operator tagged as <font face=georgia>'''d'''</font><sup>1</sup> in the analytic diagram (Figure&nbsp;33) is called the ''tangent operator'', and is usually denoted in this text as <font face=georgia>'''d'''</font> or <font face=georgia>'''T'''</font>. Because it has the properties required to qualify as a functor, namely, preserving the identity element of the composition operation and the articulated form of every composure among transformations, it also earns the title of a ''tangent functor''. According to the custom adopted here, we dissect it as <font face=georgia>'''T'''</font>&nbsp;=&nbsp;<font face=georgia>'''d'''</font>&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;d›, where d is the operator that yields the first order differential d''F'' when applied to a transformation ''F'', and whose name is legion.<br />
<br />
Figure&nbsp;34 illustrates a stage of analysis where we ignore everything but the tangent functor <font face=georgia>'''T'''</font>, and attend to it chiefly as it bears on the first order differential d''F'' in the analytic expansion of ''F''. In this situation, we often refer to the extended universes E''U''<sup>&nbsp;&bull;</sup> and E''X''<sup>&nbsp;&bull;</sup> under the equivalent designations <font face=georgia>'''T'''</font>''U''<sup>&nbsp;&bull;</sup> and <font face=georgia>'''T'''</font>''X''<sup>&nbsp;&bull;</sup>, respectively. The purpose of the tangent functor <font face=georgia>'''T'''</font> is to extract the tangent map <font face=georgia>'''T'''</font>''F'' at each point of ''U''<sup>&nbsp;&bull;</sup>, and the tangent map <font face=georgia>'''T'''</font>''F''&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;d›''F'' tells us not only what the transformation ''F'' is doing at each point of the universe ''U''<sup>&nbsp;&bull;</sup> but also what ''F'' is doing to states in the neighborhood of that point, approximately, linearly, and relatively speaking.<br />
<br />
<pre><br />
U% $T$ $T$U% $T$U%<br />
o------------------>o============o<br />
| | |<br />
| | |<br />
| | |<br />
| | |<br />
F | | $T$F = | <!e!, d> F<br />
| | |<br />
| | |<br />
| | |<br />
v v v<br />
o------------------>o============o<br />
X% $T$ $T$X% $T$X%<br />
<br />
Figure 34. Tangent Functor Diagram<br />
</pre><br />
<br />
* NB. There is one aspect of the preceding construction that remains especially problematic. Why did we define the operators W in {<math>\eta</math>,&nbsp;E,&nbsp;D,&nbsp;d,&nbsp;r} so that the ranges of their resulting maps all fall within the realms of differential quality, even fabricating a variant of the tacit extension operator to have that character? Clearly, not all of the operator maps WF have equally good reasons for placing their values in differential stocks. The only explanation I can devise at present is that, without doing this, I cannot justify the comparison and combination of their values in the various analytic steps. By default, only those values in the same functional component can be brought into algebraic modes of interaction. Up till now, the only mechanism provided for their broader association has been a purely logical one, their common placement in a target universe of discourse, but the task of converting this logical circumstance into algebraic forms of application has not yet been taken up.<br />
<br />
===Transformations of Type '''B'''<sup>2</sup> &rarr; '''B'''<sup>1</sup>===<br />
<br />
To study the effects of these analytic operators in the simplest possible situation, let us revert to a still more primitive case. Consider the singular proposition ''J''‹''u'',&nbsp;''v''›&nbsp;=&nbsp;''uv'', regarded either as the functional product of the maps ''u'' and ''v'' or as the logical conjunction of the features ''u'' and ''v'', a map whose fiber of truth ''J''<sup>&ndash;1</sup>(1) picks out the single cell of that logical description in the universe of discourse ''U''<sup>&nbsp;&bull;</sup>. Thus ''J'', or ''uv'', may be treated as a pseudonym for the point whose coordinates are ‹1,&nbsp;1› in ''U''<sup>&nbsp;&bull;</sup>.<br />
<br />
====Analytic Expansion of Conjunction====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
<p>In her sufferings she read a great deal and discovered that she had lost something, the possession of which she had previously not been much aware of: a soul.</p><br />
<br />
<p>What is that? It is easily defined negatively: it is simply what curls up and hides when there is any mention of algebraic series.</p><br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 118]<br />
|}<br />
<br />
Figure&nbsp;35 pictures the form of conjunction ''J''&nbsp;:&nbsp;'''B'''<sup>2</sup>&nbsp;&rarr;&nbsp;'''B''' as a transformation from the 2-dimensional universe [''u'',&nbsp;''v''] to the 1-dimensional universe [''x'']. This is a subtle but significant change of viewpoint on the proposition, attaching an arbitrary but concrete quality to its functional value. Using the language introduced earlier, we can express this change by saying that the proposition ''J''&nbsp;:&nbsp;〈''u'',&nbsp;''v''〉&nbsp;&rarr;&nbsp;'''B''' is being recast into the thematized role of a transformation ''J''&nbsp;:&nbsp;[''u'',&nbsp;''v'']&nbsp;&rarr;&nbsp;[''x''], where the new variable ''x'' takes the part of a thematic variable ¢(''J'').<br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 35 -- A Conjunction Viewed as a Transformation.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 35. Conjunction as Transformation'''</font></center></p><br />
<br />
=====Tacit Extension of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
I teach straying from me, yet who can stray from me?<br><br />
I follow you whoever you are from the present hour;<br><br />
My words itch at your ears till you understand them.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 83]<br />
|}<br />
<br />
Earlier I defined the tacit extension operators <math>\epsilon</math>&nbsp;:&nbsp;''X''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''Y''<sup>&nbsp;&bull;</sup> as maps embedding each proposition of a given universe ''X''<sup>&nbsp;&bull;</sup> in a more generously given universe ''Y''<sup>&nbsp;&bull;</sup> containing ''X''<sup>&nbsp;&bull;</sup>. Of immediate interest are the tacit extensions <math>\epsilon</math>&nbsp;:&nbsp;''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''U''<sup>&nbsp;&bull;</sup>, that locate each proposition of ''U''<sup>&nbsp;&bull;</sup> in the enlarged context of E''U''<sup>&nbsp;&bull;</sup>. In its application to the propositional conjunction ''J''&nbsp;=&nbsp;''u''&nbsp;''v'' in [''u'',&nbsp;''v''], the tacit extension operator <math>\epsilon</math> produces the proposition <math>\epsilon</math>''J'' in E''U''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;[''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']. The extended proposition <math>\epsilon</math>''J'' may be computed according to the scheme in Table&nbsp;36, in effect, doing nothing more than conjoining a tautology of [d''u'',&nbsp;d''v''] to ''J'' in ''U''<sup>&nbsp;&bull;</sup>.<br />
<br />
<font face="courier new"><br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"<br />
|+ Table 36. Computation of <math>\epsilon</math>''J''<br />
|<br />
{| align="left" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"<br />
| width="8%" | <math>\epsilon</math>''J''<br />
| width="4%" | =<br />
| ''J''‹''u'', ''v''›<br />
|-<br />
| width="8%" | &nbsp;<br />
| width="4%" | =<br />
| ''u'' ''v''<br />
|-<br />
| width="8%" | &nbsp;<br />
| width="4%" | =<br />
| ''u'' ''v'' (d''u'')(d''v'') || +<br />
| ''u'' ''v'' (d''u'') d''v'' || +<br />
| ''u'' ''v'' d''u'' (d''v'') || +<br />
| ''u'' ''v'' d''u'' d''v''<br />
|}<br />
|-<br />
|<br />
{| align="left" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"<br />
| width="8%" | <math>\epsilon</math>''J''<br />
| width="4%" | =<br />
| width="24%" | ''u'' ''v''&nbsp;(d''u'')(d''v'')<br />
| width="4%" | +<br />
| width="60%" | &nbsp;<br />
|-<br />
| width="8%" | &nbsp;<br />
| width="4%" | &nbsp;<br />
| width="24%" | ''u'' ''v''&nbsp;(d''u'')&nbsp;d''v''<br />
| width="4%" | +<br />
| width="60%" | &nbsp;<br />
|-<br />
| width="8%" | &nbsp;<br />
| width="4%" | &nbsp;<br />
| width="24%" | ''u'' ''v''&nbsp;&nbsp;d''u''&nbsp;(d''v'')<br />
| width="4%" | +<br />
| width="60%" | &nbsp;<br />
|-<br />
| width="8%" | &nbsp;<br />
| width="4%" | &nbsp;<br />
| width="24%" | ''u'' ''v''&nbsp;&nbsp;d''u''&nbsp;&nbsp;d''v''<br />
| width="4%" | &nbsp;<br />
| width="60%" | &nbsp;<br />
|}<br />
|}<br />
</font><br><br />
<br />
The lower portion of the Table contains the dispositional features of <math>\epsilon</math>''J'' arranged in such a way that the variety of ordinary features spreads across the rows and the variety of differential features runs through the columns. This organization serves to facilitate pattern matching in the remainder of our computations. Again, the tacit extension is usually so trivial a concern that we do not always bother to make an explicit note of it, taking it for granted that any function ''F'' that is being employed in a differential context is equivalent to <math>\epsilon</math>''F'', for a suitable <math>\epsilon</math>.<br />
<br />
Figures&nbsp;37-a through 37-d present several pictures of the proposition ''J'' and its tacit extension <math>\epsilon</math>''J''. Notice in these Figures how <math>\epsilon</math>''J'' in E''U''<sup>&nbsp;&bull;</sup> visibly extends ''J'' in ''U''<sup>&nbsp;&bull;</sup>, by annexing to the indicated cells of ''J'' all of the arcs that exit from or flow out of them. In effect, this extension attaches to these cells all of the dispositions that spring from them, in other words, it attributes to these cells all of the conceivable changes that are their issue.<br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 37-a -- Tacit Extension of J.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 37-a. Tacit Extension of ''J''&nbsp;&nbsp;(Areal)'''</font></center></p><br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 37-b -- Tacit Extension of J.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 37-b. Tacit Extension of ''J''&nbsp;&nbsp;(Bundle)'''</font></center></p><br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 37-c -- Tacit Extension of J.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 37-c. Tacit Extension of ''J''&nbsp;&nbsp;(Compact)'''</font></center></p><br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 37-d -- Tacit Extension of J.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 37-d. Tacit Extension of ''J''&nbsp;&nbsp;(Digraph)'''</font></center></p><br />
<br />
The computational scheme that was shown in Table&nbsp;36 treated ''J'' as a proposition in ''U''<sup>&nbsp;&bull;</sup> and formed <math>\epsilon</math>''J'' as a proposition in E''U''<sup>&nbsp;&bull;</sup>. When ''J'' is regarded as a mapping ''J''&nbsp;:&nbsp;''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup> then <math>\epsilon</math>''J'' must be obtained as a mapping <math>\epsilon</math>''J''&nbsp;:&nbsp;E''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup>. By default, the tacit extension of the map ''J''&nbsp;:&nbsp;[''u'',&nbsp;''v'']&nbsp;&rarr;&nbsp;[''x''] is naturally taken to be a particular map, of the following form:<br />
<br />
: <math>\epsilon</math>''J''&nbsp;:&nbsp;[''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'']&nbsp;&sube;&nbsp;[''x'',&nbsp;d''x'']<br />
<br />
This is the map that looks like ''J'' when painted in the frame of the extended source universe and that takes the same thematic variable in the extended target universe as the one that ''J'' already employs.<br />
<br />
But the choice of a particular thematic variable, for example ''x'' for ¢(''J''), is a shade more arbitrary than the initial choice of variable names {''u'',&nbsp;''v''}. This means that the map I am calling the ''trope extension'', specifically:<br />
<br />
: <math>\eta</math>''J''&nbsp;:&nbsp;[''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'']&nbsp;&sube;&nbsp;[''x'',&nbsp;d''x'']<br />
<br />
since it looks just the same as <math>\epsilon</math>''J'' in the way that its fibers paint the source domain, belongs just as fully to the family of tacit extensions, generically considered.<br />
<br />
These considerations have the practical consequence that all of our computations and illustrations of <math>\epsilon</math>''J'' perform the double duty of capturing an image of <math>\eta</math>''J'' as well. In other words, we are saved the work of carrying out calculations and drawing figures for the trope extension <math>\eta</math>''J'', because the exercise would be identical to the work already done for <math>\epsilon</math>''J''. Since the computations given for <math>\epsilon</math>''J'' are expressed solely in terms of the variables {''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''}, these variables work equally well for finding <math>\eta</math>''J''. Furthermore, since each of the above Figures shows only how the level sets of <math>\epsilon</math>''J'' partition the extended source universe E''U''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;[''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''], all of them serve equally well as portraits of <math>\eta</math>''J''.<br />
<br />
=====Enlargement Map of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
No one could have established the existence of any details that might not just as well have existed in earlier times too; but all the relations between things had shifted slightly. Ideas that had once been of lean account grew fat.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 62]<br />
|}<br />
<br />
The enlargement map E''J'' is computed from the proposition ''J'' by making a particular class of formal substitutions for its variables, in this case ''u''&nbsp;+&nbsp;d''u'' for ''u'' and ''v''&nbsp;+&nbsp;d''v'' for ''v'', and subsequently expanding the result in whatever way happens to be convenient for the end in view.<br />
<br />
Table&nbsp;38 shows a typical scheme of computation, following a systematic method of exploiting boolean expansions over selected variables, and ultimately developing E''J'' over the cells of [''u'',&nbsp;''v'']. The critical step of this procedure uses the facts that (0,&nbsp;''x'')&nbsp;=&nbsp;0&nbsp;+&nbsp;''x''&nbsp;=&nbsp;''x'' and (1,&nbsp;''x'')&nbsp;=&nbsp;1&nbsp;+&nbsp;''x''&nbsp;=&nbsp;(''x'') for any boolean variable ''x''.<br />
<br />
<font face="courier new"><br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"<br />
|+ Table 38. Computation of E''J'' (Method 1)<br />
|<br />
{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"<br />
| width="8%" | E''J''<br />
| width="4%" | =<br />
| width="44%" | ''J''‹''u'' + d''u'', ''v'' + d''v''›<br />
| width="44%" | &nbsp;<br />
|-<br />
| &nbsp;<br />
|-<br />
| width="8%" | &nbsp;<br />
| width="4%" | =<br />
| width="44%" | (''u'', d''u'')(''v'', d''v'')<br />
| width="44%" | &nbsp;<br />
|-<br />
| &nbsp;<br />
|-<br />
| width="8%" | &nbsp;<br />
| width="4%" | =<br />
| width="44%" | &nbsp;''u''&nbsp;&nbsp;''v''&nbsp;&nbsp;''J''‹1 + d''u'', 1 + d''v''›<br />
| width="44%" | +<br />
|-<br />
| width="8%" | &nbsp;<br />
| width="4%" | &nbsp;<br />
| width="44%" | &nbsp;''u''&nbsp;(''v'')&nbsp;''J''‹1 + d''u'', 0 + d''v''›<br />
| width="44%" | +<br />
|-<br />
| width="8%" | &nbsp;<br />
| width="4%" | &nbsp;<br />
| width="44%" | (''u'')&nbsp;''v''&nbsp;&nbsp;''J''‹0 + d''u'', 1 + d''v''›<br />
| width="44%" | +<br />
|-<br />
| width="8%" | &nbsp;<br />
| width="4%" | &nbsp;<br />
| width="44%" | (''u'')(''v'')&nbsp;''J''‹0 + d''u'', 0 + d''v''›<br />
| width="44%" | &nbsp;<br />
|-<br />
| &nbsp;<br />
|-<br />
| width="8%" | &nbsp;<br />
| width="4%" | =<br />
| width="44%" | &nbsp;''u''&nbsp;&nbsp;''v''&nbsp;&nbsp;''J''‹(d''u''),&nbsp;(d''v'')›<br />
| width="44%" | +<br />
|-<br />
| width="8%" | &nbsp;<br />
| width="4%" | &nbsp;<br />
| width="44%" | &nbsp;''u''&nbsp;(''v'')&nbsp;''J''‹(d''u''),&nbsp;&nbsp;d''v''&nbsp;›<br />
| width="44%" | +<br />
|-<br />
| width="8%" | &nbsp;<br />
| width="4%" | &nbsp;<br />
| width="44%" | (''u'')&nbsp;''v''&nbsp;&nbsp;''J''‹&nbsp;d''u''&nbsp;, (d''v'')›<br />
| width="44%" | +<br />
|-<br />
| width="8%" | &nbsp;<br />
| width="4%" | &nbsp;<br />
| width="44%" | (''u'')(''v'')&nbsp;''J''‹&nbsp;d''u''&nbsp;,&nbsp;&nbsp;d''v''&nbsp;›<br />
| width="44%" | &nbsp;<br />
|}<br />
|-<br />
|<br />
{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"<br />
| width="8%" | E''J''<br />
| width="23%" | = ''u'' ''v'' (d''u'')(d''v'')<br />
| width="23%" | &nbsp;<br />
| width="23%" | &nbsp;<br />
| width="23%" | &nbsp;<br />
|-<br />
| width="8%" | &nbsp;<br />
| width="23%" | &nbsp;<br />
| width="23%" | + ''u'' (''v'') (d''u'') d''v''<br />
| width="23%" | &nbsp;<br />
| width="23%" | &nbsp;<br />
|-<br />
| width="8%" | &nbsp;<br />
| width="23%" | &nbsp;<br />
| width="23%" | &nbsp;<br />
| width="23%" | + (''u'') ''v'' d''u'' (d''v'')<br />
| width="23%" | &nbsp;<br />
|-<br />
| width="8%" | &nbsp;<br />
| width="23%" | &nbsp;<br />
| width="23%" | &nbsp;<br />
| width="23%" | &nbsp;<br />
| width="23%" | + (''u'')(''v'') d''u'' d''v''<br />
|}<br />
|}<br />
</font><br><br />
<br />
Table&nbsp;39 exhibits another method that happens to work quickly in this particular case, using distributive laws to multiply things out in an algebraic manner, arranging the notations of feature and fluxion according to a scale of simple character and degree. Proceeding this way leads through an intermediate step which, in chiming the changes of ordinary calculus, should take on a familiar ring. Consequential properties of exclusive disjunction then carry us on to the concluding line.<br />
<br />
<font face="courier new"><br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"<br />
|+ Table 39. Computation of E''J'' (Method 2)<br />
|<br />
{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"<br />
| width="8%" | E''J''<br />
| colspan="2" | = ‹''u'' + d''u''› <math>\cdot</math> ‹''v'' + d''v''›<br />
| width="23%" | &nbsp;<br />
| width="23%" | &nbsp;<br />
|-<br />
| &nbsp;<br />
|-<br />
| width="8%" | &nbsp;<br />
| colspan="2" | = ''u'' ''v'' + ''u'' d''v'' + ''v'' d''u'' + d''u'' d''v''<br />
| width="23%" | &nbsp;<br />
| width="23%" | &nbsp;<br />
|-<br />
| &nbsp;<br />
|-<br />
| width="8%" | E''J''<br />
| width="23%" | = ''u'' ''v'' (d''u'')(d''v'')<br />
| width="23%" | + ''u'' (''v'') (d''u'') d''v''<br />
| width="23%" | + (''u'') ''v'' d''u'' (d''v'')<br />
| width="23%" | + (''u'')(''v'') d''u'' d''v''<br />
|}<br />
|}<br />
</font><br><br />
<br />
Figures&nbsp;40-a through 40-d present several views of the enlarged proposition E''J''.<br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 40-a -- Enlargement of J.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 40-a. Enlargement of ''J''&nbsp;&nbsp;(Areal)'''</font></center></p><br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 40-b -- Enlargement of J.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 40-b. Enlargement of ''J''&nbsp;&nbsp;(Bundle)'''</font></center></p><br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 40-c -- Enlargement of J.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 40-c. Enlargement of ''J''&nbsp;&nbsp;(Compact)'''</font></center></p><br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 40-d -- Enlargement of J.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 40-d. Enlargement of ''J''&nbsp;&nbsp;(Digraph)'''</font></center></p><br />
<br />
An intuitive reading of the proposition E''J'' becomes available at this point, and may be useful. Recall that propositions in the extended universe E''U''<sup>&nbsp;&bull;</sup> express the ''dispositions'' of system and the constraints that are placed on them. In other words, a differential proposition in E''U''<sup>&nbsp;&bull;</sup> can be read as referring to various changes that a system might undergo in and from its various states. In particular, we can understand E''J'' as a statement that tells us what changes need to be made with regard to each state in the universe of discourse in order to reach the ''truth'' of ''J'', that is, the region of the universe where ''J'' is true. This interpretation is visibly clear in the Figures above, and appeals to the imagination in a satisfying way, but it has the added benefit of giving fresh meaning to the original name of the shift operator E. Namely, E''J'' can be read as a proposition that ''enlarges'' on the meaning of ''J'', in the sense of explaining its practical bearings and clarifying what it means in terms of the available options for differential action and the consequential effects that result from each choice.<br />
<br />
Treated this way, the enlargement E''J'' has strong ties to the normal use of ''J'', no matter whether it is understood as a proposition or a function, namely, to act as a figurative device for indicating the models of ''J'', in effect, pointing to the interpretive elements in its fiber of truth ''J''<sup>&ndash;1</sup>(1). It is this kind of ''use'' that is often compared with the ''mention'' of a proposition, and thereby hangs a tale.<br />
<br />
=====Digression : Reflection on Use and Mention=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Reflection is turning a topic over in various aspects and in various lights so that nothing significant about it shall be overlooked &mdash; almost as one might turn a stone over to see what its hidden side is like or what is covered by it.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 57]<br />
|}<br />
<br />
The contrast drawn in logic between the ''use'' and the ''mention'' of a proposition corresponds to the difference that we observe in functional terms between using "''J''&nbsp;" to indicate the region ''J''<sup>&ndash;1</sup>(1) and using "''J''&nbsp;" to indicate the function ''J''. You may think that one of these uses ought to be proscribed, and logicians are quick to prescribe against their confusion. But there seems to be no likelihood in practice that their interactions can be avoided. If the name "''J''&nbsp;" is used as a sign of the function ''J'', and if the function ''J'' has its use in signifying something else, as would constantly be the case when some future theory of signs has given a functional meaning to every sign whatsoever, then is not "''J''&nbsp;" by transitivity a sign of the thing itself? There are, of course, two answers to this question. Not every act of signifying or referring need be transitive. Not every warrant or guarantee or certificate is automatically transferable, indeed, not many. Not every feature of a feature is a feature of the featuree. Otherwise we have an inference like the following: If a buffalo is white, and white is a color, then a buffalo is a color. But a buffalo is not, only buff is.<br />
<br />
The logical or pragmatic distinction between use and mention is cogent and necessary, and so is the analogous functional distinction between determining a value and determining what determines that value, but so are the normal techniques that we use to make these distinctions apply flexibly in practice. The way that the hue and cry about use and mention is raised in logical discussions, you might be led to think that this single dimension of choices embraces the only kinds of use worth mentioning and the only kinds of mention worth using. It will constitute the expeditionary and taxonomic tasks of that future theory of signs to explore and to classify the many other constellations and dimensions of use and mention that are yet to be opened up by the generative potential of full-fledged sign relations.<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
The well-known capacity that thoughts have &mdash; as doctors have discovered &mdash; for dissolving and dispersing those hard lumps of deep, ingrowing, morbidly entangled conflict that arise out of gloomy regions of the self probably rests on nothing other than their social and worldly nature, which links the individual being with other people and things; but unfortunately what gives them their power of healing seems to be the same as what diminishes the quality of personal experience in them.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 130]<br />
|}<br />
<br />
=====Difference Map of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
"It doesn't matter what one does", the Man Without Qualities said to himself, shrugging his shoulders. "In a tangle of forces like this it doesn't make a scrap of difference." He turned away like a man who has learned renunciation, almost indeed like a sick man who shrinks from any intensity of contact. And then, striding through his adjacent dressing-room, he passed a punching-ball that hung there; he gave it a blow far swifter and harder than is usual in moods of resignation or states of weakness.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 8]<br />
|}<br />
<br />
With the tacit extension map <math>\epsilon</math>''J'' and the enlargement map E''J'' well in place, the difference map D''J'' can be computed along the lines displayed in Table&nbsp;41, ending up, in this instance, with an expansion of D''J'' over the cells of [''u'',&nbsp;''v''].<br />
<br />
<font face="courier new"><br />
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"<br />
|+ Table 41. Computation of D''J'' (Method 1)<br />
|<br />
{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"<br />
| width="8%" | D''J''<br />
| width="4%" | =<br />
| width="42%" | E''J''<br />
| width="4%" | +<br />
| width="42%" | <math>\epsilon</math>''J''<br />
|-<br />
| width="8%" | &nbsp; <br />
| width="4%" | =<br />
| width="42%" | ''J''‹''u'' + d''u'', ''v'' + d''v''›<br />
| width="4%" | +<br />
| width="42%" | ''J''‹''u'', ''v''›<br />
|-<br />
| width="8%" | &nbsp;<br />
| width="4%" | =<br />
| width="42%" | (''u'', d''u'')(''v'', d''v'')<br />
| width="4%" | +<br />
| width="42%" | ''u'' ''v''<br />
|}<br />
|-<br />
|<br />
{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"<br />
| width="8%" | D''J''<br />
| width="3%" | =<br />
| width="20%" align=center | 0<br />
| width="23%" | &nbsp;<br />
| width="23%" | &nbsp;<br />
| width="23%" | &nbsp;<br />
|-<br />
| width="8%" | &nbsp;<br />
| width="3%" | +<br />
| width="20%" | ''u'' ''v'' (d''u'') d''v''<br />
| width="23%" | + ''u'' (''v'')(d''u'') d''v''<br />
| width="23%" | &nbsp;<br />
| width="23%" | &nbsp;<br />
|-<br />
| width="8%" | &nbsp;<br />
| width="3%" | +<br />
| width="20%" | ''u'' ''v''&nbsp;&nbsp;d''u'' (d''v'')<br />
| width="23%" | &nbsp;<br />
| width="23%" | + (''u'') ''v'' d''u'' (d''v'')<br />
| width="23%" | &nbsp;<br />
|-<br />
| width="8%" | &nbsp;<br />
| width="3%" | +<br />
| width="20%" | ''u'' ''v''&nbsp;&nbsp;d''u''&nbsp;&nbsp;d''v''<br />
| width="23%" | &nbsp;<br />
| width="23%" | &nbsp;<br />
| width="23%" | + (''u'')(''v'') d''u'' d''v''<br />
|}<br />
|-<br />
|<br />
{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"<br />
| width="8%" | D''J''<br />
| width="3%" | =<br />
| width="23%" | ''u'' ''v'' ((d''u'')(d''v''))<br />
| width="22%" | + ''u'' (''v'')(d''u'') d''v''<br />
| width="22%" | + (''u'') ''v'' d''u'' (d''v'')<br />
| width="22%" | + (''u'')(''v'') d''u'' d''v''<br />
|}<br />
|}<br />
</font><br><br />
<br />
Alternatively, the difference map D''J'' can be expanded over the cells of [d''u'',&nbsp;d''v''] to arrive at the formulation shown in Table&nbsp;42. The same development would be obtained from the previous Table by collecting terms in an alternate manner, along the rows rather than the columns of the middle portion of the Table.<br />
<br />
<font face="courier new"><br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"<br />
|+ Table 42. Computation of D''J'' (Method 2)<br />
|<br />
{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"<br />
| width="8%" | D''J''<br />
| width="4%" | =<br />
| width="16%" | <math>\epsilon</math>''J''<br />
| width="4%" | +<br />
| colspan="5" | E''J''<br />
|-<br />
| width="8%" | &nbsp; <br />
| width="4%" | =<br />
| width="16%" | ''J''‹''u'', ''v''›<br />
| width="4%" | +<br />
| colspan="5" | ''J''‹''u'' + d''u'', ''v'' + d''v''›<br />
|-<br />
| width="8%" | &nbsp;<br />
| width="4%" | =<br />
| width="16%" | ''u'' ''v''<br />
| width="4%" | +<br />
| colspan="5" | (''u'', d''u'')(''v'', d''v'')<br />
|-<br />
| width="8%" | &nbsp;<br />
| width="4%" | =<br />
| width="16%" | 0<br />
| width="4%" | +<br />
| width="16%" | ''u'' d''v''<br />
| width="4%" | +<br />
| width="16%" | ''v'' d''u''<br />
| width="4%" | +<br />
| width="28%" | d''u'' d''v''<br />
|-<br />
| width="8%" | D''J''<br />
| width="4%" | =<br />
| width="16%" | 0<br />
| width="4%" | +<br />
| width="16%" | ''u'' (d''u'') d''v''<br />
| width="4%" | +<br />
| width="16%" | ''v'' d''u'' (d''v'')<br />
| width="4%" | +<br />
| width="28%" | ((''u'', ''v'')) d''u'' d''v''<br />
|}<br />
|}<br />
</font><br><br />
<br />
Even more simply, the same result is reached by matching up the propositional coefficients of <math>\epsilon</math>''J'' and E''J'' along the cells of [d''u'',&nbsp;d''v''] and adding the pairs under boolean sums (that is, "mod 2", where 1&nbsp;+&nbsp;1&nbsp;=&nbsp;0), as shown in Table&nbsp;43.<br />
<br />
<font face="courier new"><br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"<br />
|+ Table 43. Computation of D''J'' (Method 3)<br />
|<br />
{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"<br />
| width="6%" | D''J''<br />
| width="3%" | =<br />
| width="20%" | <math>\epsilon</math>''J''<br />
| width="3%" | +<br />
| width="20%" | E''J''<br />
| width="48%" | &nbsp;<br />
|}<br />
|-<br />
|<br />
{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"<br />
| width="6%" | <math>\epsilon</math>''J''<br />
| width="23%" | =&nbsp;''u''&nbsp;''v''&nbsp;(d''u'')(d''v'')<br />
| width="23%" | +&nbsp;''u''&nbsp;&nbsp;''v''&nbsp;(d''u'')&nbsp;d''v''<br />
| width="23%" | +&nbsp;&nbsp;''u''&nbsp;&nbsp;''v''&nbsp;d''u''&nbsp;(d''v'')<br />
| width="25%" | +&nbsp;&nbsp;''u''&nbsp;&nbsp;''v''&nbsp;&nbsp;d''u''&nbsp;d''v''<br />
|-<br />
| width="6%" | E''J''<br />
| width="23%" | = ''u'' ''v'' (d''u'')(d''v'')<br />
| width="23%" | + ''u'' (''v'')(d''u'') d''v''<br />
| width="23%" | + (''u'') ''v'' d''u'' (d''v'')<br />
| width="25%" | + (''u'')(''v'') d''u'' d''v''<br />
|}<br />
|-<br />
|<br />
{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"<br />
| width="6%" | D''J''<br />
| width="23%" | = 0 <math>\cdot</math> (d''u'')(d''v'')<br />
| width="23%" | + ''u'' <math>\cdot</math> (d''u'') d''v''<br />
| width="23%" | + ''v'' <math>\cdot</math> d''u'' (d''v'')<br />
| width="25%" | + ((''u'', ''v'')) d''u'' d''v''<br />
|}<br />
|}<br />
</font><br><br />
<br />
The difference map D''J'' can also be given a ''dispositional'' interpretation. First, recall that <math>\epsilon</math>''J'' exhibits the dispositions to change from anywhere in ''J'' to anywhere at all, and E''J'' enumerates the dispositions to change from anywhere at all to anywhere in ''J''. Next, observe that each of these classes of dispositions may be divided in accordance with the case of ''J'' versus (''J'') that applies to their points of departure and destination, as shown below. Then, since the dispositions corresponding to <math>\epsilon</math>''J'' and E''J'' have in common the dispositions to preserve ''J'', their symmetric difference (<math>\epsilon</math>''J'',&nbsp;E''J'') is made up of all the remaining dispositions, which are in fact disposed to cross the boundary of ''J'' in one direction or the other. In other words, we may conclude that D''J'' expresses the collective disposition to make a definite change with respect to ''J'', no matter what value it holds in the current state of affairs.<br />
<br />
<br><font face="courier new"><br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"<br />
|<br />
{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"<br />
| width="6%" | <math>\epsilon</math>''J''<br />
| width="47%" | = {Dispositions from ''J'' to ''J'' }<br />
| width="47%" | + {Dispositions from ''J'' to (''J'') }<br />
|-<br />
| &nbsp;<br />
|-<br />
| width="6%" | E''J''<br />
| width="47%" | = {Dispositions from ''J'' to ''J'' }<br />
| width="47%" | + {Dispositions from (''J'') to ''J'' }<br />
|-<br />
| &nbsp;<br />
|-<br />
| width="6%" | D''J''<br />
| width="47%" | = (<math>\epsilon</math>''J'', E''J'')<br />
| width="47%" | &nbsp;<br />
|-<br />
| &nbsp;<br />
|-<br />
| width="6%" | D''J''<br />
| width="47%" | = {Dispositions from ''J'' to (''J'') }<br />
| width="47%" | + {Dispositions from (''J'') to ''J'' }<br />
|}<br />
|}<br />
</font><br><br />
<br />
Figures&nbsp;44-a through 44-d illustrate the difference proposition D''J''.<br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 44-a -- Difference Map of J.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 44-a. Difference Map of ''J''&nbsp;&nbsp;(Areal)'''</font></center></p><br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 44-b -- Difference Map of J.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 44-b. Difference Map of ''J''&nbsp;&nbsp;(Bundle)'''</font></center></p><br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 44-c -- Difference Map of J.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 44-c. Difference Map of ''J''&nbsp;&nbsp;(Compact)'''</font></center></p><br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 44-d -- Difference Map of J.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 44-d. Difference Map of ''J''&nbsp;&nbsp;(Digraph)'''</font></center></p><br />
<br />
=====Differential of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
By deploying discourse throughout a calendar, and by giving a date to each of its elements, one does not obtain a definitive hierarchy of precessions and originalities; this hierarchy is never more than relative to the systems of discourse that it sets out to evaluate.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Archaeology of Knowledge'', [Fou, 143]<br />
|}<br />
<br />
Finally, at long last, the differential proposition d''J'' can be gleaned from the difference proposition D''J'' by ranging over the cells of [''u'',&nbsp;''v''] and picking out the linear proposition of [d''u'',&nbsp;d''v''] that is "closest" to the portion of D''J'' that touches on each point. The idea of distance that would give this definition unequivocal sense has been referred to in cautionary quotes, the kind we use to distance ourselves from taking a final position. There are obvious notions of approximation that suggest themselves, but finding one that can be justified as ultimately correct is not as straightforward as it seems.<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
He had drifted into the very heart of the world. From him to the distant beloved was as far as to the next tree.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 144]<br />
|}<br />
<br />
Let us venture a guess about where these developments might be heading. From the present vantage point, it appears that the ultimate answer to the quandary of distances and the question of a fitting measure may be that, rather than having the constitution of an analytic series depend on our familiar notions of approach, proximity, and approximation, it will be found preferable, and perhaps unavoidable, to turn the tables and let the orders of approximation be defined in terms of our favored and operative notions of formal analysis. Only the aftermath of this conversion, if it does converge, could be hoped to prove whether this hortatory form of analysis and the cohort idea of an analytic form — the limitary concept of a self-corrective process and the coefficient concept of a completable product — are truly (in practical reality) the more inceptive and persistent of principles and really (for all practical purposes) the more effective and regulative of ideas.<br />
<br />
Awaiting that determination, I proceed with what seems like the obvious course, and compute d''J'' according to the pattern in Table&nbsp;45.<br />
<br />
<font face="courier new"><br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"<br />
|+ Table 45. Computation of d''J''<br />
|<br />
{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"<br />
| width="6%" | D''J''<br />
| width="25%" | = ''u'' ''v'' ((d''u'')(d''v''))<br />
| width="23%" | + ''u'' (''v'')(d''u'') d''v''<br />
| width="23%" | + (''u'') ''v'' d''u'' (d''v'')<br />
| width="23%" | + (''u'')(''v'') d''u'' d''v''<br />
|-<br />
| width="6%" | &rArr;<br />
|-<br />
| width="6%" | d''J''<br />
| width="25%" | = ''u'' ''v'' (d''u'', d''v'')<br />
| width="23%" | + ''u'' (''v'') d''v''<br />
| width="23%" | + (''u'') ''v'' d''u''<br />
| width="23%" | + (''u'')(''v'') <math>\cdot</math> 0<br />
|}<br />
|}<br />
</font><br><br />
<br />
Figures&nbsp;46-a through 46-d illustrate the proposition d''J'', rounded out in our usual array of prospects. This proposition of E''U''<sup>&nbsp;&bull;</sup> is what we refer to as the (first order) differential of ''J'', and normally regard as ''the'' differential proposition corresponding to ''J''.<br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 46-a -- Differential of J.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 46-a. Differential of ''J''&nbsp;&nbsp;(Areal)'''</font></center></p><br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 46-b -- Differential of J.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 46-b. Differential of ''J''&nbsp;&nbsp;(Bundle)'''</font></center></p><br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 46-c -- Differential of J.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 46-c. Differential of ''J''&nbsp;&nbsp;(Compact)'''</font></center></p><br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 46-d -- Differential of J.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 46-d. Differential of ''J''&nbsp;&nbsp;(Digraph)'''</font></center></p><br />
<br />
=====Remainder of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
<p>I bequeath myself to the dirt to grow from the grass I love,<br><br />
If you want me again look for me under your bootsoles.</p><br />
<br />
<p>You will hardly know who I am or what I mean,<br><br />
But I shall be good health to you nevertheless,<br><br />
And filter and fibre your blood.</p><br />
<br />
<p>Failing to fetch me at first keep encouraged,<br><br />
Missing me one place search another,<br><br />
I stop some where waiting for you</p><br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]<br />
|}<br />
<br />
Let us now recapitulate the story so far. In effect, we have been carrying out a decomposition of the enlarged proposition E''J'' in a series of stages. First, we considered the equation E''J''&nbsp;=&nbsp;<math>\epsilon</math>''J''&nbsp;+&nbsp;D''J'', which was involved in the definition of D''J'' as the difference E''J''&nbsp;&ndash;&nbsp;<math>\epsilon</math>''J''. Next, we contemplated the equation D''J''&nbsp;=&nbsp;d''J''&nbsp;+&nbsp;r''J'', which expresses D''J'' in terms of two components, the differential d''J'' that was just extracted and the residual component r''J''&nbsp;=&nbsp;D''J''&nbsp;&ndash;&nbsp;d''J''. This remaining proposition r''J'' can be computed as shown in Table&nbsp;47.<br />
<br />
<font face="courier new"><br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"<br />
|+ Table 47. Computation of r''J''<br />
|<br />
{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"<br />
| width="6%" | r''J''<br />
| width="5%" | =<br />
| align="center" width="20%" | D''J''<br />
| width="3%" | +<br />
| align="center" width="20%" | d''J''<br />
| width="46%" | &nbsp;<br />
|}<br />
|-<br />
|<br />
{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"<br />
| width="6%" | D''J''<br />
| width="25%" | = ''u'' ''v'' ((d''u'')(d''v''))<br />
| width="23%" | + ''u'' (''v'')(d''u'') d''v''<br />
| width="23%" | + (''u'') ''v'' d''u'' (d''v'')<br />
| width="23%" | + (''u'')(''v'') d''u'' d''v''<br />
|-<br />
| width="6%" | d''J''<br />
| width="25%" | = ''u'' ''v''&nbsp;&nbsp;(d''u'', d''v'')<br />
| width="23%" | + ''u'' (''v'') d''v''<br />
| width="23%" | + (''u'') ''v'' d''u''<br />
| width="23%" | + (''u'')(''v'') <math>\cdot</math> 0<br />
|}<br />
|-<br />
|<br />
{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"<br />
| width="6%" | r''J''<br />
| width="25%" | = ''u'' ''v''&nbsp;&nbsp;&nbsp;d''u'' d''v''<br />
| width="23%" | + ''u'' (''v'') d''u'' d''v''<br />
| width="23%" | + (''u'') ''v'' d''u'' d''v''<br />
| width="23%" | + (''u'')(''v'') d''u'' d''v''<br />
|}<br />
|}<br />
</font><br><br />
<br />
As it happens, the remainder r''J'' falls under the description of a second order differential r''J''&nbsp;=&nbsp;d<sup>2</sup>''J''. This means that the expansion of E''J'' in the form:<br />
<br />
:{| cellpadding=2<br />
| E''J''<br />
| =<br />
| <math>\epsilon</math>''J''<br />
| +<br />
| D''J''<br />
|-<br />
| &nbsp;<br />
| =<br />
| <math>\epsilon</math>''J''<br />
| +<br />
| d''J''<br />
| +<br />
| r''J''<br />
|-<br />
| &nbsp;<br />
| =<br />
| d<sup>0</sup>''J''<br />
| +<br />
| d<sup>1</sup>''J''<br />
| +<br />
| d<sup>2</sup>''J''<br />
|}<br />
<br />
which is nothing other than the propositional analogue of a Taylor series, is a decomposition that terminates in a finite number of steps.<br />
<br />
Figures&nbsp;48-a through 48-d illustrate the proposition r''J''&nbsp;=&nbsp;d<sup>2</sup>''J'', which forms the remainder map of ''J'' and also, in this instance, the second order differential of ''J''.<br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 48-a -- Remainder of J.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 48-a. Remainder of ''J''&nbsp;&nbsp;(Areal)'''</font></center></p><br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 48-b -- Remainder of J.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 48-b. Remainder of ''J''&nbsp;&nbsp;(Bundle)'''</font></center></p><br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 48-c -- Remainder of J.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 48-c. Remainder of ''J''&nbsp;&nbsp;(Compact)'''</font></center></p><br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 48-d -- Remainder of J.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 48-d. Remainder of ''J''&nbsp;&nbsp;(Digraph)'''</font></center></p><br />
<br />
=====Summary of Conjunction=====<br />
<br />
To establish a convenient reference point for further discussion, Table&nbsp;49 summarizes the operator actions that have been computed for the form of conjunction, as exemplified by the proposition ''J''.<br />
<br />
<font face="courier new"><br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"<br />
|+ Table 49. Computation Summary for ''J''<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| <math>\epsilon</math>''J''<br />
| = || ''uv'' || <math>\cdot</math> || 1<br />
| + || ''u''(''v'') || <math>\cdot</math> || 0<br />
| + || (''u'')''v'' || <math>\cdot</math> || 0<br />
| + || (''u'')(''v'') || <math>\cdot</math> || 0<br />
|-<br />
| E''J''<br />
| = || ''uv'' || <math>\cdot</math> || (d''u'')(d''v'')<br />
| + || ''u''(''v'') || <math>\cdot</math> || (d''u'')d''v''<br />
| + || (''u'')''v'' || <math>\cdot</math> || d''u''(d''v'')<br />
| + || (''u'')(''v'') || <math>\cdot</math> || d''u'' d''v''<br />
|-<br />
| D''J''<br />
| = || ''uv'' || <math>\cdot</math> || ((d''u'')(d''v''))<br />
| + || ''u''(''v'') || <math>\cdot</math> || (d''u'')d''v''<br />
| + || (''u'')''v'' || <math>\cdot</math> || d''u''(d''v'')<br />
| + || (''u'')(''v'') || <math>\cdot</math> || d''u'' d''v''<br />
|-<br />
| d''J''<br />
| = || ''uv'' || <math>\cdot</math> || (d''u'', d''v'')<br />
| + || ''u''(''v'') || <math>\cdot</math> || d''v''<br />
| + || (''u'')''v'' || <math>\cdot</math> || d''u''<br />
| + || (''u'')(''v'') || <math>\cdot</math> || 0<br />
|-<br />
| r''J''<br />
| = || ''uv'' || <math>\cdot</math> || d''u'' d''v''<br />
| + || ''u''(''v'') || <math>\cdot</math> || d''u'' d''v''<br />
| + || (''u'')''v'' || <math>\cdot</math> || d''u'' d''v''<br />
| + || (''u'')(''v'') || <math>\cdot</math> || d''u'' d''v''<br />
|}<br />
|}<br />
</font><br><br />
<br />
====Analytic Series : Coordinate Method====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
And if he is told that something ''is'' the way it is, then he thinks: Well, it could probably just as easily be some other way. So the sense of possibility might be defined outright as the capacity to think how everything could "just as easily" be, and to attach no more importance to what is than to what is not.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 12]<br />
|}<br />
<br />
Table&nbsp;50 exhibits a truth table method for computing the analytic series (or the differential expansion) of a proposition in terms of coordinates.<br />
<br />
<br><br />
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"<br />
|+ Table 50. Computation of an Analytic Series in Terms of Coordinates<br />
|<br />
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| ''u''<br />
| ''v''<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| d''u''<br />
| d''v''<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| ''u''<font face="courier new">’</font><br />
| ''v''<font face="courier new">’</font><br />
|}<br />
|-<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|-<br />
| &nbsp; || &nbsp;<br />
|-<br />
| &nbsp; || &nbsp;<br />
|-<br />
| &nbsp; || &nbsp;<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|-<br />
| 0 || 1<br />
|-<br />
| 1 || 0<br />
|-<br />
| 1 || 1<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|-<br />
| 0 || 1<br />
|-<br />
| 1 || 0<br />
|-<br />
| 1 || 1<br />
|}<br />
|-<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 1<br />
|-<br />
| &nbsp; || &nbsp;<br />
|-<br />
| &nbsp; || &nbsp;<br />
|-<br />
| &nbsp; || &nbsp;<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|-<br />
| 0 || 1<br />
|-<br />
| 1 || 0<br />
|-<br />
| 1 || 1<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 1<br />
|-<br />
| 0 || 0<br />
|-<br />
| 1 || 1<br />
|-<br />
| 1 || 0<br />
|}<br />
|-<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 1 || 0<br />
|-<br />
| &nbsp; || &nbsp;<br />
|-<br />
| &nbsp; || &nbsp;<br />
|-<br />
| &nbsp; || &nbsp;<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|-<br />
| 0 || 1<br />
|-<br />
| 1 || 0<br />
|-<br />
| 1 || 1<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 1 || 0<br />
|-<br />
| 1 || 1<br />
|-<br />
| 0 || 0<br />
|-<br />
| 0 || 1<br />
|}<br />
|-<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 1 || 1<br />
|-<br />
| &nbsp; || &nbsp;<br />
|-<br />
| &nbsp; || &nbsp;<br />
|-<br />
| &nbsp; || &nbsp;<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 1 || 1<br />
|-<br />
| 1 || 0<br />
|-<br />
| 0 || 1<br />
|-<br />
| 0 || 0<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|-<br />
| 0 || 1<br />
|-<br />
| 1 || 0<br />
|-<br />
| 1 || 1<br />
|}<br />
|}<br />
|<br />
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| <math>\epsilon</math>''J''<br />
| E''J''<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| D''J''<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| d''J''<br />
| d<sup>2</sup>''J''<br />
|}<br />
|-<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|-<br />
| &nbsp; || 0<br />
|-<br />
| &nbsp; || 0<br />
|-<br />
| &nbsp; || 1<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0<br />
|-<br />
| 0<br />
|-<br />
| 0<br />
|-<br />
| 1<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|-<br />
| 0 || 0<br />
|-<br />
| 0 || 0<br />
|-<br />
| 0 || 1<br />
|}<br />
|-<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|-<br />
| &nbsp; || 0<br />
|-<br />
| &nbsp; || 1<br />
|-<br />
| &nbsp; || 0<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0<br />
|-<br />
| 0<br />
|-<br />
| 1<br />
|-<br />
| 0<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|-<br />
| 0 || 0<br />
|-<br />
| 1 || 0<br />
|-<br />
| 1 || 1<br />
|}<br />
|-<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|-<br />
| &nbsp; || 1<br />
|-<br />
| &nbsp; || 0<br />
|-<br />
| &nbsp; || 0<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0<br />
|-<br />
| 1<br />
|-<br />
| 0<br />
|-<br />
| 0<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|-<br />
| 1 || 0<br />
|-<br />
| 0 || 0<br />
|-<br />
| 1 || 1<br />
|}<br />
|-<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 1<br />
|-<br />
| &nbsp; || 0<br />
|-<br />
| &nbsp; || 0<br />
|-<br />
| &nbsp; || 0<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0<br />
|-<br />
| 1<br />
|-<br />
| 1<br />
|-<br />
| 1<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|-<br />
| 1 || 0<br />
|-<br />
| 1 || 0<br />
|-<br />
| 0 || 1<br />
|}<br />
|}<br />
|}<br />
<br><br />
<br />
The first six columns of the Table, taken as a whole, represent the variables of a construct that I describe as the ''contingent universe'' [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'',&nbsp;''u''&prime;,&nbsp;''v''&prime;&nbsp;], or the bundle of ''contingency spaces'' [d''u'',&nbsp;d''v'',&nbsp;''u''&prime;,&nbsp;''v''&prime;&nbsp;] over the universe [''u'',&nbsp;''v'']. Their placement to the left of the double bar indicates that all of them amount to independent variables, but there is a co-dependency among them, as described<br />
by the following equations:<br />
<br />
<br><font face="courier new"><br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"<br />
|<br />
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| &nbsp; || ''u''’ || = || ''u'' + d''u'' || = || (''u'', d''u'') || &nbsp;<br />
|-<br />
| &nbsp; || ''v''’ || = || ''v'' + d''u'' || = || (''v'', d''v'') || &nbsp;<br />
|}<br />
|}<br />
</font><br><br />
<br />
These relations correspond to the formal substitutions that are made in defining E''J'' and D''J''. For now, the whole rigamarole of contingency spaces can be regarded as a technical device for achieving the effect of these substitutions, adapted to a setting where functional compositions and other symbolic manipulations are difficult to contemplate and execute.<br />
<br />
The five columns to the right of the double bar in Table&nbsp;50 contain the values of the dependent variables {<math>\epsilon</math>''J'',&nbsp;E''J'',&nbsp;D''J'',&nbsp;d''J'',&nbsp;d<sup>2</sup>''J''}. These are normally interpreted as values of functions W''J''&nbsp;:&nbsp;E''U''&nbsp;&rarr;&nbsp;'''B''' or as values of propositions in the extended universe [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''], but the dependencies prevailing in the contingent universe make it possible to regard these same final values as arising via functions on alternative lists of arguments, say, ‹''u'',&nbsp;''v'',&nbsp;''u''&prime;,&nbsp;''v''&prime;›.<br />
<br />
The column for <math>\epsilon</math>''J'' is computed as ''J''‹''u'',&nbsp;''v''›&nbsp;=&nbsp;''uv''. This, along with the columns for ''u'' and ''v'', illustrates the Table's ''structure-sharing'' scheme, listing only the initial entries of each constant block.<br />
<br />
The column for E''J'' is computed by means of the following chain of identities, where the contingent variables ''u''&prime; and ''v''&prime; are defined as ''u''&prime;&nbsp;=&nbsp;''u''&nbsp;+&nbsp;d''u'' and ''v''&prime;&nbsp;=&nbsp;''v''&nbsp;+&nbsp;d''v''.<br />
<br />
<br><font face="courier new"><br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"<br />
|<br />
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| E''J''‹''u'', ''v'', d''u'', d''v''›<br />
| =<br />
| ''J''‹''u'' + d''u'', ''v'' + d''v''›<br />
| =<br />
| ''J''‹''u''’, ''v''’›<br />
|}<br />
|}<br />
</font><br><br />
<br />
This makes it easy to determine E''J'' by inspection, computing the conjunction ''J''‹''u''&prime;,&nbsp;''v''&prime;›&nbsp;=&nbsp;''u''&prime;&nbsp;''v''&prime; from the columns headed ''u''&prime; and ''v''&prime;. Since all of these forms express the same proposition E''J'' in E''U''<sup>&nbsp;&bull;</sup>, the dependence on d''u'' and d''v'' is still present but merely left implicit in the final variant ''J''‹''u''&prime;,&nbsp;''v''&prime;›.<br />
<br />
* NB. On occasion, it is tempting to use the further notation ''J''&prime;‹''u'',&nbsp;''v''›&nbsp;=&nbsp;''J''‹''u''&prime;,&nbsp;''v''&prime;›, especially to suggest a transformation that acts on whole propositions, for example, taking the proposition ''J'' into the proposition ''J''&prime;&nbsp;=&nbsp;E''J''. The prime [&prime;] then signifies an action that is mediated by a field of choices, namely, the values that are picked out for the contingent variables in sweeping through the initial universe. But this heaps an unwieldy lot of construed intentions on a rather slight character, and puts too high a premium on the constant correctness of its interpretation. In practice, therefore, it is best to avoid this usage.<br />
<br />
Given the values of <math>\epsilon</math>''J'' and E''J'', the columns for the remaining functions can be filled in quickly. The difference map is computed according to the relation D''J''&nbsp;=&nbsp;<math>\epsilon</math>''J''&nbsp;+&nbsp;E''J''. The first order differential d''J'' is found by looking in each block of constant ‹''u'',&nbsp;''v''› and choosing the linear function of ‹d''u'',&nbsp;d''v''› that best approximates D''J'' in that block. Finally, the remainder is computed as r''J''&nbsp;=&nbsp;D''J''&nbsp;+&nbsp;d''J'', in this case yielding the second order differential d<sup>2</sup>''J''.<br />
<br />
====Analytic Series : Recap====<br />
<br />
Let us now summarize the results of Table&nbsp;50 by writing down for each column, and for each block of constant ‹''u'',&nbsp;''v''›, a reasonably canonical symbolic expression for the function of ‹d''u'',&nbsp;d''v''› that appears there. The synopsis formed in this way is presented in Table&nbsp;51. As one has a right to expect, it confirms the results that were obtained previously by operating solely in terms of the formal calculus.<br />
<br />
<font face="courier new"><br />
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"<br />
|+ Table 51. Computation of an Analytic Series in Symbolic Terms<br />
|<br />
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| ''u'' || ''v''<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| ''J''<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| E''J''<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| D''J''<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| d''J''<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| d<sup>2</sup>''J''<br />
|}<br />
|-<br />
|<br />
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|-<br />
| 0 || 1<br />
|-<br />
| 1 || 0<br />
|-<br />
| 1 || 1<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0<br />
|-<br />
| 0<br />
|-<br />
| 0<br />
|-<br />
| 1<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| &nbsp;d''u''&nbsp;&nbsp;d''v''&nbsp;<br />
|-<br />
| &nbsp;d''u''&nbsp;(d''v'')<br />
|-<br />
| (d''u'')&nbsp;d''v''&nbsp;<br />
|-<br />
| (d''u'')(d''v'')<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| &nbsp;&nbsp;d''u''&nbsp;&nbsp;d''v''&nbsp;&nbsp;<br />
|-<br />
| &nbsp;&nbsp;d''u''&nbsp;(d''v'')&nbsp;<br />
|-<br />
| &nbsp;(d''u'')&nbsp;d''v''&nbsp;&nbsp;<br />
|-<br />
| ((d''u'')(d''v''))<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| ()<br />
|-<br />
| d''u''<br />
|-<br />
| d''v''<br />
|-<br />
| (d''u'', d''v'')<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| d''u'' d''v''<br />
|-<br />
| d''u'' d''v''<br />
|-<br />
| d''u'' d''v''<br />
|-<br />
| d''u'' d''v''<br />
|}<br />
|}<br />
</font><br><br />
<br />
Figures&nbsp;52 and 53 provide a quick overview of the analysis performed so far, giving the successive decompositions of E''J''&nbsp;=&nbsp;''J''&nbsp;+&nbsp;D''J'' and D''J''&nbsp;=&nbsp;d''J''&nbsp;+&nbsp;r''J'' in two different styles of diagram.<br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 52 -- Decomposition of EJ.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 52. Decomposition of E''J'''''</font></center></p><br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 53 -- Decomposition of DJ.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 53. Decomposition of D''J'''''</font></center></p><br />
<br />
====Terminological Interlude====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Lastly, my attention was especially attracted, not so much to the scene, as to the mirrors that produced it. These mirrors were broken in parts. Yes, they were marked and scratched; they had been "starred", in spite of their solidity &hellip;<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 230]<br />
|}<br />
<br />
At this point several issues of terminology have accrued enough substance to intrude on our discussion. The remarks of this Section are intended to accomplish two goals. First, I call attention to important aspects of the previous series of Figures, translating into literal terms what they depict in iconic forms, and I restress the most important structural elements that they indicate. Next, I prepare the way for taking on more complex examples of transformations, whose target universes have more than a single dimension.<br />
<br />
In talking about the actions of operators it is important to keep in mind the distinctions between the operators per se, their operands, and their results. Furthermore, in working with composite forms of operators W&nbsp;=&nbsp;‹W<sub>1</sub>,&nbsp;&hellip;,&nbsp;W<sub>''n''</sub>›&nbsp;, transformations F&nbsp;=&nbsp;‹F<sub>1</sub>,&nbsp;&hellip;,&nbsp;F<sub>''n''</sub>›&nbsp;, and target domains ''X''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;[''x''<sub>1</sub>,&nbsp;&hellip;,&nbsp;''x''<sub>''n''</sub>], we need to preserve a clear distinction between the compound entity of each given type and any one of its separate components. It is curious, given the usefulness of the concepts ''operator'' and ''operand'', that we seem to lack a generic term, formed on the same root, for the corresponding result of an operation. Following the obvious paradigm would lead on to words like ''opus'', ''opera'', and ''operant'', but these words are too affected with clang associations to work well at present, though they might be adapted in time. One current usage gets around this problem by using the substantive ''map'' as a systematic epithet to express the result of each operator's action. I am following this practice as far as possible, for example, using the phrase ''tangent map'' to denote the end product of the tangent functor acting on its operand map.<br />
<br />
* Scholium. See [JGH, 6-9] for a good account of tangent functors and tangent maps in ordinary analysis, and for examples of their use in mechanics. This work as a whole is a model of clarity in applying functorial principles to problems in physical dynamics.<br />
<br />
Whenever we focus on isolated propositions, on single components of composite operators, or on the portions of transformations that have 1-dimensional ranges, we are free to shift between the native form of a proposition ''J''&nbsp;:&nbsp;''U''&nbsp;&rarr;&nbsp;'''B''' and the thematized form of a mapping ''J''&nbsp;:&nbsp;''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;[''x''] without much trouble. In these cases we are able to tolerate a higher degree of ambiguity about the precise nature of the input and output domains of an operator than we otherwise might. For example, in the preceding treatment of the example ''J'', and for each operator W in the set {<math>\epsilon</math>,&nbsp;<math>\eta</math>,&nbsp;E,&nbsp;D,&nbsp;d,&nbsp;r}, both the operand ''J'' and the result W''J'' could be viewed in either one of two ways. On the one hand, we could regard them as propositions ''J''&nbsp;:&nbsp;''U''&nbsp;&rarr;&nbsp;'''B''' and W''J''&nbsp;:&nbsp;E''U''&nbsp;&rarr;&nbsp;''B'', ignoring the qualitative distinction between the range [''x'']&nbsp;<math>\cong</math>&nbsp;'''B''' of <math>\epsilon</math>''J'' and the range [d''x'']&nbsp;<math>\cong</math>&nbsp;'''D''' of the other types of W''J''. This is what we usually do when we content ourselves simply with coloring in regions of venn diagrams. On the other hand, we could view these entities as maps ''J''&nbsp;:&nbsp;''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;[''x'']&nbsp;=&nbsp;''X''<sup>&nbsp;&bull;</sup> and <math>\epsilon</math>''J''&nbsp;:&nbsp;E''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;[''x'']&nbsp;&sube;&nbsp;E''X''<sup>&nbsp;&bull;</sup> or W''J''&nbsp;:&nbsp;E''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;[d''x'']&nbsp;&sube;&nbsp;E''X''<sup>&nbsp;&bull;</sup>, in which case the qualitative characters of the output features are not allowed to go without saying, nor thus at the risk of being forgotten.<br />
<br />
At the beginning of this Division I recast the natural form of a proposition ''J''&nbsp;:&nbsp;''U''&nbsp;&rarr;&nbsp;'''B''' into the thematic role of a transformation ''J''&nbsp;:&nbsp;''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;[''x''], where ''x'' was a variable recruited to express the newly independent ¢(''J''). However, in my computations and representations of operator actions I immediately lapsed back to viewing the results as native elements of the extended universe E''U''<sup>&nbsp;&bull;</sup>, in other words, as propositions W''J''&nbsp;:&nbsp;E''U''&nbsp;&rarr;&nbsp;'''B''', where W ranged over the set {<math>\epsilon</math>,&nbsp;E,&nbsp;D,&nbsp;d,&nbsp;r}. That is as it should be. In fact, I have worked hard to devise a language that gives us all of these competing advantages, the flexibility to exchange terms and types that bear equal information value, and the capacity to reflect as quickly and as wittingly as a controlled reflex on the fibers of our propositions, independently of whether they express amusements, beliefs, or conjectures.<br />
<br />
As we take on target spaces of increasing dimension, however, these types of confusions (and confusions of types) become less and less permissible. For this reason, Tables&nbsp;54 and 55 present a rather detailed summary of the notation and the terminology that I am using here, as applied to the case of ''J''&nbsp;=&nbsp;''uv''. The rationale of these Tables is not so much to train more elephant guns on this poor drosophila of an example, but to establish the general paradigm with enough solidity to bear the weight of abstraction that is coming on down the road.<br />
<br />
Table&nbsp;54 provides basic notation and descriptive information for the objects and operators that are used used in this Example, giving the generic type (or broadest defined type) for each entity. Here, the operators <font face=georgia>'''W'''</font> in {<font face=georgia>'''e'''</font>,&nbsp;<font face=georgia>'''E'''</font>,&nbsp;<font face=georgia>'''D'''</font>,&nbsp;<font face=georgia>'''d'''</font>,&nbsp;<font face=georgia>'''r'''</font>} and their components W in {<math>\epsilon</math>,&nbsp;<math>\eta</math>,&nbsp;E,&nbsp;D,&nbsp;d,&nbsp;r} both have the same broad type <font face=georgia>'''W'''</font>,&nbsp;W&nbsp;:&nbsp;(''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup>)&nbsp;&rarr;&nbsp;(E''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''X''<sup>&nbsp;&bull;</sup>), as would be expected of operators that map transformations ''J''&nbsp;:&nbsp;''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup> to extended transformations <font face=georgia>'''W'''</font>''J'',&nbsp;W''J''&nbsp;:&nbsp;E''U<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''X''<sup>&nbsp;&bull;</sup>.<br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"<br />
|+ '''Table 54. Cast of Characters: Expansive Subtypes of Objects and Operators'''<br />
|- style="background:paleturquoise"<br />
! Item<br />
! Notation<br />
! Description<br />
! Type<br />
|-<br />
| ''U''<sup>&nbsp;&bull;</sup><br />
| = [''u'', ''v'']<br />
| Source Universe<br />
| ['''B'''<sup>2</sup>]<br />
|-<br />
| ''X''<sup>&nbsp;&bull;</sup><br />
| = [''x'']<br />
| Target Universe<br />
| ['''B'''<sup>1</sup>]<br />
|-<br />
| E''U''<sup>&nbsp;&bull;</sup><br />
| = [''u'', ''v'', d''u'', d''v'']<br />
| Extended Source Universe<br />
| ['''B'''<sup>2</sup> &times; '''D'''<sup>2</sup>]<br />
|-<br />
| E''X''<sup>&nbsp;&bull;</sup><br />
| = [''x'', d''x'']<br />
| Extended Target Universe<br />
| ['''B'''<sup>1</sup> &times; '''D'''<sup>1</sup>]<br />
|-<br />
| ''J''<br />
| ''J'' : ''U'' &rarr; '''B'''<br />
| Proposition<br />
| ('''B'''<sup>2</sup> &rarr; '''B''') &isin; ['''B'''<sup>2</sup>]<br />
|-<br />
| ''J''<br />
| ''J'' : ''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup><br />
| Transformation, or Mapping<br />
| ['''B'''<sup>2</sup>] &rarr; ['''B'''<sup>1</sup>]<br />
|-<br />
| valign="top" |<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| W<br />
|}<br />
| valign="top" |<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| W&nbsp;:<br />
|-<br />
| ''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''U''<sup>&nbsp;&bull;</sup>&nbsp;,<br />
|-<br />
| ''X''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''X''<sup>&nbsp;&bull;</sup>&nbsp;,<br />
|-<br />
| (''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup>)<br />
|-<br />
| &rarr;<br />
|-<br />
| (E''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''X''<sup>&nbsp;&bull;</sup>)&nbsp;,<br />
|-<br />
| for each W in the set:<br />
|-<br />
| {<math>\epsilon</math>,&nbsp;<math>\eta</math>,&nbsp;E,&nbsp;D,&nbsp;d}<br />
|}<br />
| valign="top" |<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| Operator<br />
|}<br />
| valign="top" |<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100"<br />
| &nbsp;<br />
|-<br />
| ['''B'''<sup>2</sup>] &rarr; ['''B'''<sup>2</sup> &times; '''D'''<sup>2</sup>]&nbsp;,<br />
|-<br />
| ['''B'''<sup>1</sup>] &rarr; ['''B'''<sup>1</sup> &times; '''D'''<sup>1</sup>]&nbsp;,<br />
|-<br />
| (['''B'''<sup>2</sup>] &rarr; ['''B'''<sup>1</sup>])<br />
|-<br />
| &rarr;<br />
|-<br />
| (['''B'''<sup>2</sup> &times; '''D'''<sup>2</sup>] &rarr; ['''B'''<sup>1</sup> &times; '''D'''<sup>1</sup>])<br />
|-<br />
| &nbsp;<br />
|-<br />
| &nbsp;<br />
|}<br />
|-<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <math>\epsilon</math><br />
|-<br />
| <math>\eta</math><br />
|-<br />
| E<br />
|-<br />
| D<br />
|-<br />
| d<br />
|}<br />
| valign="top" | &nbsp;<br />
| colspan="2" |<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:60%"<br />
| Tacit Extension Operator || <math>\epsilon</math><br />
|-<br />
| Trope Extension Operator || <math>\eta</math><br />
|-<br />
| Enlargement Operator || E<br />
|-<br />
| Difference Operator || D<br />
|-<br />
| Differential Operator || d<br />
|}<br />
|-<br />
| valign="top" |<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <font face=georgia>'''W'''</font><br />
|}<br />
| valign="top" |<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <font face=georgia>'''W'''</font>&nbsp;:<br />
|-<br />
| ''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;<font face=georgia>'''T'''</font>''U''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;E''U''<sup>&nbsp;&bull;</sup>&nbsp;,<br />
|-<br />
| ''X''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;<font face=georgia>'''T'''</font>''X''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;E''X''<sup>&nbsp;&bull;</sup>&nbsp;,<br />
|-<br />
| (''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup>)<br />
|-<br />
| &rarr;<br />
|-<br />
| (<font face=georgia>'''T'''</font>''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;<font face=georgia>'''T'''</font>''X''<sup>&nbsp;&bull;</sup>)&nbsp;,<br />
|-<br />
| for each <font face=georgia>'''W'''</font> in the set:<br />
|-<br />
| {<font face=georgia>'''e'''</font>,&nbsp;<font face=georgia>'''E'''</font>,&nbsp;<font face=georgia>'''D'''</font>,&nbsp;<font face=georgia>'''T'''</font>}<br />
|}<br />
| valign="top" |<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| Operator<br />
|}<br />
| valign="top" |<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100"<br />
| &nbsp;<br />
|-<br />
| ['''B'''<sup>2</sup>] &rarr; ['''B'''<sup>2</sup> &times; '''D'''<sup>2</sup>]&nbsp;,<br />
|-<br />
| ['''B'''<sup>1</sup>] &rarr; ['''B'''<sup>1</sup> &times; '''D'''<sup>1</sup>]&nbsp;,<br />
|-<br />
| (['''B'''<sup>2</sup>] &rarr; ['''B'''<sup>1</sup>])<br />
|-<br />
| &rarr;<br />
|-<br />
| (['''B'''<sup>2</sup> &times; '''D'''<sup>2</sup>] &rarr; ['''B'''<sup>1</sup> &times; '''D'''<sup>1</sup>])<br />
|-<br />
| &nbsp;<br />
|-<br />
| &nbsp;<br />
|}<br />
|-<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <font face=georgia>'''e'''</font><br />
|-<br />
| <font face=georgia>'''E'''</font><br />
|-<br />
| <font face=georgia>'''D'''</font><br />
|-<br />
| <font face=georgia>'''T'''</font><br />
|}<br />
| valign="top" | &nbsp;<br />
| colspan="2" |<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:60%"<br />
| Radius Operator || <font face=georgia>'''e'''</font>&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;<math>\eta</math>›<br />
|-<br />
| Secant Operator || <font face=georgia>'''E'''</font>&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;E›<br />
|-<br />
| Chord Operator || <font face=georgia>'''D'''</font>&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;D›<br />
|-<br />
| Tangent Functor || <font face=georgia>'''T'''</font>&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;d›<br />
|}<br />
|}<br><br />
<br />
Table&nbsp;55 supplies a more detailed outline of terminology for operators and their results. Here, I list the restrictive subtype (or narrowest defined subtype) that applies to each entity, and I indicate across the span of the Table the whole spectrum of alternative types that color the interpretation of each symbol. Accordingly, each of the component operator maps W''J'', since their ranges are 1-dimensional (of type '''B'''<sup>1</sup> or '''D'''<sup>1</sup>), can be regarded either as propositions W''J''&nbsp;:&nbsp;E''U''&nbsp;&rarr;&nbsp;'''B''' or as logical transformations W''J''&nbsp;:&nbsp;E''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup>. As a rule, the plan of the Table allows us to name each entry by detaching the adjective at the left of its row and prefixing it to the generic noun at the top of its column. In one case, however, it is customary to depart from this scheme. Because the phrase ''differential proposition'', applied to the result d''J''&nbsp;:&nbsp;E''U''&nbsp;&rarr;&nbsp;'''D''', does not distinguish it from the general run of differential propositions ''G''&nbsp;:&nbsp;E''U''&nbsp;&rarr;&nbsp;'''B''', it is usual to single out d''J'' as the ''tangent proposition'' of ''J''.<br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"<br />
|+ '''Table 55. Synopsis of Terminology: Restrictive and Alternative Subtypes'''<br />
|- style="background:paleturquoise"<br />
! &nbsp;<br />
! Operator<br />
! Proposition<br />
! Map<br />
|-<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| Tacit<br />
|-<br />
| Extension<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <math>\epsilon</math> :<br />
|-<br />
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,<br />
|-<br />
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>)<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <math>\epsilon</math>''J'' :<br />
|-<br />
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''B'''<br />
|-<br />
| '''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>&nbsp;&rarr;&nbsp;'''B'''<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <math>\epsilon</math>''J'' :<br />
|-<br />
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'']<br />
|-<br />
| ['''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''B'''<sup>1</sup>]<br />
|}<br />
|-<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| Trope<br />
|-<br />
| Extension<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <math>\eta</math> :<br />
|-<br />
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,<br />
|-<br />
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; d''X''<sup>&nbsp;&bull;</sup>)<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <math>\eta</math>''J'' :<br />
|-<br />
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''<br />
|-<br />
| '''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>&nbsp;&rarr;&nbsp;'''D'''<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <math>\eta</math>''J'' :<br />
|-<br />
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'']<br />
|-<br />
| ['''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>1</sup>]<br />
|}<br />
|-<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| Enlargement<br />
|-<br />
| Operator<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| E :<br />
|-<br />
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,<br />
|-<br />
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; d''X''<sup>&nbsp;&bull;</sup>)<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| E''J'' :<br />
|-<br />
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''<br />
|-<br />
| '''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>&nbsp;&rarr;&nbsp;'''D'''<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| E''J'' :<br />
|-<br />
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'']<br />
|-<br />
| ['''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>1</sup>]<br />
|}<br />
|-<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| Difference<br />
|-<br />
| Operator<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| D :<br />
|-<br />
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,<br />
|-<br />
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; d''X''<sup>&nbsp;&bull;</sup>)<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| D''J'' :<br />
|-<br />
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''<br />
|-<br />
| '''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>&nbsp;&rarr;&nbsp;'''D'''<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| D''J'' :<br />
|-<br />
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'']<br />
|-<br />
| ['''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>1</sup>]<br />
|}<br />
|-<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| Differential<br />
|-<br />
| Operator<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| d :<br />
|-<br />
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,<br />
|-<br />
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; d''X''<sup>&nbsp;&bull;</sup>)<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| d''J'' :<br />
|-<br />
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''<br />
|-<br />
| '''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>&nbsp;&rarr;&nbsp;'''D'''<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| d''J'' :<br />
|-<br />
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'']<br />
|-<br />
| ['''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>1</sup>]<br />
|}<br />
|-<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| Remainder<br />
|-<br />
| Operator<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| r :<br />
|-<br />
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,<br />
|-<br />
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; d''X''<sup>&nbsp;&bull;</sup>)<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| r''J'' :<br />
|-<br />
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''<br />
|-<br />
| '''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>&nbsp;&rarr;&nbsp;'''D'''<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| r''J'' :<br />
|-<br />
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'']<br />
|-<br />
| ['''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>1</sup>]<br />
|}<br />
|-<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| Radius<br />
|-<br />
| Operator<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <font face=georgia>'''e'''</font> = ‹<math>\epsilon</math>, <math>\eta</math>› :<br />
|-<br />
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,<br />
|-<br />
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>)<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| &nbsp;<br />
|-<br />
| &nbsp;<br />
|-<br />
| &nbsp;<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <font face=georgia>'''e'''</font>''J'' :<br />
|-<br />
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'',&nbsp;d''x'']<br />
|-<br />
| ['''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''B'''&nbsp;&times;&nbsp;'''D''']<br />
|}<br />
|-<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| Secant<br />
|-<br />
| Operator<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <font face=georgia>'''E'''</font> = ‹<math>\epsilon</math>, E› :<br />
|-<br />
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,<br />
|-<br />
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>)<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| &nbsp;<br />
|-<br />
| &nbsp;<br />
|-<br />
| &nbsp;<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <font face=georgia>'''E'''</font>''J'' :<br />
|-<br />
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'',&nbsp;d''x'']<br />
|-<br />
| ['''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''B'''&nbsp;&times;&nbsp;'''D''']<br />
|}<br />
|-<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| Chord<br />
|-<br />
| Operator<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <font face=georgia>'''D'''</font> = ‹<math>\epsilon</math>, D› :<br />
|-<br />
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,<br />
|-<br />
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>)<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| &nbsp;<br />
|-<br />
| &nbsp;<br />
|-<br />
| &nbsp;<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <font face=georgia>'''D'''</font>''J'' :<br />
|-<br />
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'',&nbsp;d''x'']<br />
|-<br />
| ['''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''B'''&nbsp;&times;&nbsp;'''D''']<br />
|}<br />
|-<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| Tangent<br />
|-<br />
| Functor<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <font face=georgia>'''T'''</font> = ‹<math>\epsilon</math>, d› :<br />
|-<br />
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,<br />
|-<br />
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>)<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| d''J'' :<br />
|-<br />
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''<br />
|-<br />
| '''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>&nbsp;&rarr;&nbsp;'''D'''<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <font face=georgia>'''T'''</font>''J'' :<br />
|-<br />
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'',&nbsp;d''x'']<br />
|-<br />
| ['''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''B'''&nbsp;&times;&nbsp;'''D''']<br />
|}<br />
|}<br><br />
<br />
====End of Perfunctory Chatter : Time to Roll the Clip!====<br />
<br />
Two steps remain to finish the analysis of ''J'' that I began so long ago. First, I need to paste the accumulated heap of flat pictures into the frames of transformations, filling out the shapes of the operator maps <font face=georgia>'''W'''</font>''J''&nbsp;:&nbsp;E''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''X''<sup>&nbsp;&bull;</sup>. This scheme is executed in two styles, using the ''areal views'' in Figures&nbsp;56-a and the ''box views'' in Figures&nbsp;56-b. Finally, in Figures&nbsp;57-1 to 57-4 I put all the pieces together to construct the full operator diagrams for <font face=georgia>'''W'''</font>&nbsp;:&nbsp;''J''&nbsp;&rarr;&nbsp;<font face=georgia>'''W'''</font>''J''. There is a large amount of redundancy in these three series of figures. At this early stage of exposition I thought that it would be better not to tax the reader's imagination, and to guarantee that the author, at least, has worked through the relevant exercises. I hope the reader will excuse the flagrant use of space and try to view these snapshots as successive frames in the animation of logic that they are meant to become.<br />
<br />
=====Operator Maps : Areal Views=====<br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 56-a1 -- Radius Map of J.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 56-a1. Radius Map of the Conjunction ''J'' = ''uv'''''</font></center></p><br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 56-a2 -- Secant Map of J.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 56-a2. Secant Map of the Conjunction ''J'' = ''uv'''''</font></center></p><br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 56-a3 -- Chord Map of J.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 56-a3. Chord Map of the Conjunction ''J'' = ''uv'''''</font></center></p><br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 56-a4 -- Tangent Map of J.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 56-a4. Tangent Map of the Conjunction ''J'' = ''uv'''''</font></center></p><br />
<br />
=====Operator Maps : Box Views=====<br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 56-b1 -- Radius Map of J.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 56-b1. Radius Map of the Conjunction ''J'' = ''uv'''''</font></center></p><br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 56-b2 -- Secant Map of J.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 56-b2. Secant Map of the Conjunction ''J'' = ''uv'''''</font></center></p><br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 56-b3 -- Chord Map of J.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 56-b3. Chord Map of the Conjunction ''J'' = ''uv'''''</font></center></p><br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 56-b4 -- Tangent Map of J.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 56-b4. Tangent Map of the Conjunction ''J'' = ''uv'''''</font></center></p><br />
<br />
=====Operator Diagrams for the Conjunction J = uv=====<br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 57-1 -- Radius Operator Diagram for J.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 57-1. Radius Operator Diagram for the Conjunction ''J'' = ''uv'''''</font></center></p><br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 57-2 -- Secant Operator Diagram for J.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 57-2. Secant Operator Diagram for the Conjunction ''J'' = ''uv'''''</font></center></p><br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 57-3 -- Chord Operator Diagram for J.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 57-3. Chord Operator Diagram for the Conjunction ''J'' = ''uv'''''</font></center></p><br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 57-4 -- Tangent Functor Diagram for J.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 57-4. Tangent Functor Diagram for the Conjunction ''J'' = ''uv'''''</font></center></p><br />
<br />
===Taking Aim at Higher Dimensional Targets===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
The past and present wilt . . . . I have filled them and<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;emptied them,<br><br />
And proceed to fill my next fold of the future.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 87]<br />
|}<br />
<br />
In the next Subdivision I consider a logical transformation ''F'' that has the concrete type ''F''&nbsp;:&nbsp;[''u'',&nbsp;''v'']&nbsp;&rarr;&nbsp;[''x'',&nbsp;''y''] and the abstract type ''F''&nbsp;:&nbsp;['''B'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''B'''<sup>2</sup>]. From the standpoint of propositional calculus, the task of understanding such a transformation is naturally approached by parsing it into component maps with 1-dimensional ranges, as follows:<br />
<br />
<br><font face="courier new"><br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"<br />
|<br />
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| align="left" | ''F''<br />
| =<br />
| ‹''f'', ''g''›<br />
| =<br />
| ‹''F''<sub>1</sub>, ''F''<sub>2</sub>›<br />
| :<br />
| <nowiki>[</nowiki>''u'', ''v''<nowiki>]</nowiki><br />
| &rarr;<br />
| <nowiki>[</nowiki>''x'', ''y''<nowiki>]</nowiki><br />
|-<br />
| align="left" colspan="2" | where<br />
| ''f''<br />
| =<br />
| ''F''<sub>1</sub><br />
| :<br />
| <nowiki>[</nowiki>''u'', ''v''<nowiki>]</nowiki><br />
| &rarr;<br />
| <nowiki>[</nowiki>''x''<nowiki>]</nowiki><br />
|-<br />
| align="left" colspan="2" | and<br />
| ''g''<br />
| =<br />
| ''F''<sub>2</sub><br />
| :<br />
| <nowiki>[</nowiki>''u'', ''v''<nowiki>]</nowiki><br />
| &rarr;<br />
| <nowiki>[</nowiki>''y''<nowiki>]</nowiki><br />
|}<br />
|}<br />
</font><br><br />
<br />
Then one tackles the separate components, now viewed as propositions ''F''<sub>''i''</sub>&nbsp;:&nbsp;''U''&nbsp;&rarr;&nbsp;'''B''', one at a time. At the completion of this analytic phase, one returns to the task of synthesizing all of these partial and transient impressions into an agile form of integrity, a solidly coordinated and deeply integrated comprehension of the ongoing transformation. (Very often, of course, in tangling with refractory cases, one never gets as far as the beginning again.)<br />
<br />
Let us now refer to the dimension of the target space or codomain as the ''toll'' (or ''tole'') of a transformation, as distinguished from the dimension of the range or image that is customarily called the ''rank''. When we keep to transformations with a toll of 1, as ''J''&nbsp;:&nbsp;[''u'',&nbsp;''v'']&nbsp;&rarr;&nbsp;[''x''], we tend to get lazy about distinguishing a logical transformation from its component propositions. However, if we deal with transformations of a higher toll, this form of indolence can no longer be tolerated.<br />
<br />
Well, perhaps we can carry it a little further. After all, the operator result W''J''&nbsp;:&nbsp;E''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''X''<sup>&nbsp;&bull;</sup> is a map of toll 2, and cannot be unfolded in one piece as a proposition. But when a map has rank 1, like <math>\epsilon</math>''J''&nbsp;:&nbsp;E''U''&nbsp;&rarr;&nbsp;''X''&nbsp;&sube;&nbsp;E''X'' or d''J''&nbsp;:&nbsp;E''U''&nbsp;&rarr;&nbsp;d''X''&nbsp;&sube;&nbsp;E''X'', we naturally choose to concentrate on the 1-dimensional range of the operator result W''J'', ignoring the final difference in quality between the spaces ''X'' and d''X'', and view W''J'' as a proposition about E''U''.<br />
<br />
In this way, an initial ambivalence about the role of the operand ''J'' conveys a double duty to the result W''J''. The pivot that is formed by our focus of attention is essential to the linkage that transfers this double moment, as the whole process takes its bearing and wheels around the precise measure of a narrow bead that we can draw on the range of W''J''. This is the escapement that it takes to get away with what may otherwise seem to be a simple duplicity, and this is the tolerance that is needed to counterbalance a certain arrogance of equivocation, by all of which machinations we make ourselves free to indicate the operator results W''J'' as propositions or as transformations, indifferently.<br />
<br />
But that's it, and no further. Neglect of these distinctions in range and target universes of higher dimensions is bound to cause a hopeless confusion. To guard against these adverse prospects, Tables&nbsp;58 and 59 lay the groundwork for discussing a typical map ''F''&nbsp;:&nbsp;['''B'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''B'''<sup>2</sup>], and begin to pave the way, to some extent, for discussing any transformation of the form ''F''&nbsp;:&nbsp;['''B'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''B'''<sup>''k''</sup>].<br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"<br />
|+ '''Table 58. Cast of Characters: Expansive Subtypes of Objects and Operators'''<br />
|- style="background:paleturquoise"<br />
! Item<br />
! Notation<br />
! Description<br />
! Type<br />
|-<br />
| valign="top" | ''U''<sup>&nbsp;&bull;</sup><br />
| valign="top" | <font face="courier new">=&nbsp;</font>[''u'', ''v'']<br />
| valign="top" | Source Universe<br />
| valign="top" | ['''B'''<sup>''n''</sup>]<br />
|-<br />
| valign="top" | ''X''<sup>&nbsp;&bull;</sup><br />
| valign="top" |<br />
{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <font face="courier new">=&nbsp;</font>[''x'', ''y'']<br />
|-<br />
| <font face="courier new">=&nbsp;</font>[''f'', ''g'']<br />
|}<br />
| valign="top" | Target Universe<br />
| valign="top" | ['''B'''<sup>''k''</sup>]<br />
|-<br />
| valign="top" | E''U''<sup>&nbsp;&bull;</sup><br />
| valign="top" | <font face="courier new">=&nbsp;</font>[''u'', ''v'', d''u'', d''v'']<br />
| valign="top" | Extended Source Universe<br />
| valign="top" | ['''B'''<sup>''n''</sup> &times; '''D'''<sup>''n''</sup>]<br />
|-<br />
| valign="top" | E''X''<sup>&nbsp;&bull;</sup><br />
| valign="top" |<br />
{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <font face="courier new">=&nbsp;</font>[''x'', ''y'', d''x'', d''y'']<br />
|-<br />
| <font face="courier new">=&nbsp;</font>[''f'', ''g'', d''f'', d''g'']<br />
|}<br />
| valign="top" | Extended Target Universe<br />
| valign="top" | ['''B'''<sup>''k''</sup> &times; '''D'''<sup>''k''</sup>]<br />
|-<br />
| ''F''<br />
| ''F''&nbsp;=&nbsp;‹''f'',&nbsp;''g''›&nbsp;:&nbsp;''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup><br />
| Transformation, or Mapping<br />
| ['''B'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''B'''<sup>''k''</sup>]<br />
|-<br />
| valign="top" |<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| &nbsp;<br />
|-<br />
| ''f''<br />
|-<br />
| ''g''<br />
|}<br />
| valign="top" |<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| ''f'', ''g'' : ''U'' &rarr; '''B'''<br />
|-<br />
| ''f'' : ''U'' &rarr; [''x''] &sube; ''X''<sup>&nbsp;&bull;</sup><br />
|-<br />
| ''g'' : ''U'' &rarr; [''y''] &sube; ''X''<sup>&nbsp;&bull;</sup><br />
|}<br />
| valign="top" |<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| Proposition<br />
|}<br />
| valign="top" |<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100"<br />
| '''B'''<sup>''n''</sup> &rarr; '''B'''<br />
|-<br />
| &isin; ('''B'''<sup>''n''</sup>, '''B'''<sup>''n''</sup> &rarr; '''B''')<br />
|-<br />
| = ('''B'''<sup>''n''</sup> +&rarr; '''B''') = ['''B'''<sup>''n''</sup>]<br />
|}<br />
|-<br />
| valign="top" |<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| W<br />
|}<br />
| valign="top" |<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| W&nbsp;:<br />
|-<br />
| ''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''U''<sup>&nbsp;&bull;</sup>&nbsp;,<br />
|-<br />
| ''X''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''X''<sup>&nbsp;&bull;</sup>&nbsp;,<br />
|-<br />
| (''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup>)<br />
|-<br />
| &rarr;<br />
|-<br />
| (E''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''X''<sup>&nbsp;&bull;</sup>)&nbsp;,<br />
|-<br />
| for each W in the set:<br />
|-<br />
| {<math>\epsilon</math>,&nbsp;<math>\eta</math>,&nbsp;E,&nbsp;D,&nbsp;d}<br />
|}<br />
| valign="top" |<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| Operator<br />
|}<br />
| valign="top" |<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100"<br />
| &nbsp;<br />
|-<br />
| ['''B'''<sup>''n''</sup>] &rarr; ['''B'''<sup>''n''</sup> &times; '''D'''<sup>''n''</sup>]&nbsp;,<br />
|-<br />
| ['''B'''<sup>''k''</sup>] &rarr; ['''B'''<sup>''k''</sup> &times; '''D'''<sup>''k''</sup>]&nbsp;,<br />
|-<br />
| (['''B'''<sup>''n''</sup>] &rarr; ['''B'''<sup>''k''</sup>])<br />
|-<br />
| &rarr;<br />
|-<br />
| (['''B'''<sup>''n''</sup> &times; '''D'''<sup>''n''</sup>] &rarr; ['''B'''<sup>''k''</sup> &times; '''D'''<sup>''k''</sup>])<br />
|-<br />
| &nbsp;<br />
|-<br />
| &nbsp;<br />
|}<br />
|-<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <math>\epsilon</math><br />
|-<br />
| <math>\eta</math><br />
|-<br />
| E<br />
|-<br />
| D<br />
|-<br />
| d<br />
|}<br />
| valign="top" | &nbsp;<br />
| colspan="2" |<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:60%"<br />
| Tacit Extension Operator || <math>\epsilon</math><br />
|-<br />
| Trope Extension Operator || <math>\eta</math><br />
|-<br />
| Enlargement Operator || E<br />
|-<br />
| Difference Operator || D<br />
|-<br />
| Differential Operator || d<br />
|}<br />
|-<br />
| valign="top" |<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <font face=georgia>'''W'''</font><br />
|}<br />
| valign="top" |<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <font face=georgia>'''W'''</font>&nbsp;:<br />
|-<br />
| ''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;<font face=georgia>'''T'''</font>''U''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;E''U''<sup>&nbsp;&bull;</sup>&nbsp;,<br />
|-<br />
| ''X''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;<font face=georgia>'''T'''</font>''X''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;E''X''<sup>&nbsp;&bull;</sup>&nbsp;,<br />
|-<br />
| (''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup>)<br />
|-<br />
| &rarr;<br />
|-<br />
| (<font face=georgia>'''T'''</font>''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;<font face=georgia>'''T'''</font>''X''<sup>&nbsp;&bull;</sup>)&nbsp;,<br />
|-<br />
| for each <font face=georgia>'''W'''</font> in the set:<br />
|-<br />
| {<font face=georgia>'''e'''</font>,&nbsp;<font face=georgia>'''E'''</font>,&nbsp;<font face=georgia>'''D'''</font>,&nbsp;<font face=georgia>'''T'''</font>}<br />
|}<br />
| valign="top" |<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| Operator<br />
|}<br />
| valign="top" |<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100"<br />
| &nbsp;<br />
|-<br />
| ['''B'''<sup>''n''</sup>] &rarr; ['''B'''<sup>''n''</sup> &times; '''D'''<sup>''n''</sup>]&nbsp;,<br />
|-<br />
| ['''B'''<sup>''k''</sup>] &rarr; ['''B'''<sup>''k''</sup> &times; '''D'''<sup>''k''</sup>]&nbsp;,<br />
|-<br />
| (['''B'''<sup>''n''</sup>] &rarr; ['''B'''<sup>''k''</sup>])<br />
|-<br />
| &rarr;<br />
|-<br />
| (['''B'''<sup>''n''</sup> &times; '''D'''<sup>''n''</sup>] &rarr; ['''B'''<sup>''k''</sup> &times; '''D'''<sup>''k''</sup>])<br />
|-<br />
| &nbsp;<br />
|-<br />
| &nbsp;<br />
|}<br />
|-<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <font face=georgia>'''e'''</font><br />
|-<br />
| <font face=georgia>'''E'''</font><br />
|-<br />
| <font face=georgia>'''D'''</font><br />
|-<br />
| <font face=georgia>'''T'''</font><br />
|}<br />
| valign="top" | &nbsp;<br />
| colspan="2" |<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:60%"<br />
| Radius Operator || <font face=georgia>'''e'''</font>&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;<math>\eta</math>›<br />
|-<br />
| Secant Operator || <font face=georgia>'''E'''</font>&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;E›<br />
|-<br />
| Chord Operator || <font face=georgia>'''D'''</font>&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;D›<br />
|-<br />
| Tangent Functor || <font face=georgia>'''T'''</font>&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;d›<br />
|}<br />
|}<br><br />
<br />
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"<br />
|+ '''Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes'''<br />
|- style="background:paleturquoise"<br />
| &nbsp;<br />
| align="center" | '''Operator<br>or<br>Operand'''<br />
| align="center" | '''Proposition<br>or<br>Component'''<br />
| align="center" | '''Transformation<br>or<br>Mapping'''<br />
|-<br />
| Operand<br />
| valign="top" |<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| ''F'' = ‹''F''<sub>1</sub>, ''F''<sub>2</sub>›<br />
|-<br />
| ''F'' = ‹''f'', ''g''› : ''U'' &rarr; ''X''<br />
|}<br />
| valign="top" |<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| ''F''<sub>''i''</sub> : 〈''u'', ''v''〉 &rarr; '''B'''<br />
|-<br />
| ''F''<sub>''i''</sub> : '''B'''<sup>''n''</sup> &rarr; '''B'''<br />
|}<br />
| valign="top" |<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100"<br />
| ''F'' : [''u'', ''v''] &rarr; [''x'', ''y'']<br />
|-<br />
| ''F'' : '''B'''<sup>''n''</sup> &rarr; '''B'''<sup>''k''</sup><br />
|}<br />
|-<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| Tacit<br />
|-<br />
| Extension<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <math>\epsilon</math> :<br />
|-<br />
| ''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;''X''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''X''<sup>&nbsp;&bull;</sup>&nbsp;,<br />
|-<br />
| (''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup>)&nbsp;&rarr;&nbsp;(E''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup>)<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <math>\epsilon</math>''F''<sub>''i''</sub> :<br />
|-<br />
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''B'''<br />
|-<br />
| '''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''B'''<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <math>\epsilon</math>''F'' :<br />
|-<br />
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'', ''y'']<br />
|-<br />
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''B'''<sup>''k''</sup>]<br />
|}<br />
|-<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| Trope<br />
|-<br />
| Extension<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <math>\eta</math> :<br />
|-<br />
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,<br />
|-<br />
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; d''X''<sup>&nbsp;&bull;</sup>)<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <math>\eta</math>''F''<sub>''i''</sub> :<br />
|-<br />
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''<br />
|-<br />
| '''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''D'''<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <math>\eta</math>''F'' :<br />
|-<br />
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'', d''y'']<br />
|-<br />
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>''k''</sup>]<br />
|}<br />
|-<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| Enlargement<br />
|-<br />
| Operator<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| E :<br />
|-<br />
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,<br />
|-<br />
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; d''X''<sup>&nbsp;&bull;</sup>)<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| E''F''<sub>''i''</sub> :<br />
|-<br />
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''<br />
|-<br />
| '''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''D'''<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| E''F'' :<br />
|-<br />
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'', d''y'']<br />
|-<br />
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>''k''</sup>]<br />
|}<br />
|-<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| Difference<br />
|-<br />
| Operator<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| D :<br />
|-<br />
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,<br />
|-<br />
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; d''X''<sup>&nbsp;&bull;</sup>)<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| D''F''<sub>''i''</sub> :<br />
|-<br />
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''<br />
|-<br />
| '''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''D'''<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| D''F'' :<br />
|-<br />
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'', d''y'']<br />
|-<br />
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>''k''</sup>]<br />
|}<br />
|-<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| Differential<br />
|-<br />
| Operator<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| d :<br />
|-<br />
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,<br />
|-<br />
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; d''X''<sup>&nbsp;&bull;</sup>)<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| d''F''<sub>''i''</sub> :<br />
|-<br />
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''<br />
|-<br />
| '''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''D'''<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| d''F'' :<br />
|-<br />
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'', d''y'']<br />
|-<br />
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>''k''</sup>]<br />
|}<br />
|-<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| Remainder<br />
|-<br />
| Operator<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| r :<br />
|-<br />
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,<br />
|-<br />
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; d''X''<sup>&nbsp;&bull;</sup>)<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| r''F''<sub>''i''</sub> :<br />
|-<br />
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''<br />
|-<br />
| '''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''D'''<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| r''F'' :<br />
|-<br />
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'', d''y'']<br />
|-<br />
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>''k''</sup>]<br />
|}<br />
|-<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| Radius<br />
|-<br />
| Operator<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <font face=georgia>'''e'''</font> = ‹<math>\epsilon</math>, <math>\eta</math>› :<br />
|-<br />
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,<br />
|-<br />
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>)<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| &nbsp;<br />
|-<br />
| &nbsp;<br />
|-<br />
| &nbsp;<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <font face=georgia>'''e'''</font>''F'' :<br />
|-<br />
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'',&nbsp;''y'',&nbsp;d''x'',&nbsp;d''y'']<br />
|-<br />
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''B'''<sup>''k''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''k''</sup>]<br />
|}<br />
|-<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| Secant<br />
|-<br />
| Operator<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <font face=georgia>'''E'''</font> = ‹<math>\epsilon</math>, E› :<br />
|-<br />
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,<br />
|-<br />
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>)<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| &nbsp;<br />
|-<br />
| &nbsp;<br />
|-<br />
| &nbsp;<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <font face=georgia>'''E'''</font>''F'' :<br />
|-<br />
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'',&nbsp;''y'',&nbsp;d''x'',&nbsp;d''y'']<br />
|-<br />
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''B'''<sup>''k''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''k''</sup>]<br />
|}<br />
|-<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| Chord<br />
|-<br />
| Operator<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <font face=georgia>'''D'''</font> = ‹<math>\epsilon</math>, D› :<br />
|-<br />
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,<br />
|-<br />
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>)<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| &nbsp;<br />
|-<br />
| &nbsp;<br />
|-<br />
| &nbsp;<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <font face=georgia>'''D'''</font>''F'' :<br />
|-<br />
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'',&nbsp;''y'',&nbsp;d''x'',&nbsp;d''y'']<br />
|-<br />
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''B'''<sup>''k''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''k''</sup>]<br />
|}<br />
|-<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| Tangent<br />
|-<br />
| Functor<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <font face=georgia>'''T'''</font> = ‹<math>\epsilon</math>, d› :<br />
|-<br />
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,<br />
|-<br />
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>)<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| d''F''<sub>''i''</sub> :<br />
|-<br />
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''<br />
|-<br />
| '''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''D'''<br />
|}<br />
|<br />
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"<br />
| <font face=georgia>'''T'''</font>''F'' :<br />
|-<br />
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'',&nbsp;''y'',&nbsp;d''x'',&nbsp;d''y'']<br />
|-<br />
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''B'''<sup>''k''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''k''</sup>]<br />
|}<br />
|}<br><br />
<br />
===Transformations of Type '''B'''<sup>2</sup> &rarr; '''B'''<sup>2</sup>===<br />
<br />
To take up a slightly more complex example, but one that remains simple enough to pursue through a complete series of developments, consider the transformation from ''U''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;[''u'',&nbsp;''v''] to ''X''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;[''x'',&nbsp;''y''] that is defined by the following system of equations:<br />
<br />
<br><font face="courier new"><br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"<br />
|<br />
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| &nbsp;<br />
| ''x''<br />
| =<br />
| ''f''‹''u'', ''v''›<br />
| =<br />
| ((''u'')(''v''))<br />
| &nbsp;<br />
|-<br />
| &nbsp;<br />
| ''y''<br />
| =<br />
| ''g''‹''u'', ''v''›<br />
| =<br />
| ((''u'', ''v''))<br />
| &nbsp;<br />
|}<br />
|}<br />
</font><br><br />
<br />
The component notation ''F''&nbsp;=&nbsp;‹''F''<sub>1</sub>,&nbsp;''F''<sub>2</sub>›&nbsp;=&nbsp;‹''f'',&nbsp;''g''›&nbsp;:&nbsp;''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup> allows us to give a name and a type to this transformation, and permits us to define it by means of the compact description that follows:<br />
<br />
<br><font face="courier new"><br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"<br />
|<br />
{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| &nbsp;<br />
| ‹''x'', ''y''›<br />
| =<br />
| ''F''‹''u'', ''v''›<br />
| =<br />
| ‹((''u'')(''v'')), ((''u'', ''v''))›<br />
| &nbsp;<br />
|}<br />
|}<br />
</font><br><br />
<br />
The information that defines the logical transformation ''F'' can be represented in the form of a truth table, as in Table&nbsp;60. To cut down on subscripts in this example I continue to use plain letter equivalents for all components of spaces and maps.<br />
<br />
<font face="courier new"><br />
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"<br />
|+ '''Table 60. Propositional Transformation'''<br />
|- style="background:paleturquoise"<br />
| width="25%" | ''u''<br />
| width="25%" | ''v''<br />
| width="25%" | ''f''<br />
| width="25%" | ''g''<br />
|-<br />
| width="25%" |<br />
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0<br />
|-<br />
| 0<br />
|-<br />
| 1<br />
|-<br />
| 1<br />
|}<br />
| width="25%" |<br />
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0<br />
|-<br />
| 1<br />
|-<br />
| 0<br />
|-<br />
| 1<br />
|}<br />
| width="25%" |<br />
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0<br />
|-<br />
| 1<br />
|-<br />
| 1<br />
|-<br />
| 1<br />
|}<br />
| width="25%" |<br />
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 1<br />
|-<br />
| 0<br />
|-<br />
| 0<br />
|-<br />
| 1<br />
|}<br />
|-<br />
| width="25%" | &nbsp;<br />
| width="25%" | &nbsp;<br />
| width="25%" | ((''u'')(''v''))<br />
| width="25%" | ((''u'', ''v''))<br />
|}<br />
</font><br><br />
<br />
Figure&nbsp;61 shows how one might paint a picture of the logical transformation ''F'' on the canvass that was earlier primed for this purpose (way back in Figure&nbsp;30).<br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 61 -- Propositional Transformation.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 61. Propositional Transformation'''</font></center></p><br />
<br />
Figure&nbsp;62 extracts the gist of Figure&nbsp;61, exemplifying a style of diagram that is adequate for most purposes.<br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 62 -- Propositional Transformation (Short Form).gif|center]]</p><br />
<p><center><font size="+1">'''Figure 62. Propositional Transformation (Short Form)'''</font></center></p><br />
<br />
Figure&nbsp;63 give a more complete picture of the transformation ''F'', showing how the points of ''U''<sup>&nbsp;&bull;</sup> are transformed into points of ''X''<sup>&nbsp;&bull;</sup>. The lines that cross from one universe to the other trace the action that ''F'' induces on points, in other words, they depict the aspect of the transformation that acts as a mapping from points to points, and chart its effects on the elements that are variously called cells, points, positions, or singular propositions.<br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 63 -- Transformation of Positions.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 63. Transformation of Positions'''</font></center></p><br />
<br />
Table&nbsp;64 shows how the action of the transformation ''F'' on cells or points is computed in terms of coordinates.<br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"<br />
|+ '''Table 64. Transformation of Positions'''<br />
|- style="background:paleturquoise"<br />
| ''u''&nbsp;&nbsp;''v''<br />
| ''x''<br />
| ''y''<br />
| ''x''&nbsp;''y''<br />
| ''x''&nbsp;(''y'')<br />
| (''x'')&nbsp;''y''<br />
| (''x'')(''y'')<br />
| ''X''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;[''x'',&nbsp;''y''&nbsp;]<br />
|-<br />
| width="12%" |<br />
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0&nbsp;&nbsp;0<br />
|-<br />
| 0&nbsp;&nbsp;1<br />
|-<br />
| 1&nbsp;&nbsp;0<br />
|-<br />
| 1&nbsp;&nbsp;1<br />
|}<br />
| width="12%" |<br />
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0<br />
|-<br />
| 1<br />
|-<br />
| 1<br />
|-<br />
| 1<br />
|}<br />
| width="12%" |<br />
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 1<br />
|-<br />
| 0<br />
|-<br />
| 0<br />
|-<br />
| 1<br />
|}<br />
| width="12%" |<br />
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0<br />
|-<br />
| 0<br />
|-<br />
| 0<br />
|-<br />
| 1<br />
|}<br />
| width="12%" |<br />
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0<br />
|-<br />
| 1<br />
|-<br />
| 1<br />
|-<br />
| 0<br />
|}<br />
| width="12%" |<br />
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 1<br />
|-<br />
| 0<br />
|-<br />
| 0<br />
|-<br />
| 0<br />
|}<br />
| width="12%" |<br />
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0<br />
|-<br />
| 0<br />
|-<br />
| 0<br />
|-<br />
| 0<br />
|}<br />
| width="12%" |<br />
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| &uarr;<br />
|-<br />
| ''F''<br />
|-<br />
| ‹''f'',&nbsp;''g''&nbsp;›<br />
|-<br />
| &uarr;<br />
|}<br />
|-<br />
| &nbsp;<br />
| ((''u'')(''v''))<br />
| ((''u'',&nbsp;''v''))<br />
| ''u''&nbsp;''v''<br />
| (''u'',&nbsp;''v'')<br />
| (''u'')(''v'')<br />
| (&nbsp;)<br />
| ''U''<sup>&nbsp;&bull;</sup>&nbsp;=&nbsp;[''u'',&nbsp;''v''&nbsp;]<br />
|}<br />
<br><br />
<br />
Table&nbsp;65 extends this scheme from single cells to arbitrary regions of the source and target universes, and illustrates a form of computation that can be used to determine how a logical transformation acts on all of the propositions in the universe. The way that a transformation of positions affects the propositions, or any other structure that can be built on top of the positions, is normally called the ''induced action'' of the given transformation on the system of structures in question.<br />
<br />
<br><font face="courier new"><br />
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"<br />
|+ '''Table 65. Induced Transformation on Propositions'''<br />
|- style="background:paleturquoise"<br />
| ''X''<sup>&nbsp;&bull;</sup><br />
| colspan="3" |<br />
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:80%"<br />
| &larr;<br />
| ''F''&nbsp;=&nbsp;‹''f''&nbsp;,&nbsp;''g''›<br />
| &larr;<br />
|}<br />
| ''U''<sup>&nbsp;&bull;</sup><br />
|- style="background:paleturquoise"<br />
| rowspan="2" | ''f''<sub>''i''</sub>‹''x'',&nbsp;''y''›<br />
|<br />
{| align="right" style="background:paleturquoise; text-align:right"<br />
| ''u'' =<br />
|-<br />
| ''v'' =<br />
|}<br />
|<br />
{| align="center" style="background:paleturquoise; text-align:center"<br />
| 1 1 0 0<br />
|-<br />
| 1 0 1 0<br />
|}<br />
|<br />
{| align="left" style="background:paleturquoise; text-align:left"<br />
| = ''u''<br />
|-<br />
| = ''v''<br />
|}<br />
| rowspan="2" | ''f''<sub>''j''</sub>‹''u'',&nbsp;''v''›<br />
|- style="background:paleturquoise"<br />
|<br />
{| align="right" style="background:paleturquoise; text-align:right"<br />
| ''x'' =<br />
|-<br />
| ''y'' =<br />
|}<br />
|<br />
{| align="center" style="background:paleturquoise; text-align:center"<br />
| 1 1 1 0<br />
|-<br />
| 1 0 0 1<br />
|}<br />
|<br />
{| align="left" style="background:paleturquoise; text-align:left"<br />
| = ''f''‹''u'',&nbsp;''v''›<br />
|-<br />
| = ''g''‹''u'',&nbsp;''v''›<br />
|}<br />
|-<br />
|<br />
{| cellpadding="2" style="background:lightcyan"<br />
| ''f''<sub>0</sub><br />
|-<br />
| ''f''<sub>1</sub><br />
|-<br />
| ''f''<sub>2</sub><br />
|-<br />
| ''f''<sub>3</sub><br />
|-<br />
| ''f''<sub>4</sub><br />
|-<br />
| ''f''<sub>5</sub><br />
|-<br />
| ''f''<sub>6</sub><br />
|-<br />
| ''f''<sub>7</sub><br />
|}<br />
|<br />
{| cellpadding="2" style="background:lightcyan"<br />
| ()<br />
|-<br />
| &nbsp;(''x'')(''y'')&nbsp;<br />
|-<br />
| &nbsp;(''x'')&nbsp;''y''&nbsp;&nbsp;<br />
|-<br />
| &nbsp;(''x'')&nbsp;&nbsp;&nbsp;&nbsp;<br />
|-<br />
| &nbsp;&nbsp;''x''&nbsp;(''y'')&nbsp;<br />
|-<br />
| &nbsp;&nbsp;&nbsp;&nbsp;(''y'')&nbsp;<br />
|-<br />
| &nbsp;(''x'',&nbsp;''y'')&nbsp;<br />
|-<br />
| &nbsp;(''x''&nbsp;&nbsp;''y'')&nbsp;<br />
|}<br />
|<br />
{| cellpadding="2" style="background:lightcyan"<br />
| 0 0 0 0<br />
|-<br />
| 0 0 0 1<br />
|-<br />
| 0 0 1 0<br />
|-<br />
| 0 0 1 1<br />
|-<br />
| 0 1 0 0<br />
|-<br />
| 0 1 0 1<br />
|-<br />
| 0 1 1 0<br />
|-<br />
| 0 1 1 1<br />
|}<br />
|<br />
{| cellpadding="2" style="background:lightcyan"<br />
| ()<br />
|-<br />
| ()<br />
|-<br />
| &nbsp;(''u'')(''v'')&nbsp;<br />
|-<br />
| &nbsp;(''u'')(''v'')&nbsp;<br />
|-<br />
| &nbsp;(''u'',&nbsp;''v'')&nbsp;<br />
|-<br />
| &nbsp;(''u'',&nbsp;''v'')&nbsp;<br />
|-<br />
| &nbsp;(''u''&nbsp;&nbsp;''v'')&nbsp;<br />
|-<br />
| &nbsp;(''u''&nbsp;&nbsp;''v'')&nbsp;<br />
|}<br />
|<br />
{| cellpadding="2" style="background:lightcyan"<br />
| ''f''<sub>0</sub><br />
|-<br />
| ''f''<sub>0</sub><br />
|-<br />
| ''f''<sub>1</sub><br />
|-<br />
| ''f''<sub>1</sub><br />
|-<br />
| ''f''<sub>6</sub><br />
|-<br />
| ''f''<sub>6</sub><br />
|-<br />
| ''f''<sub>7</sub><br />
|-<br />
| ''f''<sub>7</sub><br />
|}<br />
|-<br />
|<br />
{| cellpadding="2" style="background:lightcyan"<br />
| ''f''<sub>8</sub><br />
|-<br />
| ''f''<sub>9</sub><br />
|-<br />
| ''f''<sub>10</sub><br />
|-<br />
| ''f''<sub>11</sub><br />
|-<br />
| ''f''<sub>12</sub><br />
|-<br />
| ''f''<sub>13</sub><br />
|-<br />
| ''f''<sub>14</sub><br />
|-<br />
| ''f''<sub>15</sub><br />
|}<br />
|<br />
{| cellpadding="2" style="background:lightcyan"<br />
| &nbsp;&nbsp;''x''&nbsp;&nbsp;''y''&nbsp;&nbsp;<br />
|-<br />
| ((''x'',&nbsp;''y''))<br />
|-<br />
| &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;''y''&nbsp;&nbsp;<br />
|-<br />
| &nbsp;(''x''&nbsp;(''y''))<br />
|-<br />
| &nbsp;&nbsp;''x''&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<br />
|-<br />
| ((''x'')&nbsp;''y'')&nbsp;<br />
|-<br />
| ((''x'')(''y''))<br />
|-<br />
| (())<br />
|}<br />
|<br />
{| cellpadding="2" style="background:lightcyan"<br />
| 1 0 0 0<br />
|-<br />
| 1 0 0 1<br />
|-<br />
| 1 0 1 0<br />
|-<br />
| 1 0 1 1<br />
|-<br />
| 1 1 0 0<br />
|-<br />
| 1 1 0 1<br />
|-<br />
| 1 1 1 0<br />
|-<br />
| 1 1 1 1<br />
|}<br />
|<br />
{| cellpadding="2" style="background:lightcyan"<br />
| &nbsp;&nbsp;''u''&nbsp;&nbsp;''v''&nbsp;&nbsp;<br />
|-<br />
| &nbsp;&nbsp;''u''&nbsp;&nbsp;''v''&nbsp;&nbsp;<br />
|-<br />
| ((''u'',&nbsp;''v''))<br />
|-<br />
| ((''u'',&nbsp;''v''))<br />
|-<br />
| ((''u'')(''v''))<br />
|-<br />
| ((''u'')(''v''))<br />
|-<br />
| (())<br />
|-<br />
| (())<br />
|}<br />
|<br />
{| cellpadding="2" style="background:lightcyan"<br />
| ''f''<sub>8</sub><br />
|-<br />
| ''f''<sub>8</sub><br />
|-<br />
| ''f''<sub>9</sub><br />
|-<br />
| ''f''<sub>9</sub><br />
|-<br />
| ''f''<sub>14</sub><br />
|-<br />
| ''f''<sub>14</sub><br />
|-<br />
| ''f''<sub>15</sub><br />
|-<br />
| ''f''<sub>15</sub><br />
|}<br />
|}<br />
</font><br><br />
<br />
Given the alphabets <font face="lucida calligraphy">U</font>&nbsp;=&nbsp;{''u'',&nbsp;''v''} and <font face="lucida calligraphy">X</font>&nbsp;=&nbsp;{''x'',&nbsp;''y''}, along with the corresponding universes of discourse ''U''<sup>&nbsp;&bull;</sup> and ''X''<sup>&nbsp;&bull;</sup>&nbsp;<math>\cong</math>&nbsp;['''B'''<sup>2</sup>], how many logical transformations of the general form ''G''&nbsp;=&nbsp;‹''G''<sub>1</sub>,&nbsp;''G''<sub>2</sub>›&nbsp;:&nbsp;''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup> are there?<br />
<br />
Since ''G''<sub>1</sub> and ''G''<sub>2</sub> can be any propositions of the type '''B'''<sup>2</sup>&nbsp;&rarr;&nbsp;'''B''', there are 2<sup>4</sup>&nbsp;=&nbsp;16 choices for each of the maps ''G''<sub>1</sub> and ''G''<sub>2</sub>, and thus there are 2<sup>4</sup><math>\cdot</math>2<sup>4</sup>&nbsp;=&nbsp;2<sup>8</sup>&nbsp;=&nbsp;256 different mappings altogether of the form ''G''&nbsp;:&nbsp;''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup>. The set of all functions of a given type is customarily denoted by placing its type indicator in parentheses, in the present instance writing (''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup>) = {''G''&nbsp;:&nbsp;''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup>}, and so the cardinality of this ''function space'' can be most conveniently summed up by writing |(''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup>)| = |('''B'''<sup>2</sup>&nbsp;&rarr;&nbsp;'''B'''<sup>2</sup>)| = 4<sup>4</sup> = 256.<br />
<br />
Given any transformation ''G'' = ‹''G''<sub>1</sub>,&nbsp;''G''<sub>2</sub>›&nbsp;:&nbsp;''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup> of this type, one can define a couple of further transformations, related to ''G'', that operate between the extended universes, E''U''<sup>&nbsp;&bull;</sup> and E''X''<sup>&nbsp;&bull;</sup>, of its source and target domains.<br />
<br />
First, the enlargement map (or the secant transformation) E''G'' = ‹E''G''<sub>1</sub>,&nbsp;E''G''<sub>2</sub>›&nbsp;:&nbsp;E''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''X''<sup>&nbsp;&bull;</sup> is defined by the following set of component equations:<br />
<br />
<br><font face="courier new"><br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"<br />
|<br />
{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"<br />
| width="8%" | E''G''<sub>''i''</sub><br />
| width="4%" | =<br />
| width="88%" | ''G''<sub>''i''</sub>‹''u'' + d''u'', ''v'' + d''v''›<br />
|}<br />
|}<br />
</font><br><br />
<br />
Second, the difference map (or the chordal transformation) D''G'' = ‹D''G''<sub>1</sub>,&nbsp;D''G''<sub>2</sub>›&nbsp;:&nbsp;E''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''X''<sup>&nbsp;&bull;</sup> is defined in component-wise fashion as the boolean sum of the initial proposition ''G''<sub>''i''</sub> and the enlarged proposition E''G''<sub>''i''</sub>, for ''i'' = 1, 2, according to the following set of equations:<br />
<br />
<br><font face="courier new"><br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"<br />
|<br />
{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"<br />
| width="8%" | D''G''<sub>''i''</sub><br />
| width="4%" | =<br />
| width="20%" | ''G''<sub>''i''</sub>‹''u'', ''v''›<br />
| width="4%" | +<br />
| width="64%" | E''G''<sub>''i''</sub>‹''u'', ''v'', d''u'', d''v''›<br />
|-<br />
| width="8%" | &nbsp;<br />
| width="4%" | =<br />
| width="20%" | ''G''<sub>''i''</sub>‹''u'', ''v''›<br />
| width="4%" | +<br />
| width="64%" | ''G''<sub>''i''</sub>‹''u'' + d''u'', ''v'' + d''v''›<br />
|}<br />
|}<br />
</font><br><br />
<br />
Maintaining a strict analogy with ordinary difference calculus would perhaps have us write D''G''<sub>''i''</sub>&nbsp;=&nbsp;E''G''<sub>''i''</sub>&nbsp;&ndash;&nbsp;''G''<sub>''i''</sub>, but the sum and difference operations are the same thing in boolean arithmetic. It is more often natural in the logical context to consider an initial proposition ''q'', then to compute the enlargement E''q'', and finally to determine the difference D''q''&nbsp;=&nbsp;''q''&nbsp;+&nbsp;E''q'', so we let the variant order of terms reflect this sequence of considerations.<br />
<br />
Viewed in this light the difference operator D is imagined to be a function of very wide scope and polymorphic application, one that is able to realize the association between each transformation ''G'' and its difference map D''G'', for instance, taking the function space (''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup>) into (E''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''X''<sup>&nbsp;&bull;</sup>). Given the interpretive flexibility of contexts in which we are allowing a proposition to appear, it should be clear that an operator of this scope is not at all a trivial matter to define properly, and may take some trouble to work out. For the moment, let's content ourselves with returning to particular cases.<br />
<br />
In their application to the present example, namely, the logical transformation ''F''&nbsp;=&nbsp;‹''f'',&nbsp;''g''›&nbsp;=&nbsp;‹((''u'')(''v'')),&nbsp;((''u'',&nbsp;''v''))›, the operators E and D respectively produce the enlarged map E''F''&nbsp;=&nbsp;‹E''f'',&nbsp;E''g''› and the difference map D''F''&nbsp;=&nbsp;‹D''f'',&nbsp;D''g''›, whose components can be given as follows, if the reader, in lieu of a special font for the logical parentheses, can forgive a syntactically bilingual formulation:<br />
<br />
<br><font face="courier new"><br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"<br />
|<br />
{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"<br />
| width="8%" | E''f''<br />
| width="4%" | =<br />
| width="88%" | ((''u'' + d''u'')(''v'' + d''v''))<br />
|-<br />
| width="8%" | E''g''<br />
| width="4%" | =<br />
| width="88%" | ((''u'' + d''u'', ''v'' + d''v''))<br />
|}<br />
|}<br />
</font><br />
<br><br />
<font face="courier new"><br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"<br />
|<br />
{| align="left" border="0" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"<br />
| width="8%" | D''f''<br />
| width="4%" | =<br />
| width="20%" | ((''u'')(''v''))<br />
| width="4%" | +<br />
| width="64%" | ((''u'' + d''u'')(''v'' + d''v''))<br />
|-<br />
| width="8%" | D''g''<br />
| width="4%" | =<br />
| width="20%" | ((''u'', ''v''))<br />
| width="4%" | +<br />
| width="64%" | ((''u'' + d''u'', ''v'' + d''v''))<br />
|}<br />
|}<br />
</font><br><br />
<br />
But these initial formulas are purely definitional, and help us little in understanding either the purpose of the operators or the meaning of their results. Working symbolically, let us apply the same method to the separate components ''f'' and ''g'' that we earlier used on ''J''. This work is recorded in Appendix&nbsp;1 and a summary of the results is presented in Tables&nbsp;66-i and 66-ii.<br />
<br />
<font face="courier new"><br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"<br />
|+ '''Table 66-i. Computation Summary for ''f''‹''u'', ''v''› = ((''u'')(''v''))'''<br />
|<br />
{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| <math>\epsilon</math>''f''<br />
| = || ''uv'' || <math>\cdot</math> || 1<br />
| + || ''u''(''v'') || <math>\cdot</math> || 1<br />
| + || (''u'')''v'' || <math>\cdot</math> || 1<br />
| + || (''u'')(''v'') || <math>\cdot</math> || 0<br />
|-<br />
| E''f''<br />
| = || ''uv'' || <math>\cdot</math> || (d''u'' d''v'')<br />
| + || ''u''(''v'') || <math>\cdot</math> || (d''u (d''v''))<br />
| + || (''u'')''v'' || <math>\cdot</math> || ((d''u'') d''v'')<br />
| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'')(d''v''))<br />
|-<br />
| D''f''<br />
| = || ''uv'' || <math>\cdot</math> || d''u'' d''v''<br />
| + || ''u''(''v'') || <math>\cdot</math> || d''u'' (d''v'')<br />
| + || (''u'')''v'' || <math>\cdot</math> || (d''u'') d''v''<br />
| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'')(d''v''))<br />
|-<br />
| d''f''<br />
| = || ''uv'' || <math>\cdot</math> || 0<br />
| + || ''u''(''v'') || <math>\cdot</math> || d''u''<br />
| + || (''u'')''v'' || <math>\cdot</math> || d''v''<br />
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')<br />
|-<br />
| r''f''<br />
| = || ''uv'' || <math>\cdot</math> || d''u'' d''v''<br />
| + || ''u''(''v'') || <math>\cdot</math> || d''u'' d''v''<br />
| + || (''u'')''v'' || <math>\cdot</math> || d''u'' d''v''<br />
| + || (''u'')(''v'') || <math>\cdot</math> || d''u'' d''v''<br />
|}<br />
|}<br />
</font><br><br />
<br />
<font face="courier new"><br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"<br />
|+ '''Table 66-ii. Computation Summary for g‹''u'', ''v''› = ((''u'', ''v''))'''<br />
|<br />
{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| <math>\epsilon</math>''g''<br />
| = || ''uv'' || <math>\cdot</math> || 1<br />
| + || ''u''(''v'') || <math>\cdot</math> || 0<br />
| + || (''u'')''v'' || <math>\cdot</math> || 0<br />
| + || (''u'')(''v'') || <math>\cdot</math> || 1<br />
|-<br />
| E''g''<br />
| = || ''uv'' || <math>\cdot</math> || ((d''u'', d''v''))<br />
| + || ''u''(''v'') || <math>\cdot</math> || (d''u'', d''v'')<br />
| + || (''u'')''v'' || <math>\cdot</math> || (d''u'', d''v'')<br />
| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'', d''v''))<br />
|-<br />
| D''g''<br />
| = || ''uv'' || <math>\cdot</math> || (d''u'', d''v'')<br />
| + || ''u''(''v'') || <math>\cdot</math> || (d''u'', d''v'')<br />
| + || (''u'')''v'' || <math>\cdot</math> || (d''u'', d''v'')<br />
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')<br />
|-<br />
| d''g''<br />
| = || ''uv'' || <math>\cdot</math> || (d''u'', d''v'')<br />
| + || ''u''(''v'') || <math>\cdot</math> || (d''u'', d''v'')<br />
| + || (''u'')''v'' || <math>\cdot</math> || (d''u'', d''v'')<br />
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')<br />
|-<br />
| r''g''<br />
| = || ''uv'' || <math>\cdot</math> || 0<br />
| + || ''u''(''v'') || <math>\cdot</math> || 0<br />
| + || (''u'')''v'' || <math>\cdot</math> || 0<br />
| + || (''u'')(''v'') || <math>\cdot</math> || 0<br />
|}<br />
|}<br />
</font><br><br />
<br />
Table&nbsp;67 shows how to compute the analytic series for ''F''&nbsp;=&nbsp;‹''f'',&nbsp;''g''›&nbsp;=&nbsp;‹((''u'')(''v'')),&nbsp;((''u'',&nbsp;''v''))› in terms of coordinates, and Table&nbsp;68 recaps these results in symbolic terms, agreeing with earlier derivations.<br />
<br />
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"<br />
|+ '''Table 67. Computation of an Analytic Series in Terms of Coordinates'''<br />
|<br />
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| ''u''<br />
| ''v''<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| d''u''<br />
| d''v''<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| ''u''<font face="courier new">’</font><br />
| ''v''<font face="courier new">’</font><br />
|}<br />
|-<br />
| valign="top" |<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|-<br />
| 0 || 1<br />
|-<br />
| 1 || 0<br />
|-<br />
| 1 || 1<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|-<br />
| 0 || 1<br />
|-<br />
| 1 || 0<br />
|-<br />
| 1 || 1<br />
|}<br />
|-<br />
| valign="top" |<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 1<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|-<br />
| 0 || 1<br />
|-<br />
| 1 || 0<br />
|-<br />
| 1 || 1<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 1<br />
|-<br />
| 0 || 0<br />
|-<br />
| 1 || 1<br />
|-<br />
| 1 || 0<br />
|}<br />
|-<br />
| valign="top" |<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 1 || 0<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|-<br />
| 0 || 1<br />
|-<br />
| 1 || 0<br />
|-<br />
| 1 || 1<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 1 || 0<br />
|-<br />
| 1 || 1<br />
|-<br />
| 0 || 0<br />
|-<br />
| 0 || 1<br />
|}<br />
|-<br />
| valign="top" |<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 1 || 1<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 1 || 1<br />
|-<br />
| 1 || 0<br />
|-<br />
| 0 || 1<br />
|-<br />
| 0 || 0<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|-<br />
| 0 || 1<br />
|-<br />
| 1 || 0<br />
|-<br />
| 1 || 1<br />
|}<br />
|}<br />
|<br />
{| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| <math>\epsilon</math>''f''<br />
| <math>\epsilon</math>''g''<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| E''f''<br />
| E''g''<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| D''f''<br />
| D''g''<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| d''f''<br />
| d''g''<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"<br />
| d<sup>2</sup>''f''<br />
| d<sup>2</sup>''g''<br />
|}<br />
|-<br />
| valign="top" |<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 1<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 1<br />
|-<br />
| 1 || 0<br />
|-<br />
| 1 || 0<br />
|-<br />
| 1 || 1<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|-<br />
| 1 || 1<br />
|-<br />
| 1 || 1<br />
|-<br />
| 1 || 0<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|-<br />
| 1 || 1<br />
|-<br />
| 1 || 1<br />
|-<br />
| 0 || 0<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|-<br />
| 0 || 0<br />
|-<br />
| 0 || 0<br />
|-<br />
| 1 || 0<br />
|}<br />
|-<br />
| valign="top" |<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 1 || 0<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 1 || 0<br />
|-<br />
| 0 || 1<br />
|-<br />
| 1 || 1<br />
|-<br />
| 1 || 0<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|-<br />
| 1 || 1<br />
|-<br />
| 0 || 1<br />
|-<br />
| 0 || 0<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|-<br />
| 1 || 1<br />
|-<br />
| 0 || 1<br />
|-<br />
| 1 || 0<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|-<br />
| 0 || 0<br />
|-<br />
| 0 || 0<br />
|-<br />
| 1 || 0<br />
|}<br />
|-<br />
| valign="top" |<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 1 || 0<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 1 || 0<br />
|-<br />
| 1 || 1<br />
|-<br />
| 0 || 1<br />
|-<br />
| 1 || 0<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|-<br />
| 0 || 1<br />
|-<br />
| 1 || 1<br />
|-<br />
| 0 || 0<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|-<br />
| 0 || 1<br />
|-<br />
| 1 || 1<br />
|-<br />
| 1 || 0<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|-<br />
| 0 || 0<br />
|-<br />
| 0 || 0<br />
|-<br />
| 1 || 0<br />
|}<br />
|-<br />
| valign="top" |<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 1 || 1<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 1 || 1<br />
|-<br />
| 1 || 0<br />
|-<br />
| 1 || 0<br />
|-<br />
| 0 || 1<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|-<br />
| 0 || 1<br />
|-<br />
| 0 || 1<br />
|-<br />
| 1 || 0<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|-<br />
| 0 || 1<br />
|-<br />
| 0 || 1<br />
|-<br />
| 0 || 0<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0 || 0<br />
|-<br />
| 0 || 0<br />
|-<br />
| 0 || 0<br />
|-<br />
| 1 || 0<br />
|}<br />
|}<br />
|}<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"<br />
|+ '''Table 68. Computation of an Analytic Series in Symbolic Terms'''<br />
|- style="background:paleturquoise"<br />
| ''u''&nbsp;&nbsp;''v''<br />
| ''f''&nbsp;&nbsp;''g''<br />
| D''f''<br />
| D''g''<br />
| d''f''<br />
| d''g''<br />
| d<sup>2</sup>''f''<br />
| d<sup>2</sup>''g''<br />
|-<br />
|<br />
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0&nbsp;&nbsp;0<br />
|-<br />
| 0&nbsp;&nbsp;1<br />
|-<br />
| 1&nbsp;&nbsp;0<br />
|-<br />
| 1&nbsp;&nbsp;1<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| 0&nbsp;&nbsp;1<br />
|-<br />
| 1&nbsp;&nbsp;0<br />
|-<br />
| 1&nbsp;&nbsp;0<br />
|-<br />
| 1&nbsp;&nbsp;1<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| ((d''u'')(d''v''))<br />
|-<br />
| (d''u'')&nbsp;d''v''&nbsp;<br />
|-<br />
| &nbsp;d''u''&nbsp;(d''v'')<br />
|-<br />
| d''u''&nbsp;&nbsp;d''v''<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| (d''u'', d''v'')<br />
|-<br />
| (d''u'', d''v'')<br />
|-<br />
| (d''u'', d''v'')<br />
|-<br />
| (d''u'', d''v'')<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| (d''u'', d''v'')<br />
|-<br />
| d''v''<br />
|-<br />
| d''u''<br />
|-<br />
| (&nbsp;)<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| (d''u'', d''v'')<br />
|-<br />
| (d''u'', d''v'')<br />
|-<br />
| (d''u'', d''v'')<br />
|-<br />
| (d''u'', d''v'')<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| d''u'' d''v''<br />
|-<br />
| d''u'' d''v''<br />
|-<br />
| d''u'' d''v''<br />
|-<br />
| d''u'' d''v''<br />
|}<br />
|<br />
{| align="center" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| (&nbsp;)<br />
|-<br />
| (&nbsp;)<br />
|-<br />
| (&nbsp;)<br />
|-<br />
| (&nbsp;)<br />
|}<br />
|}<br />
<br><br />
<br />
Figure&nbsp;69 gives a graphical picture of the difference map D''F''&nbsp;=&nbsp;‹D''f'',&nbsp;D''g''› for the transformation ''F''&nbsp;=&nbsp;‹''f'',&nbsp;''g''›&nbsp;=&nbsp;‹((''u'')(''v'')),&nbsp;((''u'',&nbsp;''v''))›. This depicts the same information about D''f'' and D''g'' that was given in the corresponding rows of the computation summary in Tables&nbsp;66-i and 66-ii, excerpted here:<br />
<br />
<br><font face="courier new"><br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"<br />
|<br />
{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| &nbsp;<br />
|-<br />
| D''f''<br />
| = || ''uv'' || <math>\cdot</math> || d''u'' d''v''<br />
| + || ''u''(''v'') || <math>\cdot</math> || d''u'' (d''v'')<br />
| + || (''u'')''v'' || <math>\cdot</math> || (d''u'') d''v''<br />
| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'')(d''v''))<br />
|-<br />
| &nbsp;<br />
|-<br />
| D''g''<br />
| = || ''uv'' || <math>\cdot</math> || (d''u'', d''v'')<br />
| + || ''u''(''v'') || <math>\cdot</math> || (d''u'', d''v'')<br />
| + || (''u'')''v'' || <math>\cdot</math> || (d''u'', d''v'')<br />
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')<br />
|-<br />
| &nbsp;<br />
|}<br />
|}<br />
</font><br><br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 69 -- Difference Map (Short Form).gif|center]]</p><br />
<p><center><font size="+1">'''Figure 69. Difference Map of F = ‹f,&nbsp;g› = ‹((u)(v)),&nbsp;((u,&nbsp;v))›'''</font></center></p><br />
<br />
Figure&nbsp;70-a shows a graphical way of picturing the tangent functor map d''F''&nbsp;=&nbsp;‹d''f'',&nbsp;d''g''› for the transformation ''F''&nbsp;=&nbsp;‹''f'',&nbsp;''g''›&nbsp;=&nbsp;›((u)(v)),&nbsp;((u,&nbsp;v))›. This amounts to the same information about d''f'' and d''g'' that was given in the computation summary of Tables&nbsp;66-i and 66-ii, the relevant rows of which are repeated here:<br />
<br />
<br><font face="courier new"><br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"<br />
|<br />
{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"<br />
| &nbsp;<br />
|-<br />
| d''f''<br />
| = || ''uv'' || <math>\cdot</math> || 0<br />
| + || ''u''(''v'') || <math>\cdot</math> || d''u''<br />
| + || (''u'')''v'' || <math>\cdot</math> || d''v''<br />
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')<br />
|-<br />
| &nbsp;<br />
|-<br />
| d''g''<br />
| = || ''uv'' || <math>\cdot</math> || (d''u'', d''v'')<br />
| + || ''u''(''v'') || <math>\cdot</math> || (d''u'', d''v'')<br />
| + || (''u'')''v'' || <math>\cdot</math> || (d''u'', d''v'')<br />
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')<br />
|-<br />
| &nbsp;<br />
|}<br />
|}<br />
</font><br><br />
<br />
<br><br />
<p>[[Image:Diff Log Dyn Sys -- Figure 70-a -- Tangent Functor Diagram.gif|center]]</p><br />
<p><center><font size="+1">'''Figure 70-a. Tangent Functor Diagram for F‹u,&nbsp;v› = ‹((u)(v)),&nbsp;((u,&nbsp;v))›'''</font></center></p><br />
<br />
Figure 70-b shows another way to picture the action of the tangent functor on the logical transformation ''F''‹''u'',&nbsp;''v''›&nbsp;=&nbsp;‹((''u'')(''v'')),&nbsp;((''u'',&nbsp;''v''))›, roughly in the style of the ''bundle of universes'' type of diagram.<br />
<br />
<p>[[Image:Tangent_Functor_Ferris_Wheel.gif|center|frame|<font size="3">'''Figure 70-b. Tangent Functor Ferris Wheel for F‹u, v› = ‹((u)(v)), ((u, v))›'''</font>]]</p><br />
<br />
* '''Nota Bene.''' The original Figure&nbsp;70-b lost some of its labeling in a succession of platform metamorphoses over the years, so I have included an Ascii version below to indicate where the missing labels go.<br />
<br />
<pre><br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| /////\ /////\ | | /XXXX\ /XXXX\ | | /\\\\\ /\\\\\ |<br />
| ///////o//////\ | | /XXXXXXoXXXXXX\ | | /\\\\\\o\\\\\\\ |<br />
| //////// \//////\ | | /XXXXXX/ \XXXXXX\ | | /\\\\\\/ \\\\\\\\ |<br />
| o/////// \//////o | | oXXXXXX/ \XXXXXXo | | o\\\\\\/ \\\\\\\o |<br />
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |<br />
| |/du//| |//dv/| | | |XXXXX| |XXXXX| | | |\du\\| |\\dv\| |<br />
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |<br />
| o//////\ ///////o | | oXXXXXX\ /XXXXXXo | | o\\\\\\\ /\\\\\\o |<br />
| \//////\ //////// | | \XXXXXX\ /XXXXXX/ | | \\\\\\\\ /\\\\\\/ |<br />
| \//////o/////// | | \XXXXXXoXXXXXX/ | | \\\\\\\o\\\\\\/ |<br />
| \///// \///// | | \XXXX/ \XXXX/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ (u)(v) o-----------------------o dv' @ (u)(v) =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| / \ /////\ | | /\\\\\ /XXXX\ | | /\\\\\ /\\\\\ |<br />
| / o//////\ | | /\\\\\\oXXXXXX\ | | /\\\\\\o\\\\\\\ |<br />
| / //\//////\ | | /\\\\\\//\XXXXXX\ | | /\\\\\\/ \\\\\\\\ |<br />
| o ////\//////o | | o\\\\\\////\XXXXXXo | | o\\\\\\/ \\\\\\\o |<br />
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |<br />
| | du |/////|//dv/| | | |\\\\\|/////|XXXXX| | | |\du\\| |\\dv\| |<br />
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |<br />
| o \//////////o | | o\\\\\\\////XXXXXXo | | o\\\\\\\ /\\\\\\o |<br />
| \ \///////// | | \\\\\\\\//XXXXXX/ | | \\\\\\\\ /\\\\\\/ |<br />
| \ o/////// | | \\\\\\\oXXXXXX/ | | \\\\\\\o\\\\\\/ |<br />
| \ / \///// | | \\\\\/ \XXXX/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ (u) v o-----------------------o dv' @ (u) v =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| /////\ / \ | | /XXXX\ /\\\\\ | | /\\\\\ /\\\\\ |<br />
| ///////o \ | | /XXXXXXo\\\\\\\ | | /\\\\\\o\\\\\\\ |<br />
| /////////\ \ | | /XXXXXX//\\\\\\\\ | | /\\\\\\/ \\\\\\\\ |<br />
| o//////////\ o | | oXXXXXX////\\\\\\\o | | o\\\\\\/ \\\\\\\o |<br />
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |<br />
| |/du//|/////| dv | | | |XXXXX|/////|\\\\\| | | |\du\\| |\\dv\| |<br />
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |<br />
| o//////\//// o | | oXXXXXX\////\\\\\\o | | o\\\\\\\ /\\\\\\o |<br />
| \//////\// / | | \XXXXXX\//\\\\\\/ | | \\\\\\\\ /\\\\\\/ |<br />
| \//////o / | | \XXXXXXo\\\\\\/ | | \\\\\\\o\\\\\\/ |<br />
| \///// \ / | | \XXXX/ \\\\\/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ u (v) o-----------------------o dv' @ u (v) =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| / \ / \ | | /\\\\\ /\\\\\ | | /\\\\\ /\\\\\ |<br />
| / o \ | | /\\\\\\o\\\\\\\ | | /\\\\\\o\\\\\\\ |<br />
| / / \ \ | | /\\\\\\/ \\\\\\\\ | | /\\\\\\/ \\\\\\\\ |<br />
| o / \ o | | o\\\\\\/ \\\\\\\o | | o\\\\\\/ \\\\\\\o |<br />
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |<br />
| | du | | dv | | | |\\\\\| |\\\\\| | | |\du\\| |\\dv\| |<br />
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |<br />
| o \ / o | | o\\\\\\\ /\\\\\\o | | o\\\\\\\ /\\\\\\o |<br />
| \ \ / / | | \\\\\\\\ /\\\\\\/ | | \\\\\\\\ /\\\\\\/ |<br />
| \ o / | | \\\\\\\o\\\\\\/ | | \\\\\\\o\\\\\\/ |<br />
| \ / \ / | | \\\\\/ \\\\\/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ u v o-----------------------o dv' @ u v =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| U | |\U\\\\\\\\\\\\\\\\\\\\\| |\U\\\\\\\\\\\\\\\\\\\\\|<br />
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|<br />
| /////\ /////\ | |\\\\\/////\\/////\\\\\\| |\\\\\/ \\/ \\\\\\|<br />
| ///////o//////\ | |\\\\///////o//////\\\\\| |\\\\/ o \\\\\|<br />
| /////////\//////\ | |\\\////////X\//////\\\\| |\\\/ /\\ \\\\|<br />
| o//////////\//////o | |\\o///////XXX\//////o\\| |\\o /\\\\ o\\|<br />
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|<br />
| |//u//|/////|//v//| | |\\|//u//|XXXXX|//v//|\\| |\\| u |\\\\\| v |\\|<br />
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|<br />
| o//////\//////////o | |\\o//////\XXX///////o\\| |\\o \\\\/ o\\|<br />
| \//////\///////// | |\\\\//////\X////////\\\| |\\\\ \\/ /\\\|<br />
| \//////o/////// | |\\\\\//////o///////\\\\| |\\\\\ o /\\\\|<br />
| \///// \///// | |\\\\\\/////\\/////\\\\\| |\\\\\\ /\\ /\\\\\|<br />
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|<br />
| | |\\\\\\\\\\\\\\\\\\\\\\\| |\\\\\\\\\\\\\\\\\\\\\\\|<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= u' o-----------------------o v' =<br />
= | U' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/u'//|XXXXX|\\v'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
Figure 70-b. Tangent Functor Ferris Wheel for F<u, v> = <((u)(v)), ((u, v))><br />
</pre><br />
<br />
==Epilogue, Enchoiry, Exodus==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
It is time to explain myself . . . . let us stand up.<br />
| width="4%" | &nbsp;<br />
|- <br />
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 79]<br />
|}<br />
<br />
==References==<br />
<br />
===Works Cited===<br />
<br />
{| cellpadding=3<br />
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|-<br />
| valign=top | [BiG]<br />
| Bishop, R.L., and Goldberg, S.I., ''Tensor Analysis on Manifolds'', Macmillan, 1968. Reprinted, Dover, New York, NY, 1980.<br />
|-<br />
| valign=top | [Boo]<br />
| Boole, G., ''An Investigation of The Laws of Thought'', Macmillan, 1854. Reprinted, Dover, New York, NY, 1958.<br />
|-<br />
| valign=top | [BoT]<br />
| Bott, R., and Tu, L.W., ''Differential Forms in Algebraic Topology'', Springer-Verlag, New York, NY, 1982.<br />
|-<br />
| valign=top | [dCa]<br />
| do Carmo, M.P., ''Riemannian Geometry''. Originally published in Portuguese, 1st editiom 1979, 2nd edition 1988. Translated by F. Flaherty, Birkhäuser, Boston, MA, 1992.<br />
|-<br />
| valign=top | [Che46]<br />
| Chevalley, C., ''Theory of Lie Groups'', Princeton University Press, Princeton, NJ, 1946.<br />
|-<br />
| valign=top | [Che56]<br />
| Chevalley, C., ''Fundamental Concepts of Algebra'', Academic Press, 1956.<br />
|-<br />
| valign=top | [Cho86]<br />
| Chomsky, N., ''Knowledge of Language : Its Nature, Origin, and Use'', Praeger, New York, NY, 1986.<br />
|-<br />
| valign=top | [Cho93]<br />
| Chomsky, N., ''Language and Thought'', Moyer Bell, Wakefield, RI, 1993.<br />
|-<br />
| valign=top | [DoM]<br />
| Doolin, B.F., and Martin, C.F., ''Introduction to Differential Geometry for Engineers'', Marcel Dekker, New York, NY, 1990.<br />
|-<br />
| valign=top | [Fuji]<br />
| Fujiwara, H., ''Logic Testing and Design for Testability'', MIT Press, Cambridge, MA, 1985.<br />
|-<br />
| valign=top | [Hic]<br />
| Hicks, N.J., ''Notes on Differential Geometry'', Van Nostrand, Princeton, NJ, 1965.<br />
|-<br />
| valign=top | [Hir]<br />
| Hirsch, M.W., ''Differential Topology'', Springer-Verlag, New York, NY, 1976.<br />
|-<br />
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| Howard, W.A., "The Formulae-as-Types Notion of Construction", Notes circulated from 1969. Reprinted in [SeH, 479-490].<br />
|-<br />
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| Jones, A., Gray, A., and Hutton, R., ''Manifolds and Mechanics'', Cambridge University Press, Cambridge, UK, 1987.<br />
|-<br />
| valign=top | [KoA]<br />
| Kosinski, A.A., ''Differential Manifolds'', Academic Press, San Diego, CA, 1993.<br />
|-<br />
| valign=top | [Koh]<br />
| Kohavi, Z., ''Switching and Finite Automata Theory'', 2nd edition, McGraw-Hill, New York, NY, 1978.<br />
|-<br />
| valign=top | [LaS]<br />
| Lambek, J., and Scott, P.J., ''Introduction to Higher Order Categorical Logic'', Cambridge University Press, Cambridge, UK, 1986.<br />
|-<br />
| valign=top | [La83]<br />
| Lang, S., ''Real Analysis'', 2nd edition, Addison-Wesley, Reading, MA, 1983.<br />
|-<br />
| valign=top | [La84]<br />
| Lang, S., ''Algebra'', 2nd edition, Addison-Wesley, Menlo Park, CA, 1984.<br />
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| Lang, S., ''Differential Manifolds'', Springer-Verlag, New York, NY, 1985.<br />
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| Lang, S., ''Real and Functional Analysis'', 3rd edition, Springer-Verlag, New York, NY, 1993.<br />
|-<br />
| valign=top | [Lie80]<br />
| Lie, S., "Sophus Lie's 1880 Transformation Group Paper", in ''Lie Groups : History, Frontiers, and Applications, Volume 1'', translated by M. Ackerman, comments by R. Hermann, Math Sci Press, Brookline, MA, 1975. Original paper 1880.<br />
|-<br />
| valign=top | [Lie84]<br />
| Lie, S., "Sophus Lie's 1884 Differential Invariant Paper", in ''Lie Groups : History, Frontiers, and Applications, Volume 3'', translated by M. Ackerman, comments by R. Hermann, Math Sci Press, Brookline, MA, 1976. Original paper 1884.<br />
|-<br />
| valign=top | [LoS]<br />
| Loomis, L.H., and Sternberg, S., ''Advanced Calculus'', Addison-Wesley, Reading, MA, 1968.<br />
|-<br />
| valign=top | [Mel]<br />
| Melzak, Z.A., ''Companion to Concrete Mathematics, Volume 2 : Mathematical Ideas, Modeling, and Applications'', John Wiley amd Sons, New York, NY, 1976.<br />
|-<br />
| valign=top | [Men]<br />
| Menabrea, L.F., "Sketch of the Analytical Engine Invented by Charles Babbage" with Notes by the Translator, Ada Augusta (Byron), Countess of Lovelace'', in [M&M, 225–297]. Originally published 1842.<br />
|-<br />
| valign=top | [M&M]<br />
| Morrison, P., and Morrison, E. (eds.), ''Charles Babbage on the Principles and Development of the Calculator, and Other Seminal Writings by Charles Babbage and Others, With an Introduction by the Editors'', Dover, Mineola, NY, 1961.<br />
|-<br />
| valign=top | [P1]<br />
| Peirce, C.S., ''Collected Papers of Charles Sanders Peirce'', vols. 1–8, C. Hartshorne, P. Weiss, and A.W. Burks (eds.), Harvard University Press, Cambridge, MA, 1931–1960. Cited as CP [volume].[paragraph].<br />
|-<br />
| valign=top | [P2]<br />
| Peirce, C.S., "Qualitative Logic", in ''The New Elements of Mathematics, Volume 4'', C. Eisele (ed.), Mouton, The Hague, 1976. Cited as NE [volume], [page].<br />
|-<br />
| valign=top | [Rob]<br />
| Roberts, D.D., ''The Existential Graphs of Charles S. Peirce'', Mouton, The Hague, 1973.<br />
|-<br />
| valign=top | [SeH]<br />
| Seldin, J.P., and Hindley, J.R. (eds.), ''To H.B. Curry : Essays on Combinatory Logic, Lambda Calculus, and Formalism'', Academic Press, London, UK, 1980.<br />
|-<br />
| valign=top | [SpB]<br />
| Spencer-Brown, G., ''Laws of Form'', George Allen and Unwin, London, UK, 1969.<br />
|-<br />
| valign=top | [Sp65]<br />
| Spivak, M., ''Calculus on Manifolds : A Modern Approach to Classical Theorems of Advanced Calculus'', W.A. Benjamin, New York, NY, 1965.<br />
|-<br />
| valign=top | [Sp79]<br />
| Spivak, M., ''A Comprehensive Introduction to Differential Geometry'', vols. 1–2. 1st edition 1970. 2nd edition, Publish or Perish Inc., Houston, TX, 1979.<br />
|-<br />
| valign=top | [Sty]<br />
| Styazhkin, N.I., ''History of Mathematical Logic from Leibniz to Peano'', 1st published in Russian, Nauka, Moscow, 1964. MIT Press, Cambridge, MA, 1969.<br />
|-<br />
| valign=top | [Wie]<br />
| Wiener, N., ''Cybernetics : or Control and Communication in the Animal and the Machine'', 1st edition 1948. 2nd edition, MIT Press, Cambridge, MA, 1961.<br />
|}<br />
<br />
===Works Consulted===<br />
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{| cellpadding=3<br />
| valign=top | [Ami]<br />
| Amit, D.J., ''Modeling Brain Function : The World of Attractor Neural Networks'', Cambridge University Press, Cambridge, UK, 1989.<br />
|-<br />
| valign=top | [Ed87]<br />
| Edelman, G.M., ''Neural Darwinism : The Theory of Neuronal Group Selection'', Basic Books, New York, NY, 1987.<br />
|-<br />
| valign=top | [Ed88]<br />
| Edelman, G.M., ''Topobiology : An Introduction to Molecular Embryology'', Basic Books, New York, NY, 1988.<br />
|-<br />
| valign=top | [Fla]<br />
| Flanders, H., ''Differential Forms with Applications to the Physical Sciences'', Academic Press, 1963. Reprinted, Dover, Mineola, NY, 1989. <br />
|-<br />
| valign=top | [Has]<br />
| Hassoun, M.H. (ed.), ''Associative Neural Memories : Theory and Implementation'', Oxford University Press, New York, NY, 1993.<br />
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| valign=top | [KoB]<br />
| Kosko, B., ''Neural Networks and Fuzzy Systems : A Dynamical Systems Approach to Machine Intelligence'', Prentice-Hall, Englewood Cliffs, NJ, 1992.<br />
|-<br />
| valign=top | [MaB]<br />
| Mac Lane, S., and Birkhoff, G., ''Algebra'', 3rd edition, Chelsea, New York, NY, 1993.<br />
|-<br />
| valign=top | [Mac]<br />
| Mac Lane, S., ''Categories for the Working Mathematician'', Springer-Verlag, New York, NY, 1971.<br />
|-<br />
| valign=top | [McC]<br />
| McCulloch, W.S., ''Embodiments of Mind'', MIT Press, Cambridge, MA, 1965.<br />
|-<br />
| valign=top | [Mc1]<br />
| McCulloch, W.S., "A Heterarchy of Values Determined by the Topology of Nervous Nets", Bulletin of Mathematical Biophysics, vol. 7 (1945), pp. 89–93. Reprinted in [McC].<br />
|-<br />
| valign=top | [MiP]<br />
| Minsky, M.L., and Papert, S.A., ''Perceptrons : An Introduction to Computational Geometry'', MIT Press, Cambridge, MA, 1969. 2nd printing 1972. Expanded edition 1988.<br />
|-<br />
| valign=top | [Rum]<br />
| Rumelhart, D.E., Hinton, G.E., and McClelland, J.L., "A General Framework for Parallel Distributed Processing" = Chapter 2 in Rumelhart, McClelland, and the PDP Research Group, ''Parallel Distributed Processing, Explorations in the Microstructure of Cognition, Volume 1 : Foundations'', MIT Press, Cambridge, MA, 1986.<br />
|}<br />
<br />
===Incidental Works===<br />
<br />
{| cellpadding=3<br />
| valign=top | [Dew]<br />
| Dewey, John, ''How We Think'', D.C. Heath, Lexington, MA, 1910. Reprinted, Prometheus Books, Buffalo, NY, 1991.<br />
|-<br />
| valign=top | [Fou]<br />
| Foucault, Michel, ''The Archaeology of Knowledge and The Discourse on Language'', A.M. Sheridan-Smith and Rupert Swyer (trans.), Pantheon, New York, NY, 1972. Originally published as ''L´Archéologie du Savoir et L´ordre du discours'', Editions Gallimard, 1969 & 1971.<br />
|-<br />
| valign=top | [Hom]<br />
| Homer, ''The Odyssey'', with an English translation by A.T. Murray, Loeb Classical Library, Harvard University Press, Cambridge, MA, 1980. First printed 1919.<br />
|-<br />
| valign=top | [Jam]<br />
| James, William, ''Pragmatism : A New Name for Some Old Ways of Thinking'', Longmans, Green, and Company, New York, NY, 1907.<br />
|-<br />
| valign=top | [Ler]<br />
| Leroux, Gaston, ''The Phantom of the Opera'', foreword by P. Haining, Dorset Press, New York, NY, 1988. Originally published in French, 1911.<br />
|-<br />
| valign=top | [Mus]<br />
| Musil, Robert, ''The Man Without Qualities'', 3 volumes, translated with a foreword by Eithne Wilkins and Ernst Kaiser, Pan Books, London, UK, 1979. English edition first published by Secker and Warburg, 1954. Originally published in German, ''Der Mann ohne Eigenschaften'', 1930 & 1932.<br />
|-<br />
| valign=top | [PlaR]<br />
| Plato, ''The Republic'', with an English translation by Paul Shorey, Loeb Classical Library, Harvard University Press, Cambridge, MA, 1980. First printed 1930 & 1935.<br />
|-<br />
| valign=top | [PlaS]<br />
| Plato, ''The Sophist'', Loeb Classical Library, William Heinemann, London, 1921, 1987.<br />
|-<br />
| valign=top | [Qui]<br />
| Quine, W.V., ''Mathematical Logic'', 1st edition, 1940. Revised edition, 1951. Harvard University Press, Cambridge, MA, 1981.<br />
|-<br />
| valign=top | [SaD]<br />
| de Santillana, Giorgio, and von Dechend, Hertha, ''Hamlet's Mill : An Essay on Myth and the Frame of Time'', David R. Godine, Publisher, Boston, MA, 1977. 1st published 1969.<br />
|-<br />
| valign=top | [Sha]<br />
| Shakespeare, William, '' William Shakespeare : The Complete Works'', Compact Edition, S. Wells and G. Taylor (eds.), Oxford University Press, Oxford, UK, 1988.<br />
|-<br />
| valign=top | [Sh1]<br />
| Shakespeare, William, ''A Midsummer Night's Dream'', Washington Square Press, New York, NY, 1958.<br />
|-<br />
| valign=top | [Sh2]<br />
| Shakespeare, William, ''The Tragedy of Hamlet, Prince of Denmark'', In [Sha], pp. 654–690.<br />
|-<br />
| valign=top | [Sh3]<br />
| Shakespeare, William, ''Measure for Measure'', Washington Square Press, New York, NY, 1965.<br />
|-<br />
| valign=top | [Web]<br />
| ''Webster's Ninth New Collegiate Dictionary'', Merriam-Webster, Springfield, MA, 1983.<br />
|-<br />
| valign=top | [Whi]<br />
| Whitman, Walt, ''Leaves of Grass'', Vintage Books / The Library of America, New York, NY, 1992. Originally published in numerous editions, 1855–1892.<br />
|-<br />
| valign=top | [Wil]<br />
| Wilhelm, R., and Baynes, C.F. (trans.), ''The I Ching, or Book of Changes'', foreword by C.G. Jung, preface by H. Wilhelm, 3rd edition, Bollingen Series XIX, Princeton University Press, Princeton, NJ, 1967.<br />
|}<br />
<br />
==Document History==<br />
<br />
<p align="center"><math>\begin{array}{lcr}<br />
& \text{Differential Logic and Dynamic Systems} &<br />
\\<br />
\text{Author:} & \text{Jon Awbrey} & \text{October 20, 1994}<br />
\\<br />
\text{Course:} & \text{Engineering 690, Graduate Project} & \text{Winter Term 1994}<br />
\\<br />
\text{Supervisor:} & \text{M.A. Zohdy} & \text{Oakland University}<br />
\\<br />
\text{Created:} && \text{16 Dec 1993}<br />
\\<br />
\text{Relayed:} && \text{31 Oct 1994}<br />
\\<br />
\text{Revised:} && \text{03 Jun 2003}<br />
\\<br />
\text{Recoded:} && \text{03 Jun 2007}<br />
\end{array}</math></p><br />
<br />
[[Category:Adaptive Systems]]<br />
[[Category:Artificial Intelligence]]<br />
[[Category:Combinatorics]]<br />
[[Category:Computer Science]]<br />
[[Category:Cybernetics]]<br />
[[Category:Differential Logic]]<br />
[[Category:Discrete Systems]]<br />
[[Category:Dynamical Systems]]<br />
[[Category:Formal Languages]]<br />
[[Category:Formal Sciences]]<br />
[[Category:Formal Systems]]<br />
[[Category:Functional Logic]]<br />
[[Category:Graph Theory]]<br />
[[Category:Group Theory]]<br />
[[Category:Inquiry]]<br />
[[Category:Knowledge Representation]]<br />
[[Category:Linguistics]]<br />
[[Category:Logic]]<br />
[[Category:Logical Graphs]]<br />
[[Category:Mathematics]]<br />
[[Category:Mathematical Systems Theory]]<br />
[[Category:Science]]<br />
[[Category:Semiotics]]<br />
[[Category:Philosophy]]<br />
[[Category:Systems Science]]<br />
[[Category:Visualization]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Directory:Jon_Awbrey/Poetry/Questionable_Verses&diff=480533Directory:Jon Awbrey/Poetry/Questionable Verses2022-07-05T19:45:08Z<p>Jon Awbrey: /* For the Master of the Games on His Birthday */</p>
<hr />
<div>{{DISPLAYTITLE:Questionable Verses}}<br />
<br />
==Ethos Anthropoi Daimon==<br />
<br />
<pre><br />
deeming done dun a man doom<br />
a man's manner is'm demoon<br />
<br />
a man's cant is his dome :<br />
a woman's master in that<br />
enchanted domain<br />
where conscience is the demon ,<br />
and the code of his duel is decrypted<br />
with a ringing in the ears<br />
and a tire iron coming<br />
from the corner of an eye .<br />
<br />
a man's manner is'm demoon<br />
deeming done din a man doom<br />
<br />
jon awbrey<br />
scribed in a volume of magritte plates<br />
for sue on her birthday 1980 with love<br />
</pre><br />
<br />
==Cassandroid Anxieties Revisited==<br />
<br />
<pre><br />
And the Ire came over Cassandra,<br />
And Cassandra went out the City,<br />
And Cassandra came to that Rock,<br />
And thus spake the Rock, I hear:<br />
<br />
And why are you thus speaking to a Rock, It says,<br />
And when the State of your City is Dire, It says,<br />
<br />
And it is Just your Monu-Mental Agility, She says,<br />
And it is Just your Ex-Orbital Capacity, She says,<br />
<br />
And should you not Ride Over Our Moment, It says,<br />
And should you not Ventilate Our Hyphen, It says,<br />
<br />
And should you Muse Over Our Reflection, She says,<br />
And Imp Etat "I Travail" As "A Trivial", She says,<br />
<br />
And after that they settled down a Bit, I hear,<br />
And after that they counted more Amens, I hear,<br />
<br />
And the Rock will be open to criticism, She says,<br />
And the Rock may reflect on its moment, She says,<br />
<br />
And still the Men of the City will vie, It says,<br />
And still the Men of the City will die, It says,<br />
<br />
And they are Persians all, so they say, She says,<br />
And they are wary of Geeks' boring GIF, She says,<br />
<br />
And still the Men of the City will lie, It says,<br />
And still the Men of the City will lie, It says,<br />
<br />
And all of that Need need come to pass, She says,<br />
And must pass muster to bear Invention, She says,<br />
<br />
And the Rock will be open to criticism, It says,<br />
And the Rock may reflect on its moment, It says,<br />
<br />
And that is just Men, the Sphinx knows, She says,<br />
And that is just Ending-Where-He-Begat, It says,<br />
And that is all there is to say, or so, I hear.<br />
<br />
Jon Awbrey<br />
21 Nov 2000<br />
</pre><br />
<br />
==Peerage==<br />
<br />
<pre><br />
there's no presumption in peer,<br />
much less peremptory emptiness,<br />
to whom can crack a dictionary.<br />
but who presumes to umpireship,<br />
is a nonpeer captain of umpiry.<br />
crack this code, crack a smile,<br />
and put the e bak in crackpoet.<br />
<br />
thus spake 0*<br />
10 Dec 2005<br />
</pre><br />
<br />
==For the Master of the Games on His Birthday==<br />
<br />
<pre><br />
Do we feign understanding of what underlies,<br />
And so become victim to the figmentations<br />
The mind posts up to the lintels of sense?<br />
<br />
Or do we but array the Data Of The Senses<br />
Along the lines that they themselves suggest?<br />
<br />
Then again, on third thought, what's the diff?<br />
<br />
— Jon Awbrey • 16 December 2007<br />
</pre><br />
<br />
<br><br />
<br />
[[Category:Poetry]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Directory:Jon_Awbrey/Poetry/Questionable_Verses&diff=480532Directory:Jon Awbrey/Poetry/Questionable Verses2022-07-05T19:43:01Z<p>Jon Awbrey: /* For the Master of the Games on His Birthday */</p>
<hr />
<div>{{DISPLAYTITLE:Questionable Verses}}<br />
<br />
==Ethos Anthropoi Daimon==<br />
<br />
<pre><br />
deeming done dun a man doom<br />
a man's manner is'm demoon<br />
<br />
a man's cant is his dome :<br />
a woman's master in that<br />
enchanted domain<br />
where conscience is the demon ,<br />
and the code of his duel is decrypted<br />
with a ringing in the ears<br />
and a tire iron coming<br />
from the corner of an eye .<br />
<br />
a man's manner is'm demoon<br />
deeming done din a man doom<br />
<br />
jon awbrey<br />
scribed in a volume of magritte plates<br />
for sue on her birthday 1980 with love<br />
</pre><br />
<br />
==Cassandroid Anxieties Revisited==<br />
<br />
<pre><br />
And the Ire came over Cassandra,<br />
And Cassandra went out the City,<br />
And Cassandra came to that Rock,<br />
And thus spake the Rock, I hear:<br />
<br />
And why are you thus speaking to a Rock, It says,<br />
And when the State of your City is Dire, It says,<br />
<br />
And it is Just your Monu-Mental Agility, She says,<br />
And it is Just your Ex-Orbital Capacity, She says,<br />
<br />
And should you not Ride Over Our Moment, It says,<br />
And should you not Ventilate Our Hyphen, It says,<br />
<br />
And should you Muse Over Our Reflection, She says,<br />
And Imp Etat "I Travail" As "A Trivial", She says,<br />
<br />
And after that they settled down a Bit, I hear,<br />
And after that they counted more Amens, I hear,<br />
<br />
And the Rock will be open to criticism, She says,<br />
And the Rock may reflect on its moment, She says,<br />
<br />
And still the Men of the City will vie, It says,<br />
And still the Men of the City will die, It says,<br />
<br />
And they are Persians all, so they say, She says,<br />
And they are wary of Geeks' boring GIF, She says,<br />
<br />
And still the Men of the City will lie, It says,<br />
And still the Men of the City will lie, It says,<br />
<br />
And all of that Need need come to pass, She says,<br />
And must pass muster to bear Invention, She says,<br />
<br />
And the Rock will be open to criticism, It says,<br />
And the Rock may reflect on its moment, It says,<br />
<br />
And that is just Men, the Sphinx knows, She says,<br />
And that is just Ending-Where-He-Begat, It says,<br />
And that is all there is to say, or so, I hear.<br />
<br />
Jon Awbrey<br />
21 Nov 2000<br />
</pre><br />
<br />
==Peerage==<br />
<br />
<pre><br />
there's no presumption in peer,<br />
much less peremptory emptiness,<br />
to whom can crack a dictionary.<br />
but who presumes to umpireship,<br />
is a nonpeer captain of umpiry.<br />
crack this code, crack a smile,<br />
and put the e bak in crackpoet.<br />
<br />
thus spake 0*<br />
10 Dec 2005<br />
</pre><br />
<br />
==For the Master of the Games on His Birthday==<br />
<br />
<pre><br />
Do we feign understanding of what underlies,<br />
And so become victim to the figmentations<br />
The mind posts up to the lintels of sense?<br />
<br />
Or do we but array the Data Of The Senses<br />
Along the lines that they themselves suggest?<br />
<br />
Then again, on third thought, what's the diff?<br />
<br />
— Jon Awbrey, 16 December 2007<br />
</pre><br />
<br />
<br><br />
<br />
[[Category:Poetry]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Differential_Logic_and_Dynamic_Systems&diff=469887Differential Logic and Dynamic Systems2021-01-14T21:56:44Z<p>Jon Awbrey: /* A Differential Extension of Propositional Calculus */ update</p>
<hr />
<div>'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''<br />
<br />
{| align="center" cellpadding="10"<br />
| [[File:Tangent Functor Ferris Wheel.jpg]]<br />
|}<br />
<br />
{| style="height:36px; width:100%"<br />
| align="left" | ''Stand and unfold yourself.''<br />
| align="right" | Hamlet: Francsico&mdash;1.1.2<br />
|}<br />
<br />
This article develops a differential extension of propositional calculus and applies it to a context of problems arising in dynamic systems.&nbsp; The work pursued here is coordinated with a parallel application that focuses on neural network systems, but the dependencies are arranged to make the present article the main and the more self-contained work, to serve as a conceptual frame and a technical background for the network project.<br />
<br />
==Review and Transition==<br />
<br />
This note continues a previous discussion on the problem of dealing with change and diversity in logic-based intelligent systems. It is useful to begin by summarizing essential material from previous reports.<br />
<br />
Table 1 outlines a notation for propositional calculus based on two types of logical connectives, both of variable <math>k</math>-ary scope.<br />
<br />
* A bracketed list of propositional expressions in the form <math>\texttt{(} e_1, e_2, \ldots, e_{k-1}, e_k \texttt{)}</math> indicates that exactly one of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> is false.<br />
<br />
* A concatenation of propositional expressions in the form <math>e_1 ~ e_2 ~ \ldots ~ e_{k-1} ~ e_k</math> indicates that all of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> are true, in other words, that their [[logical conjunction]] is true.<br />
<br />
All other propositional connectives can be obtained in a very efficient style of representation through combinations of these two forms. Strictly speaking, the concatenation form is dispensable in light of the bracketed form, but it is convenient to maintain it as an abbreviation of more complicated bracket expressions.<br />
<br />
This treatment of propositional logic is derived from the work of C.S. Peirce [P1, P2], who gave this approach an extensive development in his graphical systems of predicate, relational, and modal logic [Rob]. More recently, these ideas were revived and supplemented in an alternative interpretation by George Spencer-Brown [SpB]. Both of these authors used other forms of enclosure where I use parentheses, but the structural topologies of expression and the functional varieties of interpretation are fundamentally the same.<br />
<br />
While working with expressions solely in propositional calculus, it is easiest to use plain parentheses for logical connectives. In contexts where parentheses are needed for other purposes &ldquo;teletype&rdquo; parentheses <math>\texttt{(} \ldots \texttt{)}</math> or barred parentheses <math>(\!| \ldots |\!)</math> may be used for logical operators.<br />
<br />
The briefest expression for logical truth is the empty word, usually denoted by <math>{}^{\backprime\backprime} \boldsymbol\varepsilon {}^{\prime\prime}</math> or <math>{}^{\backprime\backprime} \boldsymbol\lambda {}^{\prime\prime}</math> in formal languages, where it forms the identity element for concatenation. To make it visible in this text, it may be denoted by the equivalent expression <math>{}^{\backprime\backprime} \texttt{((} ~ \texttt{))} {}^{\prime\prime},</math> or, especially if operating in an algebraic context, by a simple <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}.</math> Also when working in an algebraic mode, the plus sign <math>{}^{\backprime\backprime} + {}^{\prime\prime}</math> may be used for [[exclusive disjunction]]. For example, we have the following paraphrases of algebraic expressions by bracket expressions:<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
|<br />
<math>\begin{matrix}<br />
x + y ~=~ \texttt{(} x, y \texttt{)}<br />
\\[6pt]<br />
x + y + z ~=~ \texttt{((} x, y \texttt{)}, z \texttt{)} ~=~ \texttt{(} x, \texttt{(} y, z \texttt{))}<br />
\end{matrix}</math><br />
|}<br />
<br />
It is important to note that the last expressions are not equivalent to the triple bracket <math>\texttt{(} x, y, z \texttt{)}.</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 1.} ~~ \text{Syntax and Semantics of a Calculus for Propositional Logic}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Expression}</math><br />
| <math>\text{Interpretation}</math><br />
| <math>\text{Other Notations}</math><br />
|-<br />
| &nbsp;<br />
| <math>\text{True}</math><br />
| <math>1</math><br />
|-<br />
| <math>\texttt{(} ~ \texttt{)}</math><br />
| <math>\text{False}</math><br />
| <math>0</math><br />
|-<br />
| <math>x</math><br />
| <math>x</math><br />
| <math>x</math><br />
|-<br />
| <math>\texttt{(} x \texttt{)}</math><br />
| <math>\text{Not}~ x</math><br />
|<br />
<math>\begin{matrix}<br />
x'<br />
\\<br />
\tilde{x}<br />
\\<br />
\lnot x<br />
\end{matrix}</math><br />
|-<br />
| <math>x~y~z</math><br />
| <math>x ~\text{and}~ y ~\text{and}~ z</math><br />
| <math>x \land y \land z</math><br />
|-<br />
| <math>\texttt{((} x \texttt{)(} y \texttt{)(} z \texttt{))}</math><br />
| <math>x ~\text{or}~ y ~\text{or}~ z</math><br />
| <math>x \lor y \lor z</math><br />
|-<br />
| <math>\texttt{(} x ~ \texttt{(} y \texttt{))}</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{implies}~ y<br />
\\<br />
\mathrm{If}~ x ~\text{then}~ y<br />
\end{matrix}</math><br />
| <math>x \Rightarrow y</math><br />
|-<br />
| <math>\texttt{(} x \texttt{,} y \texttt{)}</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{not equal to}~ y<br />
\\<br />
x ~\text{exclusive or}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x \ne y<br />
\\<br />
x + y<br />
\end{matrix}</math><br />
|-<br />
| <math>\texttt{((} x \texttt{,} y \texttt{))}</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{is equal to}~ y<br />
\\<br />
x ~\text{if and only if}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x = y<br />
\\<br />
x \Leftrightarrow y<br />
\end{matrix}</math><br />
|-<br />
| <math>\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Just one of}<br />
\\<br />
x, y, z<br />
\\<br />
\text{is false}.<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x'y~z~ & \lor<br />
\\<br />
x~y'z~ & \lor<br />
\\<br />
x~y~z' &<br />
\end{matrix}</math><br />
|-<br />
| <math>\texttt{((} x \texttt{),(} y \texttt{),(} z \texttt{))}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Just one of}<br />
\\<br />
x, y, z<br />
\\<br />
\text{is true}.<br />
\\<br />
&<br />
\\<br />
\text{Partition all}<br />
\\<br />
\text{into}~ x, y, z.<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x~y'z' & \lor<br />
\\<br />
x'y~z' & \lor<br />
\\<br />
x'y'z~ &<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,} y \texttt{),} z \texttt{)}<br />
\\<br />
&<br />
\\<br />
\texttt{(} x \texttt{,(} y \texttt{,} z \texttt{))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Oddly many of}<br />
\\<br />
x, y, z<br />
\\<br />
\text{are true}.<br />
\end{matrix}</math><br />
|<br />
<p><math>x + y + z</math></p><br />
<br><br />
<p><math>\begin{matrix}<br />
x~y~z~ & \lor<br />
\\<br />
x~y'z' & \lor<br />
\\<br />
x'y~z' & \lor<br />
\\<br />
x'y'z~ &<br />
\end{matrix}</math></p><br />
|-<br />
| <math>\texttt{(} w \texttt{,(} x \texttt{),(} y \texttt{),(} z \texttt{))}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Partition}~ w<br />
\\<br />
\text{into}~ x, y, z.<br />
\\<br />
&<br />
\\<br />
\text{Genus}~ w ~\text{comprises}<br />
\\<br />
\text{species}~ x, y, z.<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
w'x'y'z' & \lor<br />
\\<br />
w~x~y'z' & \lor<br />
\\<br />
w~x'y~z' & \lor<br />
\\<br />
w~x'y'z~ &<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
'''Note.''' The usage that one often sees, of a plus sign "<math>+</math>" to represent inclusive disjunction, and the reference to this operation as ''boolean addition'', is a misnomer on at least two counts. Boole used the plus sign to represent exclusive disjunction (at any rate, an operation of aggregation restricted in its logical interpretation to cases where the represented sets are disjoint (Boole, 32)), as any mathematician with a sensitivity to the ring and field properties of algebra would do:<br />
<br />
<blockquote><br />
The expression <math>x + y</math> seems indeed uninterpretable, unless it be assumed that the things represented by <math>x</math> and the things represented by <math>y</math> are entirely separate; that they embrace no individuals in common. (Boole, 66).<br />
</blockquote><br />
<br />
It was only later that Peirce and Jevons treated inclusive disjunction as a fundamental operation, but these authors, with a respect for the algebraic properties that were already associated with the plus sign, used a variety of other symbols for inclusive disjunction (Sty, 177, 189). It seems to have been Schröder who later reassigned the plus sign to inclusive disjunction (Sty, 208). Additional information, discussion, and references can be found in (Boole) and (Sty, 177&ndash;263). Aside from these historical points, which never really count against a current practice that has gained a life of its own, this usage does have a further disadvantage of cutting or confounding the lines of communication between algebra and logic. For this reason, it will be avoided here.<br />
<br />
==A Functional Conception of Propositional Calculus==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Out of the dimness opposite equals advance . . . .<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Always substance and increase,<br><br />
Always a knit of identity . . . . always distinction . . . .<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;always a breed of life.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]<br />
|}<br />
<br />
In the general case, we start with a set of logical features <math>\{a_1, \ldots, a_n\}</math> that represent properties of objects or propositions about the world. In concrete examples the features <math>\{a_i\}</math> commonly appear as capital letters from an ''alphabet'' like <math>\{A, B, C, \ldots\}</math> or as meaningful words from a linguistic ''vocabulary'' of codes. This language can be drawn from any sources, whether natural, technical, or artificial in character and interpretation. In the application to dynamic systems we tend to use the letters <math>\{x_1, \ldots, x_n\}</math> as our coordinate propositions, and to interpret them as denoting properties of a system's ''state'', that is, as propositions about its location in configuration space. Because I have to consider non-deterministic systems from the outset, I often use the word ''state'' in a loose sense, to denote the position or configuration component of a contemplated state vector, whether or not it ever gets a deterministic completion.<br />
<br />
The set of logical features <math>\{a_1, \ldots, a_n\}</math> provides a basis for generating an <math>n</math>-dimensional ''[[universe of discourse]]'' that I denote as <math>[a_1, \ldots, a_n].</math> It is useful to consider each universe of discourse as a unified categorical object that incorporates both the set of points <math>\langle a_1, \ldots, a_n \rangle</math> and the set of propositions <math>f : \langle a_1, \ldots, a_n \rangle \to \mathbb{B}</math> that are implicit with the ordinary picture of a venn diagram on <math>n</math> features. Thus, we may regard the universe of discourse <math>[a_1, \ldots, a_n]</math> as an ordered pair having the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}),</math> and we may abbreviate this last type designation as <math>\mathbb{B}^n\ +\!\to \mathbb{B},</math> or even more succinctly as <math>[\mathbb{B}^n].</math> (Used this way, the angle brackets <math>\langle\ldots\rangle</math> are referred to as ''generator brackets''.)<br />
<br />
Table 2 exhibits the scheme of notation I use to formalize the domain of propositional calculus, corresponding to the logical content of truth tables and venn diagrams. Although it overworks the square brackets a bit, I also use either one of the equivalent notations <math>[n]</math> or <math>\mathbf{n}</math> to denote the data type of a finite set on <math>n</math> elements.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 2.} ~~ \text{Propositional Calculus : Basic Notation}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Symbol}</math><br />
| <math>\text{Notation}</math><br />
| <math>\text{Description}</math><br />
| <math>\text{Type}</math><br />
|-<br />
| <math>\mathfrak{A}</math><br />
| <math>\{ {}^{\backprime\backprime} a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} a_n {}^{\prime\prime} \}</math><br />
| <math>\text{Alphabet}</math><br />
| <math>[n] = \mathbf{n}</math><br />
|-<br />
| <math>\mathcal{A}</math><br />
| <math>\{ a_1, \ldots, a_n \}</math><br />
| <math>\text{Basis}</math><br />
| <math>[n] = \mathbf{n}</math><br />
|-<br />
| <math>A_i</math><br />
| <math>\{ \texttt{(} a_i \texttt{)}, a_i \}</math><br />
| <math>\text{Dimension}~ i</math><br />
| <math>\mathbb{B}</math><br />
|-<br />
| <math>A</math><br />
|<br />
<math>\begin{matrix}<br />
\langle \mathcal{A} \rangle<br />
\\[2pt]<br />
\langle a_1, \ldots, a_n \rangle<br />
\\[2pt]<br />
\{ (a_1, \ldots, a_n) \}<br />
\\[2pt]<br />
A_1 \times \ldots \times A_n<br />
\\[2pt]<br />
\textstyle \prod_{i=1}^n A_i<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Set of cells},<br />
\\[2pt]<br />
\text{coordinate tuples},<br />
\\[2pt]<br />
\text{points, or vectors}<br />
\\[2pt]<br />
\text{in the universe}<br />
\\[2pt]<br />
\text{of discourse}<br />
\end{matrix}</math><br />
| <math>\mathbb{B}^n</math><br />
|-<br />
| <math>A^*</math><br />
| <math>(\mathrm{hom} : A \to \mathbb{B})</math><br />
| <math>\text{Linear functions}</math><br />
| <math>(\mathbb{B}^n)^* \cong \mathbb{B}^n</math><br />
|-<br />
| <math>A^\uparrow</math><br />
| <math>(A \to \mathbb{B})</math><br />
| <math>\text{Boolean functions}</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}</math><br />
|-<br />
| <math>A^\bullet</math><br />
|<br />
<math>\begin{matrix}<br />
[\mathcal{A}]<br />
\\[2pt]<br />
(A, A^\uparrow)<br />
\\[2pt]<br />
(A ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
(A, (A \to \mathbb{B}))<br />
\\[2pt]<br />
[a_1, \ldots, a_n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Universe of discourse}<br />
\\[2pt]<br />
\text{based on the features}<br />
\\[2pt]<br />
\{ a_1, \ldots, a_n \}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))<br />
\\[2pt]<br />
(\mathbb{B}^n ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
[\mathbb{B}^n]<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
===Qualitative Logic and Quantitative Analogy===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
''Logical'', however, is used in a third sense, which is at once more vital and more practical; to denote, namely, the systematic care, negative and positive, taken to safeguard reflection so that it may yield the best results under the given conditions.<br />
| width="4%" | &nbsp;<br />
|- <br />
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 56]<br />
|}<br />
<br />
These concepts and notations may now be explained in greater detail. In order to begin as simply as possible, let us distinguish two levels of analysis and set out initially on the easier path. On the first level of analysis we take spaces like <math>\mathbb{B},</math> <math>\mathbb{B}^n,</math> and <math>(\mathbb{B}^n \to \mathbb{B})</math> at face value and treat them as the primary objects of interest. On the second level of analysis we use these spaces as coordinate charts for talking about points and functions in more fundamental spaces.<br />
<br />
A pair of spaces, of types <math>\mathbb{B}^n</math> and <math>(\mathbb{B}^n \to \mathbb{B}),</math> give typical expression to everything we commonly associate with the ordinary picture of a venn diagram. The dimension, <math>n,</math> counts the number of &ldquo;circles&rdquo; or simple closed curves that are inscribed in the universe of discourse, corresponding to its relevant logical features or basic propositions. Elements of type <math>\mathbb{B}^n</math> correspond to what are often called propositional ''interpretations'' in logic, that is, the different assignments of truth values to sentence letters. Relative to a given universe of discourse, these interpretations are visualized as its ''cells'', in other words, the smallest enclosed areas or undivided regions of the venn diagram. The functions <math>f : \mathbb{B}^n \to \mathbb{B}</math> correspond to the different ways of shading the venn diagram to indicate arbitrary propositions, regions, or sets. Regions included under a shading indicate the ''models'', and regions excluded represent the ''non-models'' of a proposition. To recognize and formalize the natural cohesion of these two layers of concepts into a single universe of discourse, we introduce the type notations <math>[\mathbb{B}^n] = \mathbb{B}^n\ +\!\to \mathbb{B}</math> to stand for the pair of types <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})).</math> The resulting &ldquo;stereotype&rdquo; serves to frame the universe of discourse as a unified categorical object, and makes it subject to prescribed sets of evaluations and transformations (categorical morphisms or ''arrows'') that affect the universe of discourse as an integrated whole.<br />
<br />
Most of the time we can serve the algebraic, geometric, and logical interests of our study without worrying about their occasional conflicts and incidental divergences. The conventions and definitions already set down will continue to cover most of the algebraic and functional aspects of our discussion, but to handle the logical and qualitative aspects we will need to add a few more. In general, abstract sets may be denoted by gothic, greek, or script capital variants of <math>A, B, C,</math> and so on, with their elements being denoted by a corresponding set of subscripted letters in plain lower case, for example, <math>\mathcal{A} = \{a_i\}.</math> Most of the time, a set such as <math>\mathcal{A} = \{a_i\}</math> will be employed as the ''alphabet'' of a [[formal language]]. These alphabet letters serve to name the logical features (properties or propositions) that generate a particular universe of discourse. When we want to discuss the particular features of a universe of discourse, beyond the abstract designation of a type like <math>(\mathbb{B}^n\ +\!\to \mathbb{B}),</math> then we may use the following notations. If <math>\mathcal{A} = \{a_1, \ldots, a_n\}</math> is an alphabet of logical features, then <math>A = \langle \mathcal{A} \rangle = \langle a_1, \ldots, a_n \rangle</math> is the set of interpretations, <math>A^\uparrow = (A \to \mathbb{B})</math> is the set of propositions, and <math>A^\bullet = [\mathcal{A}] = [a_1, \ldots, a_n]</math> is the combination of these interpretations and propositions into the universe of discourse that is based on the features <math>\{a_1, \ldots, a_n\}.</math><br />
<br />
As always, especially in concrete examples, these rules may be dropped whenever necessary, reverting to a free assortment of feature labels. However, when we need to talk about the logical aspects of a space that is already named as a vector space, it will be necessary to make special provisions. At any rate, these elaborations can be deferred until actually needed.<br />
<br />
===Philosophy of Notation : Formal Terms and Flexible Types===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Where number is irrelevant, regimented mathematical technique has hitherto tended to be lacking. Thus it is that the progress of natural science has depended so largely upon the discernment of measurable quantity of one sort or another.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]<br />
|}<br />
<br />
For much of our discussion propositions and boolean functions are treated as the same formal objects, or as different interpretations of the same formal calculus. This rule of interpretation has exceptions, though. There is a distinctively logical interest in the use of propositional calculus that is not exhausted by its functional interpretation. It is part of our task in this study to deal with these uniquely logical characteristics as they present themselves both in our subject matter and in our formal calculus. Just to provide a hint of what's at stake: In logic, as opposed to the more imaginative realms of mathematics, we consider it a good thing to always know what we are talking about. Where mathematics encourages tolerance for uninterpreted symbols as intermediate terms, logic exerts a keener effort to interpret directly each oblique carrier of meaning, no matter how slight, and to unfold the complicities of every indirection in the flow of information. Translated into functional terms, this means that we want to maintain a continual, immediate, and persistent sense of both the converse relation <math>f^{-1} \subseteq \mathbb{B} \times \mathbb{B}^n,</math> or what is the same thing, <math>f^{-1} : \mathbb{B} \to \mathcal{P}(\mathbb{B}^n),</math> and the ''fibers'' or inverse images <math>f^{-1}(0)</math> and <math>f^{-1}(1),</math> associated with each boolean function <math>f : \mathbb{B}^n \to \mathbb{B}</math> that we use. In practical terms, the desired implementation of a propositional interpreter should incorporate our intuitive recognition that the induced partition of the functional domain into level sets <math>f^{-1}(b),</math> for <math>b \in \mathbb{B},</math> is part and parcel of understanding the denotative uses of each propositional function <math>f.</math><br />
<br />
===Special Classes of Propositions===<br />
<br />
It is important to remember that the coordinate propositions <math>\{a_i\},</math> besides being projection maps <math>a_i : \mathbb{B}^n \to \mathbb{B},</math> are propositions on an equal footing with all others, even though employed as a basis in a particular moment. This set of <math>n</math> propositions may sometimes be referred to as the ''basic propositions'', the ''coordinate propositions'', or the ''simple propositions'' that found a universe of discourse. Either one of the equivalent notations, <math>\{a_i : \mathbb{B}^n \to \mathbb{B}\}</math> or <math>(\mathbb{B}^n \xrightarrow{i} \mathbb{B}),</math> may be used to indicate the adoption of the propositions <math>a_i</math> as a basis for describing a universe of discourse.<br />
<br />
Among the <math>2^{2^n}</math> propositions in <math>[a_1, \ldots, a_n]</math> are several families of <math>2^n</math> propositions each that take on special forms with respect to the basis <math>\{ a_1, \ldots, a_n \}.</math> Three of these families are especially prominent in the present context, the ''linear'', the ''positive'', and the ''singular'' propositions. Each family is naturally parameterized by the coordinate <math>n</math>-tuples in <math>\mathbb{B}^n</math> and falls into <math>n + 1</math> ranks, with a binomial coefficient <math>\tbinom{n}{k}</math> giving the number of propositions that have rank or weight <math>k.</math><br />
<br />
<ul><br />
<br />
<li><br />
<p>The ''linear propositions'', <math>\{ \ell : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{\ell} \mathbb{B}),</math> may be written as sums:</p><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\sum_{i=1}^n e_i ~=~ e_1 + \ldots + e_n<br />
~\text{where}~<br />
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 0 \end{matrix}\right\}<br />
~\text{for}~ i = 1 ~\text{to}~ n.</math><br />
|}<br />
</li><br />
<br />
<li><br />
<p>The ''positive propositions'', <math>\{ p : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{p} \mathbb{B}),</math> may be written as products:</p><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\prod_{i=1}^n e_i ~=~ e_1 \cdot \ldots \cdot e_n<br />
~\text{where}~<br />
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 1 \end{matrix}\right\}<br />
~\text{for}~ i = 1 ~\text{to}~ n.</math><br />
|}<br />
</li><br />
<br />
<li><br />
<p>The ''singular propositions'', <math>\{ \mathbf{x} : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{s} \mathbb{B}),</math> may be written as products:</p><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\prod_{i=1}^n e_i ~=~ e_1 \cdot \ldots \cdot e_n<br />
~\text{where}~<br />
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = \texttt{(} a_i \texttt{)} \end{matrix}\right\}<br />
~\text{for}~ i = 1 ~\text{to}~ n.</math><br />
|}<br />
</li><br />
<br />
</ul><br />
<br />
In each case the rank <math>k</math> ranges from <math>0</math> to <math>n</math> and counts the number of positive appearances of the coordinate propositions <math>a_1, \ldots, a_n</math> in the resulting expression. For example, for <math>{n = 3},</math> the linear proposition of rank <math>0</math> is <math>0,</math> the positive proposition of rank <math>0</math> is <math>1,</math> and the singular proposition of rank <math>0</math> is <math>\texttt{(} a_1 \texttt{)(} a_2 \texttt{)(} a_3\texttt{)}.</math><br />
<br />
The basic propositions <math>a_i : \mathbb{B}^n \to \mathbb{B}</math> are both linear and positive. So these two kinds of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions.<br />
<br />
Linear propositions and positive propositions are generated by taking boolean sums and products, respectively, over selected subsets of basic propositions, so both families of propositions are parameterized by the powerset <math>\mathcal{P}(\mathcal{I}),</math> that is, the set of all subsets <math>J</math> of the basic index set <math>\mathcal{I} = \{1, \ldots, n\}.</math><br />
<br />
Let us define <math>\mathcal{A}_J</math> as the subset of <math>\mathcal{A}</math> that is given by <math>\{a_i : i \in J\}.</math> Then we may comprehend the action of the linear and the positive propositions in the following terms:<br />
<br />
* The linear proposition <math>\ell_J : \mathbb{B}^n \to \mathbb{B}</math> evaluates each cell <math>\mathbf{x}</math> of <math>\mathbb{B}^n</math> by looking at the coefficients of <math>\mathbf{x}</math> with respect to the features that <math>\ell_J</math> "likes", namely those in <math>\mathcal{A}_J,</math> and then adds them up in <math>\mathbb{B}.</math> Thus, <math>\ell_J(\mathbf{x})</math> computes the parity of the number of features that <math>\mathbf{x}</math> has in <math>\mathcal{A}_J,</math> yielding one for odd and zero for even. Expressed in this idiom, <math>\ell_J(\mathbf{x}) = 1</math> says that <math>\mathbf{x}</math> seems ''odd'' (or ''oddly true'') to <math>\mathcal{A}_J,</math> whereas <math>\ell_J(\mathbf{x}) = 0</math> says that <math>\mathbf{x}</math> seems ''even'' (or ''evenly true'') to <math>\mathcal{A}_J,</math> so long as we recall that ''zero times'' is evenly often, too.<br />
<br />
* The positive proposition <math>p_J : \mathbb{B}^n \to \mathbb{B}</math> evaluates each cell <math>\mathbf{x}</math> of <math>\mathbb{B}^n</math> by looking at the coefficients of <math>\mathbf{x}</math> with regard to the features that <math>p_J</math> "likes", namely those in <math>\mathcal{A}_J,</math> and then takes their product in <math>\mathbb{B}.</math> Thus, <math>p_J(\mathbf{x})</math> assesses the unanimity of the multitude of features that <math>\mathbf{x}</math> has in <math>\mathcal{A}_J,</math> yielding one for all and aught for else. In these consensual or contractual terms, <math>p_J(\mathbf{x}) = 1</math> means that <math>\mathbf{x}</math> is ''AOK'' or congruent with all of the conditions of <math>\mathcal{A}_J,</math> while <math>p_J(\mathbf{x}) = 0</math> means that <math>\mathbf{x}</math> defaults or dissents from some condition of <math>\mathcal{A}_J.</math><br />
<br />
===Basis Relativity and Type Ambiguity===<br />
<br />
Finally, two things are important to keep in mind with regard to the simplicity, linearity, positivity, and singularity of propositions.<br />
<br />
First, all of these properties are relative to a particular basis. For example, a singular proposition with respect to a basis <math>\mathcal{A}</math> will not remain singular if <math>\mathcal{A}</math> is extended by a number of new and independent features. Even if we stick to the original set of pairwise options <math>\{a_i\} \cup \{ \texttt{(} a_i \texttt{)} \}</math> to select a new basis, the sets of linear and positive propositions are determined by the choice of simple propositions, and this determination is tantamount to the conventional choice of a cell as origin.<br />
<br />
Second, the singular propositions <math>\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B},</math> picking out as they do a single cell or a coordinate tuple <math>\mathbf{x}</math> of <math>\mathbb{B}^n,</math> become the carriers or the vehicles of a certain type-ambiguity that vacillates between the dual forms <math>\mathbb{B}^n</math> and <math>(\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B})</math> and infects the whole hierarchy of types built on them. In other words, the terms that signify the interpretations <math>\mathbf{x} : \mathbb{B}^n</math> and the singular propositions <math>\mathbf{x} : \mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B}</math> are fully equivalent in information, and this means that every token of the type <math>\mathbb{B}^n</math> can be reinterpreted as an appearance of the subtype <math>\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B}.</math> And vice versa, the two types can be exchanged with each other everywhere that they turn up. In practical terms, this allows the use of singular propositions as a way of denoting points, forming an alternative to coordinate tuples.<br />
<br />
For example, relative to the universe of discourse <math>[a_1, a_2, a_3]</math> the singular proposition <math>a_1 a_2 a_3 : \mathbb{B}^3 \xrightarrow{s} \mathbb{B}</math> could be explicitly retyped as <math>a_1 a_2 a_3 : \mathbb{B}^3</math> to indicate the point <math>(1, 1, 1)</math> but in most cases the proper interpretation could be gathered from context. Both notations remain dependent on a particular basis, but the code that is generated under the singular option has the advantage in its self-commenting features, in other words, it constantly reminds us of its basis in the process of denoting points. When the time comes to put a multiplicity of different bases into play, and to search for objects and properties that remain invariant under the transformations between them, this infinitesimal potential advantage may well evolve into an overwhelming practical necessity.<br />
<br />
===The Analogy Between Real and Boolean Types===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Measurement consists in correlating our subject matter with the series of real numbers; and such correlations are desirable because, once they are set up, all the well-worked theory of numerical mathematics lies ready at hand as a tool for our further reasoning.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]<br />
|}<br />
<br />
There are two further reasons why it useful to spend time on a careful treatment of types, and they both have to do with our being able to take full computational advantage of certain dimensions of flexibility in the types that apply to terms. First, the domains of differential geometry and logic programming are connected by analogies between real and boolean types of the same pattern. Second, the types involved in these patterns have important isomorphisms connecting them that apply on both the real and the boolean sides of the picture.<br />
<br />
Amazingly enough, these isomorphisms are themselves schematized by the axioms and theorems of propositional logic. This fact is known as the ''propositions as types'' analogy or the Curry&ndash;Howard isomorphism [How]. In another formulation it says that terms are to types as proofs are to propositions. See [LaS, 42&ndash;46] and [SeH] for a good discussion and further references. To anticipate the bearing of these issues on our immediate topic, Table&nbsp;3 sketches a partial overview of the Real to Boolean analogy that may serve to illustrate the paradigm in question.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 3.} ~~ \text{Analogy Between Real and Boolean Types}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Real Domain} ~ \mathbb{R}</math><br />
| <math>\longleftrightarrow</math><br />
| <math>\text{Boolean Domain} ~ \mathbb{B}</math><br />
|-<br />
| <math>\mathbb{R}^n</math><br />
| <math>\text{Basic Space}</math><br />
| <math>\mathbb{B}^n</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}</math><br />
| <math>\text{Function Space}</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R}</math><br />
| <math>\text{Tangent Vector}</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math><br />
|-<br />
| <math>\mathbb{R}^n \to ((\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R})</math><br />
| <math>\text{Vector Field}</math><br />
| <math>\mathbb{B}^n \to ((\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B})</math><br />
|-<br />
| <math>(\mathbb{R}^n \times (\mathbb{R}^n \to \mathbb{R})) \to \mathbb{R}</math><br />
| '''<font size="4">"</font>'''<br />
| <math>(\mathbb{B}^n \times (\mathbb{B}^n \to \mathbb{B})) \to \mathbb{B}</math><br />
|-<br />
| <math>((\mathbb{R}^n \to \mathbb{R}) \times \mathbb{R}^n) \to \mathbb{R}</math><br />
| '''<font size="4">"</font>'''<br />
| <math>((\mathbb{B}^n \to \mathbb{B}) \times \mathbb{B}^n) \to \mathbb{B}</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^n \to \mathbb{R})</math><br />
| <math>\text{Derivation}</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^n \to \mathbb{B})</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}^m</math><br />
| <math>\begin{matrix}\text{Basic}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}^m</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^m \to \mathbb{R})</math><br />
| <math>\begin{matrix}\text{Function}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^m \to \mathbb{B})</math><br />
|}<br />
<br />
<br><br />
<br />
The Table exhibits a sample of likely parallels between the real and boolean domains. The central column gives a selection of terminology that is borrowed from differential geometry and extended in its meaning to the logical side of the Table. These are the varieties of spaces that come up when we turn to analyzing the dynamics of processes that pursue their courses through the states of an arbitrary space <math>X.</math> Moreover, when it becomes necessary to approach situations of overwhelming dynamic complexity in a succession of qualitative reaches, then the methods of logic that are afforded by the boolean domains, with their declarative means of synthesis and deductive modes of analysis, supply a natural battery of tools for the task.<br />
<br />
It is usually expedient to take these spaces two at a time, in dual pairs of the form <math>X</math> and <math>(X \to \mathbb{K}).</math> In general, one creates pairs of type schemas by replacing any space <math>X</math> with its dual <math>(X \to \mathbb{K}),</math> for example, pairing the type <math>X \to Y</math> with the type <math>(X \to \mathbb{K}) \to (Y \to \mathbb{K}),</math> and <math>X \times Y</math> with <math>(X \to \mathbb{K}) \times (Y \to \mathbb{K}).</math> The word ''dual'' is used here in its broader sense to mean all of the functionals, not just the linear ones. Given any function <math>f : X \to \mathbb{K},</math> the ''converse'' or inverse relation corresponding to <math>f</math> is denoted <math>f^{-1},</math> and the subsets of <math>X</math> that are defined by <math>f^{-1}(k),</math> taken over <math>k</math> in <math>\mathbb{K},</math> are called the ''fibers'' or the ''level sets'' of the function <math>f.</math><br />
<br />
===Theory of Control and Control of Theory===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
You will hardly know who I am or what I mean,<br><br />
But I shall be good health to you nevertheless,<br><br />
And filter and fibre your blood.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]<br />
|}<br />
<br />
In the boolean context a function <math>f : X \to \mathbb{B}</math> is tantamount to a ''proposition'' about elements of <math>X,</math> and the elements of <math>X</math> constitute the ''interpretations'' of that proposition. The fiber <math>f^{-1}(1)</math> comprises the set of ''models'' of <math>f,</math> or examples of elements in <math>X</math> satisfying the proposition <math>f.</math> The fiber <math>f^{-1}(0)</math> collects the complementary set of ''anti-models'', or the exceptions to the proposition <math>f</math> that exist in <math>X.</math> Of course, the space of functions <math>(X \to \mathbb{B})</math> is isomorphic to the set of all subsets of <math>X,</math> called the ''power set'' of <math>X,</math> and often denoted <math>\mathcal{P}(X)</math> or <math>2^X.</math><br />
<br />
The operation of replacing <math>X</math> by <math>(X \to \mathbb{B})</math> in a type schema corresponds to a certain shift of attitude towards the space <math>X,</math> in which one passes from a focus on the ostensibly individual elements of <math>X</math> to a concern with the states of information and uncertainty that one possesses about objects and situations in <math>X.</math> The conceptual obstacles in the path of this transition can be smoothed over by using singular functions <math>(\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B})</math> as stepping stones. First of all, it's an easy step from an element <math>\mathbf{x}</math> of type <math>\mathbb{B}^n</math> to the equivalent information of a singular proposition <math>\mathbf{x} : X \xrightarrow{s} \mathbb{B}, </math> and then only a small jump of generalization remains to reach the type of an arbitrary proposition <math>f : X \to \mathbb{B},</math> perhaps understood to indicate a relaxed constraint on the singularity of points or a neighborhood circumscribing the original <math>\mathbf{x}.</math> This is frequently a useful transformation, communicating between the ''objective'' and the ''intentional'' perspectives, in spite perhaps of the open objection that this distinction is transient in the mean time and ultimately superficial.<br />
<br />
It is hoped that this measure of flexibility, allowing us to stretch a point into a proposition, can be useful in the examination of inquiry driven systems, where the differences between empirical, intentional, and theoretical propositions constitute the discrepancies and the distributions that drive experimental activity. I can give this model of inquiry a cybernetic cast by realizing that theory change and theory evolution, as well as the choice and the evaluation of experiments, are actions that are taken by a system or its agent in response to the differences that are detected between observational contents and theoretical coverage.<br />
<br />
All of the above notwithstanding, there are several points that distinguish these two tasks, namely, the ''theory of control'' and the ''control of theory'', features that are often obscured by too much precipitation in the quickness with which we understand their similarities. In the control of uncertainty through inquiry, some of the actuators that we need to be concerned with are axiom changers and theory modifiers, operators with the power to compile and to revise the theories that generate expectations and predictions, effectors that form and edit our grammars for the languages of observational data, and agencies that rework the proposed model to fit the actual sequences of events and the realized relationships of values that are observed in the environment. Moreover, when steps must be taken to carry out an experimental action, there must be something about the particular shape of our uncertainty that guides us in choosing what directions to explore, and this impression is more than likely influenced by previous accumulations of experience. Thus it must be anticipated that much of what goes into scientific progress, or any sustainable effort toward a goal of knowledge, is necessarily predicated on long term observation and modal expectations, not only on the more local or short term prediction and correction.<br />
<br />
===Propositions as Types and Higher Order Types===<br />
<br />
The types collected in Table&nbsp;3 (repeated below) serve to illustrate the themes of ''higher order propositional expressions'' and the ''propositions as types'' (PAT) analogy.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 3.} ~~ \text{Analogy Between Real and Boolean Types}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Real Domain} ~ \mathbb{R}</math><br />
| <math>\longleftrightarrow</math><br />
| <math>\text{Boolean Domain} ~ \mathbb{B}</math><br />
|-<br />
| <math>\mathbb{R}^n</math><br />
| <math>\text{Basic Space}</math><br />
| <math>\mathbb{B}^n</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}</math><br />
| <math>\text{Function Space}</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R}</math><br />
| <math>\text{Tangent Vector}</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math><br />
|-<br />
| <math>\mathbb{R}^n \to ((\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R})</math><br />
| <math>\text{Vector Field}</math><br />
| <math>\mathbb{B}^n \to ((\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B})</math><br />
|-<br />
| <math>(\mathbb{R}^n \times (\mathbb{R}^n \to \mathbb{R})) \to \mathbb{R}</math><br />
| '''<font size="4">"</font>'''<br />
| <math>(\mathbb{B}^n \times (\mathbb{B}^n \to \mathbb{B})) \to \mathbb{B}</math><br />
|-<br />
| <math>((\mathbb{R}^n \to \mathbb{R}) \times \mathbb{R}^n) \to \mathbb{R}</math><br />
| '''<font size="4">"</font>'''<br />
| <math>((\mathbb{B}^n \to \mathbb{B}) \times \mathbb{B}^n) \to \mathbb{B}</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^n \to \mathbb{R})</math><br />
| <math>\text{Derivation}</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^n \to \mathbb{B})</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}^m</math><br />
| <math>\begin{matrix}\text{Basic}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}^m</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^m \to \mathbb{R})</math><br />
| <math>\begin{matrix}\text{Function}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^m \to \mathbb{B})</math><br />
|}<br />
<br />
<br><br />
<br />
First, observe that the type of a ''tangent vector at a point'', also known as a ''directional derivative'' at that point, has the form <math>(\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K},</math> where <math>\mathbb{K}</math> is the chosen ground field, in the present case either <math>\mathbb{R}</math> or <math>\mathbb{B}.</math> At a point in a space of type <math>\mathbb{K}^n,</math> a directional derivative operator <math>\vartheta</math> takes a function on that space, an <math>f</math> of type <math>(\mathbb{K}^n \to \mathbb{K}),</math> and maps it to a ground field value of type <math>\mathbb{K}.</math> This value is known as the ''derivative'' of <math>f</math> in the direction <math>\vartheta</math> [Che46, 76&ndash;77]. In the boolean case <math>\vartheta : (\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math> has the form of a proposition about propositions, in other words, a proposition of the next higher type.<br />
<br />
Next, by way of illustrating the propositions as types idea, consider a proposition of the form <math>X \Rightarrow (Y \Rightarrow Z).</math> One knows from propositional calculus that this is logically equivalent to a proposition of the form <math>(X \land Y) \Rightarrow Z.</math> But this equivalence should remind us of the functional isomorphism that exists between a construction of the type <math>X \to (Y \to Z)</math> and a construction of the type <math>(X \times Y) \to Z.</math> The propositions as types analogy permits us to take a functional type like this and, under the right conditions, replace the functional arrows &ldquo;<math>\to</math>&rdquo; and products &ldquo;<math>\times</math>&rdquo; with the respective logical arrows &ldquo;<math>\Rightarrow</math>&rdquo; and products &ldquo;<math>\land</math>&rdquo;. Accordingly, viewing the result as a proposition, we can employ axioms and theorems of propositional calculus to suggest appropriate isomorphisms among the categorical and functional constructions.<br />
<br />
Finally, examine the middle four rows of Table&nbsp;3. These display a series of isomorphic types that stretch from the categories that are labeled ''Vector Field'' to those that are labeled ''Derivation''. A ''vector field'', also known as an ''infinitesimal transformation'', associates a tangent vector at a point with each point of a space. In symbols, a vector field is a function of the form <math>\textstyle \xi : X \to \bigcup_{x \in X} \xi_x</math> that assigns to each point <math>x</math> of the space <math>X</math> a tangent vector to <math>X</math> at that point, namely, the tangent vector <math>\xi_x</math> [Che46, 82&ndash;83]. If <math>X</math> is of the type <math>\mathbb{K}^n,</math> then <math>\xi</math> is of the type <math>\mathbb{K}^n \to ((\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K}).</math> This has the pattern <math>X \to (Y \to Z),</math> with <math>X = \mathbb{K}^n,</math> <math>Y = (\mathbb{K}^n \to \mathbb{K}),</math> and <math>Z = \mathbb{K}.</math><br />
<br />
Applying the propositions as types analogy, one can follow this pattern through a series of metamorphoses from the type of a vector field to the type of a derivation, as traced out in Table&nbsp;4. Observe how the function <math>f : X \to \mathbb{K},</math> associated with the place of <math>Y</math> in the pattern, moves through its paces from the second to the first position. In this way, the vector field <math>\xi,</math> initially viewed as attaching each tangent vector <math>\xi_x</math> to the site <math>x</math> where it acts in <math>X,</math> now comes to be seen as acting on each scalar potential <math>f : X \to \mathbb{K}</math> like a generalized species of differentiation, producing another function <math>\xi f : X \to \mathbb{K}</math> of the same type.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 4.} ~~ \text{An Equivalence Based on the Propositions as Types Analogy}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Pattern}</math><br />
| <math>\text{Construct}</math><br />
| <math>\text{Instance}</math><br />
|-<br />
| <math>X \to (Y \to Z)</math><br />
| <math>\text{Vector Field}</math><br />
| <math>\mathbb{K}^n \to ((\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K})</math><br />
|-<br />
| <math>(X \times Y) \to Z</math><br />
| <math>\Uparrow</math><br />
| <math>(\mathbb{K}^n \times (\mathbb{K}^n \to \mathbb{K})) \to \mathbb{K}</math><br />
|-<br />
| <math>(Y \times X) \to Z</math><br />
| <math>\Downarrow</math><br />
| <math>((\mathbb{K}^n \to \mathbb{K}) \times \mathbb{K}^n) \to \mathbb{K}</math><br />
|-<br />
| <math>Y \to (X \to Z)</math><br />
| <math>\text{Derivation}</math><br />
| <math>(\mathbb{K}^n \to \mathbb{K}) \to (\mathbb{K}^n \to \mathbb{K})</math><br />
|}<br />
<br />
<br><br />
<br />
===Reality at the Threshold of Logic===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
But no science can rest entirely on measurement, and many scientific investigations are quite out of reach of that device. To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]<br />
|}<br />
<br />
Table 5 accumulates an array of notation that I hope will not be too distracting. Some of it is rarely needed, but has been filled in for the sake of completeness. Its purpose is simple, to give literal expression to the visual intuitions that come with venn diagrams, and to help build a bridge between our qualitative and quantitative outlooks on dynamic systems.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 5.} ~~ \text{A Bridge Over Troubled Waters}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| align="center" | <math>\text{Linear Space}</math><br />
| align="center" | <math>\text{Liminal Space}</math><br />
| align="center" | <math>\text{Logical Space}</math><br />
|-<br />
| <math>\begin{matrix}\mathcal{X} & = & \{ x_1, \ldots, x_n \}\end{matrix}</math><br />
| <math>\begin{matrix}\underline{\mathcal{X}} & = & \{ \underline{x}_1, \ldots, \underline{x}_n \}\end{matrix}</math><br />
| <math>\begin{matrix}\mathcal{A} & = & \{ a_1, \ldots, a_n \}\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X_i & = & \langle x_i \rangle<br />
\\<br />
& \cong & \mathbb{K}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}_i & = & \{ \texttt{(} \underline{x}_i \texttt{)}, \underline{x}_i \}<br />
\\<br />
& \cong & \mathbb{B}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A_i & = & \{ \texttt{(} a_i \texttt{)}, a_i \}<br />
\\<br />
& \cong & \mathbb{B}<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X<br />
\\<br />
= & \langle \mathcal{X} \rangle<br />
\\<br />
= & \langle x_1, \ldots, x_n \rangle<br />
\\<br />
= & X_1 \times \ldots \times X_n<br />
\\<br />
= & \prod_{i=1}^n X_i<br />
\\<br />
\cong & \mathbb{K}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}<br />
\\<br />
= & \langle \underline{\mathcal{X}} \rangle<br />
\\<br />
= & \langle \underline{x}_1, \ldots, \underline{x}_n \rangle<br />
\\<br />
= & \underline{X}_1 \times \ldots \times \underline{X}_n<br />
\\<br />
= & \prod_{i=1}^n \underline{X}_i<br />
\\<br />
\cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A<br />
\\<br />
= & \langle \mathcal{A} \rangle<br />
\\<br />
= & \langle a_1, \ldots, a_n \rangle<br />
\\<br />
= & A_1 \times \ldots \times A_n<br />
\\<br />
= & \prod_{i=1}^n A_i<br />
\\<br />
\cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X^* & = & (\ell : X \to \mathbb{K})<br />
\\<br />
& \cong & \mathbb{K}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}^* & = & (\ell : \underline{X} \to \mathbb{B})<br />
\\<br />
& \cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A^* & = & (\ell : A \to \mathbb{B})<br />
\\<br />
& \cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X^\uparrow & = & (X \to \mathbb{K})<br />
\\<br />
& \cong & (\mathbb{K}^n \to \mathbb{K})<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}^\uparrow & = & (\underline{X} \to \mathbb{B})<br />
\\<br />
& \cong & (\mathbb{B}^n \to \mathbb{B})<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A^\uparrow & = & (A \to \mathbb{B})<br />
\\<br />
& \cong & (\mathbb{B}^n \to \mathbb{B})<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X^\bullet<br />
\\<br />
= & [\mathcal{X}]<br />
\\<br />
= & [x_1, \ldots, x_n]<br />
\\<br />
= & (X, X^\uparrow)<br />
\\<br />
= & (X ~+\!\to \mathbb{K})<br />
\\<br />
= & (X, (X \to \mathbb{K}))<br />
\\<br />
\cong & (\mathbb{K}^n, (\mathbb{K}^n \to \mathbb{K}))<br />
\\<br />
= & (\mathbb{K}^n ~+\!\to \mathbb{K})<br />
\\<br />
= & [\mathbb{K}^n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}^\bullet<br />
\\<br />
= & [\underline{\mathcal{X}}]<br />
\\<br />
= & [\underline{x}_1, \ldots, \underline{x}_n]<br />
\\<br />
= & (\underline{X}, \underline{X}^\uparrow)<br />
\\<br />
= & (\underline{X} ~+\!\to \mathbb{B})<br />
\\<br />
= & (\underline{X}, (\underline{X} \to \mathbb{B}))<br />
\\<br />
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))<br />
\\<br />
= & (\mathbb{B}^n ~+\!\to \mathbb{B})<br />
\\<br />
= & [\mathbb{B}^n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A^\bullet<br />
\\<br />
= & [\mathcal{A}]<br />
\\<br />
= & [a_1, \ldots, a_n]<br />
\\<br />
= & (A, A^\uparrow)<br />
\\<br />
= & (A ~+\!\to \mathbb{B})<br />
\\<br />
= & (A, (A \to \mathbb{B}))<br />
\\<br />
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))<br />
\\<br />
= & (\mathbb{B}^n ~+\!\to \mathbb{B})<br />
\\<br />
= & [\mathbb{B}^n]<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
The left side of the Table collects mostly standard notation for an <math>n</math>-dimensional vector space over a field <math>\mathbb{K}.</math> The right side of the table repeats the first elements of a notation that I sketched above, to be used in further developments of propositional calculus. (I plan to use this notation in the logical analysis of neural network systems.) The middle column of the table is designed as a transitional step from the case of an arbitrary field <math>\mathbb{K},</math> with a special interest in the continuous line <math>\mathbb{R},</math> to the qualitative and discrete situations that are instanced and typified by <math>\mathbb{B}.</math><br />
<br />
I now proceed to explain these concepts in more detail. The most important ideas developed in Table&nbsp;5 are these:<br />
<br />
* The idea of a universe of discourse, which includes both a space of ''points'' and a space of ''maps'' on those points.<br />
<br />
* The idea of passing from a more complex universe to a simpler universe by a process of ''thresholding'' each dimension of variation down to a single bit of information.<br />
<br />
For the sake of concreteness, let us suppose that we start with a continuous <math>n</math>-dimensional vector space like <math>X = \langle x_1, \ldots, x_n \rangle \cong \mathbb{R}^n.</math> The coordinate system <math>\mathcal{X} = \{x_i\}</math> is a set of maps <math>x_i : \mathbb{R}^n \to \mathbb{R},</math> also known as the ''coordinate projections''. Given a "dataset" of points <math>\mathbf{x}</math> in <math>\mathbb{R}^n,</math> there are numerous ways of sensibly reducing the data down to one bit for each dimension. One strategy that is general enough for our present purposes is as follows. For each <math>i</math> we choose an <math>n</math>-ary relation <math>L_i</math> on <math>\mathbb{R}^n,</math> that is, a subset of <math>\mathbb{R}^n,</math> and then we define the <math>i^\mathrm{th}</math> threshold map, or ''limen'' <math>\underline{x}_i</math> as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\underline{x}_i : \mathbb{R}^n \to \mathbb{B}\ \text{such that:}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
\underline{x}_i(\mathbf{x}) = 1 & \text{if} & \mathbf{x} \in L_i,<br />
\\[4pt]<br />
\underline{x}_i(\mathbf{x}) = 0 & \text{if} & \mathbf{x} \not\in L_i.<br />
\end{matrix}</math><br />
|}<br />
<br />
In other notations that are sometimes used, the operator <math>\chi (\ldots)</math> or the corner brackets <math>\lceil\ldots\rceil</math> can be used to denote a ''characteristic function'', that is, a mapping from statements to their truth values in <math>\mathbb{B}.</math> Finally, it is not uncommon to use the name of the relation itself as a predicate that maps <math>n</math>-tuples into truth values. Thus we have the following notational variants of the above definition:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
\underline{x}_i (\mathbf{x}) & = & \chi (\mathbf{x} \in L_i) & = & \lceil \mathbf{x} \in L_i \rceil & = & L_i (\mathbf{x}).<br />
\end{matrix}</math><br />
|}<br />
<br />
Notice that, as defined here, there need be no actual relation between the <math>n</math>-dimensional subsets <math>\{L_i\}</math> and the coordinate axes corresponding to <math>\{x_i\},</math> aside from the circumstance that the two sets have the same cardinality. In concrete cases, though, one usually has some reason for associating these "volumes" with these "lines", for instance, <math>L_i</math> is bounded by some hyperplane that intersects the <math>i^\text{th}</math> axis at a unique threshold value <math>r_i \in \mathbb{R}.</math> Often, the hyperplane is chosen normal to the axis. In recognition of this motive, let us make the following convention. When the set <math>L_i</math> has points on the <math>i^\text{th}</math> axis, that is, points of the form <math>(0, \ldots, 0, r_i, 0, \ldots, 0)</math> where only the <math>x_i</math> coordinate is possibly non-zero, we may pick any one of these coordinate values as a parametric index of the relation. In this case we say that the indexing is ''real'', otherwise the indexing is ''imaginary''. For a knowledge based system <math>X,</math> this should serve once again to mark the distinction between ''acquaintance'' and ''opinion''.<br />
<br />
States of knowledge about the location of a system or about the distribution of a population of systems in a state space <math>X = \mathbb{R}^n</math> can now be expressed by taking the set <math>\underline{\mathcal{X}} = \{\underline{x}_i\}</math> as a basis of logical features. In picturesque terms, one may think of the underscore and the subscript as combining to form a subtextual spelling for the <math>i^\text{th}</math> threshold map. This can help to remind us that the ''threshold operator'' <math>(\underline{~})_i</math> acts on <math>\mathbf{x}</math> by setting up a kind of a &ldquo;hurdle&rdquo; for it. In this interpretation the coordinate proposition <math>\underline{x}_i</math> asserts that the representative point <math>\mathbf{x}</math> resides ''above'' the <math>i^\mathrm{th}</math> threshold.<br />
<br />
Primitive assertions of the form <math>\underline{x}_i (\mathbf{x})</math> may then be negated and joined by means of propositional connectives in the usual ways to provide information about the state <math>\mathbf{x}</math> of a contemplated system or a statistical ensemble of systems. Parentheses <math>\texttt{(} \ldots \texttt{)}</math> may be used to indicate logical negation. Eventually one discovers the usefulness of the <math>k</math>-ary ''just one false'' operators of the form <math>\texttt{(} a_1 \texttt{,} \ldots \texttt{,} a_k \texttt{)},</math> as treated in earlier reports. This much tackle generates a space of points (cells, interpretations), <math>\underline{X} \cong \mathbb{B}^n,</math> and a space of functions (regions, propositions), <math>\underline{X}^\uparrow \cong (\mathbb{B}^n \to \mathbb{B}).</math> Together these form a new universe of discourse <math>\underline{X}^\bullet</math> of the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})),</math> which we may abbreviate as <math>\mathbb{B}^n\ +\!\to \mathbb{B}</math> or most succinctly as <math>[\mathbb{B}^n].</math><br />
<br />
The square brackets have been chosen to recall the rectangular frame of a venn diagram. In thinking about a universe of discourse it is a good idea to keep this picture in mind, graphically illustrating the links among the elementary cells <math>\underline{\mathbf{x}},</math> the defining features <math>\underline{x}_i,</math> and the potential shadings <math>f : \underline{X} \to \mathbb{B}</math> all at the same time, not to mention the arbitrariness of the way we choose to inscribe our distinctions in the medium of a continuous space.<br />
<br />
Finally, let <math>X^*</math> denote the space of linear functions, <math>(\ell : X \to \mathbb{K}),</math> which has in the finite case the same dimensionality as <math>X,</math> and let the same notation be extended across the Table.<br />
<br />
We have just gone through a lot of work, apparently doing nothing more substantial than spinning a complex spell of notational devices through a labyrinth of baffled spaces and baffling maps. The reason for doing this was to bind together and to constitute the intuitive concept of a universe of discourse into a coherent categorical object, the kind of thing, once grasped, that can be turned over in the mind and considered in all its manifold changes and facets. The effort invested in these preliminary measures is intended to pay off later, when we need to consider the state transformations and the time evolution of neural network systems.<br />
<br />
===Tables of Propositional Forms===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope. It provides explicit techniques for manipulating the most basic ingredients of discourse.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7&ndash;8]<br />
|}<br />
<br />
To prepare for the next phase of discussion, Tables&nbsp;6 and 7 collect and summarize all of the propositional forms on one and two variables. These propositional forms are represented over bases of boolean variables as complete sets of boolean-valued functions. Adjacent to their names and specifications are listed what are roughly the simplest expressions in the ''[[Cactus_Language_&bull;_Overview|cactus language]]'', the particular syntax for propositional calculus that I use in formal and computational contexts. For the sake of orientation, the English paraphrases and the more common notations are listed in the last two columns. As simple and circumscribed as these low-dimensional universes may appear to be, a careful exploration of their differential extensions will involve us in complexities sufficient to demand our attention for some time to come.<br />
<br />
Propositional forms on one variable correspond to boolean functions <math>f : \mathbb{B}^1 \to \mathbb{B}.</math> In Table&nbsp;6 these functions are listed in a variant form of [[truth table]], one in which the axes of the usual arrangement are rotated through a right angle. Each function <math>f_i</math> is indexed by the string of values that it takes on the points of the universe <math>X^\bullet = [x] \cong \mathbb{B}^1.</math> The binary index generated in this way is converted to its decimal equivalent and these are used as conventional names for the <math>f_i,</math> as shown in the first column of the Table. In their own right the <math>2^1</math> points of the universe <math>X^\bullet</math> are coordinated as a space of type <math>\mathbb{B}^1,</math> this in light of the universe <math>X^\bullet</math> being a functional domain where the coordinate projection <math>x</math> takes on its values in <math>\mathbb{B}.</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 6.} ~~ \text{Propositional Forms on One Variable}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon</math><br />
| <math>1~0</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_0</math><br />
| <math>f_{00}</math><br />
| <math>0~0</math><br />
| <math>\texttt{(} ~ \texttt{)}</math><br />
| <math>\text{false}</math><br />
| <math>0</math><br />
|-<br />
| <math>f_1</math><br />
| <math>f_{01}</math><br />
| <math>0~1</math><br />
| <math>\texttt{(} x \texttt{)}</math><br />
| <math>\text{not}~ x</math><br />
| <math>\lnot x</math><br />
|-<br />
| <math>f_2</math><br />
| <math>f_{10}</math><br />
| <math>1~0</math><br />
| <math>x</math><br />
| <math>x</math><br />
| <math>x</math><br />
|-<br />
| <math>f_3</math><br />
| <math>f_{11}</math><br />
| <math>1~1</math><br />
| <math>\texttt{((} ~ \texttt{))}</math><br />
| <math>\text{true}</math><br />
| <math>1</math><br />
|}<br />
<br />
<br><br />
<br />
Propositional forms on two variables correspond to boolean functions <math>f : \mathbb{B}^2 \to \mathbb{B}.</math> In Table&nbsp;7 each function <math>f_i</math> is indexed by the values that it takes on the points of the universe <math>X^\bullet = [x, y] \cong \mathbb{B}^2.</math> Converting the binary index thus generated to a decimal equivalent, we obtain the functional nicknames that are listed in the first column. The <math>2^2</math> points of the universe <math>X^\bullet</math> are coordinated as a space of type <math>\mathbb{B}^2,</math> as indicated under the heading of the Table, where the coordinate projections <math>x</math> and <math>y</math> run through the various combinations of their values in <math>\mathbb{B}.</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 7-a.} ~~ \text{Propositional Forms on Two Variables}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}</math><br />
| style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon</math><br />
| <math>1~1~0~0</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon</math><br />
| <math>1~0~1~0</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[4pt]<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\\[4pt]<br />
f_{3}<br />
\\[4pt]<br />
f_{4}<br />
\\[4pt]<br />
f_{5}<br />
\\[4pt]<br />
f_{6}<br />
\\[4pt]<br />
f_{7}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0000}<br />
\\[4pt]<br />
f_{0001}<br />
\\[4pt]<br />
f_{0010}<br />
\\[4pt]<br />
f_{0011}<br />
\\[4pt]<br />
f_{0100}<br />
\\[4pt]<br />
f_{0101}<br />
\\[4pt]<br />
f_{0110}<br />
\\[4pt]<br />
f_{0111}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~0<br />
\\[4pt]<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~0~1~1<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
0~1~0~1<br />
\\[4pt]<br />
0~1~1~0<br />
\\[4pt]<br />
0~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{)} ~ y ~<br />
\\[4pt]<br />
\texttt{(} x \texttt{)}<br />
\\[4pt]<br />
~ x ~ \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{,} ~ y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x ~ y \texttt{)}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{false}<br />
\\[4pt]<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\[4pt]<br />
y ~\text{without}~ x<br />
\\[4pt]<br />
\text{not}~ x<br />
\\[4pt]<br />
x ~\text{without}~ y<br />
\\[4pt]<br />
\text{not}~ y<br />
\\[4pt]<br />
x ~\text{not equal to}~ y<br />
\\[4pt]<br />
\text{not both}~ x ~\text{and}~ y<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
\lnot x \land \lnot y<br />
\\[4pt]<br />
\lnot x \land y<br />
\\[4pt]<br />
\lnot x<br />
\\[4pt]<br />
x \land \lnot y<br />
\\[4pt]<br />
\lnot y<br />
\\[4pt]<br />
x \ne y<br />
\\[4pt]<br />
\lnot x \lor \lnot y<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{8}<br />
\\[4pt]<br />
f_{9}<br />
\\[4pt]<br />
f_{10}<br />
\\[4pt]<br />
f_{11}<br />
\\[4pt]<br />
f_{12}<br />
\\[4pt]<br />
f_{13}<br />
\\[4pt]<br />
f_{14}<br />
\\[4pt]<br />
f_{15}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1000}<br />
\\[4pt]<br />
f_{1001}<br />
\\[4pt]<br />
f_{1010}<br />
\\[4pt]<br />
f_{1011}<br />
\\[4pt]<br />
f_{1100}<br />
\\[4pt]<br />
f_{1101}<br />
\\[4pt]<br />
f_{1110}<br />
\\[4pt]<br />
f_{1111}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
1~0~0~0<br />
\\[4pt]<br />
1~0~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\\[4pt]<br />
1~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x ~ y<br />
\\[4pt]<br />
\texttt{((} x \texttt{,} ~ y \texttt{))}<br />
\\[4pt]<br />
y<br />
\\[4pt]<br />
\texttt{(} x ~ \texttt{(} y \texttt{))}<br />
\\[4pt]<br />
x<br />
\\[4pt]<br />
\texttt{((} x \texttt{)} ~ y \texttt{)}<br />
\\[4pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
\texttt{((} ~ \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x ~\text{and}~ y<br />
\\[4pt]<br />
x ~\text{equal to}~ y<br />
\\[4pt]<br />
y<br />
\\[4pt]<br />
\text{not}~ x ~\text{without}~ y<br />
\\[4pt]<br />
x<br />
\\[4pt]<br />
\text{not}~ y ~\text{without}~ x<br />
\\[4pt]<br />
x ~\text{or}~ y<br />
\\[4pt]<br />
\text{true}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x \land y<br />
\\[4pt]<br />
x = y<br />
\\[4pt]<br />
y<br />
\\[4pt]<br />
x \Rightarrow y<br />
\\[4pt]<br />
x<br />
\\[4pt]<br />
x \Leftarrow y<br />
\\[4pt]<br />
x \lor y<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 7-b.} ~~ \text{Propositional Forms on Two Variables}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}</math><br />
| style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon</math><br />
| <math>1~1~0~0</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon</math><br />
| <math>1~0~1~0</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_{0}</math><br />
| <math>f_{0000}</math><br />
| <math>0~0~0~0</math><br />
| <math>\texttt{(} ~ \texttt{)}</math><br />
| <math>\text{false}</math><br />
| <math>0</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\\[4pt]<br />
f_{4}<br />
\\[4pt]<br />
f_{8}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0001}<br />
\\[4pt]<br />
f_{0010}<br />
\\[4pt]<br />
f_{0100}<br />
\\[4pt]<br />
f_{1000}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
1~0~0~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{)} ~ y ~<br />
\\[4pt]<br />
~ x ~ \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
~ x ~~ y ~<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\[4pt]<br />
y ~\text{without}~ x<br />
\\[4pt]<br />
x ~\text{without}~ y<br />
\\[4pt]<br />
x ~\text{and}~ y<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot x \land \lnot y<br />
\\[4pt]<br />
\lnot x \land y<br />
\\[4pt]<br />
x \land \lnot y<br />
\\[4pt]<br />
x \land y<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{3}<br />
\\[4pt]<br />
f_{12}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0011}<br />
\\[4pt]<br />
f_{1100}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\[4pt]<br />
x<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{not}~ x<br />
\\[4pt]<br />
x<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot x<br />
\\[4pt]<br />
x<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[4pt]<br />
f_{9}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0110}<br />
\\[4pt]<br />
f_{1001}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[4pt]<br />
1~0~0~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{,} y \texttt{)}<br />
\\[4pt]<br />
\texttt{((} x \texttt{,} y \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x ~\text{not equal to}~ y<br />
\\[4pt]<br />
x ~\text{equal to}~ y<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x \ne y<br />
\\[4pt]<br />
x = y<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{5}<br />
\\[4pt]<br />
f_{10}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0101}<br />
\\[4pt]<br />
f_{1010}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\[4pt]<br />
y<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{not}~ y<br />
\\[4pt]<br />
y<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot y<br />
\\[4pt]<br />
y<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[4pt]<br />
f_{11}<br />
\\[4pt]<br />
f_{13}<br />
\\[4pt]<br />
f_{14}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0111}<br />
\\[4pt]<br />
f_{1011}<br />
\\[4pt]<br />
f_{1101}<br />
\\[4pt]<br />
f_{1110}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} ~ x ~~ y ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{(} ~ x ~ \texttt{(} y \texttt{))}<br />
\\[4pt]<br />
\texttt{((} x \texttt{)} ~ y ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{not both}~ x ~\text{and}~ y<br />
\\[4pt]<br />
\text{not}~ x ~\text{without}~ y<br />
\\[4pt]<br />
\text{not}~ y ~\text{without}~ x<br />
\\[4pt]<br />
x ~\text{or}~ y<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot x \lor \lnot y<br />
\\[4pt]<br />
x \Rightarrow y<br />
\\[4pt]<br />
x \Leftarrow y<br />
\\[4pt]<br />
x \lor y<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}</math><br />
| <math>f_{1111}</math><br />
| <math>1~1~1~1</math><br />
| <math>\texttt{((} ~ \texttt{))}</math><br />
| <math>\text{true}</math><br />
| <math>1</math><br />
|}<br />
<br />
<br><br />
<br />
==A Differential Extension of Propositional Calculus==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Fire over water:<br><br />
The image of the condition before transition.<br><br />
Thus the superior man is careful<br><br />
In the differentiation of things,<br><br />
So that each finds its place.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; ''I Ching'', Hexagram 64, [Wil, 249]<br />
|}<br />
<br />
This much preparation is enough to begin introducing my subject, if I excuse myself from giving full arguments for my definitional choices until some later stage. I am trying to develop a ''differential theory of qualitative equations'' that parallels the application of differential geometry to dynamic systems. The idea of a tangent vector is key to this work and a major goal is to find the right logical analogues of tangent spaces, bundles, and functors. The strategy is taken of looking for the simplest versions of these constructions that can be discovered within the realm of propositional calculus, so long as they serve to fill out the general theme.<br />
<br />
===Differential Propositions : Qualitative Analogues of Differential Equations===<br />
<br />
In order to define the differential extension of a universe of discourse <math>[\mathcal{A}],</math> the initial alphabet <math>\mathcal{A}</math> must be extended to include a collection of symbols for ''differential features'', or basic ''changes'' that are capable of occurring in <math>[\mathcal{A}].</math> Intuitively, these symbols may be construed as denoting primitive features of change, qualitative attributes of motion, or propositions about how things or points in the universe may be changing or moving with respect to the features that are noted in the initial alphabet.<br />
<br />
Therefore, let us define the corresponding ''differential alphabet'' or ''tangent alphabet'' as <math>\mathrm{d}\mathcal{A}</math> <math>=</math> <math>\{\mathrm{d}a_1, \ldots, \mathrm{d}a_n\},</math> in principle just an arbitrary alphabet of symbols, disjoint from the initial alphabet <math>\mathcal{A}</math> <math>=</math> <math>\{ a_1, \ldots, a_n \},</math> that is intended to be interpreted in the way just indicated. It only remains to be understood that the precise interpretation of the symbols in <math>\mathrm{d}\mathcal{A}</math> is often conceived to be changeable from point to point of the underlying space <math>A.</math> Indeed, for all we know, the state space <math>A</math> might well be the state space of a language interpreter, one that is concerned, among other things, with the idiomatic meanings of the dialect generated by <math>\mathcal{A}</math> and <math>\mathrm{d}\mathcal{A}.</math><br />
<br />
The ''tangent space'' to <math>A</math> at one of its points <math>x,</math> sometimes written <math>\mathrm{T}_x(A),</math> takes the form <math>\mathrm{d}A</math> <math>=</math> <math>\langle \mathrm{d}\mathcal{A} \rangle</math> <math>=</math> <math>\langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle.</math> Strictly speaking, the name ''cotangent space'' is probably more correct for this construction, but the fact that we take up spaces and their duals in pairs to form our universes of discourse allows our language to be pliable here.<br />
<br />
Proceeding as we did with the base space <math>A,</math> the tangent space <math>\mathrm{d}A</math> at a point of <math>A</math> can be analyzed as a product of distinct and independent factors:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\mathrm{d}A ~=~ \prod_{i=1}^n \mathrm{d}A_i ~=~ \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n.</math><br />
|}<br />
<br />
Here, <math>\mathrm{d}A_i</math> is a set of two differential propositions, <math>\mathrm{d}A_i = \{ (\mathrm{d}a_i), \mathrm{d}a_i \},</math> where <math>\texttt{(} \mathrm{d}a_i \texttt{)}</math> is a proposition with the logical value of <math>\text{not} ~ \mathrm{d}a_i.</math> Each component <math>\mathrm{d}A_i</math> has the type <math>\mathbb{B},</math> operating under the ordered correspondence <math>\{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \} \cong \{ 0, 1 \}.</math> However, clarity is often served by acknowledging this differential usage with a superficially distinct type <math>\mathbb{D},</math> whose intension may be indicated as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\mathbb{D} = \{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \} = \{ \text{same}, \text{different} \} = \{ \text{stay}, \text{change} \} = \{ \text{stop}, \text{step} \}.</math><br />
|}<br />
<br />
Viewed within a coordinate representation, spaces of type <math>\mathbb{B}^n</math> and <math>\mathbb{D}^n</math> may appear to be identical sets of binary vectors, but taking a view at this level of abstraction would be like ignoring the qualitative units and the diverse dimensions that distinguish position and momentum, or the different roles of quantity and impulse.<br />
<br />
===An Interlude on the Path===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
There would have been no beginnings:&nbsp; instead, speech would proceed from me, while I stood in its path &ndash; a slender gap &ndash; the point of its possible disappearance.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
A sense of the relation between <math>\mathbb{B}</math> and <math>\mathbb{D}</math> may be obtained by considering the ''path classifier'' (or the ''equivalence class of curves'') approach to tangent vectors.&nbsp; Consider a universe <math>[\mathcal{X}].</math>&nbsp; Given the boolean value system, a path in the space <math>X = \langle \mathcal{X} \rangle</math> is a map <math>q : \mathbb{B} \to X.</math>&nbsp; In this context the set of paths <math>(\mathbb{B} \to X)</math> is isomorphic to the cartesian square <math>X^2 = X \times X,</math> or the set of ordered pairs chosen from <math>X.</math><br />
<br />
We may analyze <math>X^2 = \{ (u, v) : u, v \in X \}</math> into two parts, specifically, the ordered pairs <math>(u, v)</math> that lie on and off the diagonal:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}X^2 & = & \{ (u, v) : u = v \} & \cup & \{ (u, v) : u \ne v \}.\end{matrix}</math><br />
|}<br />
<br />
This partition may also be expressed in the following symbolic form:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}X^2 & \cong & \operatorname{diag} (X) & + & 2 \binom{X}{2}.\end{matrix}</math><br />
|}<br />
<br />
The separate terms of this formula are defined as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}\operatorname{diag} (X) & = & \{ (x, x) : x \in X \}.\end{matrix}</math><br />
|}<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}\binom{X}{k} & = & X ~\text{choose}~ k & = & \{ k\text{-sets from}~ X \}.\end{matrix}</math><br />
|}<br />
<br />
Thus we have:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}\binom{X}{2} & = & \{ \{ u, v \} : u, v \in X \}.\end{matrix}</math><br />
|}<br />
<br />
We may now use the features in <math>\mathrm{d}\mathcal{X} = \{ \mathrm{d}x_i \} = \{ \mathrm{d}x_1, \ldots, \mathrm{d}x_n \}</math> to classify the paths of <math>(\mathbb{B} \to X)</math> by way of the pairs in <math>X^2.</math>&nbsp; If <math>X \cong \mathbb{B}^n,</math> then a path <math>q</math> in <math>X</math> has the following form:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
q : (\mathbb{B} \to \mathbb{B}^n) & \cong & \mathbb{B}^n \times \mathbb{B}^n & \cong & \mathbb{B}^{2n} & \cong & (\mathbb{B}^2)^n.<br />
\end{matrix}</math><br />
|}<br />
<br />
Intuitively, we want to map this <math>(\mathbb{B}^2)^n</math> onto <math>\mathbb{D}^n</math> by mapping each component <math>\mathbb{B}^2</math> onto a copy of <math>\mathbb{D}.</math>&nbsp; But in the presenting context <math>{}^{\backprime\backprime} \mathbb{D} {}^{\prime\prime}</math> is just a name associated with, or an incidental quality attributed to, coefficient values in <math>\mathbb{B}</math> when they are attached to features in <math>\mathrm{d}\mathcal{X}.</math><br />
<br />
Taking these intentions into account, define <math>\mathrm{d}x_i : X^2 \to \mathbb{B}</math> in the following manner:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcrcl}<br />
\mathrm{d}x_i(u, v)<br />
& = & \texttt{(} ~ x_i(u) & \texttt{,} & x_i(v) ~ \texttt{)}<br />
\\<br />
& = & x_i(u) & + & x_i(v)<br />
\\<br />
& = & x_i(v) & - & x_i(u).<br />
\end{array}</math><br />
|}<br />
<br />
In the above transcription, the operator bracket of the form <math>\texttt{(} \ldots \texttt{,} \ldots \texttt{)}</math> is a ''cactus lobe'', in general signifying that just one of the arguments listed is false.&nbsp; In the case of two arguments this is the same thing as saying that the arguments are not equal.&nbsp; The plus sign signifies boolean addition, in the sense of addition in <math>\mathrm{GF}(2),</math> and thus means the same thing in this context as the minus sign, in the sense of adding the additive inverse.<br />
<br />
The above definition of <math>\mathrm{d}x_i : X^2 \to \mathbb{B}</math> is equivalent to defining <math>\mathrm{d}x_i : (\mathbb{B} \to X) \to \mathbb{B}</math> in the following way:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcrcl}<br />
\mathrm{d}x_i (q)<br />
& = & \texttt{(} ~ x_i(q_0) & \texttt{,} & x_i(q_1) ~ \texttt{)}<br />
\\<br />
& = & x_i(q_0) & + & x_i(q_1)<br />
\\<br />
& = & x_i(q_1) & - & x_i(q_0).<br />
\end{array}</math><br />
|}<br />
<br />
In this definition <math>q_b = q(b),</math> for each <math>b</math> in <math>\mathbb{B}.</math>&nbsp; Thus, the proposition <math>\mathrm{d}x_i</math> is true of the path <math>q = (u, v)</math> exactly if the terms of <math>q,</math> the endpoints <math>u</math> and <math>v,</math> lie on different sides of the question <math>x_i.</math><br />
<br />
The language of features in <math>\langle \mathrm{d}\mathcal{X} \rangle,</math> indeed the whole calculus of propositions in <math>[\mathrm{d}\mathcal{X}],</math> may now be used to classify paths and sets of paths.&nbsp; In other words, the paths can be taken as models of the propositions <math>g : \mathrm{d}X \to \mathbb{B}.</math>&nbsp; For example, the paths corresponding to <math>\mathrm{diag}(X)</math> fall under the description <math>\texttt{(} \mathrm{d}x_1 \texttt{)} \cdots \texttt{(} \mathrm{d}x_n \texttt{)},</math> which says that nothing changes against the backdrop of the coordinate frame <math>\{ x_1, \ldots, x_n \}.</math><br />
<br />
Finally, a few words of explanation may be in order.&nbsp; If this concept of a path appears to be described in a roundabout fashion, it is because I am trying to avoid using any assumption of vector space properties for the space <math>X</math> that contains its range.&nbsp; In many ways the treatment is still unsatisfactory, but improvements will have to wait for the introduction of substitution operators acting on singular propositions.<br />
<br />
===The Extended Universe of Discourse===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
At the moment of speaking, I would like to have perceived a nameless voice, long preceding me, leaving me merely to enmesh myself in it, taking up its cadence, and to lodge myself, when no one was looking, in its interstices as if it had paused an instant, in suspense, to beckon to me.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
Next we define the ''extended alphabet'' or ''bundled alphabet'' <math>\mathrm{E}\mathcal{A}</math> as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lclcl}<br />
\mathrm{E}\mathcal{A}<br />
& = & \mathcal{A} \cup \mathrm{d}\mathcal{A}<br />
& = & \{ a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}.<br />
\end{array}</math><br />
|}<br />
<br />
This supplies enough material to construct the ''differential extension'' <math>\mathrm{E}A,</math> or the ''tangent bundle'' over the initial space <math>A,</math> in the following fashion:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcl}<br />
\mathrm{E}A<br />
& = & \langle \mathrm{E}\mathcal{A} \rangle<br />
\\[4pt]<br />
& = & \langle \mathcal{A} \cup \mathrm{d}\mathcal{A} \rangle<br />
\\[4pt]<br />
& = & \langle a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle,<br />
\end{array}</math><br />
|}<br />
<br />
and also:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcl}<br />
\mathrm{E}A<br />
& = & A \times \mathrm{d}A<br />
\\[4pt]<br />
& = & A_1 \times \ldots \times A_n \times \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n.<br />
\end{array}</math><br />
|}<br />
<br />
This gives <math>\mathrm{E}A</math> the type <math>\mathbb{B}^n \times \mathbb{D}^n.</math><br />
<br />
Finally, the tangent universe <math>\mathrm{E}A^\bullet = [\mathrm{E}\mathcal{A}]</math> is constituted from the totality of points and maps, or interpretations and propositions, that are based on the extended set of features <math>\mathrm{E}\mathcal{A},</math> and this fact is summed up in the following notation:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lclcl}<br />
\mathrm{E}A^\bullet<br />
& = & [\mathrm{E}\mathcal{A}]<br />
& = & [a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n].<br />
\end{array}</math><br />
|}<br />
<br />
This gives the tangent universe <math>\mathrm{E}A^\bullet</math> the type:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcl}<br />
(\mathbb{B}^n \times \mathbb{D}^n\ +\!\to \mathbb{B})<br />
& = & (\mathbb{B}^n \times \mathbb{D}^n, (\mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B})).<br />
\end{array}</math><br />
|}<br />
<br />
A proposition in the tangent universe <math>[\mathrm{E}\mathcal{A}]</math> is called a ''differential proposition'' and forms the analogue of a system of differential equations, constraints, or relations in ordinary calculus.<br />
<br />
With these constructions, the differential extension <math>\mathrm{E}A</math> and the space of differential propositions <math>(\mathrm{E}A \to \mathbb{B}),</math> we have arrived, in main outline, at one of the major subgoals of this study. Table&nbsp;8 summarizes the concepts that have been introduced for working with differentially extended universes of discourse.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 8.} ~~ \text{Differential Extension : Basic Notation}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Symbol}</math><br />
| <math>\text{Notation}</math><br />
| <math>\text{Description}</math><br />
| <math>\text{Type}</math><br />
|-<br />
| <math>\mathrm{d}\mathfrak{A}</math><br />
| <math>\{ {}^{\backprime\backprime} \mathrm{d}a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} \mathrm{d}a_n {}^{\prime\prime} \}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Alphabet of}<br />
\\[2pt]<br />
\text{differential symbols}<br />
\end{matrix}</math><br />
| <math>[n] = \mathbf{n}</math><br />
|-<br />
| <math>\mathrm{d}\mathcal{A}</math><br />
| <math>\{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Basis of}<br />
\\[2pt]<br />
\text{differential features}<br />
\end{matrix}</math><br />
| <math>[n] = \mathbf{n}</math><br />
|-<br />
| <math>\mathrm{d}A_i</math><br />
| <math>\{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \}</math><br />
| <math>\text{Differential dimension}~ i</math><br />
| <math>\mathbb{D}</math><br />
|-<br />
| <math>\mathrm{d}A</math><br />
|<br />
<math>\begin{matrix}<br />
\langle \mathrm{d}\mathcal{A} \rangle<br />
\\[2pt]<br />
\langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle<br />
\\[2pt]<br />
\{ (\mathrm{d}a_1, \ldots, \mathrm{d}a_n) \}<br />
\\[2pt]<br />
\mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n<br />
\\[2pt]<br />
\textstyle \prod_i \mathrm{d}A_i<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Tangent space at a point:}<br />
\\[2pt]<br />
\text{Set of changes, motions,}<br />
\\[2pt]<br />
\text{steps, tangent vectors}<br />
\\[2pt]<br />
\text{at a point}<br />
\end{matrix}</math><br />
| <math>\mathbb{D}^n</math><br />
|-<br />
| <math>\mathrm{d}A^*</math><br />
| <math>(\mathrm{hom} : \mathrm{d}A \to \mathbb{B})</math><br />
| <math>\text{Linear functions on}~ \mathrm{d}A</math><br />
| <math>(\mathbb{D}^n)^* \cong \mathbb{D}^n</math><br />
|-<br />
| <math>\mathrm{d}A^\uparrow</math><br />
| <math>(\mathrm{d}A \to \mathbb{B})</math><br />
| <math>\text{Boolean functions on}~ \mathrm{d}A</math><br />
| <math>\mathbb{D}^n \to \mathbb{B}</math><br />
|-<br />
| <math>\mathrm{d}A^\bullet</math><br />
|<br />
<math>\begin{matrix}<br />
[\mathrm{d}\mathcal{A}]<br />
\\[2pt]<br />
(\mathrm{d}A, \mathrm{d}A^\uparrow)<br />
\\[2pt]<br />
(\mathrm{d}A ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
(\mathrm{d}A, (\mathrm{d}A \to \mathbb{B}))<br />
\\[2pt]<br />
[\mathrm{d}a_1, \ldots, \mathrm{d}a_n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Tangent universe at a point of}~ A^\bullet,<br />
\\[2pt]<br />
\text{based on the tangent features}<br />
\\[2pt]<br />
\{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
(\mathbb{D}^n, (\mathbb{D}^n \to \mathbb{B}))<br />
\\[2pt]<br />
(\mathbb{D}^n ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
[\mathbb{D}^n]<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
The adjectives ''differential'' or ''tangent'' are systematically attached to every construct based on the differential alphabet <math>\mathrm{d}\mathfrak{A},</math> taken by itself. Strictly speaking, we probably ought to call <math>\mathrm{d}\mathcal{A}</math> the set of ''cotangent'' features derived from <math>\mathcal{A},</math> but the only time this distinction really seems to matter is when we need to distinguish the tangent vectors as maps of type <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math> from cotangent vectors as elements of type <math>\mathbb{D}^n.</math> In like fashion, having defined <math>\mathrm{E}\mathcal{A} = \mathcal{A} \cup \mathrm{d}\mathcal{A},</math> we can systematically attach the adjective ''extended'' or the substantive ''bundle'' to all of the constructs associated with this full complement of <math>{2n}</math> features.<br />
<br />
It eventually becomes necessary to extend the initial alphabet even further, to allow for the discussion of higher order differential expressions. Table&nbsp;9 provides a suggestion of how these further extensions can be carried out.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 9.} ~~ \text{Higher Order Differential Features}</math><br />
|<br />
<p><math>\begin{array}{lllll}<br />
\mathrm{d}^0 \mathcal{A} & = & \{ a_1, \ldots, a_n \} & = & \mathcal{A}<br />
\\<br />
\mathrm{d}^1 \mathcal{A} & = & \{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \} & = & \mathrm{d}\mathcal{A}<br />
\end{array}</math></p><br />
<br />
<p><math>\begin{array}{lll}<br />
\mathrm{d}^k \mathcal{A} & = & \{ \mathrm{d}^k a_1, \ldots, \mathrm{d}^k a_n \}<br />
\\<br />
\mathrm{d}^* \mathcal{A} & = & \{ \mathrm{d}^0 \mathcal{A}, \ldots, \mathrm{d}^k \mathcal{A}, \ldots \}<br />
\end{array}</math></p><br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}^0 \mathcal{A} & = & \mathrm{d}^0 \mathcal{A}<br />
\\<br />
\mathrm{E}^1 \mathcal{A} & = & \mathrm{d}^0 \mathcal{A} ~\cup~ \mathrm{d}^1 \mathcal{A}<br />
\\<br />
\mathrm{E}^k \mathcal{A} & = & \mathrm{d}^0 \mathcal{A} ~\cup~ \ldots ~\cup~ \mathrm{d}^k \mathcal{A}<br />
\\<br />
\mathrm{E}^\infty \mathcal{A} & = & \bigcup~ \mathrm{d}^* \mathcal{A}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
===Intentional Propositions===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Do you guess I have some intricate purpose?<br><br />
Well I have . . . . for the April rain has, and the mica on<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the side of a rock has.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 45]<br />
|}<br />
<br />
In order to analyze the behavior of a system at successive moments in time, while staying within the limitations of propositional logic, it is necessary to create independent alphabets of logical features for each moment of time that we contemplate using in our discussion. These moments have reference to typical instances and relative intervals, not actual or absolute times. For example, to discuss ''velocities'' (first order rates of change) we need to consider points of time in pairs. There are a number of natural ways of doing this. Given an initial alphabet, we could use its symbols as a lexical basis to generate successive alphabets of compound symbols, say, with temporal markers appended as suffixes.<br />
<br />
As a standard way of dealing with these situations, the following scheme of notation suggests a way of extending any alphabet of logical features through as many temporal moments as a particular order of analysis may demand. The lexical operators <math>\mathrm{p}^k</math> and <math>\mathrm{Q}^k</math> are convenient in many contexts where the accumulation of prime symbols and union symbols would otherwise be cumbersome.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 10.} ~~ \text{A Realm of Intentional Features}</math><br />
|<br />
<p><math>\begin{array}{lllll}<br />
\mathrm{p}^0 \mathcal{A} & = & \{ a_1, \ldots, a_n \} & = & \mathcal{A}<br />
\\<br />
\mathrm{p}^1 \mathcal{A} & = & \{ a_1^\prime, \ldots, a_n^\prime \} & = & \mathcal{A}^\prime<br />
\\<br />
\mathrm{p}^2 \mathcal{A} & = & \{ a_1^{\prime\prime}, \ldots, a_n^{\prime\prime} \} & = & \mathcal{A}^{\prime\prime}<br />
\\<br />
\cdots & & \cdots &<br />
\end{array}</math></p><br />
<br />
<p><math>\begin{array}{lll}<br />
\mathrm{p}^k \mathcal{A} & = & \{ \mathrm{p}^k a_1, \ldots, \mathrm{p}^k a_n \}<br />
\end{array}</math></p><br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{Q}^0 \mathcal{A} & = & \mathcal{A}<br />
\\<br />
\mathrm{Q}^1 \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}'<br />
\\<br />
\mathrm{Q}^2 \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}' \cup \mathcal{A}''<br />
\\<br />
\cdots & & \cdots<br />
\\<br />
\mathrm{Q}^k \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}' \cup \ldots \cup \mathrm{p}^k \mathcal{A}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
The resulting augmentations of our logical basis determine a series of discursive universes that may be called the ''intentional extension'' of propositional calculus. This extension follows a pattern analogous to the differential extension, which was developed in terms of the operators <math>\mathrm{d}^k</math> and <math>\mathrm{E}^k,</math> and there is a natural relation between these two extensions that bears further examination. In contexts displaying this pattern, where a sequence of domains stretches from an anchoring domain <math>X</math> through an indefinite number of higher reaches, a particular collection of domains based on <math>X</math> will be referred to as a ''realm'' of <math>X,</math> and when the succession exhibits a temporal aspect, as a ''reign'' of <math>X.</math><br />
<br />
For the purposes of this discussion, an ''intentional proposition'' is defined as a proposition in the universe of discourse <math>\mathrm{Q}X^\bullet = [\mathrm{Q}\mathcal{X}],</math> in other words, a map <math>q : \mathrm{Q}X \to \mathbb{B}.</math> The sense of this definition may be seen if we consider the following facts. First, the equivalence <math>\mathrm{Q}X = X \times X'</math> motivates the following chain of isomorphisms between spaces:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lllcl}<br />
(\mathrm{Q}X \to \mathbb{B})<br />
& \cong & (X & \times & ~X' \to \mathbb{B})<br />
\\[4pt]<br />
& \cong & (X & \to & (X' \to \mathbb{B}))<br />
\\[4pt]<br />
& \cong & (X' & \to & (X~ \to \mathbb{B})).<br />
\end{array}</math><br />
|}<br />
<br />
Viewed in this light, an intentional proposition <math>q</math> may be rephrased as a map <math>q : X \times X' \to \mathbb{B},</math> which judges the juxtaposition of states in <math>X</math> from one moment to the next. Alternatively, <math>q</math> may be parsed in two stages in two different ways, as <math>q : X \to (X' \to \mathbb{B})</math> and as <math>q : X' \to (X \to \mathbb{B}),</math> which associate to each point of <math>X</math> or <math>X'</math> a proposition about states in <math>X'</math> or <math>X,</math> respectively. In this way, an intentional proposition embodies a type of value system, in effect, a proposal that places a value on a collection of ends-in-view, or a project that evaluates a set of goals as regarded from each point of view in the state space of a system.<br />
<br />
In sum, the intentional proposition <math>q</math> indicates a method for the systematic selection of local goals. As a general form of description, a map of the type <math>q : \mathrm{Q}^i X \to \mathbb{B}</math> may be referred to as an "<math>i^\text{th}</math> order intentional proposition". Naturally, when we speak of intentional propositions without qualification, we usually mean first order intentions.<br />
<br />
Many different realms of discourse have the same structure as the extensions that have been indicated here. From a strictly logical point of view, each new layer of terms is composed of independent logical variables that are no different in kind from those that go before, and each further course of logical atoms is treated like so many individual, but otherwise indifferent bricks by the prototype computer program that I use as a propositional interpreter. Thus, the names that I use to single out the differential and the intentional extensions, and the lexical paradigms that I follow to construct them, are meant to suggest the interpretations that I have in mind, but they can only hint at the extra meanings that human communicators may pack into their terms and inflections.<br />
<br />
As applied here, the word ''intentional'' is drawn from common use and may have little bearing on its technical use in other, more properly philosophical, contexts. I am merely using the complex of intentional concepts &mdash; aims, ends, goals, objectives, purposes, and so on &mdash; metaphorically to flesh out and vividly to represent any situation where one needs to contemplate a system in multiple aspects of state and destination, that is, its being in certain states and at the same time acting as if headed through certain states. If confusion arises, more neutral words like ''conative'', ''contingent'', ''discretionary'', ''experimental'', ''kinetic'', ''progressive'', ''tentative'', or ''trial'' would probably serve as well.<br />
<br />
===Life on Easy Street===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Failing to fetch me at first keep encouraged,<br><br />
Missing me one place search another,<br><br />
I stop some where waiting for you<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]<br />
|}<br />
<br />
The finite character of the extended universe <math>[\mathrm{E}\mathcal{A}]</math> makes the problem of solving differential propositions relatively straightforward, at least, in principle. The solution set of the differential proposition <math>q : \mathrm{E}A \to \mathbb{B}</math> is the set of models <math>q^{-1}(1)</math> in <math>\mathrm{E}A.</math> Finding all the models of <math>q,</math> the extended interpretations in <math>\mathrm{E}A</math> that satisfy <math>q,</math> can be carried out by a finite search. Being in possession of complete algorithms for propositional calculus modeling, theorem checking, or theorem proving makes the analytic task fairly simple in principle, though the question of efficiency in the face of arbitrary complexity may always remain another matter entirely. While the fact that propositional satisfiability is NP-complete may be discouraging for the prospects of a single efficient algorithm that covers the whole space <math>[\mathrm{E}\mathcal{A}]</math> with equal facility, there appears to be much room for improvement in classifying special forms and in developing algorithms that are tailored to their practical processing.<br />
<br />
In view of these constraints and contingencies, our focus shifts to the tasks of approximation and interpretation that support intuition, especially in dealing with the natural kinds of differential propositions that arise in applications, and in the effort to understand, in succinct and adaptive forms, their dynamic implications. In the absence of direct insights, these tasks are partially carried out by forging analogies with the familiar situations and customary routines of ordinary calculus. But the indirect approach, going by way of specious analogy and intuitive habit, forces us to remain on guard against the circumstance that occurs when the word ''forging'' takes on its shadier nuance, indicting the constant risk of a counterfeit in the proportion.<br />
<br />
==Back to the Beginning : Exemplary Universes==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
I would have preferred to be enveloped in words, borne way beyond all possible beginnings.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
To anchor our understanding of differential logic, let us look at how the various concepts apply in the simplest possible concrete cases, where the initial dimension is only 1 or 2. In spite of the obvious simplicity of these cases, it is possible to observe how central difficulties of the subject begin to arise already at this stage.<br />
<br />
===A One-Dimensional Universe===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
There was never any more inception than there is now,<br><br />
Nor any more youth or age than there is now;<br><br />
And will never be any more perfection than there is now,<br><br />
Nor any more heaven or hell than there is now.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]<br />
|}<br />
<br />
Let <math>\mathcal{X} = \{ x_1 \} = \{ A \}</math> be an alphabet that represents one boolean variable or a single logical feature. In this example the capital letter <math>{}^{\backprime\backprime} A {}^{\prime\prime}\!</math> is used usual informally, to name a feature and not a space, in departure from our formerly stated formal conventions. At any rate, the basis element <math>A = x_1\!</math> may be interpreted as a simple proposition or a coordinate projection <math>A = x_1 : \mathbb{B} \xrightarrow{i} \mathbb{B}.</math> The space <math>X = \langle A \rangle = \{ \texttt{(} A \texttt{)}, A \}</math> of points (cells, vectors, interpretations) has cardinality <math>2^n = 2^1 = 2\!</math> and is isomorphic to <math>\mathbb{B} = \{ 0, 1 \}.</math> Moreover, <math>X\!</math> may be identified with the set of singular propositions <math>\{ x : \mathbb{B} \xrightarrow{s} \mathbb{B} \}.</math> The space of linear propositions <math>X^* = \{ \mathrm{hom} : \mathbb{B} \xrightarrow{\ell} \mathbb{B} \} = \{ 0, A \}</math> is algebraically dual to <math>X\!</math> and also has cardinality <math>2.\!</math> Here, <math>{}^{\backprime\backprime} 0 {}^{\prime\prime}\!</math> is interpreted as denoting the constant function <math>0 : \mathbb{B} \to \mathbb{B},</math> amounting to the linear proposition of rank <math>0,\!</math> while <math>A\!</math> is the linear proposition of rank <math>1.\!</math> Last but not least we have the positive propositions <math>\{ \mathrm{pos} : \mathbb{B} \xrightarrow{p} \mathbb{B} \} = \{ A, 1 \},\!</math> of rank <math>1\!</math> and <math>0,\!</math> respectively, where <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}\!</math> is understood as denoting the constant function <math>1 : \mathbb{B} \to \mathbb{B}.</math> In sum, there are <math>2^{2^n} = 2^{2^1} = 4</math> propositions altogether in the universe of discourse, comprising the set <math>X^\uparrow = \{ f : X \to \mathbb{B} \} = \{ 0, \texttt{(} A \texttt{)}, A, 1 \} \cong (\mathbb{B} \to \mathbb{B}).</math><br />
<br />
The first order differential extension of <math>\mathcal{X}</math> is <math>\mathrm{E}\mathcal{X} = \{ x_1, \mathrm{d}x_1 \} = \{ A, \mathrm{d}A \}.</math> If the feature <math>A\!</math> is understood as applying to some object or state, then the feature <math>\mathrm{d}A</math> may be interpreted as an attribute of the same object or state that says that it is changing ''significantly'' with respect to the property <math>A,\!</math> or that it has an ''escape velocity'' with respect to the state <math>A.\!</math> In practice, differential features acquire their logical meaning through a class of ''temporal inference rules''.<br />
<br />
For example, relative to a frame of observation that is left implicit for now, one is permitted to make the following sorts of inference: From the fact that <math>A\!</math> and <math>\mathrm{d}A</math> are true at a given moment one may infer that <math>\texttt{(} A \texttt{)}\!</math> will be true in the next moment of observation. Altogether in the present instance, there is the fourfold scheme of inference that is shown below:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:50%"<br />
|<br />
<math>\begin{matrix}<br />
\text{From} & \texttt{(} A \texttt{)}<br />
& \text{and} & \texttt{(} \mathrm{d}A \texttt{)}<br />
& \text{infer} & \texttt{(} A \texttt{)} & \text{next.}<br />
\\[8pt]<br />
\text{From} & \texttt{(} A \texttt{)}<br />
& \text{and} & \mathrm{d}A<br />
& \text{infer} & A & \text{next.}<br />
\\[8pt]<br />
\text{From} & A<br />
& \text{and} & \texttt{(} \mathrm{d}A \texttt{)}<br />
& \text{infer} & A & \text{next.}<br />
\\[8pt]<br />
\text{From} & A<br />
& \text{and} & \mathrm{d}A<br />
& \text{infer} & \texttt{(} A \texttt{)} & \text{next.}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
It might be thought that an independent time variable needs to be brought in at this point, but it is an insight of fundamental importance that the idea of process is logically prior to the notion of time. A time variable is a reference to a ''clock'' &mdash; a canonical, conventional process that is accepted or established as a standard of measurement, but in essence no different than any other process. This raises the question of how different subsystems in a more global process can be brought into comparison, and what it means for one process to serve the function of a local standard for others. But these inquiries only wrap up puzzles in further riddles, and are obviously too involved to be handled at our current level of approximation.<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
The clock indicates the moment . . . . but what does<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;eternity indicate?<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 79]<br />
|}<br />
<br />
Observe that the secular inference rules, used by themselves, involve a loss of information, since nothing in them can tell us whether the momenta <math>\{ \texttt{(} \mathrm{d}A \texttt{)}, \mathrm{d}A \}\!</math> are changed or unchanged in the next instance. In order to know this, one would have to determine <math>\mathrm{d}^2 A,\!</math> and so on, pursuing an infinite regress. Ultimately, in order to rest with a finitely determinate system, it is necessary to make an infinite assumption, for example, that <math>\mathrm{d}^k A = 0\!</math> for all <math>k\!</math> greater than some fixed value <math>M.\!</math> Another way to escape the regress is through the provision of a dynamic law, in typical form making higher order differentials dependent on lower degrees and estates.<br />
<br />
===Example 1. A Square Rigging===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Urge and urge and urge,<br><br />
Always the procreant urge of the world.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]<br />
|}<br />
<br />
By way of example, suppose that we are given the initial condition <math>A = \mathrm{d}A\!</math> and the law <math>\mathrm{d}^2 A = \texttt{(} A \texttt{)}.\!</math> Since the equation <math>A = \mathrm{d}A\!</math> is logically equivalent to the disjunction <math>A ~ \mathrm{d}A ~\text{or}~ \texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)},\!</math> we may infer two possible trajectories, as displayed in Table&nbsp;11. In either case the state <math>A ~ \texttt{(} \mathrm{d}A \texttt{)(} \mathrm{d}^2 A \texttt{)}\!</math> is a stable attractor or a terminal condition for both starting points.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 11.} ~~ \text{A Pair of Commodious Trajectories}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Time}\!</math><br />
| <math>\text{Trajectory 1}\!</math><br />
| <math>\text{Trajectory 2}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
2<br />
\\[4pt]<br />
3<br />
\\[4pt]<br />
4<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A & \mathrm{d}A & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
\texttt{(} A \texttt{)} & \mathrm{d}A & \mathrm{d}^2 A<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
{}^{\shortparallel} & {}^{\shortparallel} & {}^{\shortparallel}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{)} & \texttt{(} \mathrm{d}A \texttt{)} & \mathrm{d}^2 A<br />
\\[4pt]<br />
\texttt{(} A \texttt{)} & \mathrm{d}A & \mathrm{d}^2 A<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
{}^{\shortparallel} & {}^{\shortparallel} & {}^{\shortparallel}<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Because the initial space <math>X = \langle A \rangle\!</math> is one-dimensional, we can easily fit the second order extension <math>\mathrm{E}^2 X = \langle A, \mathrm{d}A, \mathrm{d}^2 A \rangle\!</math> within the compass of a single venn diagram, charting the couple of converging trajectories as shown in Figure&nbsp;12.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 12 -- The Anchor.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 12.} ~~ \text{The Anchor}\!</math><br />
|}<br />
<br />
If we eliminate from view the regions of <math>\mathrm{E}^2 X\!</math> that are ruled out by the dynamic law <math>\mathrm{d}^2 A = \texttt{(} A \texttt{)},\!</math> then what remains is the quotient structure that is shown in Figure&nbsp;13. This picture makes it easy to see that the dynamically allowable portion of the universe is partitioned between the properties <math>A\!</math> and <math>\mathrm{d}^2 A\!.</math> As it happens, this fact might have been expressed &ldquo;right off the bat&rdquo; by an equivalent formulation of the differential law, one that uses the exclusive disjunction to state the law as <math>\texttt{(} A \texttt{,} \mathrm{d}^2 A \texttt{)}\!.</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 13 -- The Tiller.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 13.} ~~ \text{The Tiller}\!</math><br />
|}<br />
<br />
What we have achieved in this example is to give a differential description of a simple dynamic process. In effect, we did this by embedding a directed graph, which can be taken to represent the state transitions of a finite automaton, in a dynamically allotted quotient structure that is created from a boolean lattice or an <math>n\!</math>-cube by nullifying all of the regions that the dynamics outlaws. With growth in the dimensions of our contemplated universes, it becomes essential, both for human comprehension and for computer implementation, that the dynamic structures of interest to us be represented not actually, by acquaintance, but virtually, by description. In our present study, we are using the language of propositional calculus to express the relevant descriptions, and to comprehend the structure that is implicit in the subsets of a <math>n\!</math>-cube without necessarily being forced to actualize all of its points.<br />
<br />
One of the reasons for engaging in this kind of extremely reduced, but explicitly controlled case study is to throw light on the general study of languages, formal and natural, in their full array of syntactic, semantic, and pragmatic aspects. Propositional calculus is one of the last points of departure where we can view these three aspects interacting in a non-trivial way without being immediately and totally overwhelmed by the complexity they generate. Often this complexity causes investigators of formal and natural languages to adopt the strategy of focusing on a single aspect and to abandon all hope of understanding the whole, whether it's the still living natural language or the dynamics of inquiry that lies crystallized in formal logic.<br />
<br />
From the perspective that I find most useful here, a language is a syntactic system that is designed or evolved in part to express a set of descriptions. When the explicit symbols of a language have extensions in its object world that are actually infinite, or when the implicit categories and generative devices of a linguistic theory have extensions in its subject matter that are potentially infinite, then the finite characters of terms, statements, arguments, grammars, logics, and rhetorics force an excess of intension to reside in all these symbols and functions, across the spectrum from the object language to the metalinguistic uses. In the aphorism from W. von Humboldt that Chomsky often cites, for example, in [Cho86, 30] and [Cho93, 49], language requires &ldquo;the infinite use of finite means&rdquo;. This is necessarily true when the extensions are infinite, when the referential symbols and grammatical categories of a language possess infinite sets of models and instances. But it also voices a practical truth when the extensions, though finite at every stage, tend to grow at exponential rates.<br />
<br />
This consequence of dealing with extensions that are &ldquo;practically infinite&rdquo; becomes crucial when one tries to build neural network systems that learn, since the learning competence of any intelligent system is limited to the objects and domains that it is able to represent. If we want to design systems that operate intelligently with the full deck of propositions dealt by intact universes of discourse, then we must supply them with succinct representations and efficient transformations in this domain. Furthermore, in the project of constructing inquiry driven systems, we find ourselves forced to contemplate the level of generality that is embodied in propositions, because the dynamic evolution of these systems is driven by the measurable discrepancies that occur among their expectations, intentions, and observations, and because each of these subsystems or components of knowledge constitutes a propositional modality that can take on the fully generic character of an empirical summary or an axiomatic theory.<br />
<br />
A compression scheme by any other name is a symbolic representation, and this is what the differential extension of propositional calculus, through all of its many universes of discourse, is intended to supply. Why is this particular program of mental calisthenics worth carrying out in general? By providing a uniform logical medium for describing dynamic systems we can make the task of understanding complex systems much easier, both in looking for invariant representations of individual cases and in finding points of comparison among diverse structures that would otherwise appear as isolated systems. All of this goes to facilitate the search for compact knowledge and to adapt what is learned from individual cases to the general realm.<br />
<br />
===Back to the Feature===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
I guess it must be the flag of my disposition, out of hopeful<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;green stuff woven.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 31]<br />
|}<br />
<br />
Let us assume that the sense intended for differential features is well enough established in the intuition, for now, that we may continue with outlining the structure of the differential extension <math>[\mathrm{E}\mathcal{X}] = [A, \mathrm{d}A].\!</math> Over the extended alphabet <math>\mathrm{E}\mathcal{X} = \{ x_1, \mathrm{d}x_1 \} = \{ A, \mathrm{d}A \}\!</math> of cardinality <math>2^n = 2\!</math> we generate the set of points <math>\mathrm{E}X\!</math> of cardinality <math>2^{2n} = 4\!</math> that bears the following chain of equivalent descriptions:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}X & = & \langle A, \mathrm{d}A \rangle<br />
\\[4pt]<br />
& = & \{ \texttt{(} A \texttt{)}, A \} ~\times~ \{ \texttt{(} \mathrm{d}A \texttt{)}, \mathrm{d}A \}<br />
\\[4pt]<br />
& = &<br />
\{<br />
\texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)},~<br />
\texttt{(} A \texttt{)} \mathrm{d}A,~<br />
A \texttt{(} \mathrm{d}A \texttt{)},~<br />
A ~ \mathrm{d}A<br />
\}.<br />
\end{array}</math><br />
|}<br />
<br />
The space <math>\mathrm{E}X\!</math> may be assigned the mnemonic type <math>\mathbb{B} \times \mathbb{D},\!</math> which is really no different than <math>\mathbb{B} \times \mathbb{B} = \mathbb{B}^2.\!</math> An individual element of <math>\mathrm{E}X\!</math> may be regarded as a ''disposition at a point'' or a ''situated direction'', in effect, a singular mode of change occurring at a single point in the universe of discourse. In applications, the modality of this change can be interpreted in various ways, for example, as an expectation, an intention, or an observation with respect to the behavior of a system.<br />
<br />
To complete the construction of the extended universe of discourse <math>\mathrm{E}X^\bullet = [x_1, \mathrm{d}x_1] = [A, \mathrm{d}A]\!</math> one must add the set of differential propositions <math>\mathrm{E}X^\uparrow = \{ g : \mathrm{E}X \to \mathbb{B} \} \cong (\mathbb{B} \times \mathbb{D} \to \mathbb{B})\!</math> to the set of dispositions in <math>\mathrm{E}X.\!</math> There are <math>2^{2^{2n}} = 16\!</math> propositions in <math>\mathrm{E}X^\uparrow,\!</math> as detailed in Table&nbsp;14.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 14.} ~~ \text{Differential Propositions}\!</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>A\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>\mathrm{d}A\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>g_{0}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{1}<br />
\\[4pt]<br />
g_{2}<br />
\\[4pt]<br />
g_{4}<br />
\\[4pt]<br />
g_{8}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
1~0~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
\texttt{(} A \texttt{)} ~ \mathrm{d}A ~<br />
\\[4pt]<br />
~ A ~ \texttt{(} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
~ A ~~ \mathrm{d}A ~<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{neither}~ A ~\text{nor}~ \mathrm{d}A<br />
\\[4pt]<br />
\mathrm{d}A ~\text{and not}~ A<br />
\\[4pt]<br />
A ~\text{and not}~ \mathrm{d}A<br />
\\[4pt]<br />
A ~\text{and}~ \mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot A \land \lnot \mathrm{d}A<br />
\\[4pt]<br />
\lnot A \land \mathrm{d}A<br />
\\[4pt]<br />
A \land \lnot \mathrm{d}A<br />
\\[4pt]<br />
A \land \mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
g_{3}<br />
\\[4pt]<br />
g_{12}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{)}<br />
\\[4pt]<br />
A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ A<br />
\\[4pt]<br />
A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot A<br />
\\[4pt]<br />
A<br />
\end{matrix}\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{6}<br />
\\[4pt]<br />
g_{9}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[4pt]<br />
1~0~0~1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{,} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
\texttt{((} A \texttt{,} \mathrm{d}A \texttt{))}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
A ~\text{not equal to}~ \mathrm{d}A<br />
\\[4pt]<br />
A ~\text{equal to}~ \mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
A \ne \mathrm{d}A<br />
\\[4pt]<br />
A = \mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{5}<br />
\\[4pt]<br />
g_{10}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
\mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ \mathrm{d}A<br />
\\[4pt]<br />
\mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot \mathrm{d}A<br />
\\[4pt]<br />
\mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{7}<br />
\\[4pt]<br />
g_{11}<br />
\\[4pt]<br />
g_{13}<br />
\\[4pt]<br />
g_{14}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} ~ A ~~ \mathrm{d}A ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{(} ~ A ~ \texttt{(} \mathrm{d}A \texttt{))}<br />
\\[4pt]<br />
\texttt{((} A \texttt{)} ~ \mathrm{d}A ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{((} A \texttt{)(} \mathrm{d}A \texttt{))}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not both}~ A ~\text{and}~ \mathrm{d}A<br />
\\[4pt]<br />
\text{not}~ A ~\text{without}~ \mathrm{d}A<br />
\\[4pt]<br />
\text{not}~ \mathrm{d}A ~\text{without}~ A<br />
\\[4pt]<br />
A ~\text{or}~ \mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot A \lor \lnot \mathrm{d}A<br />
\\[4pt]<br />
A \Rightarrow \mathrm{d}A<br />
\\[4pt]<br />
A \Leftarrow \mathrm{d}A<br />
\\[4pt]<br />
A \lor \mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
| <math>f_{3}\!</math><br />
| <math>g_{15}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
Aside from changing the names of variables and shuffling the order of rows, this Table follows the format that was used previously for boolean functions of two variables. The rows are grouped to reflect natural similarity classes among the propositions. In a future discussion, these classes will be given additional explanation and motivation as the orbits of a certain transformation group acting on the set of 16 propositions. Notice that four of the propositions, in their logical expressions, resemble those given in the table for <math>X^\uparrow.\!</math> Thus the first set of propositions <math>\{ f_i \}\!</math> is automatically embedded in the present set <math>\{ g_j \}\!</math> and the corresponding inclusions are indicated at the far left margin of the Table.<br />
<br />
===Tacit Extensions===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
I would really like to have slipped imperceptibly into this lecture, as into all the others I shall be delivering, perhaps over the years ahead.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
Strictly speaking, however, there is a subtle distinction in type between the function <math>f_i : X \to \mathbb{B}</math> and the corresponding function <math>g_j : \mathrm{E}X \to \mathbb{B},</math> even though they share the same logical expression. Naturally, we want to maintain the logical equivalence of expressions that represent the same proposition while appreciating the full diversity of that proposition's functional and typical representatives. Both perspectives, and all the levels of abstraction extending through them, have their reasons, as will develop in time.<br />
<br />
Because this special circumstance points up an important general theme, it is a good idea to discuss it more carefully. Whenever there arises a situation like this, where one alphabet <math>\mathcal{X}</math> is a subset of another alphabet <math>\mathcal{Y},</math> then we say that any proposition <math>f : \langle \mathcal{X} \rangle \to \mathbb{B}</math> has a ''tacit extension'' to a proposition <math>\boldsymbol\varepsilon f : \langle \mathcal{Y} \rangle \to \mathbb{B},\!</math> and that the space <math>(\langle \mathcal{X} \rangle \to \mathbb{B})</math> has an ''automatic embedding'' within the space <math>(\langle \mathcal{Y} \rangle \to \mathbb{B}).</math> The extension is defined in such a way that <math>\boldsymbol\varepsilon f\!</math> puts the same constraint on the variables of <math>\mathcal{X}</math> that are contained in <math>\mathcal{Y}</math> as the proposition <math>f\!</math> initially did, while it puts no constraint on the variables of <math>\mathcal{Y}</math> outside of <math>\mathcal{X},</math> in effect, conjoining the two constraints.<br />
<br />
If the variables in question are indexed as <math>\mathcal{X} = \{ x_1, \ldots, x_n \}</math> and <math>\mathcal{Y} = \{ x_1, \ldots, x_n, \ldots, x_{n+k} \},</math> then the definition of the tacit extension from <math>\mathcal{X}</math> to <math>\mathcal{Y}</math> may be expressed in the form of an equation:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\boldsymbol\varepsilon f(x_1, \ldots, x_n, \ldots, x_{n+k}) ~=~ f(x_1, \ldots, x_n).\!</math><br />
|}<br />
<br />
On formal occasions, such as the present context of definition, the tacit extension from <math>\mathcal{X}</math> to <math>\mathcal{Y}</math> is explicitly symbolized by the operator <math>\boldsymbol\varepsilon : (\langle \mathcal{X} \rangle \to \mathbb{B}) \to (\langle \mathcal{Y} \rangle \to \mathbb{B}),</math> where the appropriate alphabets <math>\mathcal{X}</math> and <math>\mathcal{Y}</math> are understood from context, but normally one may leave the "<math>\boldsymbol\varepsilon\!</math>" silent.<br />
<br />
Let's explore what this means for the present Example. Here, <math>\mathcal{X} = \{ A \}</math> and <math>\mathcal{Y} = \mathrm{E}\mathcal{X} = \{ A, \mathrm{d}A \}.</math> For each of the propositions <math>f_i\!</math> over <math>X\!,</math> specifically, those whose expression <math>e_i\!</math> lies in the collection <math>\{ 0, \texttt{(} A \texttt{)}, A, 1 \},\!</math> the tacit extension <math>\boldsymbol\varepsilon f\!</math> of <math>f\!</math> to <math>\mathrm{E}X</math> can be phrased as a logical conjunction of two factors, <math>f_i = e_i \cdot \tau ~ ,\!</math> where <math>\tau\!</math> is a logical tautology that uses all the variables of <math>\mathcal{Y} - \mathcal{X}.</math> Working in these terms, the tacit extensions <math>\boldsymbol\varepsilon f\!</math> of <math>f\!</math> to <math>\mathrm{E}X</math> may be explicated as shown in Table&nbsp;15.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:60%"<br />
|+ style="height:30px" | <math>\text{Table 15.} ~~ \text{Tacit Extension of}~ [A] ~\text{to}~ [A, \mathrm{d}A]\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0<br />
& = & 0 & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))} & = & & 0<br />
\\[8pt]<br />
\texttt{(} A \texttt{)}<br />
& = & \texttt{(} A \texttt{)} & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))}<br />
& = & \texttt{(} A \texttt{)} \, \mathrm{d}A ~ & + & \texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)}<br />
\\[8pt]<br />
A<br />
& = & ~A~ & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))}<br />
& = & ~A~ ~\mathrm{d}A~ & + & ~A~ \texttt{(} \mathrm{d}A \texttt{)}<br />
\\[8pt]<br />
1<br />
& = & 1 & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))} & = & & 1<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
In its effect on the singular propositions over <math>X,\!</math> this analysis has an interesting interpretation. The tacit extension takes us from thinking about a particular state, like <math>A\!</math> or <math>\texttt{(} A \texttt{)},\!</math> to considering the collection of outcomes, the outgoing changes or the singular dispositions, that spring from that state.<br />
<br />
===Example 2. Drives and Their Vicissitudes===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
I open my scuttle at night and see the far-sprinkled systems,<br><br />
And all I see, multiplied as high as I can cipher, edge but<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the rim of the farther systems.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 81]<br />
|}<br />
<br />
Before we leave the one-feature case let's look at a more substantial example, one that illustrates a general class of curves that can be charted through the extended feature spaces and that provides an opportunity to discuss a number of important themes concerning their structure and dynamics.<br />
<br />
Again, let <math>\mathcal{X} = \{ x_1 \} = \{ A \}.\!</math> In the discussion that follows we will consider a class of trajectories having the property that <math>\mathrm{d}^k A = 0\!</math> for all <math>k\!</math> greater than some fixed <math>m\!</math> and we may indulge in the use of some picturesque terms that describe salient classes of such curves. Given the finite order condition, there is a highest order non-zero difference <math>\mathrm{d}^m A\!</math> exhibited at each point in the course of any determinate trajectory that one may wish to consider. With respect to any point of the corresponding orbit or curve let us call this highest order differential feature <math>\mathrm{d}^m A\!</math> the ''drive'' at that point. Curves of constant drive <math>\mathrm{d}^m A\!</math> are then referred to as ''<math>m^\text{th}\!</math>-gear curves''.<br />
<br />
* '''Scholium.''' The fact that a difference calculus can be developed for boolean functions is well known [Fuji], [Koh, &sect; 8-4] and was probably familiar to Boole, who was an expert in difference equations before he turned to logic. And of course there is the strange but true story of how the Turin machines of the 1840s prefigured the Turing machines of the 1940s [Men, 225-297]. At the very outset of general purpose, mechanized computing we find that the motive power driving the Analytical Engine of Babbage, the kernel of an idea behind all of his wheels, was exactly his notion that difference operations, suitably trained, can serve as universal joints for any conceivable computation [M&M], [Mel, ch. 4].<br />
<br />
Given this language, the Example we take up here can be described as the family of <math>4^\text{th}\!</math>-gear curves through <math>\mathrm{E}^4 X\!</math> <math>=\!</math> <math>\langle A, ~\mathrm{d}A, ~\mathrm{d}^2\!A, ~\mathrm{d}^3\!A, ~\mathrm{d}^4\!A \rangle.</math> These are the trajectories generated subject to the dynamic law <math>\mathrm{d}^4 A = 1,\!</math> where it is understood in such a statement that all higher order differences are equal to <math>0.\!</math> Since <math>\mathrm{d}^4 A\!</math> and all higher <math>\mathrm{d}^k A\!</math> are fixed, the temporal or transitional conditions (initial, mediate, terminal &mdash; transient or stable states) vary only with respect to their projections as points of <math>\mathrm{E}^3 X = \langle A, ~\mathrm{d}A, ~\mathrm{d}^2\!A, ~\mathrm{d}^3\!A \rangle.</math> Thus, there is just enough space in a planar venn diagram to plot all of these orbits and to show how they partition the points of <math>\mathrm{E}^3 X.\!</math> It turns out that there are exactly two possible orbits, of eight points each, as illustrated in Figure&nbsp;16.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 16 -- A Couple of Fourth Gear Orbits.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 16.} ~~ \text{A Couple of Fourth Gear Orbits}\!</math><br />
|}<br />
<br />
With a little thought it is possible to devise an indexing scheme for the general run of dynamic states that allows for comparing universes of discourse that weigh in on different scales of observation. With this end in sight, let us index the states <math>q \in \mathrm{E}^m X\!</math> with the dyadic rationals (or the binary fractions) in the half-open interval <math>[0, 2).\!</math> Formally and canonically, a state <math>q_r\!</math> is indexed by a fraction <math>r = \tfrac{s}{t}\!</math> whose denominator is the power of two <math>t = 2^m\!</math> and whose numerator is a binary numeral formed from the coefficients of state in a manner to be described next. The ''differential coefficients'' of the state <math>q\!</math> are just the values <math>\mathrm{d}^k\!A(q)</math> for <math>k = 0 ~\text{to}~ m,\!</math> where <math>\mathrm{d}^0\!A</math> is defined as being identical to <math>A.\!</math> To form the binary index <math>d_0.d_1 \ldots d_m\!</math> of the state <math>q\!</math> the coefficient <math>\mathrm{d}^k\!A(q)</math> is read off as the binary digit <math>d_k\!</math> associated with the place value <math>2^{-k}.\!</math> Expressed by way of algebraic formulas, the rational index <math>r\!</math> of the state <math>q\!</math> can be given by the following equivalent formulations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
r(q)<br />
& = &<br />
\displaystyle\sum_k d_k \cdot 2^{-k}<br />
& = &<br />
\displaystyle\sum_k \text{d}^k A(q) \cdot 2^{-k}<br />
\\[8pt]<br />
=<br />
\\[8pt]<br />
\displaystyle\frac{s(q)}{t}<br />
& = &<br />
\displaystyle\frac{\sum_k d_k \cdot 2^{(m-k)}}{2^m}<br />
& = &<br />
\displaystyle\frac{\sum_k \text{d}^k A(q) \cdot 2^{(m-k)}}{2^m}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Applied to the example of <math>4^\text{th}\!</math>-gear curves, this scheme results in the data of Tables&nbsp;17-a and 17-b, which exhibit one period for each orbit. The states in each orbit are listed as ordered pairs <math>(p_i, q_j),\!</math> where <math>p_i\!</math> may be read as a temporal parameter that indicates the present time of the state and where <math>j\!</math> is the decimal equivalent of the binary numeral <math>s.\!</math> Informally and more casually, the Tables exhibit the states <math>q_s\!</math> as subscripted with the numerators of their rational indices, taking for granted the constant denominators of <math>2^m\! = 2^4 = 16.\!</math> In this set-up the temporal successions of states can be reckoned as given by a kind of ''parallel round-up rule''. That is, if <math>(d_k, d_{k+1})\!</math> is any pair of adjacent digits in the state index <math>r,\!</math> then the value of <math>d_k\!</math> in the next state is <math>{d_k}' = d_k + d_{k+1}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:55%"<br />
|+ style="height:30px" | <math>\text{Table 17-a.} ~~ \text{A Couple of Orbits in Fourth Gear : Orbit 1}\!</math><br />
|- style="background:ghostwhite"<br />
| <math>\text{Time}\!</math><br />
| <math>\text{State}\!</math><br />
| <math>A\!</math><br />
| <math>\mathrm{d}A\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| <math>p_i\!</math><br />
| <math>q_j\!</math><br />
| <math>\mathrm{d}^0\!A</math><br />
| <math>\mathrm{d}^1\!A</math><br />
| <math>\mathrm{d}^2\!A</math><br />
| <math>\mathrm{d}^3\!A</math><br />
| <math>\mathrm{d}^4\!A</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
p_0<br />
\\[4pt]<br />
p_1<br />
\\[4pt]<br />
p_2<br />
\\[4pt]<br />
p_3<br />
\\[4pt]<br />
p_4<br />
\\[4pt]<br />
p_5<br />
\\[4pt]<br />
p_6<br />
\\[4pt]<br />
p_7<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
q_{01}<br />
\\[4pt]<br />
q_{03}<br />
\\[4pt]<br />
q_{05}<br />
\\[4pt]<br />
q_{15}<br />
\\[4pt]<br />
q_{17}<br />
\\[4pt]<br />
q_{19}<br />
\\[4pt]<br />
q_{21}<br />
\\[4pt]<br />
q_{31}<br />
\end{matrix}\!</math><br />
| colspan="5" |<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="text-align:center; width:100%"<br />
|<br />
<math>\begin{matrix}<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
1.<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:55%"<br />
|+ style="height:30px" | <math>\text{Table 17-b.} ~~ \text{A Couple of Orbits in Fourth Gear : Orbit 2}\!</math><br />
|- style="background:ghostwhite"<br />
| <math>\text{Time}\!</math><br />
| <math>\text{State}\!</math><br />
| <math>A\!</math><br />
| <math>\mathrm{d}A\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| <math>p_i\!</math><br />
| <math>q_j\!</math><br />
| <math>\mathrm{d}^0\!A</math><br />
| <math>\mathrm{d}^1\!A</math><br />
| <math>\mathrm{d}^2\!A</math><br />
| <math>\mathrm{d}^3\!A</math><br />
| <math>\mathrm{d}^4\!A</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
p_0<br />
\\[4pt]<br />
p_1<br />
\\[4pt]<br />
p_2<br />
\\[4pt]<br />
p_3<br />
\\[4pt]<br />
p_4<br />
\\[4pt]<br />
p_5<br />
\\[4pt]<br />
p_6<br />
\\[4pt]<br />
p_7<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
q_{25}<br />
\\[4pt]<br />
q_{11}<br />
\\[4pt]<br />
q_{29}<br />
\\[4pt]<br />
q_{07}<br />
\\[4pt]<br />
q_{09}<br />
\\[4pt]<br />
q_{27}<br />
\\[4pt]<br />
q_{13}<br />
\\[4pt]<br />
q_{23}<br />
\end{matrix}\!</math><br />
| colspan="5" |<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="text-align:center; width:100%"<br />
|<br />
<math>\begin{matrix}<br />
1.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
==Transformations of Discourse==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
It is understandable that an engineer should be completely absorbed in his speciality, instead of pouring himself out into the freedom and vastness of the world of thought, even though his machines are being sent off to the ends of the earth; for he no more needs to be capable of applying to his own personal soul what is daring and new in the soul of his subject than a machine is in fact capable of applying to itself the differential calculus on which it is based. The same thing cannot, however, be said about mathematics; for here we have the new method of thought, pure intellect, the very well-spring of the times, the ''fons et origo'' of an unfathomable transformation.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 39]<br />
|}<br />
<br />
In this section we take up the general study of logical transformations, or maps that relate one universe of discourse to another. In many ways, and especially as applied to the subject of intelligent dynamic systems, my argument develops the antithesis of the statement just quoted. Along the way, if incidental to my ends, I hope this essay can pose a fittingly irenic epitaph to the frankly ironic epigraph inscribed at its head.<br />
<br />
My goal in this section is to answer a single question: What is a propositional tangent functor? In other words, my aim is to develop a clear conception of what manner of thing would pass in the logical realm for a genuine analogue of the tangent functor, an object conceived to generalize as far as possible in the abstract terms of category theory the ordinary notions of functional differentiation and the all too familiar operations of taking derivatives.<br />
<br />
As a first step I discuss the kinds of transformations that we already know as ''extensions'' and ''projections'', and I use these special cases to illustrate several different styles of logical and visual representation that will figure heavily in the sequel.<br />
<br />
===Foreshadowing Transformations : Extensions and Projections of Discourse===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
And, despite the care which she took to look behind her at every moment, she failed to see a shadow which followed her like her own shadow, which stopped when she stopped, which started again when she did, and which made no more noise than a well-conducted shadow should.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 126]<br />
|}<br />
<br />
Many times in our discussion we have occasion to place one universe of discourse in the context of a larger universe of discourse. An embedding of the general type <math>[\mathcal{X}] \to [\mathcal{Y}]\!</math> is implied any time that we make use of one alphabet <math>[\mathcal{X}]\!</math> that happens to be included in another alphabet <math>[\mathcal{Y}].\!</math> When we are discussing differential issues we usually have in mind that the extended alphabet <math>[\mathcal{Y}]\!</math> has a special construction or a specific lexical relation with respect to the initial alphabet <math>[\mathcal{X}],\!</math> one that is marked by characteristic types of accents, indices, or inflected forms.<br />
<br />
====Extension from 1 to 2 Dimensions====<br />
<br />
Figure 18-a lays out the ''angular form'' of venn diagram for universes of 1 and 2 dimensions, indicating the embedding map of type <math>\mathbb{B}^1 \to \mathbb{B}^2\!</math> and detailing the coordinates that are associated with individual cells. Because all points, cells, or logical interpretations are represented as connected geometric areas, we can say that these pictures provide us with an ''areal view'' of each universe of discourse.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-a -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-a.} ~~ \text{Extension from 1 to 2 Dimensions : Areal}\!</math><br />
|}<br />
<br />
Figure 18-b shows the differential extension from <math>X^\bullet = [x]\!</math> to <math>\mathrm{E}X^\bullet = [x, \mathrm{d}x]\!</math> in a ''bundle of boxes'' form of venn diagram. As awkward as it may seem at first, this type of picture is often the most natural and the most easily available representation when we want to conceptualize the localized information or momentary knowledge of an intelligent dynamic system. It gives a ready picture of a ''proposition at a point'', in the present instance, of a proposition about changing states which is itself associated with a particular dynamic state of a system. It is easy to see how this application might be extended to conceive of more general types of instantaneous knowledge that are possessed by a system.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-b -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-b.} ~~ \text{Extension from 1 to 2 Dimensions : Bundle}\!</math><br />
|}<br />
<br />
Figure 18-c shows the same extension in a ''compact'' style of venn diagram, where the differential features at each position are represented by arrows extending from that position that cross or do not cross, as the case may be, the corresponding feature boundaries.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-c -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-c.} ~~ \text{Extension from 1 to 2 Dimensions : Compact}\!</math><br />
|}<br />
<br />
Figure 18-d compresses the picture of the differential extension even further, yielding a directed graph or ''digraph'' form of representation. (Notice that my definition of a digraph allows for loops or ''slings'' at individual points, in addition to arcs or ''arrows'' between the points.)<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-d -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-d.} ~~ \text{Extension from 1 to 2 Dimensions : Digraph}\!</math><br />
|}<br />
<br />
====Extension from 2 to 4 Dimensions====<br />
<br />
Figure 19-a lays out the ''areal view'' or the ''angular form'' of venn diagram for universes of 2 and 4 dimensions, indicating the embedding map of type <math>\mathbb{B}^2 \to \mathbb{B}^4.\!</math> In many ways these pictures are the best kind there is, giving full canvass to an ideal vista. Their style allows the clearest, the fairest, and the plainest view that we can form of a universe of discourse, affording equal representation to all dispositions and maintaining a balance with respect to ordinary and differential features. If only we could extend this view! Unluckily, an obvious difficulty beclouds this prospect, and that is how precipitately we run into the limits of our plane and visual intuitions. Even within the scope of the spare few dimensions that we have scanned up to this point subtle discrepancies have crept in already. The circumstances that bind us and the frameworks that block us, the flat distortion of the planar projection and the inevitable ineffability that precludes us from wrapping its rhomb figure into rings around a torus, all of these factors disguise the underlying but true connectivity of the universe of discourse.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-a -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-a.} ~~ \text{Extension from 2 to 4 Dimensions : Areal}\!</math><br />
|}<br />
<br />
Figure 19-b shows the differential extension from <math>U^\bullet = [u, v]\!</math> to <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v]\!</math> in the ''bundle of boxes'' form of venn diagram.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-b -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-b.} ~~ \text{Extension from 2 to 4 Dimensions : Bundle}\!</math><br />
|}<br />
<br />
As dimensions increase, this factorization of the extended universe along the lines that are marked out by the bundle picture begins to look more and more like a practical necessity. But whenever we use a propositional model to address a real situation in the context of nature we need to remain aware that this articulation into factors, affecting our description, may be wholly artificial in nature and cleave to nothing, no joint in nature, nor any juncture in time to be in or out of joint.<br />
<br />
Figure 19-c illustrates the extension from 2 to 4 dimensions in the ''compact'' style of venn diagram. Here, just the changes with respect to the center cell are shown.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-c -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-c.} ~~ \text{Extension from 2 to 4 Dimensions : Compact}\!</math><br />
|}<br />
<br />
Figure 19-d gives the ''digraph'' form of representation for the differential extension <math>U^\bullet \to \mathrm{E}U^\bullet,\!</math> where the 4 nodes marked with a circle <math>{}^{\bigcirc}\!</math> are the cells <math>uv,\, u \texttt{(} v \texttt{)},\, \texttt{(} u \texttt{)} v,\, \texttt{(} u \texttt{)(} v \texttt{)},\!</math> respectively, and where a 2-headed arc counts as 2 arcs of the differential digraph.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-d -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-d.} ~~ \text{Extension from 2 to 4 Dimensions : Digraph}\!</math><br />
|}<br />
<br />
===Thematization of Functions : And a Declaration of Independence for Variables===<br />
<br />
{| width="100%"<br />
| align="left" |<br />
''And as imagination bodies forth''<br><br />
''The forms of things unknown, the poet's pen''<br><br />
''Turns them to shapes, and gives to airy nothing''<br><br />
''A local habitation and a name.''<br />
| align="right" valign="bottom" | A Midsummer Night's Dream, 5.1.18<br />
|}<br />
<br />
In the representation of propositions as functions it is possible to notice different degrees of explicitness in the way their functional character is symbolized. To indicate what I mean by this, the next series of Figures illustrates a set of graphic conventions that will be put to frequent use in the remainder of this discussion, both to mark the relevant distinctions and to help us convert between related expressions at different levels of explicitness in their functionality.<br />
<br />
====Thematization : Venn Diagrams====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
The known universe has one complete lover and that is the greatest poet. He consumes an eternal passion and is indifferent which chance happens and which possible contingency of fortune or misfortune and persuades daily and hourly his delicious pay.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 11&ndash;12]<br />
|}<br />
<br />
Figure 20-i traces the first couple of steps in this order of ''thematic'' progression, that will gradually run the gamut through a complete series of degrees of functional explicitness in the expression of logical propositions. The first venn diagram represents a situation where the function is indicated by a shaded figure and a logical expression. At this stage one may be thinking of the proposition only as expressed by a formula in a particular language and its content only as a subset of the universe of discourse, as when considering the proposition <math>u\!\cdot\!v</math> in the universe <math>[u, v].\!</math> The second venn diagram depicts a situation in which two significant steps have been taken. First, one has taken the trouble to give the proposition <math>u\!\cdot\!v</math> a distinctive functional name <math>{}^{\backprime\backprime} J {}^{\prime\prime}.\!</math> Second, one has come to think explicitly about the target domain that contains the functional values of <math>J,\!</math> as when writing <math>J : \langle u, v \rangle \to \mathbb{B}.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 20-i -- Thematization of Conjunction (Stage 1).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 20-i.} ~~ \text{Thematization of Conjunction (Stage 1)}\!</math><br />
|}<br />
<br />
In Figure 20-ii the proposition <math>J\!</math> is viewed explicitly as a transformation from one universe of discourse to another.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 20-ii -- Thematization of Conjunction (Stage 2).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 20-ii.} ~~ \text{Thematization of Conjunction (Stage 2)}\!</math><br />
|}<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
o-------------------------------o o-------------------------------o<br />
| | | |<br />
| o-----o o-----o | | o-----o o-----o |<br />
| / \ / \ | | / \ / \ |<br />
| / o \ | | / o \ |<br />
| / /`\ \ | | / /`\ \ |<br />
| o o```o o | | o o```o o |<br />
| | u |```| v | | | | u |```| v | |<br />
| o o```o o | | o o```o o |<br />
| \ \`/ / | | \ \`/ / |<br />
| \ o / | | \ o / |<br />
| \ / \ / | | \ / \ / |<br />
| o-----o o-----o | | o-----o o-----o |<br />
| | | |<br />
o-------------------------------o o-------------------------------o<br />
\ / \ /<br />
\ / \ /<br />
\ / \ J /<br />
\ / \ /<br />
\ / \ /<br />
o----------\---------/----------o o----------\---------/----------o<br />
| \ / | | \ / |<br />
| \ / | | \ / |<br />
| o-----@-----o | | o-----@-----o |<br />
| /`````````````\ | | /`````````````\ |<br />
| /```````````````\ | | /```````````````\ |<br />
| /`````````````````\ | | /`````````````````\ |<br />
| o```````````````````o | | o```````````````````o |<br />
| |```````````````````| | | |```````````````````| |<br />
| |```````` J ````````| | | |```````` x ````````| |<br />
| |```````````````````| | | |```````````````````| |<br />
| o```````````````````o | | o```````````````````o |<br />
| \`````````````````/ | | \`````````````````/ |<br />
| \```````````````/ | | \```````````````/ |<br />
| \`````````````/ | | \`````````````/ |<br />
| o-----------o | | o-----------o |<br />
| | | |<br />
| | | |<br />
o-------------------------------o o-------------------------------o<br />
J = u v x = J<u, v><br />
<br />
Figure 20-ii. Thematization of Conjunction (Stage 2)<br />
</pre><br />
|}<br />
<br />
In the first venn diagram the name that is assigned to a composite proposition, function, or region in the source universe is delegated to a simple feature in the target universe. This can result in a single character or term exceeding the responsibilities it can carry off well. Allowing the name of a function <math>J : \langle u, v \rangle \to \mathbb{B}\!</math> to serve as the name of its dependent variable <math>J : \mathbb{B}\!</math> does not mean that one has to confuse a function with any of its values, but it does put one at risk for a number of obvious problems, and we should not be surprised, on numerous and limiting occasions, when quibbling arises from the attempts of a too original syntax to serve these two masters.<br />
<br />
The second venn diagram circumvents these difficulties by introducing a new variable name for each basic feature of the target universe, as when writing <math>J : \langle u, v \rangle \to \langle x \rangle,\!</math> and thereby assigns a concrete type <math>\langle x \rangle</math> to the abstract codomain <math>\mathbb{B}.\!</math> To make this induction of variables more formal one can append subscripts, as in <math>x_J,\!</math> to indicate the origin or derivation of the new characters. Or we may use a lexical modifier to convert function names into variable names, for example, associating the function name <math>J\!</math> with the variable name <math>\check{J}.\!</math> Thus we may think of <math>x = x_J = \check{J}\!</math> as the ''cache variable'' corresponding to the function <math>J\!</math> or the symbol <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> considered as a contingent variable.<br />
<br />
In Figure 20-iii we arrive at a stage where the functional equations <math>J = u\!\cdot\!v</math> and <math>x = u\!\cdot\!v</math> are regarded as propositions in their own right, reigning in and ruling over the 3-feature universes of discourse <math>[u, v, J]~\!</math> and <math>[u, v, x],\!</math> respectively. Subject to the cautions already noted, the function name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> can be reinterpreted as the name of a feature <math>\check{J}</math> and the equation <math>J = u\!\cdot\!v</math> can be read as the logical equivalence <math>\texttt{((} J, u ~ v \texttt{))}.\!</math> To give it a generic name let us call this newly expressed, collateral proposition the ''thematization'' or the ''thematic extension'' of the original proposition <math>J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 20-iii -- Thematization of Conjunction (Stage 3).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 20-iii.} ~~ \text{Thematization of Conjunction (Stage 3)}\!</math><br />
|}<br />
<br />
The first venn diagram represents the thematization of the conjunction <math>J\!</math> with shading in the appropriate regions of the universe <math>[u, v, J].\!</math> Also, it illustrates a quick way of constructing a thematic extension. First, draw a line, in practice or the imagination, that bisects every cell of the original universe, placing half of each cell under the aegis of the thematized proposition and the other half under its antithesis. Next, on the scene where the theme applies leave the shade wherever it lies, and off the stage, where it plays otherwise, stagger the pattern in a harlequin guise.<br />
<br />
In the final venn diagram of this sequence the thematic progression comes full circle and completes one round of its development. The ambiguities that were occasioned by the changing role of the name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> are resolved by introducing a new variable name <math>{}^{\backprime\backprime} x {}^{\prime\prime}</math> to take the place of <math>\check{J},\!</math> and the region that represents this fresh featured <math>x\!</math> is circumscribed in a more conventional symmetry of form and placement. Just as we once gave the name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> to the proposition <math>u\!\cdot\!v,</math> we now give the name <math>{}^{\backprime\backprime} \iota {}^{\prime\prime}</math> to its thematization <math>\texttt{((} x, u ~ v \texttt{))}.\!</math> Already, again, at this culminating stage of reflection, we begin to think of the newly named proposition as a distinctive individual, a particular function <math>\iota : \langle u, v, x \rangle \to \mathbb{B}.\!</math><br />
<br />
From now on, the terms ''thematic extension'' and ''thematization'' will be used to describe both the process and degree of explication that progresses through this series of pictures, both the operation of increasingly explicit symbolization and the dimension of variation that is swept out by it. To speak of this change in general, that takes us in our current example from <math>J\!</math> to <math>\iota,\!</math> we introduce a class of operators symbolized by the Greek letter <math>\theta,\!</math> writing <math>\iota = \theta J\!</math> in the present instance. The operator <math>\theta,\!</math> in the present situation bearing the type <math>\theta : [u, v] \to [u, v, x],\!</math> provides us with a convenient way of recapitulating and summarizing the complete cycle of thematic developments.<br />
<br />
Figure 21 shows how the thematic extension operator <math>\theta\!</math> acts on two further examples, the disjunction <math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math> and the equality <math>\texttt{((} u, v \texttt{))}.\!</math> Referring to the disjunction as <math>f(u, v)\!</math> and the equality as <math>f(u, v),\!</math> we may express the thematic extensions as <math>\varphi = \theta f\!</math> and <math>\gamma = \theta g.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 21 -- Thematization of Disjunction and Equality.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 21.} ~~ \text{Thematization of Disjunction and Equality}\!</math><br />
|}<br />
<br />
====Thematization : Truth Tables====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
That which distorts honest shapes or which creates unearthly beings or places or contingencies is a nuisance and a revolt.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 19]<br />
|}<br />
<br />
Tables 22 through 25 outline a method for computing the thematic extensions of propositions in terms of their coordinate values.<br />
<br />
A preliminary step, as illustrated in Table&nbsp;22, is to write out the truth table representations of the propositional forms whose thematic extensions one wants to compute, in the present instance, the functions <math>f(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math> and <math>g(u, v) = \texttt{((} u, v \texttt{))}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:50%"<br />
|+ style="height:30px" | <math>\text{Table 22.} ~~ \text{Disjunction}~ f ~\text{and Equality}~ g\!</math><br />
|- style="height:35px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black" | <math>f\!</math><br />
| <math>g\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Next, each propositional form is individually represented in the fashion shown in Tables&nbsp;23-i and 23-ii, using <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} g {}^{\prime\prime}\!</math> as function names and creating new variables <math>x\!</math> and <math>y\!</math> to hold the associated functional values. This pair of Tables outlines the first stage in the transition from the <math>2\!</math>-dimensional universes of <math>f\!</math> and <math>g\!</math> to the <math>3\!</math>-dimensional universes of <math>\theta f\!</math> and <math>\theta g.\!</math> The top halves of the Tables replicate the truth table patterns for <math>f\!</math> and <math>g\!</math> in the form <math>f : [u, v] \to [x]\!</math> and <math>g : [u, v] \to [y].\!</math> The bottom halves of the tables print the negatives of these pictures, as it were, and paste the truth tables for <math>\texttt{(} f \texttt{)}\!</math> and <math>\texttt{(} g \texttt{)}\!</math> under the copies for <math>f\!</math> and <math>g.\!</math> At this stage, the columns for <math>\theta f\!</math> and <math>\theta g\!</math> are appended almost as afterthoughts, amounting to indicator functions for the sets of ordered triples that make up the functions <math>f\!</math> and <math>g.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 23-i and 23-ii.} ~~ \text{Thematics of Disjunction and Equality (1)}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 23-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black" | <math>f\!</math><br />
| <math>x\!</math><br />
| <math>\varphi\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |<br />
<math>\begin{matrix}\to\\\to\\\to\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" | &nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 23-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black" | <math>g\!</math><br />
| <math>y\!</math><br />
| <math>\gamma\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |<br />
<math>\begin{matrix}\to\\\to\\\to\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" | &nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
All the data are now in place to give the truth tables for <math>\theta f\!</math> and <math>\theta g.\!</math> All that remains to be done is to permute the rows and change the roles of <math>x\!</math> and <math>y\!</math> from dependent to independent variables. In Tables&nbsp;24-i and 24-ii the rows are arranged in such a way as to put the 3-tuples <math>(u, v, x)\!</math> and <math>(u, v, y)\!</math> in binary numerical order, suitable for viewing as the arguments of the maps <math>\theta f = \varphi : [u, v, x] \to \mathbb{B}\!</math> and <math>\theta g = \gamma : [u, v, y] \to \mathbb{B}.\!</math> Moreover, the structure of the tables is altered slightly, allowing the now vestigial functions <math>\theta f\!</math> and <math>\theta g\!</math> to be passed over without further attention and shifting the heavy vertical bars a notch to the right. In effect, this clinches the fact that the thematic variables <math>x := \check{f}\!</math> and <math>y := \check{g}\!</math> are now treated as independent variables.<br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 24-i and 24-ii.} ~~ \text{Thematics of Disjunction and Equality (2)}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 24-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>f\!</math><br />
| <math>x\!</math><br />
| style="border-left:1px solid black" | <math>\varphi\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <font size="+2">&nbsp;<br></font><math>\begin{matrix}\to\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 24-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>g\!</math><br />
| <math>y\!</math><br />
| style="border-left:1px solid black" | <math>\gamma\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | &nbsp;<br><math>\begin{matrix}\to\\\to\end{matrix}\!</math><br>&nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
An optional reshuffling of the rows brings additional features of the thematic extensions to light. Leaving the columns in place for the sake of comparison, Tables&nbsp;25-i and 25-ii sort the rows in a different order, in effect treating <math>x\!</math> and <math>y\!</math> as the primary variables in their respective 3-tuples. Regarding the thematic extensions in the form <math>\varphi : [x, u, v] \to \mathbb{B}\!</math> and <math>\gamma : [y, u, v] \to \mathbb{B}\!</math> makes it easier to see in this tabular setting a property that was graphically obvious in the venn diagrams above. Specifically, when the thematic variable <math>\check{F}\!</math> is true then <math>\theta F\!</math> exhibits the pattern of the original <math>F,\!</math> and when <math>\check{F}\!</math> is false then <math>\theta F\!</math> exhibits the pattern of its negation <math>\texttt{(} F \texttt{)}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 25-i and 25-ii.} ~~ \text{Thematics of Disjunction and Equality (3)}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 25-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>f\!</math><br />
| <math>x\!</math><br />
| style="border-left:1px solid black" | <math>\varphi\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>{\to}\!</math><br><font size="+2">&nbsp;<br>&nbsp;<br>&nbsp;<br></font><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <font size="+2">&nbsp;<br></font><math>\begin{matrix}\to\\\to\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 25-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>g\!</math><br />
| <math>y\!</math><br />
| style="border-left:1px solid black" | <math>\gamma\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | &nbsp;<br><math>\begin{matrix}\to\\\to\end{matrix}\!</math><br>&nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
Finally, Tables&nbsp;26-i and 26-ii compare the tacit extensions <math>\boldsymbol\varepsilon : [u, v] \to [u, v, x]\!</math> and <math>\boldsymbol\varepsilon : [u, v] \to [u, v, y]\!</math> with the thematic extensions of the same types, as applied to the propositions <math>f\!</math> and <math>g,\!</math> respectively.<br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 26-i and 26-ii.} ~~ \text{Tacit Extension and Thematization}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 26-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>x\!</math><br />
| style="border-left:1px solid black" | <math>\boldsymbol\varepsilon f\!</math><br />
| <math>\theta f\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 26-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>y\!</math><br />
| style="border-left:1px solid black" | <math>\boldsymbol\varepsilon g\!</math><br />
| <math>\theta g\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
Table 27 summarizes the thematic extensions of all propositions on two variables. Column&nbsp;4 lists the equations of form <math>\texttt{((} \check{f_i}, f_i (u, v) \texttt{))}\!</math> and Column&nbsp;5 simplifies these equations into the form of algebraic expressions. As always, <math>{}^{\backprime\backprime} + {}^{\prime\prime}\!</math> refers to exclusive disjunction and each <math>{}^{\backprime\backprime} \check{f} {}^{\prime\prime}\!</math> appearing in the last two Columns refers to the corresponding variable name <math>{}^{\backprime\backprime} \check{f_i} {}^{\prime\prime}.~\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 27.} ~~ \text{Thematization of Bivariate Propositions}\!</math><br />
|- style="height:30px; background:ghostwhite"<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>{f}\!</math><br />
| <math>\theta f\!</math><br />
| <math>\theta f\!</math><br />
|- style="background:ghostwhite"<br />
| align="right" | <math>u\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| align="right" | <math>v\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| align="left" | <math>\texttt{((} \check{f} \texttt{,~(~)~))}\!</math><br />
| align="left" | <math>\check{f} + 1\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\\[4pt]<br />
f_{4}<br />
\\[4pt]<br />
f_{8}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
1~0~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} u \texttt{)~} v \texttt{~}<br />
\\[4pt]<br />
\texttt{~} u \texttt{~(} v \texttt{)}<br />
\\[4pt]<br />
\texttt{~} u \texttt{~~} v \texttt{~}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(u)(v)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~(u)~v~~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~u~(v)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~u~~v~~))}<br />
\end{array}</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + u + v + uv<br />
\\[4pt]<br />
\check{f} + v + uv + 1<br />
\\[4pt]<br />
\check{f} + u + uv + 1<br />
\\[4pt]<br />
\check{f} + uv + 1<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{3}<br />
\\[4pt]<br />
f_{12}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{)}<br />
\\[4pt]<br />
\texttt{~} u \texttt{~}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(u)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~u~~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + u<br />
\\[4pt]<br />
\check{f} + u + 1<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[4pt]<br />
f_{9}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[4pt]<br />
1~0~0~1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{,} v \texttt{)}<br />
\\[4pt]<br />
\texttt{((} u \texttt{,} v \texttt{))}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~~(} u \texttt{,} v \texttt{)~~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~((} u \texttt{,} v \texttt{))~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + u + v + 1<br />
\\[4pt]<br />
\check{f} + u + v<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{5}<br />
\\[4pt]<br />
f_{10}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} v \texttt{)}<br />
\\[4pt]<br />
\texttt{~} v \texttt{~}<br />
\end{matrix}</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(} v \texttt{)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~} v \texttt{~~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + v<br />
\\[4pt]<br />
\check{f} + v + 1<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[4pt]<br />
f_{11}<br />
\\[4pt]<br />
f_{13}<br />
\\[4pt]<br />
f_{14}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(~} u \texttt{~~} v \texttt{~)}<br />
\\[4pt]<br />
\texttt{(~} u \texttt{~(} v \texttt{))}<br />
\\[4pt]<br />
\texttt{((} u \texttt{)~} v \texttt{~)}<br />
\\[4pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(~} u \texttt{~~} v \texttt{~)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~(~} u \texttt{~(} v \texttt{))~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~((} u \texttt{)~} v \texttt{~)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~((} u \texttt{)(} v \texttt{))~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + uv<br />
\\[4pt]<br />
\check{f} + u + uv<br />
\\[4pt]<br />
\check{f} + v + uv<br />
\\[4pt]<br />
\check{f} + u + v + uv + 1<br />
\end{array}\!</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| align="left" | <math>\texttt{((} \check{f} \texttt{,~((~))~))}\!</math><br />
| align="left" | <math>\check{f}\!</math><br />
|}<br />
<br />
<br><br />
<br />
In order to show what all of the thematic extensions from two dimensions to three dimensions look like in terms of coordinates, Tables&nbsp;28 and 29 present ordinary truth tables for the functions <math>f_i : \mathbb{B}^2 \to \mathbb{B}\!</math> and for the corresponding thematizations <math>\theta f_i = \varphi_i : \mathbb{B}^3 \to \mathbb{B}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 28.} ~~ \text{Propositions on Two Variables}\!</math><br />
|- style="height:35px; background:ghostwhite"<br />
| style="width:5%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:5%; border-bottom:1px solid black; border-left:1px solid black" | <math>f_{0}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{1}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{2}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{3}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{4}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{5}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{6}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{7}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{8}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{9}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{10}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{11}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{12}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{13}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{14}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{15}\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 29.} ~~ \text{Thematic Extensions of Bivariate Propositions}\!</math><br />
|- style="height:35px; background:ghostwhite"<br />
| style="width:5%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\check{f}\!</math><br />
| style="width:5%; border-bottom:1px solid black; border-left:1px solid black" | <math>\varphi_{0}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{1}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{2}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{3}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{4}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{5}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{6}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{7}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{8}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{9}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{10}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{11}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{12}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{13}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{14}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{15}\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
|-<br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Propositional Transformations===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
If only the word &lsquo;artificial&rsquo; were associated with the idea of ''art'', or expert skill gained through voluntary apprenticeship (instead of suggesting the factitious and unreal), we might say that ''logical'' refers to artificial thought.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 56&ndash;57]<br />
|}<br />
<br />
In this section we develop a comprehensive set of concepts for dealing with transformations between universes of discourse. In this most general setting the source and target universes of a transformation are allowed to be different, but may be the same. When we apply these concepts to dynamic systems we focus on the important special case of transformations that map a universe into itself, regarding them as the state transitions of a discrete dynamical process and placing them among the myriad ways that a universe of discourse might change, and by that change turn into itself.<br />
<br />
====Alias and Alibi Transformations====<br />
<br />
There are customarily two modes of understanding a transformation, at least, when we try to interpret its relevance to something in reality. A transformation always refers to a changing prospect, to say it in a unified but equivocal way, but this can be taken to mean either a subjective change in the interpreting observer's point of view or an objective change in the systematic subject of discussion. In practice these variant uses of the transformation concept are distinguished in the following terms:<br />
<br />
# A ''perspectival'' or ''alias'' transformation refers to a shift in perspective or a change in language that takes place in the observer's frame of reference.<br />
# A ''transitional'' or ''alibi'' transformation refers to a change of position or an alteration of state that occurs in the object system as it falls under study.<br />
<br />
(For a recent discussion of the alias vs. alibi issue, as it relates to linear transformations in vector spaces and to other issues of an algebraic nature, see [MaB, 256, 582-4].)<br />
<br />
Naturally, when we are concerned with the dynamical properties of a system, the transitional aspect of transformation is the factor that comes to the fore, and this involves us in contemplating all of the ways of changing a universe into itself while remaining under the rule of established dynamical laws. In the prospective application to dynamic systems, and to neural networks viewed in this light, our interest lies chiefly with the transformations of a state space into itself that constitute the state transitions of a discrete dynamic process. Nevertheless, many important properties of these transformations, and some constructions that we need to see most clearly, are independent of the transitional interpretation and are likely to be confounded with irrelevant features if presented first and only in that association.<br />
<br />
In addition, and in partial contrast, intelligent systems are exactly that species of dynamic agents that have the capacity to have a point of view, and we cannot do justice to their peculiar properties without examining their ability to form and transform their own frames of reference in exposure to the elements of their own experience. In this setting, the perspectival aspect of transformation is the facet that shines most brightly, perhaps too often leaving us fascinated with mere glimmerings of its actual potential. It needs to be emphasized that nothing of the ordinary sort needs be moved in carrying out a transformation under the alias interpretation, that it may only involve a change in the forms of address, an amendment of the terms which are customed to approach and fashioned to describe the very same things in the very same world. But again, working within a discipline of realistic computation, we know how formidably complex and resource-consuming such transformations of perspective can be to implement in practice, much less to endow in the self-governed form of a nascently intelligent dynamical system.<br />
<br />
====Transformations of General Type====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
''Es ist passiert'', &ldquo;it just sort of happened&rdquo;, people said there when other people in other places thought heaven knows what had occurred. It was a peculiar phrase, not known in this sense to the Germans and with no equivalent in other languages, the very breath of it transforming facts and the bludgeonings of fate into something light as eiderdown, as thought itself.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 34]<br />
|}<br />
<br />
Consider the situation illustrated in Figure&nbsp;30, where the alphabets <math>\mathcal{U} = \{ u, v \}\!</math> and <math>\mathcal{X} = \{ x, y, z \}\!</math> are used to label basic features in two different logical universes, <math>U^\bullet = [u, v]\!</math> and <math>X^\bullet = [x, y, z].\!</math><br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
o-------------------------------------------------------o<br />
| U |<br />
| |<br />
| o-----------o o-----------o |<br />
| / \ / \ |<br />
| / o \ |<br />
| / / \ \ |<br />
| / / \ \ |<br />
| o o o o |<br />
| | | | | |<br />
| | u | | v | |<br />
| | | | | |<br />
| o o o o |<br />
| \ \ / / |<br />
| \ \ / / |<br />
| \ o / |<br />
| \ / \ / |<br />
| o-----------o o-----------o |<br />
| |<br />
| |<br />
o---------------------------o---------------------------o<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
o-------------------------o o-------------------------o o-------------------------o<br />
| U | | U | | U |<br />
| o---o o---o | | o---o o---o | | o---o o---o |<br />
| / \ / \ | | / \ / \ | | / \ / \ |<br />
| / o \ | | / o \ | | / o \ |<br />
| / / \ \ | | / / \ \ | | / / \ \ |<br />
| o o o o | | o o o o | | o o o o |<br />
| | u | | v | | | | u | | v | | | | u | | v | |<br />
| o o o o | | o o o o | | o o o o |<br />
| \ \ / / | | \ \ / / | | \ \ / / |<br />
| \ o / | | \ o / | | \ o / |<br />
| \ / \ / | | \ / \ / | | \ / \ / |<br />
| o---o o---o | | o---o o---o | | o---o o---o |<br />
| | | | | |<br />
o-------------------------o o-------------------------o o-------------------------o<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ g | \ f / | h /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ o----------|-----------\-----/-----------|----------o /<br />
\ | X | \ / | | /<br />
\ | | \ / | | /<br />
\ | | o-----o-----o | | /<br />
\| | / \ | |/<br />
\ | / \ | /<br />
|\ | / \ | /|<br />
| \ | / \ | / |<br />
| \ | / \ | / |<br />
| \ | o x o | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \| | | |/ |<br />
| o--o--------o o--------o--o |<br />
| / \ \ / / \ |<br />
| / \ \ / / \ |<br />
| / \ o / \ |<br />
| / \ / \ / \ |<br />
| / \ / \ / \ |<br />
| o o--o-----o--o o |<br />
| | | | | |<br />
| | | | | |<br />
| | | | | |<br />
| | y | | z | |<br />
| | | | | |<br />
| | | | | |<br />
| o o o o |<br />
| \ \ / / |<br />
| \ \ / / |<br />
| \ o / |<br />
| \ / \ / |<br />
| \ / \ / |<br />
| o-----------o o-----------o |<br />
| |<br />
| |<br />
o---------------------------------------------------o<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ p , q /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
o<br />
<br />
Figure 30. Generic Frame of a Logical Transformation<br />
</pre><br />
|}<br />
<br />
Enter the picture, as we usually do, in the middle of things, with features like <math>x, y , z\!</math> that present themselves to be simple enough in their own right and that form a satisfactory, if temporary foundation to provide a basis for discussion. In this universe and on these terms we find expression for various propositions and questions of principal interest to ourselves, as indicated by the maps <math>p, q : X \to \mathbb{B}.\!</math> Then we discover that the simple features <math>\{ x, y, z \}\!</math> are really more complex than we thought at first, and it becomes useful to regard them as functions <math>\{ f, g, h \}\!</math> of other features <math>\{ u, v \}\!</math> that we place in a preface to our original discourse, or suppose as topics of a preliminary universe of discourse <math>U^\bullet = [u, v].\!</math> It may happen that these late-blooming but pre-ambling features are found to lie closer, in a sense that may be our job to determine, to the central nature of the situation of interest, in which case they earn our regard as being more fundamental, but these functions and features are only required to supply a critical stance on the universe of discourse or an alternate perspective on the nature of things in order to be preserved as useful.<br />
<br />
A particular transformation <math>F : [u, v] \to [x, y, z]\!</math> may be expressed by a system of equations, as shown below. Here, <math>F\!</math> is defined by its component maps <math>F = (F_1, F_2, F_3) = (f, g, h),\!</math> where each component map in <math>\{ f, g, h \}\!</math> is a proposition of type <math>\mathbb{B}^n \to \mathbb{B}^1.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:50%"<br />
|<br />
<math>\begin{matrix}<br />
x & = & f(u, v)<br />
\\[10pt]<br />
y & = & g(u, v)<br />
\\[10pt]<br />
z & = & h(u, v)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Regarded as a logical statement, this system of equations expresses a relation between a collection of freely chosen propositions <math>\{ f, g, h \}\!</math> in one universe of discourse and the special collection of simple propositions <math>\{ x, y, z \}\!</math> on which is founded another universe of discourse. Growing familiarity with a particular transformation of discourse, and the desire to achieve a ready understanding of its implications, requires that we be able to convert this information about generals and simples into information about all the main subtypes of propositions, including the linear and singular propositions.<br />
<br />
===Analytic Expansions : Operators and Functors===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Consider what effects that might ''conceivably'' have practical bearings you ''conceive'' the objects of your ''conception'' to have. Then, your ''conception'' of those effects is the whole of your ''conception'' of the object.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; C.S. Peirce, &ldquo;The Maxim of Pragmatism&rdquo;, CP 5.438<br />
|}<br />
<br />
Given the barest idea of a logical transformation, as suggested by the sketch in Figure&nbsp;30, and having conceptualized the universe of discourse, with all of its points and propositions, as a beginning object of discussion, we are ready to enter the next phase of our investigation.<br />
<br />
====Operators on Propositions and Transformations====<br />
<br />
The next step is naturally inclined toward objects of the next higher order, namely, with operators that take in argument lists of logical transformations and that give back specified types of logical transformations as their results. For our present aims, we do not need to consider the most general class of such operators, nor any one of them for its own sake. Rather, we are interested in the special sorts of operators that arise in the study and analysis of logical transformations. Figuratively speaking, these operators serve as instruments for the live tomography (and hopefully not the vivisection) of the forms of change under view. Beyond that, they open up ways to implement the changes of view that we need to grasp all the variations on a transformational theme, or to appreciate enough of its significant features to &ldquo;get the drift&rdquo; of the change occurring, to form a passing acquaintance or a synthetic comprehension of its general character and disposition.<br />
<br />
The simplest type of operator is one that takes a single transformation as an argument and returns a single transformation as a result, and most of the operators explicitly considered in our discussion will be of this kind. Figure&nbsp;31 illustrates the typical situation.<br />
<br />
{| align="center" border="0" cellpadding="20"<br />
|<br />
<pre><br />
o---------------------------------------o<br />
| |<br />
| |<br />
| U% F X% |<br />
| o------------------>o |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| !W! | | !W! |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| v v |<br />
| o------------------>o |<br />
| !W!U% !W!F !W!X% |<br />
| |<br />
| |<br />
o---------------------------------------o<br />
Figure 31. Operator Diagram (1)<br />
</pre><br />
|}<br />
<br />
In this Figure <math>{}^{\backprime\backprime} \mathsf{W} {}^{\prime\prime}\!</math> stands for a generic operator <math>\mathsf{W},\!</math> in this case one that takes a logical transformation <math>F\!</math> of type <math>(U^\bullet \to X^\bullet)\!</math> into a logical transformation <math>\mathsf{W}F\!</math> of type <math>(\mathsf{W}U^\bullet \to \mathsf{W}X^\bullet).\!</math> Thus, the operator <math>\mathsf{W}\!</math> must be viewed as making assignments for both families of objects we have previously considered, that is, for universes of discourse like <math>{U^\bullet}\!</math> and <math>{X^\bullet}\!</math> and for logical transformations like <math>F.\!</math><br />
<br />
'''Note.''' Strictly speaking, an operator like <math>\mathsf{W}\!</math> works between two whole categories of universes and transformations, which we call the ''source'' and the ''target'' categories of <math>\mathsf{W}.\!</math> Given this setting, <math>\mathsf{W}\!</math> specifies for each universe <math>U^\bullet\!</math> in its source category a definite universe <math>\mathsf{W}U^\bullet\!</math> in its target category, and to each transformation <math>F\!</math> in its source category it assigns a unique transformation <math>\mathsf{W}F\!</math> in its target category. Naturally, this only works if <math>\mathsf{W}\!</math> takes the source <math>U^\bullet</math> and the target <math>X^\bullet</math> of the map <math>F\!</math> over to the source <math>\mathsf{W}U^\bullet\!</math> and the target <math>\mathsf{W}X^\bullet\!</math> of the map <math>\mathsf{W}F.\!</math> With luck or care enough, we can avoid ever having to put anything like that in words again, letting diagrams do the work. In the situations of present concern we are usually focused on a single transformation <math>F,\!</math> and thus we can take it for granted that the assignment of universes under <math>\mathsf{W}\!</math> is defined appropriately at the source and target ends of <math>F.\!</math> It is not always the case, though, that we need to use the particular names (like <math>{}^{\backprime\backprime} \mathsf{W}U^\bullet {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \mathsf{W}X^\bullet {}^{\prime\prime}\!</math>) that <math>\mathsf{W}\!</math> assigns by default to its operative image universes. In most contexts we will usually have a prior acquaintance with these universes under other names and it is necessary only that we can tell from the information associated with an operator <math>\mathsf{W}\!</math> what universes they are.<br />
<br />
In Figure&nbsp;31 the maps <math>F\!</math> and <math>\mathsf{W}F\!</math> are displayed horizontally, the way one normally orients functional arrows in a written text, and <math>\mathsf{W}\!</math> rolls the map <math>F\!</math> downward into the images that are associated with <math>\mathsf{W}F.\!</math> In Figure&nbsp;32 the same information is redrawn so that the maps <math>F\!</math> and <math>\mathsf{W}F\!</math> flow down the page, and <math>\mathsf{W}\!</math> unfurls the map <math>F\!</math> rightward into domains that are the eminent purview of <math>\mathsf{W}F.\!</math><br />
<br />
{| align="center" border="0" cellpadding="20"<br />
|<br />
<pre><br />
o---------------------------------------o<br />
| |<br />
| |<br />
| U% !W! !W!U% |<br />
| o------------------>o |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| F | | !W!F |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| v v |<br />
| o------------------>o |<br />
| X% !W! !W!X% |<br />
| |<br />
| |<br />
o---------------------------------------o<br />
Figure 32. Operator Diagram (2)<br />
</pre><br />
|}<br />
<br />
The latter arrangement, as exhibited in Figure&nbsp;32, is more congruent with the thinking about operators that we shall do in the rest of this discussion, since all logical transformations from here on out will be pictured vertically, after the fashion of Figure&nbsp;30.<br />
<br />
====Differential Analysis of Propositions and Transformations====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" | The resultant metaphysical problem now is this: ''Does the man go round the squirrel or not?''<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 43]<br />
|}<br />
<br />
The approach to the differential analysis of logical propositions and transformations of discourse to be pursued here is carried out in terms of particular operators <math>\mathsf{W}\!</math> that act on propositions <math>F\!</math> or on transformations <math>F\!</math> to yield the corresponding operator maps <math>\mathsf{W}F.\!</math> The operator results then become the subject of a series of further stages of analysis, which take them apart into their propositional components, rendering them as a set of purely logical constituents. After this is done, all the parts are then re-integrated to reconstruct the original object in the light of a more complete understanding, at least in ways that enable one to appreciate certain aspects of it with fresh insight.<br />
<br />
* '''Remark on Strategy.''' At this point we run into a set of conceptual difficulties that force us to make a strategic choice in how we proceed. Part of the problem can be remedied by extending our discussion of tacit extensions to the transformational context. But the troubles that remain are much more obstinate and lead us to try two different types of solution. The approach that we develop first makes use of a variant type of extension operator, the ''trope extension'', to be defined below. This method is more conservative and requires less preparation, but has features which make it seem unsatisfactory in the long run. A more radical approach, but one with a better hope of long term success, makes use of the notion of ''contingency spaces''. These are an even more generous type of extended universe than the kind we currently use, but are defined subject to certain internal constraints. The extra work needed to set up this method forces us to put it off to a later stage. However, as a compromise, and to prepare the ground for the next pass, we call attention to the various conceptual difficulties as they arise along the way and try to give an honest estimate of how well our first approach deals with them.<br />
<br />
We now describe in general terms the particular operators that are instrumental to this form of analysis. The main series of operators all have the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathsf{W}<br />
& : &<br />
( U^\bullet \to X^\bullet )<br />
& \to &<br />
( \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet )<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
If we assume that the source universe <math>U^\bullet</math> and the target universe <math>X^\bullet</math> have finite dimensions <math>n\!</math> and <math>k,\!</math> respectively, then each operator <math>\mathsf{W}\!</math> is encompassed by the same abstract type:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathsf{W}<br />
& : &<br />
( [\mathbb{B}^n] \to [\mathbb{B}^k] )<br />
& \to &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k \times \mathbb{D}^k] )<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Since the range features of the operator result <math>\mathsf{W}F : [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k \times \mathbb{D}^k]</math> can be sorted by their ordinary versus differential qualities and the component maps can be examined independently, the complete operator <math>\mathsf{W}\!</math> can be separated accordingly into two components, in the form <math>\mathsf{W} = (\boldsymbol\varepsilon, \mathrm{W}).\!</math> Given a fixed context of source and target universes, <math>\boldsymbol\varepsilon\!</math> is always the same type of operator, a multiple component version of the tacit extension operators that were described earlier. In this context <math>\boldsymbol\varepsilon\!</math> has the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{array}{lccccc}<br />
\text{Concrete type}<br />
& \boldsymbol\varepsilon<br />
& : &<br />
( U^\bullet \to X^\bullet )<br />
& \to &<br />
( \mathrm{E}U^\bullet \to X^\bullet )<br />
\\[10pt]<br />
\text{Abstract type}<br />
& \boldsymbol\varepsilon<br />
& : &<br />
( [\mathbb{B}^n] \to [\mathbb{B}^k] )<br />
& \to &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k] )<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
On the other hand, the operator <math>\mathrm{W}\!</math> is specific to each <math>\mathsf{W}.\!</math> In this context <math>\mathrm{W}\!</math> always has the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{array}{lccccc}<br />
\text{Concrete type}<br />
& W<br />
& : &<br />
( U^\bullet \to X^\bullet )<br />
& \to &<br />
( \mathrm{E}U^\bullet \to \mathrm{d}X^\bullet )<br />
\\[10pt]<br />
\text{Abstract type}<br />
& W<br />
& : &<br />
( [\mathbb{B}^n] \to [\mathbb{B}^k] )<br />
& \to &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{D}^k] )<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
In the types just assigned to <math>\boldsymbol\varepsilon\!</math> and <math>\mathrm{W}\!</math> and by implication to their results <math>\boldsymbol\varepsilon F\!</math> and <math>\mathrm{W}F,\!</math> we have listed the most restrictive ranges defined for them rather than the more expansive target spaces that subsume these ranges. When there is need to recognize both, we may use type indications like the following:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\boldsymbol\varepsilon F<br />
& : &<br />
( \mathrm{E}U^\bullet \to X^\bullet \subseteq \mathrm{E}X^\bullet )<br />
& \cong &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k] \subseteq [\mathbb{B}^k \times \mathbb{D}^k] )<br />
\\[10pt]<br />
WF<br />
& : &<br />
( \mathrm{E}U^\bullet \to \mathrm{d}X^\bullet \subseteq \mathrm{E}X^\bullet )<br />
& \cong &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{D}^k] \subseteq [\mathbb{B}^k \times \mathbb{D}^k] )<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Hopefully, though, a general appreciation of these subsumptions will prevent us from having to make such declarations more often than absolutely necessary.<br />
<br />
In giving names to these operators we try to preserve as much of the traditional nomenclature and as many of the classical associations as possible. The chief difficulty in doing this is occasioned by the distinction between the &ldquo;sans&nbsp;serif&rdquo; operators <math>\mathsf{W}\!</math> and their &ldquo;serified&rdquo; components <math>\mathrm{W},\!</math> which forces us to find two distinct but parallel sets of terminology. Here is a plan to that purpose. First, the component operators <math>\mathrm{W}\!</math> are named by analogy with the corresponding operators in the classical difference calculus. Next, the complete operators <math>\mathsf{W} = (\boldsymbol\varepsilon, \mathrm{W})</math> are assigned titles according to their roles in a geometric or trigonometric allegory, if only to ensure that the tangent functor, that belongs to this family and whose exposition we are still working toward, comes out fit with its customary name. Finally, the operator results <math>\mathsf{W}F\!</math> and <math>\mathrm{W}F\!</math> can be fixed in our frame of reference by tethering the operative adjective for <math>\mathsf{W}\!</math> or <math>\mathrm{W}\!</math> to the anchoring epithet &ldquo;map&rdquo;, in conformity with an already standard practice.<br />
<br />
=====The Secant Operator : '''E'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Mr. Peirce, after pointing out that our beliefs are really rules for action, said that, to develop a thought's meaning, we need only determine what conduct it is fitted to produce: that conduct is for us its sole significance.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 46]<br />
|}<br />
<br />
Figures&nbsp;33-i and 33-ii depict two stages in the form of analysis that will be applied to transformations throughout the remainder of this study. From now on our interest is staked on an operator denoted <math>{}^{\backprime\backprime} \mathsf{E} {}^{\prime\prime},\!</math> which receives the principal investment of analytic attention, and on the constituent parts of <math>\mathsf{E},\!</math> which derive their shares of significance as developed by the analysis. In the sequel, we refer to <math>\mathsf{E}\!</math> as the ''secant operator'', taking it for granted that a context has been chosen that defines its type. The secant operator has the component description <math>\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}),\!</math> and its active ingredient <math>\mathrm{E}\!</math> is known as the ''enlargement operator''. (Here, we name <math>\mathrm{E}\!</math> after the literal ancestor of the shift operator in the calculus of finite differences, defined so that <math>\mathrm{E}f(x) = f(x+1)\!</math> for any suitable function <math>f,\!</math> though of course the logical analogue that we take up here must have a rather different definition.)<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
U% $E$ $E$U% $E$U% $E$U%<br />
o------------------>o============o============o<br />
| | | |<br />
| | | |<br />
| | | |<br />
| | | |<br />
F | | $E$F = | $d$^0.F + | $r$^0.F<br />
| | | |<br />
| | | |<br />
| | | |<br />
v v v v<br />
o------------------>o============o============o<br />
X% $E$ $E$X% $E$X% $E$X%<br />
<br />
Figure 33-i. Analytic Diagram (1)<br />
</pre><br />
|}<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
U% $E$ $E$U% $E$U% $E$U% $E$U%<br />
o------------------>o============o============o============o<br />
| | | | |<br />
| | | | |<br />
| | | | |<br />
| | | | |<br />
F | | $E$F = | $d$^0.F + | $d$^1.F + | $r$^1.F<br />
| | | | |<br />
| | | | |<br />
| | | | |<br />
v v v v v<br />
o------------------>o============o============o============o<br />
X% $E$ $E$X% $E$X% $E$X% $E$X%<br />
<br />
Figure 33-ii. Analytic Diagram (2)<br />
</pre><br />
|}<br />
<br />
In its action on universes <math>\mathsf{E}\!</math> yields the same result as <math>\mathrm{E},\!</math> a fact that can be expressed in equational form by writing <math>\mathsf{E}U^\bullet = \mathrm{E}U^\bullet\!</math> for any universe <math>U^\bullet.\!</math> Notice that the extended universes across the top and bottom of the diagram are indicated to be strictly identical, rather than requiring a corresponding decomposition for them. In a certain sense, the functional parts of <math>\mathsf{E}F\!</math> are partitioned into separate contexts that have to be re-integrated again, but the best image to use is that of making transparent copies of each universe and then overlapping their functional contents once more at the conclusion of the analysis, as suggested by the graphic conventions that are used at the top of Figure&nbsp;30.<br />
<br />
Acting on a transformation <math>F\!</math> from universe <math>U^\bullet\!</math> to universe <math>X^\bullet,\!</math> the operator <math>\mathsf{E}\!</math> determines a transformation <math>\mathsf{E}F\!</math> from <math>\mathsf{E}U^\bullet\!</math> to <math>\mathsf{E}X^\bullet.\!</math> The map <math>\mathsf{E}F\!</math> forms the main body of evidence to be investigated in performing a differential analysis of <math>F.\!</math> Because we shall frequently be focusing on small pieces of this map for considerable lengths of time, and consequently lose sight of the &ldquo;big picture&rdquo;, it is critically important to emphasize that the map <math>\mathsf{E}F\!</math> is a transformation that determines a relation from one extended universe into another. This means that we should not be satisfied with our understanding of a transformation <math>F\!</math> until we can lay out the full &ldquo;parts diagram&rdquo; of <math>\mathsf{E}F\!</math> along the lines of the generic frame in Figure&nbsp;30.<br />
<br />
Working within the confines of propositional calculus, it is possible to give an elementary definition of <math>\mathsf{E}F\!</math> by means of a system of propositional equations, as we now describe.<br />
<br />
Given a transformation<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>F = (F_1, \ldots, F_k) : \mathbb{B}^n \to \mathbb{B}^k\!</math><br />
|}<br />
<br />
of concrete type<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>F : [u_1, \ldots, u_n] \to [x_1, \ldots, x_k],\!</math><br />
|}<br />
<br />
the transformation<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\mathsf{E}F = (F_1, \ldots, F_k, \mathrm{E}F_1, \ldots, \mathrm{E}F_k) : \mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B}^k \times \mathbb{D}^k\!</math><br />
|}<br />
<br />
of concrete type<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\mathsf{E}F : [u_1, \dots, u_n, \mathrm{d}u_1, \dots, \mathrm{d}u_n] \to [x_1, \ldots, x_k, \mathrm{d}x_1, \ldots, \mathrm{d}x_k]\!</math><br />
|}<br />
<br />
is defined by means of the following system of logical equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\\[16pt]<br />
\mathrm{d}x_1<br />
& = & \mathrm{E}F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1 + \mathrm{d}u_1, \ldots, u_n + \mathrm{d}u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
\mathrm{d}x_k<br />
& = & \mathrm{E}F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1 + \mathrm{d}u_1, \ldots, u_n + \mathrm{d}u_n)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
It is important to note that this system of equations can be read as a conjunction of equational propositions, in effect, as a single proposition in the universe of discourse generated by all the named variables. Specifically, this is the universe of discourse over <math>2(n+k)\!</math> variables denoted by:<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
\mathrm{E}[\mathcal{U} \cup \mathcal{X}]<br />
& = &<br />
[u_1, \ldots, u_n, ~ x_1, \ldots, x_k, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n, ~ \mathrm{d}x_1, \ldots, \mathrm{d}x_k].<br />
\end{matrix}</math><br />
|}<br />
<br />
In this light, it should be clear that the system of equations defining <math>\mathsf{E}F\!</math> embodies, in a higher rank and differentially extended version, an analogy with the process of thematization that we treated earlier for propositions of type <math>F : \mathbb{B}^n \to \mathbb{B}.\!</math><br />
<br />
The entire collection of constraints that is represented in the above system of equations may be abbreviated by writing <math>\mathsf{E}F = (\boldsymbol\varepsilon F, \mathrm{E}F),\!</math> for any map <math>F.\!</math> This is tantamount to regarding <math>\mathsf{E}\!</math> as a complex operator, <math>\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}),\!</math> with a form of application that distributes each component of the operator to work on each component of the operand, as follows:<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
\mathsf{E}F<br />
& = &<br />
(\boldsymbol\varepsilon, \mathrm{E})F<br />
& = &<br />
(\boldsymbol\varepsilon F, \mathrm{E}F)<br />
& = &<br />
(\boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k, ~ \mathrm{E}F_1, \ldots, \mathrm{E}F_k).<br />
\end{matrix}</math><br />
|}<br />
<br />
Quite a lot of &ldquo;thematic infrastructure&rdquo; or interpretive information is being swept under the rug in the use of such abbreviations. When confusion arises about the meaning of such constructions, one always has recourse to the defining system of equations, in its totality a purely propositional expression. This means that the parenthesized argument lists, that were used in this context to build an image of multi-component transformations, should not be expected to determine a well-defined product in themselves but only to serve as reminders of the prior thematic decisions (choices of variable names, etc.) that have to be made in order to determine one. Accordingly, the argument list notation can be regarded as a kind of ''thematic frame'', an interpretive storage device that preserves the proper associations of concrete logical features between the extended universes at the source and target of <math>\mathsf{E}F.\!</math><br />
<br />
The generic notations <math>\mathsf{d}^0\!F, \mathsf{d}^1\!F, \ldots, \mathsf{d}^m\!F\!</math> in Figure&nbsp;33 refer to the increasing orders of differentials that are extracted in the course of analyzing <math>F.\!</math> When the analysis is halted at a partial stage of development, notations like <math>\mathsf{r}^0\!F, \mathsf{r}^1\!F, \ldots, \mathsf{r}^m\!F\!</math> may be used to summarize the contributions to <math>\mathsf{E}F\!</math> that remain to be analyzed. The Figure illustrates a convention that makes <math>\mathsf{r}^m\!F,\!</math> in effect, the sum of all differentials of order strictly greater than <math>m.\!</math><br />
<br />
We next discuss the operators that figure into this form of analysis, describing their effects on transformations. In simplified or specialized contexts these operators tend to take on a variety of different names and notations, some of whose number we introduce along the way.<br />
<br />
=====The Radius Operator : '''e'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
And the tangible fact at the root of all our thought-distinctions, however subtle, is that there is no one of them so fine as to consist in anything but a possible difference of practice.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 46]<br />
|}<br />
<br />
The operator identified as <math>\mathrm{d}^0\!</math> in the analytic diagram (Figure&nbsp;33) has the sole purpose of creating a proxy for <math>F\!</math> in the appropriately extended context. Construed in terms of its broadest components, <math>\mathrm{d}^0\!</math> is equivalent to the doubly tacit extension operator <math>(\boldsymbol\varepsilon, \boldsymbol\varepsilon),\!</math> in recognition of which let us redub it as <math>{}^{\backprime\backprime} \mathsf{e} {}^{\prime\prime}.\!</math> Pursuing a geometric analogy, we may refer to <math>\mathsf{e} =(\boldsymbol\varepsilon, \boldsymbol\varepsilon) = \mathrm{d}^0\!</math> as the ''radius operator''. The operation intended by all of these forms is defined by the following equation:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathsf{e}F<br />
& = &<br />
(\boldsymbol\varepsilon, \boldsymbol\varepsilon)F<br />
\\[4pt]<br />
& = &<br />
(\boldsymbol\varepsilon F, ~ \boldsymbol\varepsilon F)<br />
\\[4pt]<br />
& = &<br />
(\boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k, ~ \boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k).<br />
\end{array}</math><br />
|}<br />
<br />
which is tantamount to the system of equations below.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\\[16pt]<br />
\mathrm{d}x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
\mathrm{d}x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
=====The Phantom of the Operators : '''&eta;'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
I was wondering what the reason could be, when I myself raised my head and everything within me seemed drawn towards the Unseen, ''which was playing the most perfect music''!<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 81]<br />
|}<br />
<br />
We now describe an operator whose persistent but elusive action behind the scenes, whose slightly twisted and ambivalent character, and whose fugitive disposition, caught somewhere in flight between the arrantly negative and the positive but errant intent, has cost us some painstaking trouble to detect. In the end we shall place it among the other extensions and projections, as a shade among shadows, of muted tones and motley hue, that adumbrates its own thematic frame and paradoxically lights the way toward a whole new spectrum of values.<br />
<br />
Given a transformation <math>F : [u_1, \ldots, u_n] \to [x_1, \dots, x_k],\!</math> we often have call to consider a family of related transformations, all having the form:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>F^\dagger : [u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n] \to [\mathrm{d}x_1, \dots, \mathrm{d}x_k].\!</math><br />
|}<br />
<br />
The operator <math>\eta</math> is introduced to deal with the simplest one of these maps:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\eta F : [u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n] \to [\mathrm{d}x_1, \ldots \mathrm{d}x_k],\!</math><br />
|}<br />
<br />
which is defined by the following equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
\mathrm{d}x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
In effect, the operator <math>\eta\!</math> is nothing but the stand-alone version of a procedure that is otherwise invoked subordinate to the work of the radius operator <math>\mathsf{e}.\!</math> Operating independently, <math>\eta\!</math> achieves precisely the same results that the second <math>\boldsymbol\varepsilon\!</math> in <math>(\boldsymbol\varepsilon, \boldsymbol\varepsilon)\!</math> accomplishes by working within the context of its ordered pair thematic frame. From this point on, because the use of <math>\boldsymbol\varepsilon\!</math> and <math>\eta\!</math> in this setting combines the aims of both the tacit and the thematic extensions, and because <math>\eta\!</math> reflects in regard to <math>\boldsymbol\varepsilon\!</math> little more than the application of a differential twist, a mere turn of phrase, we refer to <math>\eta\!</math> as the ''trope extension'' operator.<br />
<br />
=====The Chord Operator : '''D'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
What difference would it practically make to any one if this notion rather than that notion were true? If no practical difference whatever can be traced, then the alternatives mean practically the same thing, and all dispute is idle.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 45]<br />
|}<br />
<br />
Next we discuss an operator that is always immanent in this form of analysis, and remains implicitly present in the entire proceeding. It may appear once as a record: a relic or revenant that reprises the reminders of an earlier stage of development. Or it may appear always as a resource: a reserve or redoubt that caches in advance an echo of what remains to be played out, cleared up, and requited in full at a future stage. And all of this remains true whether or not we recall the key at any time, and whether or not the subtending theme is recited explicitly at any stage of play.<br />
<br />
This is the operator that is referred to as <math>\mathsf{r}^0\!</math> in the initial stage of analysis (Figure&nbsp;33-i) and that is expanded as <math>\mathsf{d}^1 + \mathsf{r}^1\!</math> in the subsequent step (Figure&nbsp;33-ii). In congruence, but not quite harmony with our allusions of analogy that are not quite geometry, we call this the ''chord operator'' and denote it <math>\mathsf{D}.\!</math> In the more casual terms that are here introduced, <math>\mathsf{D}</math> is defined as the remainder of <math>\mathsf{E}\!</math> and <math>\mathsf{e}\!</math> and it assigns a due measure to each undertone of accord or discord that is struck between the note of enterprise <math>\mathsf{E}\!</math> and the bar of exigency <math>\mathsf{e}.\!</math><br />
<br />
The tension between these counterposed notions, in balance transient but regular in stridence, may be refracted along familiar lines, though never by any such fraction resolved. In this style we write <math>\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}),\!</math> calling <math>\mathrm{D}\!</math> the ''difference operator'' and noting that it plays a role in this realm of mutable and diverse discourse that is analogous to the part taken by the discrete difference operator in the ordinary difference calculus. Finally, we should note that the chord <math>\mathsf{D}\!</math> is not one that need be lost at any stage of development. At the <math>m^\text{th}\!</math> stage of play it can always be reconstituted in the following form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathsf{D}<br />
& = & \mathsf{E} - \mathsf{e}<br />
\\[6pt]<br />
& = & \mathsf{r}^0<br />
\\[6pt]<br />
& = & \mathsf{d}^1 + \mathsf{r}^1<br />
\\[6pt]<br />
& = & \mathsf{d}^1 + \ldots + \mathsf{d}^m + \mathsf{r}^m<br />
\\[6pt]<br />
& = & \displaystyle \sum_{i=1}^m \mathsf{d}^i + \mathsf{r}^m<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====The Tangent Operator : '''T'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
They take part in scenes of whose significance they have no inkling. They are merely tangent to curves of history the beginnings and ends and forms of which pass wholly beyond their ken. So we are tangent to the wider life of things.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 300]<br />
|}<br />
<br />
The operator tagged as <math>\mathsf{d}^1\!</math> in the analytic diagram (Figure&nbsp;33) is called the ''tangent operator'' and is usually denoted in this text as <math>\mathsf{d}\!</math> or <math>\mathsf{T}.\!</math> Because it has the properties required to qualify as a functor, namely, preserving the identity element of the composition operation and the articulated form of every composition of transformations, it also earns the title of a ''tangent functor''. According to the custom adopted here, we dissect it as <math>\mathsf{T} = \mathsf{d} = (\boldsymbol\varepsilon, \mathrm{d}),\!</math> where <math>\mathrm{d}\!</math> is the operator that yields the first order differential <math>\mathrm{d}F\!</math> when applied to a transformation <math>F,\!</math> and whose name is legion.<br />
<br />
Figure&nbsp;34 illustrates a stage of analysis where we ignore everything but the tangent functor <math>\mathsf{T}\!</math> and attend to it chiefly as it bears on the first order differential <math>\mathrm{d}F\!</math> in the analytic expansion of <math>F.\!</math> In this situation we often refer to the extended universes <math>\mathrm{E}U^\bullet\!</math> and <math>\mathrm{E}X^\bullet\!</math> under the equivalent designations <math>\mathsf{T}U^\bullet\!</math> and <math>\mathsf{T}X^\bullet,\!</math> respectively. The purpose of the tangent functor <math>\mathsf{T}\!</math> is to extract the tangent map <math>\mathsf{T}F\!</math> at each point of <math>U^\bullet,\!</math> and the tangent map <math>\mathsf{T}F = (\boldsymbol\varepsilon, \mathrm{d})F\!</math> tells us not only what the transformation <math>F\!</math> is doing at each point of the universe <math>U^\bullet\!</math> but also what <math>F\!</math> is doing to states in the neighborhood of that point, approximately, linearly, and relatively speaking.<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
U% $T$ $T$U% $T$U%<br />
o------------------>o============o<br />
| | |<br />
| | |<br />
| | |<br />
| | |<br />
F | | $T$F = | <!e!, d> F<br />
| | |<br />
| | |<br />
| | |<br />
v v v<br />
o------------------>o============o<br />
X% $T$ $T$X% $T$X%<br />
<br />
Figure 34. Tangent Functor Diagram<br />
</pre><br />
|}<br />
<br />
* '''NB.''' There is one aspect of the preceding construction that remains especially problematic. Why did we define the operators <math>\mathrm{W}\!</math> in <math>\{ \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}\!</math> so that the ranges of their resulting maps all fall within the realms of differential quality, even fabricating a variant of the tacit extension operator to have that character? Clearly, not all of the operator maps <math>\mathrm{W}F\!</math> have equally good reasons for placing their values in differential stocks. The reason for it appears to be that, without doing this, we cannot justify the comparison and combination of their functional values in the various analytic steps. By default, only those values in the same functional component can be brought into algebraic modes of interaction. Up till now the only mechanism provided for their broader association has been a purely logical one, their common placement in a target universe of discourse, but the task of converting this logical circumstance into algebraic forms of application has not yet been taken up.<br />
<br />
===Transformations of Type '''B'''<sup>2</sup> &rarr; '''B'''<sup>1</sup>===<br />
<br />
To study the effects of these analytic operators in the simplest possible setting, let us revert to a still more primitive case. Consider the singular proposition <math>J(u, v)= u\!\cdot\!v,\!</math> regarded either as the functional product of the maps <math>u\!</math> and <math>v\!</math> or as the logical conjunction of the features <math>u\!</math> and <math>v,\!</math> a map whose fiber of truth <math>J^{-1}(1)\!</math> picks out the single cell of that logical description in the universe of discourse <math>U^\bullet.\!</math> Thus <math>J,\!</math> or <math>u\!\cdot\!v,\!</math> may be treated as another name for the point whose coordinates are <math>(1, 1)\!</math> in <math>U^\bullet.\!</math><br />
<br />
====Analytic Expansion of Conjunction====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
<p>In her sufferings she read a great deal and discovered that she had lost something, the possession of which she had previously not been much aware of: a&nbsp;soul.</p><br />
<br />
<p>What is that? It is easily defined negatively: it is simply what curls up and hides when there is any mention of algebraic series.</p><br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 118]<br />
|}<br />
<br />
Figure&nbsp;35 pictures the form of conjunction <math>J : \mathbb{B}^2 \to \mathbb{B}\!</math> as a transformation from the <math>2\!</math>-dimensional universe <math>[u, v]\!</math> to the <math>1\!</math>-dimensional universe <math>[x].\!</math> This is a subtle but significant change of viewpoint on the proposition, attaching an arbitrary but concrete quality to its functional value. Using the language introduced earlier, we can express this change by saying that the proposition <math>J : \langle u, v \rangle \to \mathbb{B}\!</math> is being recast into the thematized role of a transformation <math>J : [u, v] \to [x],\!</math> where the new variable <math>x\!</math> takes the part of a thematic variable <math>\check{J}.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 35 -- A Conjunction Viewed as a Transformation.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 35.} ~~ \text{Conjunction as Transformation}\!</math><br />
|}<br />
<br />
=====Tacit Extension of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
I teach straying from me, yet who can stray from me?<br><br />
I follow you whoever you are from the present hour;<br><br />
My words itch at your ears till you understand them.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 83]<br />
|}<br />
<br />
Earlier we defined the tacit extension operators <math>\boldsymbol\varepsilon : X^\bullet \to Y^\bullet\!</math> as maps embedding each proposition of a given universe <math>X^\bullet~\!</math> in a more generously given universe <math>Y^\bullet \supset X^\bullet.\!</math> Of immediate interest are the tacit extensions <math>\boldsymbol\varepsilon : U^\bullet \to \mathrm{E}U^\bullet,\!</math> that locate each proposition of <math>U^\bullet\!</math> in the enlarged context of <math>\mathrm{E}U^\bullet.\!</math> In its application to the propositional conjunction <math>J = u\!\cdot\!v</math> in <math>[u, v],\!</math> the tacit extension operator <math>\boldsymbol\varepsilon\!</math> yields the proposition <math>\boldsymbol\varepsilon J\!</math> in <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v].\!</math> The extended proposition <math>\boldsymbol\varepsilon J\!</math> may be computed according to the scheme in Table&nbsp;36, in effect doing nothing more that conjoining a tautology of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> to <math>J\!</math> in <math>U^\bullet.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 36.} ~~ \text{Computation of}~ \boldsymbol\varepsilon J\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\boldsymbol\varepsilon J & = & J {}_{^\langle} u, v {}_{^\rangle}<br />
\\[4pt]<br />
& = & u \cdot v<br />
\\[4pt]<br />
& = & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{ } \mathrm{d}v \texttt{ }<br />
& + & u \cdot v \cdot \texttt{ } \mathrm{d}u \texttt{ } \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \cdot v \cdot \texttt{ } \mathrm{d}u \texttt{ } \cdot \texttt{ } \mathrm{d}v \texttt{ }<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{array}{*{4}{l}}<br />
\boldsymbol\varepsilon J<br />
& = && u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & u \cdot v \cdot \texttt{~} \mathrm{d}u \texttt{~} \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & u \cdot v \cdot \texttt{~} \mathrm{d}u \texttt{~} \cdot \texttt{~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
The lower portion of the Table contains the dispositional features of <math>\boldsymbol\varepsilon J\!</math> arranged in such a way that the variety of ordinary features spreads across the rows and the variety of differential features runs through the columns. This organization serves to facilitate pattern matching in the remainder of our computations. Again, the tacit extension is usually so trivial a concern that we do not always bother to make an explicit note of it, taking it for granted that any function <math>F\!</math> being employed in a differential context is equivalent to <math>\boldsymbol\varepsilon F\!</math> for a suitable <math>\boldsymbol\varepsilon.\!</math><br />
<br />
Figures&nbsp;37-a through 37-d present several pictures of the proposition <math>J\!</math> and its tacit extension <math>\boldsymbol\varepsilon J.\!</math> Notice in these Figures how <math>\boldsymbol\varepsilon J\!</math> in <math>\mathrm{E}U^\bullet\!</math> visibly extends <math>J\!</math> in <math>U^\bullet\!</math> by annexing to the indicated cells of <math>J\!</math> all the arcs that exit from or flow out of them. In effect, this extension attaches to these cells all the dispositions that spring from them, in other words, it attributes to these cells all the conceivable changes that are their issue.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-a -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-a.} ~~ \text{Tacit Extension of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-b -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-b.} ~~ \text{Tacit Extension of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-c -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-c.} ~~ \text{Tacit Extension of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-d -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-d.} ~~ \text{Tacit Extension of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
The computational scheme shown in Table&nbsp;36 treated <math>J\!</math> as a proposition in <math>U^\bullet\!</math> and formed <math>\boldsymbol\varepsilon J\!</math> as a proposition in <math>\mathrm{E}U^\bullet.\!</math> When <math>J\!</math> is regarded as a mapping <math>J : U^\bullet \to X^\bullet\!</math> then <math>\boldsymbol\varepsilon J\!</math> must be obtained as a mapping <math>\boldsymbol\varepsilon J : \mathrm{E}U^\bullet \to X^\bullet.\!</math> By default, the tacit extension of the map <math>J : [u, v] \to [x]\!</math> is naturally taken to be a particular map,<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\boldsymbol\varepsilon J : [u, v, \mathrm{d}u, \mathrm{d}v] \to [x] \subseteq [x, \mathrm{d}x],\!</math><br />
|}<br />
<br />
namely, the one that looks like <math>J\!</math> when painted in the frame of the extended source universe and that takes the same thematic variable in the extended target universe as the one that <math>J\!</math> already takes.<br />
<br />
But the choice of a particular thematic variable, for example <math>x\!</math> for <math>\check{J},\!</math> is a shade more arbitrary than the choice of original variable names <math>\{ u, v \},\!</math> so the map we are calling the ''trope extension'',<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\eta J : [u, v, \mathrm{d}u, \mathrm{d}v] \to [\mathrm{d}x] \subseteq [x, \mathrm{d}x],\!</math><br />
|}<br />
<br />
since it looks just the same as <math>\boldsymbol\varepsilon J\!</math> in the way its fibers paint the source domain, belongs just as fully to the family of tacit extensions, generically considered.<br />
<br />
These considerations have the practical consequence that all of our computations and illustrations of <math>\boldsymbol\varepsilon J\!</math> perform the double duty of capturing <math>\eta J\!</math> as well. In other words, we are saved the work of carrying out calculations and drawing figures for the trope extension <math>\eta J,\!</math> because it would be identical to the work already done for <math>\boldsymbol\varepsilon J.\!</math> Since the computations given for <math>\boldsymbol\varepsilon J\!</math> are expressed solely in terms of the variables <math>\{ u, v, \mathrm{d}u, \mathrm{d}v \},\!</math> they work equally well for finding <math>\eta J.\!</math> Further, since each of the above Figures shows only how the level sets of <math>\boldsymbol\varepsilon J\!</math> partition the extended source universe <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v],\!</math> all of them serve equally well as portraits of <math>\eta J.\!</math><br />
<br />
=====Enlargement Map of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
No one could have established the existence of any details that might not just as well have existed in earlier times too; but all the relations between things had shifted slightly. Ideas that had once been of lean account grew fat.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 62]<br />
|}<br />
<br />
The enlargement map <math>\mathrm{E}J\!</math> is computed from the proposition <math>J\!</math> by making a particular class of formal substitutions for its variables, in this case <math>u + \mathrm{d}u\!</math> for <math>u\!</math> and <math>v + \mathrm{d}v\!</math> for <math>v,\!</math> and afterwards expanding the result in whatever way is found convenient.<br />
<br />
Table&nbsp;38 shows a typical scheme of computation, following a systematic method of exploiting boolean expansions over selected variables and ultimately developing <math>\mathrm{E}J\!</math> over the cells of <math>[u, v].\!</math> The critical step of this procedure uses the facts that <math>\texttt{(} 0, x \texttt{)} = 0 + x = x\!</math> and <math>\texttt{(} 1, x \texttt{)} = 1 + x = \texttt{(} x \texttt{)}\!</math> for any boolean variable <math>x.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 38.} ~~ \text{Computation of}~ \mathrm{E}J ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{E}J & = & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}<br />
\\[4pt]<br />
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
& = & \texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot J_{(1 + \mathrm{d}u, 1 + \mathrm{d}v)}<br />
& + & \texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot J_{(1 + \mathrm{d}u, \mathrm{d}v)}<br />
& + & \texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot J_{(\mathrm{d}u, 1 + \mathrm{d}v)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot J_{(\mathrm{d}u, \mathrm{d}v)}<br />
\\[4pt]<br />
& = &<br />
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot J_{(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})}<br />
& + &<br />
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot J_{(\texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v)}<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot J_{(\mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)})}<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot J_{(\mathrm{d}u, \mathrm{d}v)}<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{E}J<br />
& = &<br />
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&& + &<br />
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{ } \mathrm{d}v \texttt{ }<br />
\\[4pt]<br />
&&&&& + &<br />
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot \texttt{ } \mathrm{d}u \texttt{ } \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&&&&&& + &<br />
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \texttt{ } \mathrm{d}u \texttt{ }~\texttt{ } \mathrm{d}v \texttt{ }<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;39 exhibits another method that happens to work quickly in this particular case, using distributive laws to multiply things out in an algebraic manner, arranging the notations of feature and fluxion according to a scale of simple character and degree. Proceeding this way leads through an intermediate step which, in chiming the changes of ordinary calculus, should take on a familiar ring. Consequential properties of exclusive disjunction then carry us on to the concluding line.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 39.} ~~ \text{Computation of}~ \mathrm{E}J ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{c}}<br />
\mathrm{E}J<br />
& = & (u + \mathrm{d}u) \cdot (v + \mathrm{d}v)<br />
\\[6pt]<br />
& = & u \cdot v<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = &<br />
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{ } \mathrm{d}v \texttt{ }<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot \texttt{ } \mathrm{d}u \texttt{ } \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \texttt{ } \mathrm{d}u \texttt{ }~\texttt{ } \mathrm{d}v \texttt{ }<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;40-a through 40-d present several views of the enlarged proposition <math>\mathrm{E}J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-a -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-a.} ~~ \text{Enlargement of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-b -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-b.} ~~ \text{Enlargement of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-c -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-c.} ~~ \text{Enlargement of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-d -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-d.} ~~ \text{Enlargement of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
An intuitive reading of the proposition <math>\mathrm{E}J\!</math> becomes available at this point. Recall that propositions in the extended universe <math>\mathrm{E}U^\bullet\!</math> express the ''dispositions'' of a system and the constraints that are placed on them. In other words, a differential proposition in <math>\mathrm{E}U^\bullet\!</math> can be read as referring to various changes that a system might undergo in and from its various states. In particular, we can understand <math>\mathrm{E}J\!</math> as a statement that tells us what changes need to be made with regard to each state in the universe of discourse in order to reach the ''truth'' of <math>J,\!</math> that is, the region of the universe where <math>J\!</math> is true. This interpretation is visibly clear in the Figures above and appeals to the imagination in a satisfying way but it has the added benefit of giving fresh meaning to the original name of the shift operator <math>\mathrm{E}.\!</math> Namely, <math>\mathrm{E}J\!</math> can be read as a proposition that ''enlarges'' on the meaning of <math>J,\!</math> in the sense of explaining its practical bearings and clarifying what it means in terms of actions and effects &mdash; the available options for differential action and the consequential effects that result from each choice.<br />
<br />
Read this way, the enlargement <math>\mathrm{E}J\!</math> has strong ties to the normal use of <math>J,\!</math> no matter whether it is understood as a proposition or a function, namely, to act as a figurative device for indicating the models of <math>J,\!</math> in effect, pointing to the interpretive elements in its fiber of truth <math>J^{-1}(1).\!</math> It is this kind of &ldquo;use&rdquo; that is often contrasted with the &ldquo;mention&rdquo; of a proposition, and thereby hangs a tale.<br />
<br />
=====Digression : Reflection on Use and Mention=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Reflection is turning a topic over in various aspects and in various lights so that nothing significant about it shall be overlooked &mdash; almost as one might turn a stone over to see what its hidden side is like or what is covered by it.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 57]<br />
|}<br />
<br />
The contrast drawn in logic between the ''use'' and the ''mention'' of a proposition corresponds to the difference that we observe in functional terms between using <math>{}^{\backprime\backprime} J \, {}^{\prime\prime}\!</math> to indicate the region <math>J^{-1}(1)\!</math> and using <math>{}^{\backprime\backprime} J \, {}^{\prime\prime}\!</math> to indicate the function <math>J.\!</math> You may think that one of these uses ought to be proscribed, and logicians are quick to prescribe against their confusion. But there seems to be no likelihood in practice that their interactions can be avoided. If the name <math>{}^{\backprime\backprime} J \, {}^{\prime\prime}\!</math> is used as a sign of the function <math>J,\!</math> and if the function <math>J\!</math> has its use in signifying something else, as would constantly be the case when some future theory of signs has given a functional meaning to every sign whatsoever, then is not <math>J,\!</math> by transitivity a sign of the thing itself? There are, of course, two answers to this question. Not every act of signifying or referring need be transitive. Not every warrant or guarantee or certificate is automatically transferable, indeed, not many. Not every feature of a feature is a feature of the featuree. Otherwise, if a buffalo is white, and white is a color, then a buffalo would ''be'' a color.<br />
<br />
The logical or pragmatic distinction between use and mention is cogent and necessary, and so is the analogous functional distinction between determining a value and determining what determines that value, but so are the normal techniques that we use to make these distinctions apply flexibly in practice. The way that the hue and cry about use and mention is raised in logical discussions, you might be led to think that this single dimension of choices embraces the only kinds of use worth mentioning and the only kinds of mention worth using. It will constitute the expeditionary and taxonomic tasks of that future theory of signs to explore and to classify the many other constellations and dimensions of use and mention that are yet to be opened up by the generative potential of full-fledged sign relations.<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
The well-known capacity that thoughts have &mdash; as doctors have discovered &mdash; for dissolving and dispersing those hard lumps of deep, ingrowing, morbidly entangled conflict that arise out of gloomy regions of the self probably rests on nothing other than their social and worldly nature, which links the individual being with other people and things; but unfortunately what gives them their power of healing seems to be the same as what diminishes the quality of personal experience in them.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 130]<br />
|}<br />
<br />
=====Difference Map of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
&ldquo;It doesn't matter what one does,&rdquo; the Man Without Qualities said to himself, shrugging his shoulders. &ldquo;In a tangle of forces like this it doesn't make a scrap of difference.&rdquo; He turned away like a man who has learned renunciation, almost indeed like a sick man who shrinks from any intensity of contact. And then, striding through his adjacent dressing-room, he passed a punching-ball that hung there; he gave it a blow far swifter and harder than is usual in moods of resignation or states of weakness.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 8]<br />
|}<br />
<br />
With the tacit extension map <math>\boldsymbol\varepsilon J\!</math> and the enlargement map <math>\mathrm{E}J\!</math> well in place, the difference map <math>\mathrm{D}J\!</math> can be computed along the lines displayed in Table&nbsp;41, ending up with an expansion of <math>\mathrm{D}J\!</math> over the cells of <math>[u, v].\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 41.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & \mathrm{E}J<br />
& + & \boldsymbol\varepsilon J<br />
\\[6pt]<br />
& = & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}<br />
& + & J_{(u, v)}<br />
\\[6pt]<br />
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \cdot v<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = &<br />
u \cdot v \cdot \qquad 0<br />
\\[6pt]<br />
& + &<br />
u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v<br />
& + &<br />
u \cdot \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v<br />
\\[6pt]<br />
& + &<br />
u \cdot v \cdot \texttt{~} \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
&&& + &<br />
\texttt{(} u \texttt{)} \cdot v \cdot \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
& + &<br />
u \cdot v \cdot \texttt{~} \mathrm{d}u \;\cdot\; \mathrm{d}v \texttt{~}<br />
&&&&& + &<br />
\texttt{(} u \texttt{)} \cdot \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = &<br />
u \cdot v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
u \cdot \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v<br />
& + &<br />
\texttt{(} u \texttt{)} \cdot v \cdot \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)} \cdot \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Alternatively, the difference map <math>\mathrm{D}J\!</math> can be expanded over the cells of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> to arrive at the formulation shown in Table&nbsp;42. The same development would be obtained from the previous Table by collecting terms in an alternate manner, along the rows rather than the columns in the middle portion of the Table.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 42.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & \boldsymbol\varepsilon J<br />
& + & \mathrm{E}J<br />
\\[6pt]<br />
& = & J_{(u, v)}<br />
& + & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}<br />
\\[6pt]<br />
& = & u \cdot v<br />
& + & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
& = & 0<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}J<br />
& = & 0<br />
& + & u \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Even more simply, the same result is reached by matching up the propositional coefficients of <math>\boldsymbol\varepsilon J</math> and <math>\mathrm{E}J\!</math> along the cells of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> and adding the pairs under boolean addition, that is, &ldquo;mod 2&rdquo;, where 1&nbsp;+&nbsp;1&nbsp;=&nbsp;0, as shown in Table&nbsp;43.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 43.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 3)}\!</math><br />
|<br />
<math>\begin{array}{*{5}{l}}<br />
\mathrm{D}J & = & \boldsymbol\varepsilon J & + & \mathrm{E}J<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\boldsymbol\varepsilon J<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ u \,\cdot\, v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~ u \;\cdot\; v \;\cdot\; \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u ~ \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} ~ v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot\, \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & ~~ 0 ~~ \,\cdot\, ~ \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~~ u ~ \,\cdot\, ~~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ ~ v ~~ \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
The difference map <math>\mathrm{D}J\!</math> can also be given a ''dispositional'' interpretation. First, recall that <math>\boldsymbol\varepsilon J\!</math> exhibits the dispositions to change from anywhere in <math>J\!</math> to anywhere at all in the universe of discourse and <math>\mathrm{E}J\!</math> exhibits the dispositions to change from anywhere in the universe to anywhere in <math>J.\!</math> Next, observe that each of these classes of dispositions may be divided in accordance with the case of <math>J\!</math> versus <math>\texttt{(} J \texttt{)}\!</math> that applies to their points of departure and destination, as shown below. Then, since the dispositions corresponding to <math>\boldsymbol\varepsilon J</math> and <math>\mathrm{E}J\!</math> have in common the dispositions to preserve <math>J,\!</math> their symmetric difference <math>\texttt{(} \boldsymbol\varepsilon J, \mathrm{E}J \texttt{)}\!</math> is made up of all the remaining dispositions, which are in fact disposed to cross the boundary of <math>J\!</math> in one direction or the other. In other words, we may conclude that <math>\mathrm{D}J\!</math> expresses the collective disposition to make a definite change with respect to <math>J,\!</math> no matter what value it holds in the current state of affairs.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
\boldsymbol\varepsilon J<br />
& = & \{ \text{Dispositions from}~ J ~\text{to}~ J \}<br />
& + & \{ \text{Dispositions from}~ J ~\text{to}~ \texttt{(} J \texttt{)} \}<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = & \{ \text{Dispositions from}~ J ~\text{to}~ J \}<br />
& + & \{ \text{Dispositions from}~ \texttt{(} J \texttt{)} ~\text{to}~ J \}<br />
\\[6pt]<br />
\mathrm{D}J<br />
& = & \{ \text{Dispositions from}~ J ~\text{to}~ \texttt{(} J \texttt{)} \}<br />
& + & \{ \text{Dispositions from}~ \texttt{(} J \texttt{)} ~\text{to}~ J \}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;44-a through 44-d illustrate the difference proposition <math>\mathrm{D}J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-a -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-a.} ~~ \text{Difference Map of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-b -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-b.} ~~ \text{Difference Map of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-c -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-c.} ~~ \text{Difference Map of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-d -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-d.} ~~ \text{Difference Map of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
=====Differential of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
By deploying discourse throughout a calendar, and by giving a date to each of its elements, one does not obtain a definitive hierarchy of precessions and originalities; this hierarchy is never more than relative to the systems of discourse that it sets out to evaluate.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Archaeology of Knowledge'', [Fou, 143]<br />
|}<br />
<br />
Finally, at long last, the differential proposition <math>\mathrm{d}J\!</math> can be gleaned from the difference proposition <math>\mathrm{D}J\!</math> by ranging over the cells of <math>[u, v]\!</math> and picking out the linear proposition of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> that is &ldquo;closest&rdquo; to the portion of <math>\mathrm{D}J\!</math> that touches on each point. The idea of distance that would give this definition unequivocal sense has been referred to in cautionary quotes, the kind we use to distance ourselves from taking a final position. There are obvious notions of approximation that suggest themselves, but finding one that can be justified as ultimately correct is not as straightforward as it seems.<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
He had drifted into the very heart of the world. From him to the distant beloved was as far as to the next tree.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 144]<br />
|}<br />
<br />
Let us venture a guess as to where these developments might be heading. From the present vantage point it appears that the ultimate answer to the quandary of distances and the question of a fitting measure may be that, rather than having the constitution of an analytic series depend on our familiar notions of approach, proximity, and approximation, it will be found preferable, and perhaps unavoidable, to turn the tables and let the orders of approximation be defined in terms of our favored and operative notions of formal analysis. Only the aftermath of this conversion, if it does converge, could be hoped to prove whether this hortatory form of analysis and the cohort idea of an analytic form &mdash; the limitary concept of a self-corrective process and the coefficient concept of a completable product &mdash; are truly (in practical reality) the more inceptive and persistent of principles and really (for all practical purposes) the more effective and regulative of ideas.<br />
<br />
Awaiting that determination, I proceed with what seems like the obvious course, and compute <math>\mathrm{d}J\!</math> according to the pattern in Table&nbsp;45.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 45.} ~~ \text{Computation of}~ \mathrm{d}J\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}J<br />
& = &<br />
u\!\cdot\!v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
u \, \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \, \mathrm{d}v<br />
& + &<br />
\texttt{(} u \texttt{)} \, v \cdot \mathrm{d}u \, \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \!\cdot\! \mathrm{d}v \texttt{~}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}J<br />
& = &<br />
u\!\cdot\!v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + &<br />
u \, \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + &<br />
\texttt{(} u \texttt{)} \, v \cdot \mathrm{d}u<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;46-a through 46-d illustrate the proposition <math>{\mathrm{d}J},\!</math> rounded out in our usual array of prospects. This proposition of <math>\mathrm{E}U^\bullet\!</math> is what we refer to as the (first order) differential of <math>J,\!</math> and normally regard as ''the'' differential proposition corresponding to <math>J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-a -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-a.} ~~ \text{Differential of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-b -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-b.} ~~ \text{Differential of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-c -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-c.} ~~ \text{Differential of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-d -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-d.} ~~ \text{Differential of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
=====Remainder of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
<p>I bequeath myself to the dirt to grow from the grass I love,<br><br />
If you want me again look for me under your bootsoles.</p><br />
<br />
<p>You will hardly know who I am or what I mean,<br><br />
But I shall be good health to you nevertheless,<br><br />
And filter and fibre your blood.</p><br />
<br />
<p>Failing to fetch me at first keep encouraged,<br><br />
Missing me one place search another,<br><br />
I stop some where waiting for you</p><br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]<br />
|}<br />
<br />
<br><br />
<br />
Let us recapitulate the story so far. We have in effect been carrying out a decomposition of the enlarged proposition <math>\mathrm{E}J\!</math> in a series of stages. First, we considered the equation <math>\mathrm{E}J = \boldsymbol\varepsilon J + \mathrm{D}J,\!</math> which was involved in the definition of <math>\mathrm{D}J\!</math> as the difference <math>\mathrm{E}J - \boldsymbol\varepsilon J.\!</math> Next, we contemplated the equation <math>\mathrm{D}J = \mathrm{d}J + \mathrm{r}J,\!</math> which expresses <math>\mathrm{D}J\!</math> in terms of two components, the differential <math>\mathrm{d}J\!</math> that was just extracted and the residual component <math>\mathrm{r}J = \mathrm{D}J - \mathrm{d}J.~\!</math> This remaining proposition <math>\mathrm{r}J\!</math> can be computed as shown in Table&nbsp;47.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 47.} ~~ \text{Computation of}~ \mathrm{r}J\!</math><br />
|<br />
<math>\begin{array}{*{5}{l}}<br />
\mathrm{r}J & = & \mathrm{D}J & + & \mathrm{d}J<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}J<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{r}J ~<br />
& = & u \!\cdot\! v \cdot ~ \mathrm{d}u \cdot \mathrm{d}v ~ ~ ~ ~ ~<br />
& + & u \texttt{(} v \texttt{)} \cdot \, \mathrm{d}u \cdot \mathrm{d}v \,<br />
& + & \texttt{(} u \texttt{)} v \cdot \, \mathrm{d}u \cdot \mathrm{d}v \,<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \, \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
As it happens, the remainder <math>\mathrm{r}J\!</math> falls under the description of a second order differential <math>\mathrm{r}J = \mathrm{d}^2 J.\!</math> This means that the expansion of <math>\mathrm{E}J\!</math> in the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|<br />
<math>\begin{array}{*{7}{l}}<br />
\mathrm{E}J<br />
& = & \boldsymbol\varepsilon J<br />
& + & \mathrm{D}J<br />
\\[6pt]<br />
& = & \boldsymbol\varepsilon J<br />
& + & \mathrm{d}J<br />
& + & \mathrm{r}J<br />
\\[6pt]<br />
& = & \mathrm{d}^0 J<br />
& + & \mathrm{d}^1 J<br />
& + & \mathrm{d}^2 J<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
which is nothing other than the propositional analogue of a Taylor series, is a decomposition that terminates in a finite number of steps.<br />
<br />
Figures&nbsp;48-a through 48-d illustrate the proposition <math>\mathrm{r}J = \mathrm{d}^2 J,\!</math> which forms the remainder map of <math>J\!</math> and also, in this instance, the second order differential of <math>J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-a -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-a.} ~~ \text{Remainder of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-b -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-b.} ~~ \text{Remainder of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-c -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-c.} ~~ \text{Remainder of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-d -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-d.} ~~ \text{Remainder of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
=====Summary of Conjunction=====<br />
<br />
To establish a convenient reference point for further discussion, Table&nbsp;49 summarizes the operator actions that have been computed for the form of conjunction, as exemplified by the proposition <math>J.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 49.} ~~ \text{Computation Summary for}~ J~\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon J<br />
& = & u \!\cdot\! v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}J<br />
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}J<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{r}J<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Analytic Series : Coordinate Method====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
And if he is told that something ''is'' the way it is, then he thinks: Well, it could probably just as easily be some other way. So the sense of possibility might be defined outright as the capacity to think how everything could &ldquo;just as easily&rdquo; be, and to attach no more importance to what is than to what is not.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 12]<br />
|}<br />
<br />
Table&nbsp;50 exhibits a truth table method for computing the analytic series (or the differential expansion) of a proposition in terms of coordinates.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:70%"<br />
|+ style="height:30px" | <math>\text{Table 50.} ~~ \text{Computation of an Analytic Series in Terms of Coordinates}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:8%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:8%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:8%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}u\!</math><br />
| style="width:8%; border-bottom:1px solid black" | <math>\mathrm{d}v\!</math><br />
| style="width:8%; border-bottom:1px solid black; border-left:1px solid black" | <math>u'\!</math><br />
| style="width:8%; border-bottom:1px solid black" | <math>v'\!</math><br />
| style="width:10%; border-bottom:1px solid black; border-left:4px double black" | <math>\boldsymbol\varepsilon J\!</math><br />
| style="width:10%; border-bottom:1px solid black" | <math>\mathrm{E}J\!</math><br />
| style="width:12%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{D}J\!</math><br />
| style="width:10%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}J\!</math><br />
| style="width:10%; border-bottom:1px solid black" | <math>\mathrm{d}^2\!J\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
The first six columns of the Table, taken as a whole, represent the variables of a construct called the ''contingent universe'' <math>[u, v, \mathrm{d}u, \mathrm{d}v, u', v'],\!</math> or the bundle of ''contingency spaces'' <math>[\mathrm{d}u, \mathrm{d}v, u', v']\!</math> over the universe <math>[u, v].\!</math> Their placement to the left of the double bar indicates that all of them amount to independent variables, but there is a co-dependency among them, as described by the following equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
u' & = & u + \mathrm{d}u & = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\\[8pt]<br />
v' & = & v + \mathrm{d}v & = & \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
These relations correspond to the formal substitutions that are made in defining <math>\mathrm{E}J\!</math> and <math>\mathrm{D}J.\!</math> For now, the whole rigamarole of contingency spaces can be regarded as a technical device for achieving the effect of these substitutions, adapted to a setting where functional compositions and other symbolic manipulations are difficult to contemplate and execute.<br />
<br />
The five columns to the right of the double bar in Table&nbsp;50 contain the values of the dependent variables <math>\{ \boldsymbol\varepsilon J, ~\mathrm{E}J, ~\mathrm{D}J, ~\mathrm{d}J, ~\mathrm{d}^2\!J \}.\!</math> These are normally interpreted as values of functions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B}\!</math> or as values of propositions in the extended universe <math>[u, v, \mathrm{d}u, \mathrm{d}v]\!</math> but the dependencies prevailing in the contingent universe make it possible to regard these same final values as arising via functions on alternative lists of arguments, for example, the set <math>\{ u, v, u', v' \}.\!</math><br />
<br />
The column for <math>\boldsymbol\varepsilon J\!</math> is computed as <math>J(u, v) = uv\!</math> and together with the columns for <math>u\!</math> and <math>v\!</math> illustrates how we &ldquo;share structure&rdquo; in the Table by listing only the first entries of each constant block.<br />
<br />
The column for <math>\mathrm{E}J\!</math> is computed by means of the following chain of identities, where the contingent variables <math>u'\!</math> and <math>v'\!</math> are defined as <math>u' = u + \mathrm{d}u\!</math> and <math>v' = v + \mathrm{d}v.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathrm{E}J(u, v, \mathrm{d}u, \mathrm{d}v)<br />
& = &<br />
J(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& = &<br />
J(u', v')<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
This makes it easy to determine <math>\mathrm{E}J\!</math> by inspection, computing the conjunction <math>J(u', v') = u'v'\!</math> from the columns headed <math>u'\!</math> and <math>v'.\!</math> Since each of these forms expresses the same proposition <math>\mathrm{E}J\!</math> in <math>\mathrm{E}U^\bullet,\!</math> the dependence on <math>\mathrm{d}u\!</math> and <math>\mathrm{d}v\!</math> is still present but merely left implicit in the final variant <math>J(u', v').\!</math><br />
<br />
* '''Note.''' On occasion, it is tempting to use the further notation <math>J'(u, v) = J(u', v'),\!</math> especially to suggest a transformation that acts on whole propositions, for example, taking the proposition <math>J\!</math> into the proposition <math>J' = \mathrm{E}J.\!</math> The prime <math>( {}^{\prime} )\!</math> then signifies an action that is mediated by a field of choices, namely, the values that are picked out for the contingent variables in sweeping through the initial universe. But this heaps an unwieldy lot of construed intentions on a rather slight character and puts too high a premium on the constant correctness of its interpretation. In practice, therefore, it is best to avoid this usage.<br />
<br />
Given the values of <math>\boldsymbol\varepsilon J\!</math> and <math>\mathrm{E}J,\!</math> the columns for the remaining functions can be filled in quickly. The difference map is computed according to the relation <math>\mathrm{D}J = \boldsymbol\varepsilon J + \mathrm{E}J.\!</math> The first order differential <math>\mathrm{d}J\!</math> is found by looking in each block of constant argument pairs <math>u, v\!</math> and choosing the linear function of <math>\mathrm{d}u, \mathrm{d}v\!</math> that best approximates <math>\mathrm{D}J\!</math> in that block. Finally, the remainder is computed as <math>\mathrm{r}J = \mathrm{D}J + \mathrm{d}J,\!</math> in this case yielding the second order differential <math>\mathrm{d}^2\!J.\!</math><br />
<br />
====Analytic Series : Recap====<br />
<br />
Let us now summarize the results of Table&nbsp;50 by writing down for each column and for each block of constant argument pairs <math>u, v\!</math> a reasonably canonical symbolic expression for the function of <math>\mathrm{d}u, \mathrm{d}v\!</math> that appears there. The synopsis formed in this way is presented in Table&nbsp;51. As one has a right to expect, it confirms the results that were obtained previously by operating solely in terms of the formal calculus.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:70%"<br />
|+ style="height:30px" | <math>\text{Table 51.} ~~ \text{Computation of an Analytic Series in Symbolic Terms}\!</math><br />
|- style="height:35px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black" | <math>J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{E}J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{D}J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{d}J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{d}^2\!J\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}u \!\;\cdot\;\! \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}u \!\;\cdot\;\! \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
\mathrm{d}u<br />
\\[4pt]<br />
\mathrm{d}v<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;52 and 53 provide a quick overview of the analysis performed so far, giving the successive decompositions of <math>\mathrm{E}J = J + \mathrm{D}J\!</math> and <math>\mathrm{D}J = \mathrm{d}J + \mathrm{r}J\!</math> in two different styles of diagram.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 52 -- Decomposition of EJ.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 52.} ~~ \text{Decomposition of}~ \mathrm{E}J\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 53 -- Decomposition of DJ.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 53.} ~~ \text{Decomposition of}~ \mathrm{D}J\!</math><br />
|}<br />
<br />
====Terminological Interlude====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Lastly, my attention was especially attracted, not so much to the scene, as to the mirrors that produced it. These mirrors were broken in parts. Yes, they were marked and scratched; they had been &ldquo;starred&rdquo;, in spite of their solidity &hellip;<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 230]<br />
|}<br />
<br />
At this point several issues of terminology have accrued enough substance to intrude on our discussion. The remarks of this Subsection are intended to accomplish two goals. First, we call attention to significant aspects of the previous series of Figures, translating into literal terms what they depict in iconic forms, and we re-stress the most important structural elements they indicate. Next, we prepare the way for taking on more complex examples of transformations, those whose target universes have more than one dimension.<br />
<br />
In talking about the actions of operators it is important to keep in mind the distinctions between the operators per&nbsp;se, their operands, and their results. Furthermore, in working with composite forms of operators <math>\mathrm{W} = (\mathrm{W}_1, \ldots, \mathrm{W}_n),\!</math> transformations <math>\mathrm{F} = (\mathrm{F}_1, \ldots, \mathrm{F}_n),\!</math> and target domains <math>X^\bullet = [x_1, \ldots, x_n],\!</math> we need to preserve a clear distinction between the compound entity of each given type and any one of its separate components. It is curious, given the usefulness of the concepts ''operator'' and ''operand'', that we seem to lack a generic term, formed on the same root, for the corresponding result of an operation. Following the obvious paradigm would lead to words like ''opus'', ''opera'', and ''operant'', but these words are too affected with clang associations to work well at present, though they might be adapted in time. One current usage gets around this problem by using the substantive ''map'' as a systematic epithet to express the result of each operator's action. We will follow this practice as far as possible, for example, using the phrase ''tangent map'' to denote the end product of the tangent functor acting on its operand map.<br />
<br />
* '''Scholium.''' See [JGH, 6-9] for a good account of tangent functors and tangent maps in ordinary analysis and for examples of their use in mechanics. This work as a whole is a model of clarity in applying functorial principles to problems in physical dynamics.<br />
<br />
Whenever we focus on isolated propositions, on single components of composite operators, or on the portions of transformations that have <math>1\!</math>-dimensional ranges, we are free to shift between the native form of a proposition <math>J : U \to \mathbb{B}\!</math> and the thematized form of a mapping <math>J : U^\bullet \to [x]\!</math> without much trouble. In these cases we are able to tolerate a higher degree of ambiguity about the precise nature of an operator's input and output domains than we otherwise might. For example, in the preceding treatment of the example <math>J,\!</math> and for each operator <math>\mathrm{W}\!</math> in the set <math>\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \},\!</math> both the operand <math>J\!</math> and the result <math>\mathrm{W}J\!</math> could be viewed in either one of two ways. On one hand we may treat them as propositions <math>J : U \to \mathbb{B}\!</math> and <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B},\!</math> ignoring the distinction between the range <math>[x] \cong \mathbb{B}\!</math> of <math>\boldsymbol\varepsilon J\!</math> and the range <math>[\mathrm{d}x] \cong \mathbb{D}\!</math> of the other types of <math>\mathrm{W}J.\!</math> This is what we usually do when we content ourselves with simply coloring in regions of venn diagrams. On the other hand we may view these entities as maps <math>J : U^\bullet \to [x] = X^\bullet\!</math> and <math>\boldsymbol\varepsilon J : \mathrm{E}U^\bullet \to [x] \subseteq \mathrm{E}X^\bullet\!</math> or <math>\mathrm{W}J : \mathrm{E}U^\bullet \to [\mathrm{d}x] \subseteq \mathrm{E}X^\bullet,\!</math> in which case the qualitative characters of the output features are not ignored.<br />
<br />
At the beginning of this Section we recast the natural form of a proposition <math>J : U \to \mathbb{B}\!</math> into the thematic role of a transformation <math>J : U^\bullet \to [x],\!</math> where <math>x\!</math> was a variable recruited to express the newly independent <math>\check{J}.\!</math> However, in our computations and representations of operator actions we immediately lapsed back to viewing the results as native elements of the extended universe <math>\mathrm{E}U^\bullet,\!</math> in other words, as propositions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B},\!</math> where <math>\mathrm{W}\!</math> ranged over the set <math>\{ \boldsymbol\varepsilon, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}.\!</math> That is as it should be. We have worked hard to devise a language that gives us these advantages &mdash; the flexibility to exchange terms and types of equal information value and the capacity to reflect as quickly and as wittingly as a controlled reflex on the fibers of our propositions, independently of whether they express amusements, beliefs, or conjectures.<br />
<br />
As we take on target spaces of increasing dimension, however, these types of confusions (and confusions of types) become less and less permissible. For this reason, Tables&nbsp;54 and 55 present a rather detailed summary of the notation and the terminology we are using, as applied to the case <math>J = uv.\!</math> The rationale of these Tables is not so much to train more elephant guns on this poor drosophila of a concrete example but to invest our paradigm with enough solidity to bear the weight of abstraction to come.<br />
<br />
Table&nbsp;54 provides basic notation and descriptive information for the objects and operators used in this Example, giving the generic type (or broadest defined type) for each entity. Here, the sans&nbsp;serif operators <math>\mathsf{W} \in \{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{d}, \mathsf{r} \}\!</math> and their components <math>\mathrm{W} \in \{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}\!</math> both have the same broad type <math>\mathsf{W}, \mathrm{W} : (U^\bullet \to X^\bullet) \to (\mathrm{E}U^\bullet \to \mathrm{E}X^\bullet),\!</math> as appropriate to operators that map transformations <math>J : U^\bullet \to X^\bullet\!</math> to extended transformations <math>\mathsf{W}J, \mathrm{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 54.} ~~ \text{Cast of Characters : Expansive Subtypes of Objects and Operators}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| align="center" | <math>\text{Symbol}\!</math><br />
| align="center" | <math>\text{Notation}\!</math><br />
| align="center" | <math>\text{Description}\!</math><br />
| align="center" | <math>\text{Type}\!</math><br />
|-<br />
| align="center" | <math>U^\bullet\!</math><br />
| <math>= [u, v]\!</math><br />
| <math>\text{Source universe}\!</math><br />
| <math>[\mathbb{B}^2]\!</math><br />
|-<br />
| align="center" | <math>X^\bullet~\!</math><br />
| <math>= [x]\!</math><br />
| <math>\text{Target universe}\!</math><br />
| <math>[\mathbb{B}^1]~\!</math><br />
|-<br />
| align="center" | <math>\mathrm{E}U^\bullet\!</math><br />
| <math>= [u, v, \mathrm{d}u, \mathrm{d}v]\!</math><br />
| <math>\text{Extended source universe}\!</math><br />
| <math>[\mathbb{B}^2 \!\times\! \mathbb{D}^2]</math><br />
|-<br />
| align="center" | <math>\mathrm{E}X^\bullet\!</math><br />
| <math>= [x, \mathrm{d}x]~\!</math><br />
| <math>\text{Extended target universe}\!</math><br />
| <math>[\mathbb{B}^1 \!\times\! \mathbb{D}^1]</math><br />
|-<br />
| align="center" | <math>J\!</math><br />
| <math>J : U \!\to\! \mathbb{B}\!</math><br />
| <math>\text{Proposition}\!</math><br />
| <math>(\mathbb{B}^2 \!\to\! \mathbb{B}) \in [\mathbb{B}^2]\!</math><br />
|-<br />
| align="center" | <math>J\!</math><br />
| <math>J : U^\bullet \!\to\! X^\bullet\!</math><br />
| <math>\text{Transformation or Map}\!</math><br />
| <math>[\mathbb{B}^2] \!\to\! [\mathbb{B}^1]\!</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\boldsymbol\varepsilon<br />
\\<br />
\eta<br />
\\<br />
\mathrm{E}<br />
\\<br />
\mathrm{D}<br />
\\<br />
\mathrm{d}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{W} : U^\bullet \!\to\! \mathrm{E}U^\bullet,<br />
\\<br />
\mathrm{W} : X^\bullet \!\to\! \mathrm{E}X^\bullet,<br />
\\<br />
\mathrm{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\\<br />
\text{for each}~ \mathrm{W} ~\text{in the set:}<br />
\\<br />
\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{ll}<br />
\text{Tacit extension operator} & \boldsymbol\varepsilon<br />
\\<br />
\text{Trope extension operator} & \eta<br />
\\<br />
\text{Enlargement operator} & \mathrm{E}<br />
\\<br />
\text{Difference operator} & \mathrm{D}<br />
\\<br />
\text{Differential operator} & \mathrm{d}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^2] \!\to\! [\mathbb{B}^2 \!\times\! \mathbb{D}^2]},<br />
\\<br />
{[\mathbb{B}^1] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]},<br />
\\\\<br />
([\mathbb{B}^2] \!\to\! [\mathbb{B}^1]) \!\to\!<br />
\\<br />
([\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1])<br />
\end{array}</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\mathsf{e}<br />
\\<br />
\mathsf{E}<br />
\\<br />
\mathsf{D}<br />
\\<br />
\mathsf{T}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{W} : U^\bullet \!\to\! \mathsf{T}U^\bullet = \mathrm{E}U^\bullet,<br />
\\<br />
\mathsf{W} : X^\bullet \!\to\! \mathsf{T}X^\bullet = \mathrm{E}X^\bullet,<br />
\\<br />
\mathsf{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathsf{T}U^\bullet \!\to\! \mathsf{T}X^\bullet)<br />
\\<br />
\text{for each}~ \mathsf{W} ~\text{in the set:}<br />
\\<br />
\{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{T} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{lll}<br />
\text{Radius operator} & \mathsf{e} & = (\boldsymbol\varepsilon, \eta)<br />
\\<br />
\text{Secant operator} & \mathsf{E} & = (\boldsymbol\varepsilon, \mathrm{E})<br />
\\<br />
\text{Chord operator} & \mathsf{D} & = (\boldsymbol\varepsilon, \mathrm{D})<br />
\\<br />
\text{Tangent functor} & \mathsf{T} & = (\boldsymbol\varepsilon, \mathrm{d})<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^2] \!\to\! [\mathbb{B}^2 \!\times\! \mathbb{D}^2]},<br />
\\<br />
{[\mathbb{B}^1] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]},<br />
\\\\<br />
([\mathbb{B}^2] \!\to\! [\mathbb{B}^1]) \!\to\!<br />
\\<br />
([\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1])<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;55 supplies a more detailed outline of terminology for operators and their results. Here, we list the restrictive subtype (or narrowest defined subtype) that applies to each entity and we indicate across the span of the Table the whole spectrum of alternative types that color the interpretation of each symbol. For example, all the component operator maps <math>\mathrm{W}J\!</math> have <math>1\!</math>-dimensional ranges, either <math>\mathbb{B}^1\!</math> or <math>\mathbb{D}^1,\!</math> and so they can be viewed either as propositions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B}\!</math> or as logical transformations <math>\mathrm{W}J : \mathrm{E}U^\bullet \to X^\bullet.\!</math> As a rule, the plan of the Table allows us to name each entry by detaching the underlined adjective at the left of its row and prefixing it to the generic noun at the top of its column. In one case, however, it is customary to depart from this scheme. Because the phrase ''differential proposition'', applied to the result <math>\mathrm{d}J : \mathrm{E}U \to \mathbb{D},\!</math> does not distinguish it from the general run of differential propositions <math>\mathrm{G}: \mathrm{E}U \to \mathbb{B},\!</math> it is usual to single out <math>\mathrm{d}J\!</math> as the ''tangent proposition'' of <math>J.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 55.} ~~ \text{Synopsis of Terminology : Restrictive and Alternative Subtypes}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| &nbsp;<br />
| align="center" | <math>\text{Operator}\!</math><br />
| align="center" | <math>\text{Proposition}\!</math><br />
| align="center" | <math>\text{Map}\!</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tacit}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\boldsymbol\varepsilon : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\boldsymbol\varepsilon : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{B} \\<br />
\boldsymbol\varepsilon J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{B}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x] \\<br />
\boldsymbol\varepsilon J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Trope}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\eta : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\eta : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\eta J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\eta J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Enlargement}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{E} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{E}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{E}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Difference}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{D} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{D}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{D}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Differential}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{d} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{d} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{d}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}\!</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Remainder}}\\\text{operator}\end{matrix}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{r} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{r} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{r}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{r}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Radius}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e} = (\boldsymbol\varepsilon, \eta) \\<br />
\mathsf{e} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{e}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Secant}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}) \\<br />
\mathsf{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{E}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Chord}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}) \\<br />
\mathsf{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{D}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tangent}}\\\text{functor}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T} = (\boldsymbol\varepsilon, \mathrm{d}) \\<br />
\mathsf{T} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{T}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====End of Perfunctory Chatter : Time to Roll the Clip!====<br />
<br />
Two steps remain to finish the analysis of <math>J\!</math> that we began so long ago. First, we need to paste our accumulated heap of flat pictures into the frames of transformations, filling out the shapes of the operator maps <math>\mathsf{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet.~\!</math> This scheme is executed in two styles, using the ''areal views'' in Figures&nbsp;56-a and the ''box views'' in Figures&nbsp;56-b. Finally, in Figures&nbsp;57-1 to 57-4 we put all the pieces together to construct the full operator diagrams for <math>\mathsf{W} : J \to \mathsf{W}J.\!</math> There is a considerable amount of redundancy among the following three series of Figures but that will hopefully provide a fuller picture of the operations under review, enabling these snapshots to serve as successive frames in the animation of logic they are meant to become.<br />
<br />
=====Operator Maps : Areal Views=====<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a1 -- Radius Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a1.} ~~ \text{Radius Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a2 -- Secant Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a2.} ~~ \text{Secant Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a3 -- Chord Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a3.} ~~ \text{Chord Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a4 -- Tangent Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a4.} ~~ \text{Tangent Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
=====Operator Maps : Box Views=====<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b1 -- Radius Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b1.} ~~ \text{Radius Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b2 -- Secant Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b2.} ~~ \text{Secant Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b3 -- Chord Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b3.} ~~ \text{Chord Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b4 -- Tangent Map of J ISW.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b4.} ~~ \text{Tangent Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
=====Operator Diagrams for the Conjunction J = uv=====<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-1 -- Radius Operator Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-1.} ~~ \text{Radius Operator Diagram for the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-2 -- Secant Operator Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-2.} ~~ \text{Secant Operator Diagram for the Conjunction}~ J = uv~\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-3 -- Chord Operator Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-3.} ~~ \text{Chord Operator Diagram for the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-4 -- Tangent Functor Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-4.} ~~ \text{Tangent Functor Diagram for the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
===Taking Aim at Higher Dimensional Targets===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
The past and present wilt . . . . I have filled them and<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;emptied them,<br><br />
And proceed to fill my next fold of the future.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 87]<br />
|}<br />
<br />
In the next Section we consider a transformation <math>F\!</math> of concrete type <math>F : [u, v] \to [x, y]\!</math> and abstract type <math>F : [\mathbb{B}^2] \to [\mathbb{B}^2].\!</math> From the standpoint of propositional calculus we naturally approach the task of understanding such a transformation by parsing it into component maps with <math>1\!</math>-dimensional ranges, as follows:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{ccccccl}<br />
F & = & (F_1, F_2) & = & (f, g) & : & [u, v] \to [x, y],<br />
\\[6pt]<br />
&& F_1 & = & f & : & [u, v] \to [x],<br />
\\[6pt]<br />
&& F_2 & = & g & : & [u, v] \to [y].<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Then we tackle the separate components, now viewed as propositions <math>F_i : U \to \mathbb{B},\!</math> one at a time. At the completion of this analytic phase, we return to the task of synthesizing these partial and transient impressions into an agile form of integrity, a solidly coordinated and deeply integrated comprehension of the ongoing transformation. (Very often, of course, in tangling with refractory cases, we never get as far as the beginning again.)<br />
<br />
Let us now refer to the dimension of the target space or codomain as the ''toll'' (or ''tole'') of a transformation, as distinguished from the dimension of the range or image that is customarily called the ''rank''. When we keep to transformations with a toll of <math>1,\!</math> as <math>J : [u, v] \to [x],\!</math> we tend to get lazy about distinguishing a logical transformation from its component propositions. However, if we deal with transformations of a higher toll, this form of indolence can no longer be tolerated.<br />
<br />
Well, perhaps we can carry it a little further. After all, the operator result <math>\mathrm{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet\!</math> is a map of toll <math>2,\!</math> and cannot be unfolded in one piece as a proposition. But when a map has rank <math>1,\!</math> like <math>\boldsymbol\varepsilon J : \mathrm{E}U \to X \subseteq \mathrm{E}X\!</math> or <math>\mathrm{d}J : \mathrm{E}U \to \mathrm{d}X \subseteq \mathrm{E}X,\!</math> we naturally choose to concentrate on the <math>1\!</math>-dimensional range of the operator result <math>\mathrm{W}J,\!</math> ignoring the final difference in quality between the spaces <math>X\!</math> and <math>\mathrm{d}X,\!</math> and view <math>\mathrm{W}J\!</math> as a proposition about <math>\mathrm{E}U.\!</math><br />
<br />
In this way, an initial ambivalence about the role of the operand <math>J\!</math> conveys a double duty to the result <math>\mathrm{W}J.\!</math> The pivot that is formed by our focus of attention is essential to the linkage that transfers this double moment, as the whole process takes its bearing and wheels around the precise measure of a narrow bead that we can draw on the range of <math>\mathrm{W}J.\!</math> This is the escapement that it takes to get away with what may otherwise seem to be a simple duplicity, and this is the tolerance that is needed to counterbalance a certain arrogance of equivocation, by all of which machinations we make ourselves free to indicate the operator results <math>\mathrm{W}J\!</math> as propositions or as transformations, indifferently.<br />
<br />
But that's it, and no further. Neglect of these distinctions in range and target universes of higher dimensions is bound to cause a hopeless confusion. To guard against these adverse prospects, Tables&nbsp;58 and 59 lay the groundwork for discussing a typical map <math>F : [\mathbb{B}^2] \to [\mathbb{B}^2],\!</math> and begin to pave the way to some extent for discussing any transformation of the form <math>F : [\mathbb{B}^n] \to [\mathbb{B}^k].\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 58.} ~~ \text{Cast of Characters : Expansive Subtypes of Objects and Operators}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| align="center" | <math>\text{Symbol}\!</math><br />
| align="center" | <math>\text{Notation}\!</math><br />
| align="center" | <math>\text{Description}\!</math><br />
| align="center" | <math>\text{Type}\!</math><br />
|-<br />
| align="center" | <math>U^\bullet\!</math><br />
| <math>= [u, v]\!</math><br />
| <math>\text{Source universe}\!</math><br />
| <math>[\mathbb{B}^n]\!</math><br />
|-<br />
| align="center" | <math>X^\bullet~\!</math><br />
| <math>\begin{array}{l}<br />
= [x, y] \\<br />
= [f, g]<br />
\end{array}</math><br />
| <math>\text{Target universe}\!</math><br />
| <math>[\mathbb{B}^k]\!</math><br />
|-<br />
| align="center" | <math>\mathrm{E}U^\bullet\!</math><br />
| <math>= [u, v, \mathrm{d}u, \mathrm{d}v]\!</math><br />
| <math>\text{Extended source universe}\!</math><br />
| <math>[\mathbb{B}^n \!\times\! \mathbb{D}^n]\!</math><br />
|-<br />
| align="center" | <math>\mathrm{E}X^\bullet\!</math><br />
| <math>\begin{array}{l}<br />
= [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
= [f, g, \mathrm{d}f, \mathrm{d}g]<br />
\end{array}</math><br />
| <math>\text{Extended target universe}\!</math><br />
| <math>[\mathbb{B}^k \!\times\! \mathbb{D}^k]\!</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
f \\ g<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{ll}<br />
f : U \!\to\! [x] \cong \mathbb{B} \\<br />
g : U \!\to\! [y] \cong \mathbb{B}<br />
\end{array}</math><br />
| <math>\text{Proposition}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathbb{B}^n \!\to\! \mathbb{B} \\<br />
\in (\mathbb{B}^n, \mathbb{B}^n \!\to\! \mathbb{B}) = [\mathbb{B}^n]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>F\!</math><br />
| <math>F = (f, g) : U^\bullet \!\to\! X^\bullet\!</math><br />
| <math>\text{Transformation of Map}\!</math><br />
| <math>[\mathbb{B}^n] \!\to\! [\mathbb{B}^k]</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\boldsymbol\varepsilon<br />
\\<br />
\eta<br />
\\<br />
\mathrm{E}<br />
\\<br />
\mathrm{D}<br />
\\<br />
\mathrm{d}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{W} : U^\bullet \!\to\! \mathrm{E}U^\bullet,<br />
\\<br />
\mathrm{W} : X^\bullet \!\to\! \mathrm{E}X^\bullet,<br />
\\<br />
\mathrm{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\\<br />
\text{for each}~ \mathrm{W} ~\text{in the set:}<br />
\\<br />
\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{ll}<br />
\text{Tacit extension operator} & \boldsymbol\varepsilon<br />
\\<br />
\text{Trope extension operator} & \eta<br />
\\<br />
\text{Enlargement operator} & \mathrm{E}<br />
\\<br />
\text{Difference operator} & \mathrm{D}<br />
\\<br />
\text{Differential operator} & \mathrm{d}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^n] \!\to\! [\mathbb{B}^n \!\times\! \mathbb{D}^n]},<br />
\\<br />
{[\mathbb{B}^k] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]},<br />
\\\\<br />
([\mathbb{B}^n] \!\to\! [\mathbb{B}^k]) \!\to\!<br />
\\<br />
([\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k])<br />
\end{array}</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\mathsf{e}<br />
\\<br />
\mathsf{E}<br />
\\<br />
\mathsf{D}<br />
\\<br />
\mathsf{T}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{W} : U^\bullet \!\to\! \mathsf{T}U^\bullet = \mathrm{E}U^\bullet,<br />
\\<br />
\mathsf{W} : X^\bullet \!\to\! \mathsf{T}X^\bullet = \mathrm{E}X^\bullet,<br />
\\<br />
\mathsf{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathsf{T}U^\bullet \!\to\! \mathsf{T}X^\bullet)<br />
\\<br />
\text{for each}~ \mathsf{W} ~\text{in the set:}<br />
\\<br />
\{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{T} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{lll}<br />
\text{Radius operator} & \mathsf{e} & = (\boldsymbol\varepsilon, \eta)<br />
\\<br />
\text{Secant operator} & \mathsf{E} & = (\boldsymbol\varepsilon, \mathrm{E})<br />
\\<br />
\text{Chord operator} & \mathsf{D} & = (\boldsymbol\varepsilon, \mathrm{D})<br />
\\<br />
\text{Tangent functor} & \mathsf{T} & = (\boldsymbol\varepsilon, \mathrm{d})<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^n] \!\to\! [\mathbb{B}^n \!\times\! \mathbb{D}^n]},<br />
\\<br />
{[\mathbb{B}^k] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]},<br />
\\\\<br />
([\mathbb{B}^n] \!\to\! [\mathbb{B}^k]) \!\to\!<br />
\\<br />
([\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k])<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 59.} ~~ \text{Synopsis of Terminology : Restrictive and Alternative Subtypes}~\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| &nbsp;<br />
| align="center" | <math>\begin{matrix}\text{Operator}\\\text{or}\\\text{Operand}\end{matrix}</math><br />
| align="center" | <math>\begin{matrix}\text{Proposition}\\\text{or}\\\text{Component}\end{matrix}</math><br />
| align="center" | <math>\begin{matrix}\text{Transformation}\\\text{or}\\\text{Map}\end{matrix}</math><br />
|-<br />
| align="center" | <math>\underline{\text{Operand}}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
F = (F_1, F_2) \\<br />
F = (f, g) : U \!\to\! X<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
F_i : \langle u, v \rangle \!\to\! \mathbb{B} \\<br />
F_i : \mathbb{B}^n \!\to\! \mathbb{B}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
F : [u, v] \!\to\! [x, y] \\<br />
F : [\mathbb{B}^n] \!\to\! [\mathbb{B}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tacit}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\boldsymbol\varepsilon : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\boldsymbol\varepsilon : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{B} \\<br />
\boldsymbol\varepsilon F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{B}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y] \\<br />
\boldsymbol\varepsilon F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Trope}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\eta : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\eta : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\eta F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\eta F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Enlargement}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{E} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{E}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{E}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Difference}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{D} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{D}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{D}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Differential}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{d} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{d} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{d}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}\!</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Remainder}}\\\text{operator}\end{matrix}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{r} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{r} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{r}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{r}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Radius}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e} = (\boldsymbol\varepsilon, \eta) \\<br />
\mathsf{e} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{e}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Secant}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}) \\<br />
\mathsf{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{E}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Chord}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}) \\<br />
\mathsf{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{D}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tangent}}\\\text{functor}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T} = (\boldsymbol\varepsilon, \mathrm{d}) \\<br />
\mathsf{T} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{T}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
===Transformations of Type '''B'''<sup>2</sup> &rarr; '''B'''<sup>2</sup>===<br />
<br />
To take up a slightly more complex example, but one that remains simple enough to pursue through a complete series of developments, consider the transformation from <math>U^\bullet = [u, v]\!</math> to <math>X^\bullet = [x, y]\!</math> that is defined by the following system of equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
x<br />
& = & f(u, v)<br />
& = & \texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[8pt]<br />
y<br />
& = & g(u, v)<br />
& = & \texttt{((} u \texttt{,} v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
The component notation <math>F = (F_1, F_2) = (f, g) : U^\bullet \to X^\bullet\!</math> allows us to give a name and a type to this transformation and permits defining it by the compact description that follows:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
(x, y)<br />
& = & F(u, v)<br />
& = & (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Logical Transformations====<br />
<br />
The information that defines the logical transformation <math>F\!</math> can be represented in the form of a truth table, as shown in Table&nbsp;60. To cut down on subscripts in this example we continue to use plain letter equivalents for all components of spaces and maps.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:60%"<br />
|+ style="height:30px" | <math>\text{Table 60.} ~~ \text{A Propositional Transformation}\!</math><br />
|- style="height:40px; background:ghostwhite; width:100%"<br />
| style="width:25%" | <math>u\!</math><br />
| style="width:25%" | <math>v\!</math><br />
| style="width:25%; border-left:1px solid black" | <math>f\!</math><br />
| style="width:25%" | <math>g\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="border-top:1px solid black" | <math>u\!</math><br />
| style="border-top:1px solid black" | <math>v\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;61 shows how we might paint a picture of the transformation <math>F\!</math> in the manner of Figure&nbsp;30.<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 61 -- Propositional Transformation.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 61.} ~~ \text{A Propositional Transformation}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;62 extracts the gist of Figure&nbsp;61, exhibiting a style of diagram that is adequate for most purposes.<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 62 -- Propositional Transformation (Short Form).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 62.} ~~ \text{A Propositional Transformation (Short Form)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Local Transformations====<br />
<br />
Figure&nbsp;63 gives a more complete picture of the transformation <math>F,\!</math> showing how the points of <math>U^\bullet\!</math> are transformed into points of <math>X^\bullet.\!</math> The bold lines crossing from one universe to the other trace the action that <math>F\!</math> induces on points, in other words, they show how the transformation acts as a mapping from points to points and chart its effects on the elements that are variously called cells, points, positions, or singular propositions.<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 63 -- Transformation of Positions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 63.} ~~ \text{A Transformation of Positions}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;64 shows how the action of <math>F\!</math> on cells or points can be computed in terms of coordinates.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 64.} ~~ \text{A Transformation of Positions}\!</math><br />
|- style="height:40px; background:ghostwhite; width:100%"<br />
| style="width:8%" | <math>u\!</math><br />
| style="width:8%" | <math>v\!</math><br />
| style="width:12%; border-left:1px solid black" | <math>x\!</math><br />
| style="width:12%" | <math>y\!</math><br />
| style="width:10%; border-left:1px solid black" | <math>x~y\!</math><br />
| style="width:10%" | <math>x \texttt{(} y \texttt{)}\!</math><br />
| style="width:10%" | <math>\texttt{(} x \texttt{)} y\!</math><br />
| style="width:10%" | <math>\texttt{(} x \texttt{)(} y \texttt{)}\!</math><br />
| style="width:20%; border-left:1px solid black" | <math>X^\bullet = [x, y]\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\uparrow<br />
\\[4pt]<br />
F =<br />
\\[4pt]<br />
(f, g)<br />
\\[4pt]<br />
\uparrow<br />
\end{matrix}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="border-top:1px solid black" | <math>u\!</math><br />
| style="border-top:1px solid black" | <math>v\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
| style="border-top:1px solid black" | <math>\texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>u~v\!</math><br />
| style="border-top:1px solid black" | <math>\texttt{(} u \texttt{,} v \texttt{)}\!</math><br />
| style="border-top:1px solid black" | <math>\texttt{(} u \texttt{)(} v \texttt{)}\!</math><br />
| style="border-top:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>U^\bullet = [u, v]\!</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;65 extends this scheme from single cells to arbitrary regions, showing how we might compute the action of a logical transformation on arbitrary propositions in the universe of discourse. The effect of a point-transformation on arbitrary propositions, or any other structures erected on points, is referred to as the ''induced action'' of the transformation on the structures in question.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 65-a.} ~~ \text{An Induced Transformation on Propositions}\!</math><br />
|- style="height:50px; background:ghostwhite"<br />
| style="width:20%" | <math>X^\bullet~\!</math><br />
| style="width:20%; border-right:none" | <math>\longleftarrow\!</math><br />
| style="width:20%; border-left:none; border-right:none" | <math>F = (f, g)\!</math><br />
| style="width:20%; border-left:none" | <math>\longleftarrow\!</math><br />
| style="width:20%" | <math>U^\bullet~\!</math><br />
|- style="background:ghostwhite"<br />
| rowspan="2" | <math>f_i (x, y)\!</math><br />
| align="right" | <math>\begin{matrix}u = \\ v =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~0~0\\1~0~1~0\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= u \\ = v\end{matrix}</math><br />
| rowspan="2" style="width:20%" | <math>f_j (u, v)\!</math><br />
|- style="background:ghostwhite"<br />
| align="right" | <math>\begin{matrix}x = \\ y =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~1~0\\1~0~0~1\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= f(u, v) \\ = g(u, v)\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{2}<br />
\\[2pt]<br />
f_{3}<br />
\\[2pt]<br />
f_{4}<br />
\\[2pt]<br />
f_{5}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{7}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)~ ~}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{~ ~(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{,~} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{~~} y \texttt{)}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~0<br />
\\[2pt]<br />
0~0~0~0<br />
\\[2pt]<br />
0~0~0~1<br />
\\[2pt]<br />
0~0~0~1<br />
\\[2pt]<br />
0~1~1~0<br />
\\[2pt]<br />
0~1~1~0<br />
\\[2pt]<br />
0~1~1~1<br />
\\[2pt]<br />
0~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{,~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{,~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{~~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{~~} v \texttt{)}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[2pt]<br />
f_{0}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{7}<br />
\\[2pt]<br />
f_{7}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{8}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{10}<br />
\\[2pt]<br />
f_{11}<br />
\\[2pt]<br />
f_{12}<br />
\\[2pt]<br />
f_{13}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{15}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[2pt]<br />
\texttt{~ ~ ~ ~} y \texttt{~~}<br />
\\[2pt]<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[2pt]<br />
\texttt{~~} x \texttt{~ ~ ~ ~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
1~0~0~0<br />
\\[2pt]<br />
1~0~0~0<br />
\\[2pt]<br />
1~0~0~1<br />
\\[2pt]<br />
1~0~0~1<br />
\\[2pt]<br />
1~1~1~0<br />
\\[2pt]<br />
1~1~1~0<br />
\\[2pt]<br />
1~1~1~1<br />
\\[2pt]<br />
1~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~~} u \texttt{~~} v \texttt{~~}<br />
\\[2pt]<br />
\texttt{~~} u \texttt{~~} v \texttt{~~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{8}<br />
\\[2pt]<br />
f_{8}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{15}<br />
\\[2pt]<br />
f_{15}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 65-b.} ~~ \text{An Induced Transformation on Propositions}\!</math><br />
|- style="height:50px; background:ghostwhite"<br />
| style="width:20%" | <math>X^\bullet~\!</math><br />
| style="width:20%; border-right:none" | <math>\longleftarrow\!</math><br />
| style="width:20%; border-left:none; border-right:none" | <math>F = (f, g)\!</math><br />
| style="width:20%; border-left:none" | <math>\longleftarrow\!</math><br />
| style="width:20%" | <math>U^\bullet~\!</math><br />
|- style="background:ghostwhite"<br />
| rowspan="2" | <math>f_i (x, y)\!</math><br />
| align="right" | <math>\begin{matrix}u = \\ v =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~0~0\\1~0~1~0\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= u \\ = v\end{matrix}</math><br />
| rowspan="2" style="width:20%" | <math>f_j (u, v)\!</math><br />
|- style="background:ghostwhite"<br />
| align="right" | <math>\begin{matrix}x = \\ y =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~1~0\\1~0~0~1\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= f(u, v) \\ = g(u, v)\end{matrix}</math><br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>f_{0}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[2pt]<br />
f_{2}<br />
\\[2pt]<br />
f_{4}<br />
\\[2pt]<br />
f_{8}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~0<br />
\\[2pt]<br />
0~0~0~1<br />
\\[2pt]<br />
0~1~1~0<br />
\\[2pt]<br />
1~0~0~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{,~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{~} u \texttt{~~} v \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{8}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{3}<br />
\\[2pt]<br />
f_{12}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[2pt]<br />
1~1~1~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} u \texttt{)(} v \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[2pt]<br />
f_{14}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[2pt]<br />
f_{9}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[2pt]<br />
1~0~0~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{~~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{~} u \texttt{~~} v \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[2pt]<br />
f_{8}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{5}<br />
\\[2pt]<br />
f_{10}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[2pt]<br />
1~0~0~1<br />
\end{matrix}~\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} u \texttt{,~} v \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[2pt]<br />
f_{9}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[2pt]<br />
f_{11}<br />
\\[2pt]<br />
f_{13}<br />
\\[2pt]<br />
f_{14}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[2pt]<br />
1~0~0~1<br />
\\[2pt]<br />
1~1~1~0<br />
\\[2pt]<br />
1~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} u \texttt{~~} v \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{15}<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>f_{15}\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Difference Operators and Tangent Functors====<br />
<br />
Given the alphabets <math>\mathcal{U} = \{ u, v \}\!</math> and <math>\mathcal{X} = \{ x, y \},\!</math> along with the corresponding universes of discourse <math>U^\bullet, X^\bullet \cong [\mathbb{B}^2],\!</math> how many logical transformations of the general form <math>G = (G_1, G_2) : U^\bullet \to X^\bullet\!</math> are there? Since <math>G_1\!</math> and <math>G_2\!</math> can be any propositions of the type <math>\mathbb{B}^2 \to \mathbb{B},\!</math> there are <math>2^4 = 16\!</math> choices for each of the maps <math>G_1\!</math> and <math>G_2\!</math> and thus there are <math>2^4 \cdot 2^4 = 2^8 = 256\!</math> different mappings altogether of the form <math>G : U^\bullet \to X^\bullet.\!</math> The set of functions of a given type is denoted by placing its type indicator in parentheses, in the present instance writing <math>(U^\bullet \to X^\bullet) = \{ G : U^\bullet \to X^\bullet \},\!</math> and so the cardinality of the ''function space'' <math>(U^\bullet \to X^\bullet)\!</math> is summed up by writing <math>|(U^\bullet \to X^\bullet)| = |(\mathbb{B}^2 \to \mathbb{B}^2)| = 4^4 = 256.\!</math><br />
<br />
Given a transformation <math>G = (G_1, G_2) : U^\bullet \to X^\bullet\!</math> of this type, we proceed to define a pair of further transformations, related to <math>G,\!</math> that operate between the extended universes, <math>\mathrm{E}U^\bullet\!</math> and <math>\mathrm{E}X^\bullet,\!</math> of its source and target domains.<br />
<br />
First, the ''enlargement map'' (or ''secant transformation'') <math>\mathrm{E}G = (\mathrm{E}G_1, \mathrm{E}G_2) : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet\!</math> is defined by the following set of component equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}G_i<br />
& = & G_i (u + \mathrm{d}u, v + \mathrm{d}v)<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Next, the ''difference map'' (or ''chordal transformation'') <math>\mathrm{D}G = (\mathrm{D}G_1, \mathrm{D}G_2) : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet~\!</math> is defined in component-wise fashion as the boolean sum of the initial proposition <math>G_i\!</math> and the enlarged proposition <math>\mathrm{E}G_i,\!</math> for <math>i = 1, 2,\!</math> according to the following set of equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
\mathrm{D}G_i<br />
& = & G_i (u, v)<br />
& + & \mathrm{E}G_i (u, v, \mathrm{d}u, \mathrm{d}v)<br />
\\[8pt]<br />
& = & G_i (u, v)<br />
& + & G_i (u + \mathrm{d}u, v + \mathrm{d}v)<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Maintaining a strict analogy with ordinary difference calculus would perhaps have us write <math>\mathrm{D}G_i = \mathrm{E}G_i - G_i,\!</math> but the sum and difference operations are the same thing in boolean arithmetic. It is more often natural in the logical context to consider an initial proposition <math>q,\!</math> then to compute the enlargement <math>\mathrm{E}q,\!</math> and finally to determine the difference <math>\mathrm{D}q = q + \mathrm{E}q,\!</math> so we let the variant order of terms reflect this sequence of considerations.<br />
<br />
Viewed in this light the difference operator <math>\mathrm{D}\!</math> is imagined to be a function of very wide scope and polymorphic application, one that is able to realize the association between each transformation <math>G\!</math> and its difference map <math>\mathrm{D}G,\!</math> for example, taking the function space <math>(U^\bullet \to X^\bullet)\!</math> into <math>(\mathrm{E}U^\bullet \to \mathrm{E}X^\bullet).\!</math> When we consider the variety of interpretations permitted to propositions over the contexts in which we put them to use, it should be clear that an operator of this scope is not at all a trivial matter to define in general and that it may take some trouble to work out. For the moment we content ourselves with returning to particular cases.<br />
<br />
Acting on the logical transformation <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;),\!</math> the operators <math>\mathrm{E}\!</math> and <math>\mathrm{D}\!</math> yield the enlarged map <math>\mathrm{E}F = (\mathrm{E}f, \mathrm{E}g)\!</math> and the difference map <math>\mathrm{D}F = (\mathrm{D}f, \mathrm{D}g),\!</math> respectively, whose components are given as follows.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}f<br />
& = & \texttt{((} u + \mathrm{d}u \texttt{)(} v + \mathrm{d}v \texttt{))}<br />
\\[8pt]<br />
\mathrm{E}g<br />
& = & \texttt{((} u + \mathrm{d}u \texttt{,~} v + \mathrm{d}v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
\mathrm{D}f<br />
& = & \texttt{((} u \texttt{)(} v \texttt{))}<br />
& + & \texttt{((} u + \mathrm{d}u \texttt{)(} v + \mathrm{d}v \texttt{))}<br />
\\[8pt]<br />
\mathrm{D}g<br />
& = & \texttt{((} u \texttt{,~} v \texttt{))}<br />
& + & \texttt{((} u + \mathrm{d}u \texttt{,~} v + \mathrm{d}v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
But these initial formulas are purely definitional, and help us little in understanding either the purpose of the operators or the meaning of their results. Working symbolically, let us apply the same method to the separate components <math>f\!</math> and <math>g\!</math> that we earlier used on <math>J.\!</math> This work is recorded in Appendix&nbsp;3 and a summary of the results is presented in Tables&nbsp;66-i and 66-ii.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 66-i.} ~~ \text{Computation Summary for}~ f(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f<br />
& = & u \!\cdot\! v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 1<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}f<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \cdot \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{d}f<br />
& = & u \!\cdot\! v \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}f<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 66-ii.} ~~ \text{Computation Summary for}~ g(u, v) = \texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon g<br />
& = & u \!\cdot\! v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}g<br />
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}g<br />
& = & u \!\cdot\! v \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;67 shows how to compute the analytic series for <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math> in terms of coordinates, and Table&nbsp;68 recaps these results in symbolic terms, agreeing with earlier derivations.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 67.} ~~ \text{Computation of an Analytic Series in Terms of Coordinates}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:6%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}u\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>\mathrm{d}v\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>u'\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>v'\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:4px double black" | <math>f\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>g\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{E}f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{E}g}\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{D}f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{D}g}\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{d}f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{d}g}\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{d}^2\!f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{d}^2\!g}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 68.} ~~ \text{Computation of an Analytic Series in Symbolic Terms}\!</math><br />
|- style="height:40px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black" | <math>f\!</math><br />
| <math>g\!</math><br />
| style="border-left:1px solid black" | <math>{\mathrm{D}f}\!</math><br />
| <math>{\mathrm{D}g}\!</math><br />
| style="border-left:1px solid black" | <math>{\mathrm{d}f}\!</math><br />
| <math>{\mathrm{d}g}\!</math><br />
| style="border-left:1px solid black" | <math>{\mathrm{d}^2\!f}\!</math><br />
| <math>{\mathrm{d}^2\!g}\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[4pt]<br />
\texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
\\[4pt]<br />
\texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
\\[4pt]<br />
\texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;69 gives a graphical picture of the difference map <math>\mathrm{D}F = (\mathrm{D}f, \mathrm{D}g)\!</math> for the transformation <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math> This represents the same information about <math>\mathrm{D}f~\!</math> and <math>\mathrm{D}g~\!</math> that was given in the corresponding rows of Tables&nbsp;66-i and 66-ii, for ease of reference repeated below.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[8pt]<br />
\mathrm{D}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 69 -- Difference Map (Short Form).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 69.} ~~ \text{Difference Map of}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;70-a shows a way of visualizing the tangent functor map <math>\mathrm{d}F = (\mathrm{d}f, \mathrm{d}g)\!</math> for the transformation <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math> This amounts to the same information about <math>\mathrm{d}f~\!</math> and <math>\mathrm{d}g~\!</math> that was given in Tables&nbsp;66-i and 66-ii, the corresponding rows of which are repeated below.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{d}f<br />
& = & u \!\cdot\! v \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[8pt]<br />
\mathrm{d}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 70-a -- Tangent Functor Diagram.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 70-a.} ~~ \text{Tangent Functor Diagram for}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math><br />
|}<br />
<br />
<br><br />
<br />
Figure 70-b shows another way to picture the action of the tangent functor on the logical transformation <math>F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 70-b -- Tangent Functor Diagram.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 70-b.} ~~ \text{Tangent Functor Ferris Wheel for}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math><br />
|}<br />
<br />
<br><br />
<br />
* '''Note.''' The original Figure&nbsp;70-b lost some of its labeling in a succession of platform metamorphoses over the years, so we have included an ASCII version below to indicate where the missing labels go.<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| /////\ /////\ | | /XXXX\ /XXXX\ | | /\\\\\ /\\\\\ |<br />
| ///////o//////\ | | /XXXXXXoXXXXXX\ | | /\\\\\\o\\\\\\\ |<br />
| //////// \//////\ | | /XXXXXX/ \XXXXXX\ | | /\\\\\\/ \\\\\\\\ |<br />
| o/////// \//////o | | oXXXXXX/ \XXXXXXo | | o\\\\\\/ \\\\\\\o |<br />
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |<br />
| |/du//| |//dv/| | | |XXXXX| |XXXXX| | | |\du\\| |\\dv\| |<br />
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |<br />
| o//////\ ///////o | | oXXXXXX\ /XXXXXXo | | o\\\\\\\ /\\\\\\o |<br />
| \//////\ //////// | | \XXXXXX\ /XXXXXX/ | | \\\\\\\\ /\\\\\\/ |<br />
| \//////o/////// | | \XXXXXXoXXXXXX/ | | \\\\\\\o\\\\\\/ |<br />
| \///// \///// | | \XXXX/ \XXXX/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ (u)(v) o-----------------------o dv' @ (u)(v) =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| / \ /////\ | | /\\\\\ /XXXX\ | | /\\\\\ /\\\\\ |<br />
| / o//////\ | | /\\\\\\oXXXXXX\ | | /\\\\\\o\\\\\\\ |<br />
| / //\//////\ | | /\\\\\\//\XXXXXX\ | | /\\\\\\/ \\\\\\\\ |<br />
| o ////\//////o | | o\\\\\\////\XXXXXXo | | o\\\\\\/ \\\\\\\o |<br />
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |<br />
| | du |/////|//dv/| | | |\\\\\|/////|XXXXX| | | |\du\\| |\\dv\| |<br />
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |<br />
| o \//////////o | | o\\\\\\\////XXXXXXo | | o\\\\\\\ /\\\\\\o |<br />
| \ \///////// | | \\\\\\\\//XXXXXX/ | | \\\\\\\\ /\\\\\\/ |<br />
| \ o/////// | | \\\\\\\oXXXXXX/ | | \\\\\\\o\\\\\\/ |<br />
| \ / \///// | | \\\\\/ \XXXX/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ (u) v o-----------------------o dv' @ (u) v =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| /////\ / \ | | /XXXX\ /\\\\\ | | /\\\\\ /\\\\\ |<br />
| ///////o \ | | /XXXXXXo\\\\\\\ | | /\\\\\\o\\\\\\\ |<br />
| /////////\ \ | | /XXXXXX//\\\\\\\\ | | /\\\\\\/ \\\\\\\\ |<br />
| o//////////\ o | | oXXXXXX////\\\\\\\o | | o\\\\\\/ \\\\\\\o |<br />
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |<br />
| |/du//|/////| dv | | | |XXXXX|/////|\\\\\| | | |\du\\| |\\dv\| |<br />
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |<br />
| o//////\//// o | | oXXXXXX\////\\\\\\o | | o\\\\\\\ /\\\\\\o |<br />
| \//////\// / | | \XXXXXX\//\\\\\\/ | | \\\\\\\\ /\\\\\\/ |<br />
| \//////o / | | \XXXXXXo\\\\\\/ | | \\\\\\\o\\\\\\/ |<br />
| \///// \ / | | \XXXX/ \\\\\/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ u (v) o-----------------------o dv' @ u (v) =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| / \ / \ | | /\\\\\ /\\\\\ | | /\\\\\ /\\\\\ |<br />
| / o \ | | /\\\\\\o\\\\\\\ | | /\\\\\\o\\\\\\\ |<br />
| / / \ \ | | /\\\\\\/ \\\\\\\\ | | /\\\\\\/ \\\\\\\\ |<br />
| o / \ o | | o\\\\\\/ \\\\\\\o | | o\\\\\\/ \\\\\\\o |<br />
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |<br />
| | du | | dv | | | |\\\\\| |\\\\\| | | |\du\\| |\\dv\| |<br />
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |<br />
| o \ / o | | o\\\\\\\ /\\\\\\o | | o\\\\\\\ /\\\\\\o |<br />
| \ \ / / | | \\\\\\\\ /\\\\\\/ | | \\\\\\\\ /\\\\\\/ |<br />
| \ o / | | \\\\\\\o\\\\\\/ | | \\\\\\\o\\\\\\/ |<br />
| \ / \ / | | \\\\\/ \\\\\/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ u v o-----------------------o dv' @ u v =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| U | |\U\\\\\\\\\\\\\\\\\\\\\| |\U\\\\\\\\\\\\\\\\\\\\\|<br />
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|<br />
| /////\ /////\ | |\\\\\/////\\/////\\\\\\| |\\\\\/ \\/ \\\\\\|<br />
| ///////o//////\ | |\\\\///////o//////\\\\\| |\\\\/ o \\\\\|<br />
| /////////\//////\ | |\\\////////X\//////\\\\| |\\\/ /\\ \\\\|<br />
| o//////////\//////o | |\\o///////XXX\//////o\\| |\\o /\\\\ o\\|<br />
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|<br />
| |//u//|/////|//v//| | |\\|//u//|XXXXX|//v//|\\| |\\| u |\\\\\| v |\\|<br />
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|<br />
| o//////\//////////o | |\\o//////\XXX///////o\\| |\\o \\\\/ o\\|<br />
| \//////\///////// | |\\\\//////\X////////\\\| |\\\\ \\/ /\\\|<br />
| \//////o/////// | |\\\\\//////o///////\\\\| |\\\\\ o /\\\\|<br />
| \///// \///// | |\\\\\\/////\\/////\\\\\| |\\\\\\ /\\ /\\\\\|<br />
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|<br />
| | |\\\\\\\\\\\\\\\\\\\\\\\| |\\\\\\\\\\\\\\\\\\\\\\\|<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= u' o-----------------------o v' =<br />
= | U' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/u'//|XXXXX|\\v'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
Figure 70-b. Tangent Functor Ferris Wheel for F<u, v> = <((u)(v)), ((u, v))><br />
</pre><br />
|}<br />
<br />
==Epilogue, Enchoiry, Exodus==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
It is time to explain myself . . . . let us stand up.<br />
| width="4%" | &nbsp;<br />
|- <br />
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 79]<br />
|}<br />
<br />
==Appendices==<br />
<br />
===Appendix 1. Propositional Forms and Differential Expansions===<br />
<br />
====Table A1. Propositional Forms on Two Variables====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A1.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="background:ghostwhite"<br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}</math><br />
| width="25%" | <math>\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{0}\\f_{1}\\f_{2}\\f_{3}\\f_{4}\\f_{5}\\f_{6}\\f_{7}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0000}\\f_{0001}\\f_{0010}\\f_{0011}\\f_{0100}\\f_{0101}\\f_{0110}\\f_{0111}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~0\\0~0~0~1\\0~0~1~0\\0~0~1~1\\0~1~0~0\\0~1~0~1\\0~1~1~0\\0~1~1~1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)~ ~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~ ~(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{,~} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{~~} y \texttt{)}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{false}<br />
\\<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\<br />
y ~\text{without}~ x<br />
\\<br />
\text{not}~ x<br />
\\<br />
x ~\text{without}~ y<br />
\\<br />
\text{not}~ y<br />
\\<br />
x ~\text{not equal to}~ y<br />
\\<br />
\text{not both}~ x ~\text{and}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\<br />
\lnot x \land \lnot y<br />
\\<br />
\lnot x \land y<br />
\\<br />
\lnot x<br />
\\<br />
x \land \lnot y<br />
\\<br />
\lnot y<br />
\\<br />
x \ne y<br />
\\<br />
\lnot x \lor \lnot y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{8}\\f_{9}\\f_{10}\\f_{11}\\f_{12}\\f_{13}\\f_{14}\\f_{15}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{1000}\\f_{1001}\\f_{1010}\\f_{1011}\\f_{1100}\\f_{1101}\\f_{1110}\\f_{1111}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1~0~0~0\\1~0~0~1\\1~0~1~0\\1~0~1~1\\1~1~0~0\\1~1~0~1\\1~1~1~0\\1~1~1~1<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~ ~ ~ ~} y \texttt{~~}<br />
\\<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{~~} x \texttt{~ ~ ~ ~}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{((~))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{and}~ y<br />
\\<br />
x ~\text{equal to}~ y<br />
\\<br />
y<br />
\\<br />
\text{not}~ x ~\text{without}~ y<br />
\\<br />
x<br />
\\<br />
\text{not}~ y ~\text{without}~ x<br />
\\<br />
x ~\text{or}~ y<br />
\\<br />
\text{true}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x \land y<br />
\\<br />
x = y<br />
\\<br />
y<br />
\\<br />
x \Rightarrow y<br />
\\<br />
x<br />
\\<br />
x \Leftarrow y<br />
\\<br />
x \lor y<br />
\\<br />
1<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A2. Propositional Forms on Two Variables====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A2.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="background:ghostwhite"<br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}</math><br />
| width="25%" | <math>\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>f_{0000}\!</math><br />
| <math>0~0~0~0</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0001}\\f_{0010}\\f_{0100}\\f_{1000}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~1\\0~0~1~0\\0~1~0~0\\1~0~0~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\<br />
y ~\text{without}~ x<br />
\\<br />
x ~\text{without}~ y<br />
\\<br />
x ~\text{and}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot x \land \lnot y<br />
\\<br />
\lnot x \land y<br />
\\<br />
x \land \lnot y<br />
\\<br />
x \land y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{3}\\f_{12}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0011}\\f_{1100}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~1~1\\1~1~0~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ x<br />
\\<br />
x<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot x<br />
\\<br />
x<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{6}\\f_{9}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0110}\\f_{1001}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~0\\1~0~0~1<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{not equal to}~ y<br />
\\<br />
x ~\text{equal to}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x \ne y<br />
\\<br />
x = y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{5}\\f_{10}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0101}\\f_{1010}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~0~1\\1~0~1~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ y<br />
\\<br />
y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot y<br />
\\<br />
y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{7}\\f_{11}\\f_{13}\\f_{14}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0111}\\f_{1011}\\f_{1101}\\f_{1110}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~1\\1~0~1~1\\1~1~0~1\\1~1~1~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not both}~ x ~\text{and}~ y<br />
\\<br />
\text{not}~ x ~\text{without}~ y<br />
\\<br />
\text{not}~ y ~\text{without}~ x<br />
\\<br />
x ~\text{or}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot x \lor \lnot y<br />
\\<br />
x \Rightarrow y<br />
\\<br />
x \Leftarrow y<br />
\\<br />
x \lor y<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>f_{1111}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A3. E''f'' Expanded Over Differential Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A3.} ~~ \mathrm{E}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{T}_{11}f\\\mathrm{E}f|_{\mathrm{d}x ~ \mathrm{d}y}\end{matrix}</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{T}_{10}f\\\mathrm{E}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}\end{matrix}</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{T}_{01}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}\end{matrix}</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{T}_{00}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{3}\\f_{12}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{6}\\f_{9}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{5}\\f_{10}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{7}\\f_{11}\\f_{13}\\f_{14}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
|- style="background:ghostwhite"<br />
| style="border-top:1px solid black" colspan="2" | <math>\text{Fixed Point Total}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>4\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>4\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>4\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>16\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A4. D''f'' Expanded Over Differential Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A4.} ~~ \mathrm{D}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\mathrm{D}f|_{\mathrm{d}x ~ \mathrm{d}y}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}~\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
y<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
y<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
x<br />
\\<br />
x<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
x<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
y<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
y<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <br />
<math>\begin{matrix}<br />
x<br />
\\<br />
x<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A5. E''f'' Expanded Over Ordinary Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A5.} ~~ \mathrm{E}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\mathrm{E}f|_{xy}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{E}f|_{x \texttt{(} y \texttt{)}}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{E}f|_{\texttt{(} x \texttt{)} y}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{E}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{3}\\f_{12}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{6}\\f_{9}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{5}\\f_{10}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{7}\\f_{11}\\f_{13}\\f_{14}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A6. D''f'' Expanded Over Ordinary Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A6.} ~~ \mathrm{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\mathrm{D}f|_{xy}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{x \texttt{(} y \texttt{)}}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} x \texttt{)} y}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\texttt{(} x \texttt{)}\\\texttt{~} x \texttt{~}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Appendix 2. Differential Forms===<br />
<br />
The actions of the difference operator <math>\mathrm{D}\!</math> and the tangent operator <math>\mathrm{d}\!</math> on the 16 bivariate propositions are shown in Tables&nbsp;A7 and A8.<br />
<br />
Table A7 expands the differential forms that result over a ''logical basis'':<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
|<br />
<math>\{~ \texttt{(}\mathrm{d}x\texttt{)(}\mathrm{d}y\texttt{)}, ~\mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}, ~\texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.\!</math><br />
|}<br />
<br />
This set consists of the singular propositions in the first order differential variables, indicating mutually exclusive and exhaustive ''cells'' of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the cell-basis, point-basis, or singular differential basis. In this setting it is frequently convenient to use the following abbreviations:<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
|<br />
<math>\partial x ~=~ \mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}\!</math> &nbsp; &nbsp; and &nbsp; &nbsp; <math>\partial y ~=~ \texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y.\!</math><br />
|}<br />
<br />
Table A8 expands the differential forms that result over an ''algebraic basis'':<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
| <math>\{~ 1, ~\mathrm{d}x, ~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.\!</math><br />
|}<br />
<br />
This set consists of the ''positive propositions'' in the first order differential variables, indicating overlapping positive regions of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the ''positive differential basis''.<br />
<br />
====Table A7. Differential Forms Expanded on a Logical Basis====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A7.} ~~ \text{Differential Forms Expanded on a Logical Basis}\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| &nbsp;<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" | <math>\mathrm{D}f~\!</math><br />
| <math>\mathrm{d}f~\!</math><br />
|-<br />
| <math>f_{0}\!</math><br />
| style="border-right:none" | <math>\texttt{(~)}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x) ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\\<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\\<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\\<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\partial x<br />
\\<br />
\partial x<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
\\<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\partial x & + & \partial y<br />
\\<br />
\partial x & + & \partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\partial y<br />
\\<br />
\partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x) ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\\<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\\<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\\<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| style="border-right:none" | <math>\texttt{((~))}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A8. Differential Forms Expanded on an Algebraic Basis====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A8.} ~~ \text{Differential Forms Expanded on an Algebraic Basis}\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| &nbsp;<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" | <math>\mathrm{D}f~\!</math><br />
| <math>\mathrm{d}f~\!</math><br />
|-<br />
| <math>f_{0}\!</math><br />
| style="border-right:none" | <math>\texttt{(~)}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x<br />
\\<br />
\mathrm{d}x<br />
\end{matrix}\!</math><br />
| <math>\begin{matrix}<br />
\mathrm{d}x<br />
\\<br />
\mathrm{d}x<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\\<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\\<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}y<br />
\\<br />
\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y<br />
\\<br />
\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| style="border-right:none" | <math>\texttt{((~))}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A9. Tangent Proposition as Pointwise Linear Approximation====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A9.} ~~ \text{Tangent Proposition}~ \mathrm{d}f = \text{Pointwise Linear Approximation to the Difference Map}~ \mathrm{D}f\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}f =<br />
\\[2pt]<br />
\partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}^2\!f =<br />
\\[2pt]<br />
\partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\mathrm{d}f|_{x \, y}</math><br />
| <math>\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math><br />
|-<br />
| style="border-right:none" | <math>f_0\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{5}\\f_{10}\end{matrix}\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\end{matrix}\!</math><br />
| <math>\begin{matrix}<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>f_{15}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A10. Taylor Series Expansion Df = d''f'' + d<sup>2</sup>''f''====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" |<br />
<math>\text{Table A10.} ~~ \text{Taylor Series Expansion}~ {\mathrm{D}f = \mathrm{d}f + \mathrm{d}^2\!f}\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{D}f<br />
\\<br />
= & \mathrm{d}f & + & \mathrm{d}^2\!f<br />
\\<br />
= & \partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y & + & \partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\mathrm{d}f|_{x \, y}</math><br />
| <math>\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math><br />
|-<br />
| style="border-right:none" | <math>f_0\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>f_{15}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A11. Partial Differentials and Relative Differentials====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A11.} ~~ \text{Partial Differentials and Relative Differentials}\!</math><br />
|- style="background:ghostwhite; height:50px"<br />
| &nbsp;<br />
| <math>f\!</math><br />
| <math>\frac{\partial f}{\partial x}\!</math><br />
| <math>\frac{\partial f}{\partial y}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}f =<br />
\\[2pt]<br />
\partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\left. \frac{\partial x}{\partial y} \right| f\!</math><br />
| <math>\left. \frac{\partial y}{\partial x} \right| f\!</math><br />
|-<br />
| <math>f_0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A12. Detail of Calculation for the Difference Map====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:4px double black; border-left:4px double black; border-right:4px double black; border-top:4px double black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A12.} ~~ \text{Detail of Calculation for}~ {\mathrm{E}f + f = \mathrm{D}f}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:6%" | &nbsp;<br />
| style="width:14%; border-left:1px solid black" | <math>f\!</math><br />
| style="width:20%; border-left:4px double black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\mathrm{d}x ~ \mathrm{d}y}<br />
\\[4pt]<br />
+ & f|_{\mathrm{d}x ~ \mathrm{d}y}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\mathrm{d}x ~ \mathrm{d}y}<br />
\end{array}</math><br />
| style="width:20%; border-left:1px solid black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}<br />
\\[4pt]<br />
+ & f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}<br />
\end{array}</math><br />
| style="width:20%; border-left:1px solid black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
+ & f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}<br />
\end{array}</math><br />
| style="width:20%; border-left:1px solid black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
+ & f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}<br />
\end{array}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{0}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:4px double black; border-left:4px double black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{1}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{)(} y \texttt{)~}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{2}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{)~} y \texttt{~~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{4}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~~} x \texttt{~(} y \texttt{)~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{8}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~~} x \texttt{~~} y \texttt{~~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{3}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{(} x \texttt{)}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{12}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>x\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{6}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{,~} y \texttt{)~}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{9}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} x \texttt{,~} y \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{5}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{(} y \texttt{)}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{10}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>y\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{7}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{~~} y \texttt{)~}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{11}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{~(} y \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{13}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} x \texttt{)~} y \texttt{)~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{14}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} x \texttt{)(} y \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{15}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:4px double black; border-left:4px double black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Appendix 3. Computational Details===<br />
<br />
====Operator Maps for the Logical Conjunction ''f''<sub>8</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{8}~\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{8}<br />
& = && f_{8}(u, v)<br />
\\[4pt]<br />
& = && uv<br />
\\[4pt]<br />
& = && uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & uv \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & uv \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{8}<br />
& = && uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & uv \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & uv \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & uv \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.2-i} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{E}f_{8}<br />
& = & f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{8}(\mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{8}(\mathrm{d}u, \mathrm{d}v)<br />
\\[4pt]<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[20pt]<br />
\mathrm{E}f_{8}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
\\[4pt]<br />
&&&&& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&&&&&& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.2-ii} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{c}}<br />
\mathrm{E}f_{8}<br />
& = & (u + \mathrm{d}u) \cdot (v + \mathrm{d}v)<br />
\\[6pt]<br />
& = & u \cdot v<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}f_{8}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{8}<br />
& = && \mathrm{E}f_{8}<br />
& + & \boldsymbol\varepsilon f_{8}<br />
\\[4pt]<br />
& = && f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{8}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & uv<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{~}<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}f_{8}<br />
& = & \boldsymbol\varepsilon f_{8}<br />
& + & \mathrm{E}f_{8}<br />
\\[6pt]<br />
& = & f_{8}(u, v)<br />
& + & f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[6pt]<br />
& = & uv<br />
& + & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
& = & 0<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}f_{8}<br />
& = & 0<br />
& + & u \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.3-iii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 3)}\!</math><br />
|<br />
<math>\begin{array}{c*{9}{l}}<br />
\mathrm{D}f_{8}<br />
& = & \boldsymbol\varepsilon f_{8} ~+~ \mathrm{E}f_{8}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{8}<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ u \,\cdot\, v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~ u \;\cdot\; v \;\cdot\; \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}f_{8}<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u ~ \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} ~ v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot\, \mathrm{d}u ~ \mathrm{d}v<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = & ~ ~ 0 ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~ ~ u ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ ~ ~ v ~~ \cdot ~ \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
=====Computation of d''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.4} ~~ \text{Computation of}~ \mathrm{d}f_{8}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{8}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.5} ~~ \text{Computation of}~ \mathrm{r}f_{8}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{8} & = & \mathrm{D}f_{8} ~+~ \mathrm{d}f_{8}<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[20pt]<br />
\mathrm{r}f_{8}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Conjunction=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.6} ~~ \text{Computation Summary for}~ f_{8}(u, v) = uv\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{8}<br />
& = & uv \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}f_{8}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{r}f_{8}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Operator Maps for the Logical Equality ''f''<sub>9</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{9}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{9}<br />
& = && f_{9}(u, v)<br />
\\[4pt]<br />
& = && \texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{9}(1, 1)<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{9}(1, 0)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{9}(0, 1)<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{9}(0, 0)<br />
\\[4pt]<br />
& = && u v & + & 0 & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{9}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.2} ~~ \text{Computation of}~ \mathrm{E}f_{9}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{E}f_{9}<br />
& = && f_{9}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! f_{9}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! f_{9}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{ } \mathrm{d}v \texttt{ })<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! f_{9}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! f_{9}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{ } \mathrm{d}v \texttt{ })<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{ (} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{) }<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{ (} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{) }<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{E}f_{9}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{9}<br />
& = && \mathrm{E}f_{9}<br />
& + & \boldsymbol\varepsilon f_{9}<br />
\\[4pt]<br />
& = && f_{9}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{9}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{9}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[20pt]<br />
\mathrm{D}f_{9}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}f_{9}<br />
& = & 0 \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & 1 \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & 1 \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & 0 \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of d''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.4} ~~ \text{Computation of}~ \mathrm{d}f_{9}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.5} ~~ \text{Computation of}~ \mathrm{r}f_{9}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{9} & = & \mathrm{D}f_{9} ~+~ \mathrm{d}f_{9}<br />
\\[20pt]<br />
\mathrm{D}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[20pt]<br />
\mathrm{r}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Equality=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.6} ~~ \text{Computation Summary for}~ f_{9}(u, v) = \texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{9}<br />
& = & uv \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}f_{9}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f_{9}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{9}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}f_{9}<br />
& = & uv \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Operator Maps for the Logical Implication ''f''<sub>11</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{11}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{11}<br />
& = && f_{11}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(} u \texttt{(} v \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{11}(1, 1)<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{11}(1, 0)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{11}(0, 1)<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{11}(0, 0)<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ }<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ }<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{11}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.2} ~~ \text{Computation of}~ \mathrm{E}f_{11}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{E}f_{11}<br />
& = && f_{11}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = &&<br />
\texttt{(}<br />
\\<br />
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\\<br />
&&& \texttt{(}<br />
\\<br />
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\<br />
&&& \texttt{))}<br />
\\[4pt]<br />
& = &&<br />
u v<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)((} \mathrm{d}v \texttt{)))}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u \texttt{((} \mathrm{d}v \texttt{)))}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[4pt]<br />
& = &&<br />
u v<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{E}f_{11}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{11}<br />
& = && \mathrm{E}f_{11}<br />
& + & \boldsymbol\varepsilon f_{11}<br />
\\[4pt]<br />
& = && f_{11}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{11}(u, v)<br />
\\[4pt]<br />
& = &&<br />
\texttt{(} \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\texttt{(} \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{(} v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & u v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & 0<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & 0<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = && u v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{c*{9}{l}}<br />
\mathrm{D}f_{11}<br />
& = & \boldsymbol\varepsilon f_{11} ~+~ \mathrm{E}f_{11}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{11}<br />
& = & u v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}f_{11}<br />
& = &<br />
u v<br />
\cdot<br />
\texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\cdot<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\cdot<br />
\texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\cdot<br />
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = &<br />
u v<br />
\cdot<br />
\texttt{~(} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{~}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\cdot<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\cdot<br />
\texttt{~} \mathrm{d}u ~ \mathrm{d}v \texttt{~}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\cdot<br />
\texttt{~} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of d''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.4} ~~ \text{Computation of}~ \mathrm{d}f_{11}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{11}<br />
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{11}<br />
& = & u v \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.5} ~~ \text{Computation of}~ \mathrm{r}f_{11}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{11} & = & \mathrm{D}f_{11} ~+~ \mathrm{d}f_{11}<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{11}<br />
& = & u v \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u<br />
\\[20pt]<br />
\mathrm{r}f_{11}<br />
& = & u v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Implication=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.6} ~~ \text{Computation Summary for}~ f_{11}(u, v) = \texttt{(} u \texttt{(} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{11}<br />
& = & u v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}f_{11}<br />
& = & u v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f_{11}<br />
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{11}<br />
& = & u v \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u<br />
\\[6pt]<br />
\mathrm{r}f_{11}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Operator Maps for the Logical Disjunction ''f''<sub>14</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{14}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{14}<br />
& = && f_{14}(u, v)<br />
\\[4pt]<br />
& = && \texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{14}(1, 1)<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{14}(1, 0)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{14}(0, 1)<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{14}(0, 0)<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ }<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ }<br />
& + & 0<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{14}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.2} ~~ \text{Computation of}~ \mathrm{E}f_{14}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{E}f_{14}<br />
& = && f_{14}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = &&<br />
\texttt{((}<br />
\\<br />
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\\<br />
&&& \texttt{)(}<br />
\\<br />
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\<br />
&&& \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! f_{14}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! f_{14}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{ } \mathrm{d}v \texttt{ })<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! f_{14}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! f_{14}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{ } \mathrm{d}v \texttt{ })<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{E}f_{14}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{14}<br />
& = && \mathrm{E}f_{14}<br />
& + & \boldsymbol\varepsilon f_{14}<br />
\\[4pt]<br />
& = && f_{14}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{14}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{))((} v \texttt{,} \mathrm{d}v \texttt{)))}<br />
& + & \texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{14}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
\\[4pt]<br />
&& + & 0<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
\\[4pt]<br />
&& + & uv \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
\\[20pt]<br />
\mathrm{D}f_{14}<br />
& = && uv \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}f_{14}<br />
& = & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & 0 \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of d''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.4} ~~ \text{Computation of}~ \mathrm{d}f_{14}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.5} ~~ \text{Computation of}~ \mathrm{r}f_{14}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{14} & = & \mathrm{D}f_{14} ~+~ \mathrm{d}f_{14}<br />
\\[20pt]<br />
\mathrm{D}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{d}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[20pt]<br />
\mathrm{r}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Disjunction=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.6} ~~ \text{Computation Summary for}~ f_{14}(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{14}<br />
& = & uv \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 1<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}f_{14}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f_{14}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{d}f_{14}<br />
& = & uv \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}f_{14}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
===Appendix 4. Source Materials===<br />
<br />
===Appendix 5. Various Definitions of the Tangent Vector===<br />
<br />
==References==<br />
<br />
===Works Cited===<br />
<br />
{| cellpadding=3<br />
| valign=top | [AuM]<br />
| Auslander, L., and MacKenzie, R.E., ''Introduction to Differentiable Manifolds'', McGraw-Hill, 1963. Reprinted, Dover, New York, NY, 1977.<br />
|-<br />
| valign=top | [BiG]<br />
| Bishop, R.L., and Goldberg, S.I., ''Tensor Analysis on Manifolds'', Macmillan, 1968. Reprinted, Dover, New York, NY, 1980.<br />
|-<br />
| valign=top | [Boo]<br />
| Boole, G., ''An Investigation of The Laws of Thought'', Macmillan, 1854. Reprinted, Dover, New York, NY, 1958.<br />
|-<br />
| valign=top | [BoT]<br />
| Bott, R., and Tu, L.W., ''Differential Forms in Algebraic Topology'', Springer-Verlag, New York, NY, 1982.<br />
|-<br />
| valign=top | [dCa]<br />
| do Carmo, M.P., ''Riemannian Geometry''. Originally published in Portuguese, 1st editiom 1979, 2nd edition 1988. Translated by F. Flaherty, Birkhäuser, Boston, MA, 1992.<br />
|-<br />
| valign=top | [Che46]<br />
| Chevalley, C., ''Theory of Lie Groups'', Princeton University Press, Princeton, NJ, 1946.<br />
|-<br />
| valign=top | [Che56]<br />
| Chevalley, C., ''Fundamental Concepts of Algebra'', Academic Press, 1956.<br />
|-<br />
| valign=top | [Cho86]<br />
| Chomsky, N., ''Knowledge of Language : Its Nature, Origin, and Use'', Praeger, New York, NY, 1986.<br />
|-<br />
| valign=top | [Cho93]<br />
| Chomsky, N., ''Language and Thought'', Moyer Bell, Wakefield, RI, 1993.<br />
|-<br />
| valign=top | [DoM]<br />
| Doolin, B.F., and Martin, C.F., ''Introduction to Differential Geometry for Engineers'', Marcel Dekker, New York, NY, 1990.<br />
|-<br />
| valign=top | [Fuji]<br />
| Fujiwara, H., ''Logic Testing and Design for Testability'', MIT Press, Cambridge, MA, 1985.<br />
|-<br />
| valign=top | [Hic]<br />
| Hicks, N.J., ''Notes on Differential Geometry'', Van Nostrand, Princeton, NJ, 1965.<br />
|-<br />
| valign=top | [Hir]<br />
| Hirsch, M.W., ''Differential Topology'', Springer-Verlag, New York, NY, 1976.<br />
|-<br />
| valign=top | [How]<br />
| Howard, W.A., "The Formulae-as-Types Notion of Construction", Notes circulated from 1969. Reprinted in [SeH, 479-490].<br />
|-<br />
| valign=top | [JGH]<br />
| Jones, A., Gray, A., and Hutton, R., ''Manifolds and Mechanics'', Cambridge University Press, Cambridge, UK, 1987.<br />
|-<br />
| valign=top | [KoA]<br />
| Kosinski, A.A., ''Differential Manifolds'', Academic Press, San Diego, CA, 1993.<br />
|-<br />
| valign=top | [Koh]<br />
| Kohavi, Z., ''Switching and Finite Automata Theory'', 2nd edition, McGraw-Hill, New York, NY, 1978.<br />
|-<br />
| valign=top | [LaS]<br />
| Lambek, J., and Scott, P.J., ''Introduction to Higher Order Categorical Logic'', Cambridge University Press, Cambridge, UK, 1986.<br />
|-<br />
| valign=top | [La83]<br />
| Lang, S., ''Real Analysis'', 2nd edition, Addison-Wesley, Reading, MA, 1983.<br />
|-<br />
| valign=top | [La84]<br />
| Lang, S., ''Algebra'', 2nd edition, Addison-Wesley, Menlo Park, CA, 1984.<br />
|-<br />
| valign=top | [La85]<br />
| Lang, S., ''Differential Manifolds'', Springer-Verlag, New York, NY, 1985.<br />
|-<br />
| valign=top | [La93]<br />
| Lang, S., ''Real and Functional Analysis'', 3rd edition, Springer-Verlag, New York, NY, 1993.<br />
|-<br />
| valign=top | [Lie80]<br />
| Lie, S., "Sophus Lie's 1880 Transformation Group Paper", in ''Lie Groups : History, Frontiers, and Applications, Volume 1'', translated by M. Ackerman, comments by R. Hermann, Math Sci Press, Brookline, MA, 1975. Original paper 1880.<br />
|-<br />
| valign=top | [Lie84]<br />
| Lie, S., "Sophus Lie's 1884 Differential Invariant Paper", in ''Lie Groups : History, Frontiers, and Applications, Volume 3'', translated by M. Ackerman, comments by R. Hermann, Math Sci Press, Brookline, MA, 1976. Original paper 1884.<br />
|-<br />
| valign=top | [LoS]<br />
| Loomis, L.H., and Sternberg, S., ''Advanced Calculus'', Addison-Wesley, Reading, MA, 1968.<br />
|-<br />
| valign=top | [Mel]<br />
| Melzak, Z.A., ''Companion to Concrete Mathematics, Volume 2 : Mathematical Ideas, Modeling, and Applications'', John Wiley amd Sons, New York, NY, 1976.<br />
|-<br />
| valign=top | [Men]<br />
| Menabrea, L.F., "Sketch of the Analytical Engine Invented by Charles Babbage" with Notes by the Translator, Ada Augusta (Byron), Countess of Lovelace'', in [M&M, 225–297]. Originally published 1842.<br />
|-<br />
| valign=top | [M&M]<br />
| Morrison, P., and Morrison, E. (eds.), ''Charles Babbage on the Principles and Development of the Calculator, and Other Seminal Writings by Charles Babbage and Others, With an Introduction by the Editors'', Dover, Mineola, NY, 1961.<br />
|-<br />
| valign=top | [P1]<br />
| Peirce, C.S., ''Collected Papers of Charles Sanders Peirce'', vols. 1–8, C. Hartshorne, P. Weiss, and A.W. Burks (eds.), Harvard University Press, Cambridge, MA, 1931–1960. Cited as CP [volume].[paragraph].<br />
|-<br />
| valign=top | [P2]<br />
| Peirce, C.S., "Qualitative Logic", in ''The New Elements of Mathematics, Volume 4'', C. Eisele (ed.), Mouton, The Hague, 1976. Cited as NE [volume], [page].<br />
|-<br />
| valign=top | [Rob]<br />
| Roberts, D.D., ''The Existential Graphs of Charles S. Peirce'', Mouton, The Hague, 1973.<br />
|-<br />
| valign=top | [SeH]<br />
| Seldin, J.P., and Hindley, J.R. (eds.), ''To H.B. Curry : Essays on Combinatory Logic, Lambda Calculus, and Formalism'', Academic Press, London, UK, 1980.<br />
|-<br />
| valign=top | [SpB]<br />
| Spencer-Brown, G., ''Laws of Form'', George Allen and Unwin, London, UK, 1969.<br />
|-<br />
| valign=top | [Sp65]<br />
| Spivak, M., ''Calculus on Manifolds : A Modern Approach to Classical Theorems of Advanced Calculus'', W.A. Benjamin, New York, NY, 1965.<br />
|-<br />
| valign=top | [Sp79]<br />
| Spivak, M., ''A Comprehensive Introduction to Differential Geometry'', vols. 1–2. 1st edition 1970. 2nd edition, Publish or Perish Inc., Houston, TX, 1979.<br />
|-<br />
| valign=top | [Sty]<br />
| Styazhkin, N.I., ''History of Mathematical Logic from Leibniz to Peano'', 1st published in Russian, Nauka, Moscow, 1964. MIT Press, Cambridge, MA, 1969.<br />
|-<br />
| valign=top | [Wie]<br />
| Wiener, N., ''Cybernetics : or Control and Communication in the Animal and the Machine'', 1st edition 1948. 2nd edition, MIT Press, Cambridge, MA, 1961.<br />
|}<br />
<br />
===Works Consulted===<br />
<br />
{| cellpadding=3<br />
| valign=top | [Ami]<br />
| Amit, D.J., ''Modeling Brain Function : The World of Attractor Neural Networks'', Cambridge University Press, Cambridge, UK, 1989.<br />
|-<br />
| valign=top | [Ed87]<br />
| Edelman, G.M., ''Neural Darwinism : The Theory of Neuronal Group Selection'', Basic Books, New York, NY, 1987.<br />
|-<br />
| valign=top | [Ed88]<br />
| Edelman, G.M., ''Topobiology : An Introduction to Molecular Embryology'', Basic Books, New York, NY, 1988.<br />
|-<br />
| valign=top | [Fla]<br />
| Flanders, H., ''Differential Forms with Applications to the Physical Sciences'', Academic Press, 1963. Reprinted, Dover, Mineola, NY, 1989. <br />
|-<br />
| valign=top | [Has]<br />
| Hassoun, M.H. (ed.), ''Associative Neural Memories : Theory and Implementation'', Oxford University Press, New York, NY, 1993.<br />
|-<br />
| valign=top | [KoB]<br />
| Kosko, B., ''Neural Networks and Fuzzy Systems : A Dynamical Systems Approach to Machine Intelligence'', Prentice-Hall, Englewood Cliffs, NJ, 1992.<br />
|-<br />
| valign=top | [MaB]<br />
| Mac Lane, S., and Birkhoff, G., ''Algebra'', 3rd edition, Chelsea, New York, NY, 1993.<br />
|-<br />
| valign=top | [Mac]<br />
| Mac Lane, S., ''Categories for the Working Mathematician'', Springer-Verlag, New York, NY, 1971.<br />
|-<br />
| valign=top | [McC]<br />
| McCulloch, W.S., ''Embodiments of Mind'', MIT Press, Cambridge, MA, 1965.<br />
|-<br />
| valign=top | [Mc1]<br />
| McCulloch, W.S., "A Heterarchy of Values Determined by the Topology of Nervous Nets", Bulletin of Mathematical Biophysics, vol. 7 (1945), pp. 89–93. Reprinted in [McC].<br />
|-<br />
| valign=top | [MiP]<br />
| Minsky, M.L., and Papert, S.A., ''Perceptrons : An Introduction to Computational Geometry'', MIT Press, Cambridge, MA, 1969. 2nd printing 1972. Expanded edition 1988.<br />
|-<br />
| valign=top | [Rum]<br />
| Rumelhart, D.E., Hinton, G.E., and McClelland, J.L., "A General Framework for Parallel Distributed Processing" = Chapter 2 in Rumelhart, McClelland, and the PDP Research Group, ''Parallel Distributed Processing, Explorations in the Microstructure of Cognition, Volume 1 : Foundations'', MIT Press, Cambridge, MA, 1986.<br />
|}<br />
<br />
===Incidental Works===<br />
<br />
{| cellpadding=3<br />
| valign=top | [Dew]<br />
| Dewey, John, ''How We Think'', D.C. Heath, Lexington, MA, 1910. Reprinted, Prometheus Books, Buffalo, NY, 1991.<br />
|-<br />
| valign=top | [Fou]<br />
| Foucault, Michel, ''The Archaeology of Knowledge and The Discourse on Language'', A.M. Sheridan-Smith and Rupert Swyer (trans.), Pantheon, New York, NY, 1972. Originally published as ''L´Archéologie du Savoir et L´ordre du discours'', Editions Gallimard, 1969 & 1971.<br />
|-<br />
| valign=top | [Hom]<br />
| Homer, ''The Odyssey'', with an English translation by A.T. Murray, Loeb Classical Library, Harvard University Press, Cambridge, MA, 1980. First printed 1919.<br />
|-<br />
| valign=top | [Jam]<br />
| James, William, ''Pragmatism : A New Name for Some Old Ways of Thinking'', Longmans, Green, and Company, New York, NY, 1907.<br />
|-<br />
| valign=top | [Ler]<br />
| Leroux, Gaston, ''The Phantom of the Opera'', foreword by P. Haining, Dorset Press, New York, NY, 1988. Originally published in French, 1911.<br />
|-<br />
| valign=top | [Mus]<br />
| Musil, Robert, ''The Man Without Qualities'', 3 volumes, translated with a foreword by Eithne Wilkins and Ernst Kaiser, Pan Books, London, UK, 1979. English edition first published by Secker and Warburg, 1954. Originally published in German, ''Der Mann ohne Eigenschaften'', 1930 & 1932.<br />
|-<br />
| valign=top | [PlaR]<br />
| Plato, ''The Republic'', with an English translation by Paul Shorey, Loeb Classical Library, Harvard University Press, Cambridge, MA, 1980. First printed 1930 & 1935.<br />
|-<br />
| valign=top | [PlaS]<br />
| Plato, ''The Sophist'', Loeb Classical Library, William Heinemann, London, 1921, 1987.<br />
|-<br />
| valign=top | [Qui]<br />
| Quine, W.V., ''Mathematical Logic'', 1st edition, 1940. Revised edition, 1951. Harvard University Press, Cambridge, MA, 1981.<br />
|-<br />
| valign=top | [SaD]<br />
| de Santillana, Giorgio, and von Dechend, Hertha, ''Hamlet's Mill : An Essay on Myth and the Frame of Time'', David R. Godine, Publisher, Boston, MA, 1977. 1st published 1969.<br />
|-<br />
| valign=top | [Sha]<br />
| Shakespeare, William, '' William Shakespeare : The Complete Works'', Compact Edition, S. Wells and G. Taylor (eds.), Oxford University Press, Oxford, UK, 1988.<br />
|-<br />
| valign=top | [Sh1]<br />
| Shakespeare, William, ''A Midsummer Night's Dream'', Washington Square Press, New York, NY, 1958.<br />
|-<br />
| valign=top | [Sh2]<br />
| Shakespeare, William, ''The Tragedy of Hamlet, Prince of Denmark'', In [Sha], pp. 654&ndash;690.<br />
|-<br />
| valign=top | [Sh3]<br />
| Shakespeare, William, ''Measure for Measure'', Washington Square Press, New York, NY, 1965.<br />
|-<br />
| valign=top | [Web]<br />
| ''Webster's Ninth New Collegiate Dictionary'', Merriam-Webster, Springfield, MA, 1983.<br />
|-<br />
| valign=top | [Whi]<br />
| Whitman, Walt, ''Leaves of Grass'', Vintage Books / The Library of America, New York, NY, 1992. Originally published in numerous editions, 1855&ndash;1892.<br />
|-<br />
| valign=top | [Wil]<br />
| Wilhelm, R., and Baynes, C.F. (trans.), ''The I Ching, or Book of Changes'', foreword by C.G. Jung, preface by H. Wilhelm, 3rd edition, Bollingen Series XIX, Princeton University Press, Princeton, NJ, 1967.<br />
|}<br />
<br />
==Document History==<br />
<br />
<pre><br />
Author: Jon Awbrey<br />
Created: 16 Dec 1993<br />
Relayed: 31 Oct 1994<br />
Revised: 03 Jun 2003<br />
Recoded: 03 Jun 2007<br />
</pre><br />
<br />
[[Category:Adaptive Systems]]<br />
[[Category:Artificial Intelligence]]<br />
[[Category:Boolean Algebra]]<br />
[[Category:Boolean Functions]]<br />
[[Category:Charles Sanders Peirce]]<br />
[[Category:Combinatorics]]<br />
[[Category:Computer Science]]<br />
[[Category:Cybernetics]]<br />
[[Category:Differential Logic]]<br />
[[Category:Discrete Systems]]<br />
[[Category:Dynamical Systems]]<br />
[[Category:Formal Languages]]<br />
[[Category:Formal Sciences]]<br />
[[Category:Formal Systems]]<br />
[[Category:Functional Logic]]<br />
[[Category:Graph Theory]]<br />
[[Category:Group Theory]]<br />
[[Category:Inquiry]]<br />
[[Category:Knowledge Representation]]<br />
[[Category:Linguistics]]<br />
[[Category:Logic]]<br />
[[Category:Logical Graphs]]<br />
[[Category:Mathematics]]<br />
[[Category:Mathematical Systems Theory]]<br />
[[Category:Philosophy]]<br />
[[Category:Science]]<br />
[[Category:Semiotics]]<br />
[[Category:Systems Science]]<br />
[[Category:Visualization]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Differential_Logic_and_Dynamic_Systems&diff=469886Differential Logic and Dynamic Systems2021-01-14T21:40:52Z<p>Jon Awbrey: /* A Functional Conception of Propositional Calculus */ update</p>
<hr />
<div>'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''<br />
<br />
{| align="center" cellpadding="10"<br />
| [[File:Tangent Functor Ferris Wheel.jpg]]<br />
|}<br />
<br />
{| style="height:36px; width:100%"<br />
| align="left" | ''Stand and unfold yourself.''<br />
| align="right" | Hamlet: Francsico&mdash;1.1.2<br />
|}<br />
<br />
This article develops a differential extension of propositional calculus and applies it to a context of problems arising in dynamic systems.&nbsp; The work pursued here is coordinated with a parallel application that focuses on neural network systems, but the dependencies are arranged to make the present article the main and the more self-contained work, to serve as a conceptual frame and a technical background for the network project.<br />
<br />
==Review and Transition==<br />
<br />
This note continues a previous discussion on the problem of dealing with change and diversity in logic-based intelligent systems. It is useful to begin by summarizing essential material from previous reports.<br />
<br />
Table 1 outlines a notation for propositional calculus based on two types of logical connectives, both of variable <math>k</math>-ary scope.<br />
<br />
* A bracketed list of propositional expressions in the form <math>\texttt{(} e_1, e_2, \ldots, e_{k-1}, e_k \texttt{)}</math> indicates that exactly one of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> is false.<br />
<br />
* A concatenation of propositional expressions in the form <math>e_1 ~ e_2 ~ \ldots ~ e_{k-1} ~ e_k</math> indicates that all of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> are true, in other words, that their [[logical conjunction]] is true.<br />
<br />
All other propositional connectives can be obtained in a very efficient style of representation through combinations of these two forms. Strictly speaking, the concatenation form is dispensable in light of the bracketed form, but it is convenient to maintain it as an abbreviation of more complicated bracket expressions.<br />
<br />
This treatment of propositional logic is derived from the work of C.S. Peirce [P1, P2], who gave this approach an extensive development in his graphical systems of predicate, relational, and modal logic [Rob]. More recently, these ideas were revived and supplemented in an alternative interpretation by George Spencer-Brown [SpB]. Both of these authors used other forms of enclosure where I use parentheses, but the structural topologies of expression and the functional varieties of interpretation are fundamentally the same.<br />
<br />
While working with expressions solely in propositional calculus, it is easiest to use plain parentheses for logical connectives. In contexts where parentheses are needed for other purposes &ldquo;teletype&rdquo; parentheses <math>\texttt{(} \ldots \texttt{)}</math> or barred parentheses <math>(\!| \ldots |\!)</math> may be used for logical operators.<br />
<br />
The briefest expression for logical truth is the empty word, usually denoted by <math>{}^{\backprime\backprime} \boldsymbol\varepsilon {}^{\prime\prime}</math> or <math>{}^{\backprime\backprime} \boldsymbol\lambda {}^{\prime\prime}</math> in formal languages, where it forms the identity element for concatenation. To make it visible in this text, it may be denoted by the equivalent expression <math>{}^{\backprime\backprime} \texttt{((} ~ \texttt{))} {}^{\prime\prime},</math> or, especially if operating in an algebraic context, by a simple <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}.</math> Also when working in an algebraic mode, the plus sign <math>{}^{\backprime\backprime} + {}^{\prime\prime}</math> may be used for [[exclusive disjunction]]. For example, we have the following paraphrases of algebraic expressions by bracket expressions:<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
|<br />
<math>\begin{matrix}<br />
x + y ~=~ \texttt{(} x, y \texttt{)}<br />
\\[6pt]<br />
x + y + z ~=~ \texttt{((} x, y \texttt{)}, z \texttt{)} ~=~ \texttt{(} x, \texttt{(} y, z \texttt{))}<br />
\end{matrix}</math><br />
|}<br />
<br />
It is important to note that the last expressions are not equivalent to the triple bracket <math>\texttt{(} x, y, z \texttt{)}.</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 1.} ~~ \text{Syntax and Semantics of a Calculus for Propositional Logic}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Expression}</math><br />
| <math>\text{Interpretation}</math><br />
| <math>\text{Other Notations}</math><br />
|-<br />
| &nbsp;<br />
| <math>\text{True}</math><br />
| <math>1</math><br />
|-<br />
| <math>\texttt{(} ~ \texttt{)}</math><br />
| <math>\text{False}</math><br />
| <math>0</math><br />
|-<br />
| <math>x</math><br />
| <math>x</math><br />
| <math>x</math><br />
|-<br />
| <math>\texttt{(} x \texttt{)}</math><br />
| <math>\text{Not}~ x</math><br />
|<br />
<math>\begin{matrix}<br />
x'<br />
\\<br />
\tilde{x}<br />
\\<br />
\lnot x<br />
\end{matrix}</math><br />
|-<br />
| <math>x~y~z</math><br />
| <math>x ~\text{and}~ y ~\text{and}~ z</math><br />
| <math>x \land y \land z</math><br />
|-<br />
| <math>\texttt{((} x \texttt{)(} y \texttt{)(} z \texttt{))}</math><br />
| <math>x ~\text{or}~ y ~\text{or}~ z</math><br />
| <math>x \lor y \lor z</math><br />
|-<br />
| <math>\texttt{(} x ~ \texttt{(} y \texttt{))}</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{implies}~ y<br />
\\<br />
\mathrm{If}~ x ~\text{then}~ y<br />
\end{matrix}</math><br />
| <math>x \Rightarrow y</math><br />
|-<br />
| <math>\texttt{(} x \texttt{,} y \texttt{)}</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{not equal to}~ y<br />
\\<br />
x ~\text{exclusive or}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x \ne y<br />
\\<br />
x + y<br />
\end{matrix}</math><br />
|-<br />
| <math>\texttt{((} x \texttt{,} y \texttt{))}</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{is equal to}~ y<br />
\\<br />
x ~\text{if and only if}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x = y<br />
\\<br />
x \Leftrightarrow y<br />
\end{matrix}</math><br />
|-<br />
| <math>\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Just one of}<br />
\\<br />
x, y, z<br />
\\<br />
\text{is false}.<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x'y~z~ & \lor<br />
\\<br />
x~y'z~ & \lor<br />
\\<br />
x~y~z' &<br />
\end{matrix}</math><br />
|-<br />
| <math>\texttt{((} x \texttt{),(} y \texttt{),(} z \texttt{))}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Just one of}<br />
\\<br />
x, y, z<br />
\\<br />
\text{is true}.<br />
\\<br />
&<br />
\\<br />
\text{Partition all}<br />
\\<br />
\text{into}~ x, y, z.<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x~y'z' & \lor<br />
\\<br />
x'y~z' & \lor<br />
\\<br />
x'y'z~ &<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,} y \texttt{),} z \texttt{)}<br />
\\<br />
&<br />
\\<br />
\texttt{(} x \texttt{,(} y \texttt{,} z \texttt{))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Oddly many of}<br />
\\<br />
x, y, z<br />
\\<br />
\text{are true}.<br />
\end{matrix}</math><br />
|<br />
<p><math>x + y + z</math></p><br />
<br><br />
<p><math>\begin{matrix}<br />
x~y~z~ & \lor<br />
\\<br />
x~y'z' & \lor<br />
\\<br />
x'y~z' & \lor<br />
\\<br />
x'y'z~ &<br />
\end{matrix}</math></p><br />
|-<br />
| <math>\texttt{(} w \texttt{,(} x \texttt{),(} y \texttt{),(} z \texttt{))}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Partition}~ w<br />
\\<br />
\text{into}~ x, y, z.<br />
\\<br />
&<br />
\\<br />
\text{Genus}~ w ~\text{comprises}<br />
\\<br />
\text{species}~ x, y, z.<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
w'x'y'z' & \lor<br />
\\<br />
w~x~y'z' & \lor<br />
\\<br />
w~x'y~z' & \lor<br />
\\<br />
w~x'y'z~ &<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
'''Note.''' The usage that one often sees, of a plus sign "<math>+</math>" to represent inclusive disjunction, and the reference to this operation as ''boolean addition'', is a misnomer on at least two counts. Boole used the plus sign to represent exclusive disjunction (at any rate, an operation of aggregation restricted in its logical interpretation to cases where the represented sets are disjoint (Boole, 32)), as any mathematician with a sensitivity to the ring and field properties of algebra would do:<br />
<br />
<blockquote><br />
The expression <math>x + y</math> seems indeed uninterpretable, unless it be assumed that the things represented by <math>x</math> and the things represented by <math>y</math> are entirely separate; that they embrace no individuals in common. (Boole, 66).<br />
</blockquote><br />
<br />
It was only later that Peirce and Jevons treated inclusive disjunction as a fundamental operation, but these authors, with a respect for the algebraic properties that were already associated with the plus sign, used a variety of other symbols for inclusive disjunction (Sty, 177, 189). It seems to have been Schröder who later reassigned the plus sign to inclusive disjunction (Sty, 208). Additional information, discussion, and references can be found in (Boole) and (Sty, 177&ndash;263). Aside from these historical points, which never really count against a current practice that has gained a life of its own, this usage does have a further disadvantage of cutting or confounding the lines of communication between algebra and logic. For this reason, it will be avoided here.<br />
<br />
==A Functional Conception of Propositional Calculus==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Out of the dimness opposite equals advance . . . .<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Always substance and increase,<br><br />
Always a knit of identity . . . . always distinction . . . .<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;always a breed of life.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]<br />
|}<br />
<br />
In the general case, we start with a set of logical features <math>\{a_1, \ldots, a_n\}</math> that represent properties of objects or propositions about the world. In concrete examples the features <math>\{a_i\}</math> commonly appear as capital letters from an ''alphabet'' like <math>\{A, B, C, \ldots\}</math> or as meaningful words from a linguistic ''vocabulary'' of codes. This language can be drawn from any sources, whether natural, technical, or artificial in character and interpretation. In the application to dynamic systems we tend to use the letters <math>\{x_1, \ldots, x_n\}</math> as our coordinate propositions, and to interpret them as denoting properties of a system's ''state'', that is, as propositions about its location in configuration space. Because I have to consider non-deterministic systems from the outset, I often use the word ''state'' in a loose sense, to denote the position or configuration component of a contemplated state vector, whether or not it ever gets a deterministic completion.<br />
<br />
The set of logical features <math>\{a_1, \ldots, a_n\}</math> provides a basis for generating an <math>n</math>-dimensional ''[[universe of discourse]]'' that I denote as <math>[a_1, \ldots, a_n].</math> It is useful to consider each universe of discourse as a unified categorical object that incorporates both the set of points <math>\langle a_1, \ldots, a_n \rangle</math> and the set of propositions <math>f : \langle a_1, \ldots, a_n \rangle \to \mathbb{B}</math> that are implicit with the ordinary picture of a venn diagram on <math>n</math> features. Thus, we may regard the universe of discourse <math>[a_1, \ldots, a_n]</math> as an ordered pair having the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}),</math> and we may abbreviate this last type designation as <math>\mathbb{B}^n\ +\!\to \mathbb{B},</math> or even more succinctly as <math>[\mathbb{B}^n].</math> (Used this way, the angle brackets <math>\langle\ldots\rangle</math> are referred to as ''generator brackets''.)<br />
<br />
Table 2 exhibits the scheme of notation I use to formalize the domain of propositional calculus, corresponding to the logical content of truth tables and venn diagrams. Although it overworks the square brackets a bit, I also use either one of the equivalent notations <math>[n]</math> or <math>\mathbf{n}</math> to denote the data type of a finite set on <math>n</math> elements.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 2.} ~~ \text{Propositional Calculus : Basic Notation}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Symbol}</math><br />
| <math>\text{Notation}</math><br />
| <math>\text{Description}</math><br />
| <math>\text{Type}</math><br />
|-<br />
| <math>\mathfrak{A}</math><br />
| <math>\{ {}^{\backprime\backprime} a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} a_n {}^{\prime\prime} \}</math><br />
| <math>\text{Alphabet}</math><br />
| <math>[n] = \mathbf{n}</math><br />
|-<br />
| <math>\mathcal{A}</math><br />
| <math>\{ a_1, \ldots, a_n \}</math><br />
| <math>\text{Basis}</math><br />
| <math>[n] = \mathbf{n}</math><br />
|-<br />
| <math>A_i</math><br />
| <math>\{ \texttt{(} a_i \texttt{)}, a_i \}</math><br />
| <math>\text{Dimension}~ i</math><br />
| <math>\mathbb{B}</math><br />
|-<br />
| <math>A</math><br />
|<br />
<math>\begin{matrix}<br />
\langle \mathcal{A} \rangle<br />
\\[2pt]<br />
\langle a_1, \ldots, a_n \rangle<br />
\\[2pt]<br />
\{ (a_1, \ldots, a_n) \}<br />
\\[2pt]<br />
A_1 \times \ldots \times A_n<br />
\\[2pt]<br />
\textstyle \prod_{i=1}^n A_i<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Set of cells},<br />
\\[2pt]<br />
\text{coordinate tuples},<br />
\\[2pt]<br />
\text{points, or vectors}<br />
\\[2pt]<br />
\text{in the universe}<br />
\\[2pt]<br />
\text{of discourse}<br />
\end{matrix}</math><br />
| <math>\mathbb{B}^n</math><br />
|-<br />
| <math>A^*</math><br />
| <math>(\mathrm{hom} : A \to \mathbb{B})</math><br />
| <math>\text{Linear functions}</math><br />
| <math>(\mathbb{B}^n)^* \cong \mathbb{B}^n</math><br />
|-<br />
| <math>A^\uparrow</math><br />
| <math>(A \to \mathbb{B})</math><br />
| <math>\text{Boolean functions}</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}</math><br />
|-<br />
| <math>A^\bullet</math><br />
|<br />
<math>\begin{matrix}<br />
[\mathcal{A}]<br />
\\[2pt]<br />
(A, A^\uparrow)<br />
\\[2pt]<br />
(A ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
(A, (A \to \mathbb{B}))<br />
\\[2pt]<br />
[a_1, \ldots, a_n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Universe of discourse}<br />
\\[2pt]<br />
\text{based on the features}<br />
\\[2pt]<br />
\{ a_1, \ldots, a_n \}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))<br />
\\[2pt]<br />
(\mathbb{B}^n ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
[\mathbb{B}^n]<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
===Qualitative Logic and Quantitative Analogy===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
''Logical'', however, is used in a third sense, which is at once more vital and more practical; to denote, namely, the systematic care, negative and positive, taken to safeguard reflection so that it may yield the best results under the given conditions.<br />
| width="4%" | &nbsp;<br />
|- <br />
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 56]<br />
|}<br />
<br />
These concepts and notations may now be explained in greater detail. In order to begin as simply as possible, let us distinguish two levels of analysis and set out initially on the easier path. On the first level of analysis we take spaces like <math>\mathbb{B},</math> <math>\mathbb{B}^n,</math> and <math>(\mathbb{B}^n \to \mathbb{B})</math> at face value and treat them as the primary objects of interest. On the second level of analysis we use these spaces as coordinate charts for talking about points and functions in more fundamental spaces.<br />
<br />
A pair of spaces, of types <math>\mathbb{B}^n</math> and <math>(\mathbb{B}^n \to \mathbb{B}),</math> give typical expression to everything we commonly associate with the ordinary picture of a venn diagram. The dimension, <math>n,</math> counts the number of &ldquo;circles&rdquo; or simple closed curves that are inscribed in the universe of discourse, corresponding to its relevant logical features or basic propositions. Elements of type <math>\mathbb{B}^n</math> correspond to what are often called propositional ''interpretations'' in logic, that is, the different assignments of truth values to sentence letters. Relative to a given universe of discourse, these interpretations are visualized as its ''cells'', in other words, the smallest enclosed areas or undivided regions of the venn diagram. The functions <math>f : \mathbb{B}^n \to \mathbb{B}</math> correspond to the different ways of shading the venn diagram to indicate arbitrary propositions, regions, or sets. Regions included under a shading indicate the ''models'', and regions excluded represent the ''non-models'' of a proposition. To recognize and formalize the natural cohesion of these two layers of concepts into a single universe of discourse, we introduce the type notations <math>[\mathbb{B}^n] = \mathbb{B}^n\ +\!\to \mathbb{B}</math> to stand for the pair of types <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})).</math> The resulting &ldquo;stereotype&rdquo; serves to frame the universe of discourse as a unified categorical object, and makes it subject to prescribed sets of evaluations and transformations (categorical morphisms or ''arrows'') that affect the universe of discourse as an integrated whole.<br />
<br />
Most of the time we can serve the algebraic, geometric, and logical interests of our study without worrying about their occasional conflicts and incidental divergences. The conventions and definitions already set down will continue to cover most of the algebraic and functional aspects of our discussion, but to handle the logical and qualitative aspects we will need to add a few more. In general, abstract sets may be denoted by gothic, greek, or script capital variants of <math>A, B, C,</math> and so on, with their elements being denoted by a corresponding set of subscripted letters in plain lower case, for example, <math>\mathcal{A} = \{a_i\}.</math> Most of the time, a set such as <math>\mathcal{A} = \{a_i\}</math> will be employed as the ''alphabet'' of a [[formal language]]. These alphabet letters serve to name the logical features (properties or propositions) that generate a particular universe of discourse. When we want to discuss the particular features of a universe of discourse, beyond the abstract designation of a type like <math>(\mathbb{B}^n\ +\!\to \mathbb{B}),</math> then we may use the following notations. If <math>\mathcal{A} = \{a_1, \ldots, a_n\}</math> is an alphabet of logical features, then <math>A = \langle \mathcal{A} \rangle = \langle a_1, \ldots, a_n \rangle</math> is the set of interpretations, <math>A^\uparrow = (A \to \mathbb{B})</math> is the set of propositions, and <math>A^\bullet = [\mathcal{A}] = [a_1, \ldots, a_n]</math> is the combination of these interpretations and propositions into the universe of discourse that is based on the features <math>\{a_1, \ldots, a_n\}.</math><br />
<br />
As always, especially in concrete examples, these rules may be dropped whenever necessary, reverting to a free assortment of feature labels. However, when we need to talk about the logical aspects of a space that is already named as a vector space, it will be necessary to make special provisions. At any rate, these elaborations can be deferred until actually needed.<br />
<br />
===Philosophy of Notation : Formal Terms and Flexible Types===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Where number is irrelevant, regimented mathematical technique has hitherto tended to be lacking. Thus it is that the progress of natural science has depended so largely upon the discernment of measurable quantity of one sort or another.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]<br />
|}<br />
<br />
For much of our discussion propositions and boolean functions are treated as the same formal objects, or as different interpretations of the same formal calculus. This rule of interpretation has exceptions, though. There is a distinctively logical interest in the use of propositional calculus that is not exhausted by its functional interpretation. It is part of our task in this study to deal with these uniquely logical characteristics as they present themselves both in our subject matter and in our formal calculus. Just to provide a hint of what's at stake: In logic, as opposed to the more imaginative realms of mathematics, we consider it a good thing to always know what we are talking about. Where mathematics encourages tolerance for uninterpreted symbols as intermediate terms, logic exerts a keener effort to interpret directly each oblique carrier of meaning, no matter how slight, and to unfold the complicities of every indirection in the flow of information. Translated into functional terms, this means that we want to maintain a continual, immediate, and persistent sense of both the converse relation <math>f^{-1} \subseteq \mathbb{B} \times \mathbb{B}^n,</math> or what is the same thing, <math>f^{-1} : \mathbb{B} \to \mathcal{P}(\mathbb{B}^n),</math> and the ''fibers'' or inverse images <math>f^{-1}(0)</math> and <math>f^{-1}(1),</math> associated with each boolean function <math>f : \mathbb{B}^n \to \mathbb{B}</math> that we use. In practical terms, the desired implementation of a propositional interpreter should incorporate our intuitive recognition that the induced partition of the functional domain into level sets <math>f^{-1}(b),</math> for <math>b \in \mathbb{B},</math> is part and parcel of understanding the denotative uses of each propositional function <math>f.</math><br />
<br />
===Special Classes of Propositions===<br />
<br />
It is important to remember that the coordinate propositions <math>\{a_i\},</math> besides being projection maps <math>a_i : \mathbb{B}^n \to \mathbb{B},</math> are propositions on an equal footing with all others, even though employed as a basis in a particular moment. This set of <math>n</math> propositions may sometimes be referred to as the ''basic propositions'', the ''coordinate propositions'', or the ''simple propositions'' that found a universe of discourse. Either one of the equivalent notations, <math>\{a_i : \mathbb{B}^n \to \mathbb{B}\}</math> or <math>(\mathbb{B}^n \xrightarrow{i} \mathbb{B}),</math> may be used to indicate the adoption of the propositions <math>a_i</math> as a basis for describing a universe of discourse.<br />
<br />
Among the <math>2^{2^n}</math> propositions in <math>[a_1, \ldots, a_n]</math> are several families of <math>2^n</math> propositions each that take on special forms with respect to the basis <math>\{ a_1, \ldots, a_n \}.</math> Three of these families are especially prominent in the present context, the ''linear'', the ''positive'', and the ''singular'' propositions. Each family is naturally parameterized by the coordinate <math>n</math>-tuples in <math>\mathbb{B}^n</math> and falls into <math>n + 1</math> ranks, with a binomial coefficient <math>\tbinom{n}{k}</math> giving the number of propositions that have rank or weight <math>k.</math><br />
<br />
<ul><br />
<br />
<li><br />
<p>The ''linear propositions'', <math>\{ \ell : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{\ell} \mathbb{B}),</math> may be written as sums:</p><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\sum_{i=1}^n e_i ~=~ e_1 + \ldots + e_n<br />
~\text{where}~<br />
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 0 \end{matrix}\right\}<br />
~\text{for}~ i = 1 ~\text{to}~ n.</math><br />
|}<br />
</li><br />
<br />
<li><br />
<p>The ''positive propositions'', <math>\{ p : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{p} \mathbb{B}),</math> may be written as products:</p><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\prod_{i=1}^n e_i ~=~ e_1 \cdot \ldots \cdot e_n<br />
~\text{where}~<br />
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 1 \end{matrix}\right\}<br />
~\text{for}~ i = 1 ~\text{to}~ n.</math><br />
|}<br />
</li><br />
<br />
<li><br />
<p>The ''singular propositions'', <math>\{ \mathbf{x} : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{s} \mathbb{B}),</math> may be written as products:</p><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\prod_{i=1}^n e_i ~=~ e_1 \cdot \ldots \cdot e_n<br />
~\text{where}~<br />
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = \texttt{(} a_i \texttt{)} \end{matrix}\right\}<br />
~\text{for}~ i = 1 ~\text{to}~ n.</math><br />
|}<br />
</li><br />
<br />
</ul><br />
<br />
In each case the rank <math>k</math> ranges from <math>0</math> to <math>n</math> and counts the number of positive appearances of the coordinate propositions <math>a_1, \ldots, a_n</math> in the resulting expression. For example, for <math>{n = 3},</math> the linear proposition of rank <math>0</math> is <math>0,</math> the positive proposition of rank <math>0</math> is <math>1,</math> and the singular proposition of rank <math>0</math> is <math>\texttt{(} a_1 \texttt{)(} a_2 \texttt{)(} a_3\texttt{)}.</math><br />
<br />
The basic propositions <math>a_i : \mathbb{B}^n \to \mathbb{B}</math> are both linear and positive. So these two kinds of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions.<br />
<br />
Linear propositions and positive propositions are generated by taking boolean sums and products, respectively, over selected subsets of basic propositions, so both families of propositions are parameterized by the powerset <math>\mathcal{P}(\mathcal{I}),</math> that is, the set of all subsets <math>J</math> of the basic index set <math>\mathcal{I} = \{1, \ldots, n\}.</math><br />
<br />
Let us define <math>\mathcal{A}_J</math> as the subset of <math>\mathcal{A}</math> that is given by <math>\{a_i : i \in J\}.</math> Then we may comprehend the action of the linear and the positive propositions in the following terms:<br />
<br />
* The linear proposition <math>\ell_J : \mathbb{B}^n \to \mathbb{B}</math> evaluates each cell <math>\mathbf{x}</math> of <math>\mathbb{B}^n</math> by looking at the coefficients of <math>\mathbf{x}</math> with respect to the features that <math>\ell_J</math> "likes", namely those in <math>\mathcal{A}_J,</math> and then adds them up in <math>\mathbb{B}.</math> Thus, <math>\ell_J(\mathbf{x})</math> computes the parity of the number of features that <math>\mathbf{x}</math> has in <math>\mathcal{A}_J,</math> yielding one for odd and zero for even. Expressed in this idiom, <math>\ell_J(\mathbf{x}) = 1</math> says that <math>\mathbf{x}</math> seems ''odd'' (or ''oddly true'') to <math>\mathcal{A}_J,</math> whereas <math>\ell_J(\mathbf{x}) = 0</math> says that <math>\mathbf{x}</math> seems ''even'' (or ''evenly true'') to <math>\mathcal{A}_J,</math> so long as we recall that ''zero times'' is evenly often, too.<br />
<br />
* The positive proposition <math>p_J : \mathbb{B}^n \to \mathbb{B}</math> evaluates each cell <math>\mathbf{x}</math> of <math>\mathbb{B}^n</math> by looking at the coefficients of <math>\mathbf{x}</math> with regard to the features that <math>p_J</math> "likes", namely those in <math>\mathcal{A}_J,</math> and then takes their product in <math>\mathbb{B}.</math> Thus, <math>p_J(\mathbf{x})</math> assesses the unanimity of the multitude of features that <math>\mathbf{x}</math> has in <math>\mathcal{A}_J,</math> yielding one for all and aught for else. In these consensual or contractual terms, <math>p_J(\mathbf{x}) = 1</math> means that <math>\mathbf{x}</math> is ''AOK'' or congruent with all of the conditions of <math>\mathcal{A}_J,</math> while <math>p_J(\mathbf{x}) = 0</math> means that <math>\mathbf{x}</math> defaults or dissents from some condition of <math>\mathcal{A}_J.</math><br />
<br />
===Basis Relativity and Type Ambiguity===<br />
<br />
Finally, two things are important to keep in mind with regard to the simplicity, linearity, positivity, and singularity of propositions.<br />
<br />
First, all of these properties are relative to a particular basis. For example, a singular proposition with respect to a basis <math>\mathcal{A}</math> will not remain singular if <math>\mathcal{A}</math> is extended by a number of new and independent features. Even if we stick to the original set of pairwise options <math>\{a_i\} \cup \{ \texttt{(} a_i \texttt{)} \}</math> to select a new basis, the sets of linear and positive propositions are determined by the choice of simple propositions, and this determination is tantamount to the conventional choice of a cell as origin.<br />
<br />
Second, the singular propositions <math>\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B},</math> picking out as they do a single cell or a coordinate tuple <math>\mathbf{x}</math> of <math>\mathbb{B}^n,</math> become the carriers or the vehicles of a certain type-ambiguity that vacillates between the dual forms <math>\mathbb{B}^n</math> and <math>(\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B})</math> and infects the whole hierarchy of types built on them. In other words, the terms that signify the interpretations <math>\mathbf{x} : \mathbb{B}^n</math> and the singular propositions <math>\mathbf{x} : \mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B}</math> are fully equivalent in information, and this means that every token of the type <math>\mathbb{B}^n</math> can be reinterpreted as an appearance of the subtype <math>\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B}.</math> And vice versa, the two types can be exchanged with each other everywhere that they turn up. In practical terms, this allows the use of singular propositions as a way of denoting points, forming an alternative to coordinate tuples.<br />
<br />
For example, relative to the universe of discourse <math>[a_1, a_2, a_3]</math> the singular proposition <math>a_1 a_2 a_3 : \mathbb{B}^3 \xrightarrow{s} \mathbb{B}</math> could be explicitly retyped as <math>a_1 a_2 a_3 : \mathbb{B}^3</math> to indicate the point <math>(1, 1, 1)</math> but in most cases the proper interpretation could be gathered from context. Both notations remain dependent on a particular basis, but the code that is generated under the singular option has the advantage in its self-commenting features, in other words, it constantly reminds us of its basis in the process of denoting points. When the time comes to put a multiplicity of different bases into play, and to search for objects and properties that remain invariant under the transformations between them, this infinitesimal potential advantage may well evolve into an overwhelming practical necessity.<br />
<br />
===The Analogy Between Real and Boolean Types===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Measurement consists in correlating our subject matter with the series of real numbers; and such correlations are desirable because, once they are set up, all the well-worked theory of numerical mathematics lies ready at hand as a tool for our further reasoning.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]<br />
|}<br />
<br />
There are two further reasons why it useful to spend time on a careful treatment of types, and they both have to do with our being able to take full computational advantage of certain dimensions of flexibility in the types that apply to terms. First, the domains of differential geometry and logic programming are connected by analogies between real and boolean types of the same pattern. Second, the types involved in these patterns have important isomorphisms connecting them that apply on both the real and the boolean sides of the picture.<br />
<br />
Amazingly enough, these isomorphisms are themselves schematized by the axioms and theorems of propositional logic. This fact is known as the ''propositions as types'' analogy or the Curry&ndash;Howard isomorphism [How]. In another formulation it says that terms are to types as proofs are to propositions. See [LaS, 42&ndash;46] and [SeH] for a good discussion and further references. To anticipate the bearing of these issues on our immediate topic, Table&nbsp;3 sketches a partial overview of the Real to Boolean analogy that may serve to illustrate the paradigm in question.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 3.} ~~ \text{Analogy Between Real and Boolean Types}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Real Domain} ~ \mathbb{R}</math><br />
| <math>\longleftrightarrow</math><br />
| <math>\text{Boolean Domain} ~ \mathbb{B}</math><br />
|-<br />
| <math>\mathbb{R}^n</math><br />
| <math>\text{Basic Space}</math><br />
| <math>\mathbb{B}^n</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}</math><br />
| <math>\text{Function Space}</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R}</math><br />
| <math>\text{Tangent Vector}</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math><br />
|-<br />
| <math>\mathbb{R}^n \to ((\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R})</math><br />
| <math>\text{Vector Field}</math><br />
| <math>\mathbb{B}^n \to ((\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B})</math><br />
|-<br />
| <math>(\mathbb{R}^n \times (\mathbb{R}^n \to \mathbb{R})) \to \mathbb{R}</math><br />
| '''<font size="4">"</font>'''<br />
| <math>(\mathbb{B}^n \times (\mathbb{B}^n \to \mathbb{B})) \to \mathbb{B}</math><br />
|-<br />
| <math>((\mathbb{R}^n \to \mathbb{R}) \times \mathbb{R}^n) \to \mathbb{R}</math><br />
| '''<font size="4">"</font>'''<br />
| <math>((\mathbb{B}^n \to \mathbb{B}) \times \mathbb{B}^n) \to \mathbb{B}</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^n \to \mathbb{R})</math><br />
| <math>\text{Derivation}</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^n \to \mathbb{B})</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}^m</math><br />
| <math>\begin{matrix}\text{Basic}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}^m</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^m \to \mathbb{R})</math><br />
| <math>\begin{matrix}\text{Function}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^m \to \mathbb{B})</math><br />
|}<br />
<br />
<br><br />
<br />
The Table exhibits a sample of likely parallels between the real and boolean domains. The central column gives a selection of terminology that is borrowed from differential geometry and extended in its meaning to the logical side of the Table. These are the varieties of spaces that come up when we turn to analyzing the dynamics of processes that pursue their courses through the states of an arbitrary space <math>X.</math> Moreover, when it becomes necessary to approach situations of overwhelming dynamic complexity in a succession of qualitative reaches, then the methods of logic that are afforded by the boolean domains, with their declarative means of synthesis and deductive modes of analysis, supply a natural battery of tools for the task.<br />
<br />
It is usually expedient to take these spaces two at a time, in dual pairs of the form <math>X</math> and <math>(X \to \mathbb{K}).</math> In general, one creates pairs of type schemas by replacing any space <math>X</math> with its dual <math>(X \to \mathbb{K}),</math> for example, pairing the type <math>X \to Y</math> with the type <math>(X \to \mathbb{K}) \to (Y \to \mathbb{K}),</math> and <math>X \times Y</math> with <math>(X \to \mathbb{K}) \times (Y \to \mathbb{K}).</math> The word ''dual'' is used here in its broader sense to mean all of the functionals, not just the linear ones. Given any function <math>f : X \to \mathbb{K},</math> the ''converse'' or inverse relation corresponding to <math>f</math> is denoted <math>f^{-1},</math> and the subsets of <math>X</math> that are defined by <math>f^{-1}(k),</math> taken over <math>k</math> in <math>\mathbb{K},</math> are called the ''fibers'' or the ''level sets'' of the function <math>f.</math><br />
<br />
===Theory of Control and Control of Theory===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
You will hardly know who I am or what I mean,<br><br />
But I shall be good health to you nevertheless,<br><br />
And filter and fibre your blood.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]<br />
|}<br />
<br />
In the boolean context a function <math>f : X \to \mathbb{B}</math> is tantamount to a ''proposition'' about elements of <math>X,</math> and the elements of <math>X</math> constitute the ''interpretations'' of that proposition. The fiber <math>f^{-1}(1)</math> comprises the set of ''models'' of <math>f,</math> or examples of elements in <math>X</math> satisfying the proposition <math>f.</math> The fiber <math>f^{-1}(0)</math> collects the complementary set of ''anti-models'', or the exceptions to the proposition <math>f</math> that exist in <math>X.</math> Of course, the space of functions <math>(X \to \mathbb{B})</math> is isomorphic to the set of all subsets of <math>X,</math> called the ''power set'' of <math>X,</math> and often denoted <math>\mathcal{P}(X)</math> or <math>2^X.</math><br />
<br />
The operation of replacing <math>X</math> by <math>(X \to \mathbb{B})</math> in a type schema corresponds to a certain shift of attitude towards the space <math>X,</math> in which one passes from a focus on the ostensibly individual elements of <math>X</math> to a concern with the states of information and uncertainty that one possesses about objects and situations in <math>X.</math> The conceptual obstacles in the path of this transition can be smoothed over by using singular functions <math>(\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B})</math> as stepping stones. First of all, it's an easy step from an element <math>\mathbf{x}</math> of type <math>\mathbb{B}^n</math> to the equivalent information of a singular proposition <math>\mathbf{x} : X \xrightarrow{s} \mathbb{B}, </math> and then only a small jump of generalization remains to reach the type of an arbitrary proposition <math>f : X \to \mathbb{B},</math> perhaps understood to indicate a relaxed constraint on the singularity of points or a neighborhood circumscribing the original <math>\mathbf{x}.</math> This is frequently a useful transformation, communicating between the ''objective'' and the ''intentional'' perspectives, in spite perhaps of the open objection that this distinction is transient in the mean time and ultimately superficial.<br />
<br />
It is hoped that this measure of flexibility, allowing us to stretch a point into a proposition, can be useful in the examination of inquiry driven systems, where the differences between empirical, intentional, and theoretical propositions constitute the discrepancies and the distributions that drive experimental activity. I can give this model of inquiry a cybernetic cast by realizing that theory change and theory evolution, as well as the choice and the evaluation of experiments, are actions that are taken by a system or its agent in response to the differences that are detected between observational contents and theoretical coverage.<br />
<br />
All of the above notwithstanding, there are several points that distinguish these two tasks, namely, the ''theory of control'' and the ''control of theory'', features that are often obscured by too much precipitation in the quickness with which we understand their similarities. In the control of uncertainty through inquiry, some of the actuators that we need to be concerned with are axiom changers and theory modifiers, operators with the power to compile and to revise the theories that generate expectations and predictions, effectors that form and edit our grammars for the languages of observational data, and agencies that rework the proposed model to fit the actual sequences of events and the realized relationships of values that are observed in the environment. Moreover, when steps must be taken to carry out an experimental action, there must be something about the particular shape of our uncertainty that guides us in choosing what directions to explore, and this impression is more than likely influenced by previous accumulations of experience. Thus it must be anticipated that much of what goes into scientific progress, or any sustainable effort toward a goal of knowledge, is necessarily predicated on long term observation and modal expectations, not only on the more local or short term prediction and correction.<br />
<br />
===Propositions as Types and Higher Order Types===<br />
<br />
The types collected in Table&nbsp;3 (repeated below) serve to illustrate the themes of ''higher order propositional expressions'' and the ''propositions as types'' (PAT) analogy.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 3.} ~~ \text{Analogy Between Real and Boolean Types}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Real Domain} ~ \mathbb{R}</math><br />
| <math>\longleftrightarrow</math><br />
| <math>\text{Boolean Domain} ~ \mathbb{B}</math><br />
|-<br />
| <math>\mathbb{R}^n</math><br />
| <math>\text{Basic Space}</math><br />
| <math>\mathbb{B}^n</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}</math><br />
| <math>\text{Function Space}</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R}</math><br />
| <math>\text{Tangent Vector}</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math><br />
|-<br />
| <math>\mathbb{R}^n \to ((\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R})</math><br />
| <math>\text{Vector Field}</math><br />
| <math>\mathbb{B}^n \to ((\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B})</math><br />
|-<br />
| <math>(\mathbb{R}^n \times (\mathbb{R}^n \to \mathbb{R})) \to \mathbb{R}</math><br />
| '''<font size="4">"</font>'''<br />
| <math>(\mathbb{B}^n \times (\mathbb{B}^n \to \mathbb{B})) \to \mathbb{B}</math><br />
|-<br />
| <math>((\mathbb{R}^n \to \mathbb{R}) \times \mathbb{R}^n) \to \mathbb{R}</math><br />
| '''<font size="4">"</font>'''<br />
| <math>((\mathbb{B}^n \to \mathbb{B}) \times \mathbb{B}^n) \to \mathbb{B}</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^n \to \mathbb{R})</math><br />
| <math>\text{Derivation}</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^n \to \mathbb{B})</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}^m</math><br />
| <math>\begin{matrix}\text{Basic}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}^m</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^m \to \mathbb{R})</math><br />
| <math>\begin{matrix}\text{Function}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^m \to \mathbb{B})</math><br />
|}<br />
<br />
<br><br />
<br />
First, observe that the type of a ''tangent vector at a point'', also known as a ''directional derivative'' at that point, has the form <math>(\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K},</math> where <math>\mathbb{K}</math> is the chosen ground field, in the present case either <math>\mathbb{R}</math> or <math>\mathbb{B}.</math> At a point in a space of type <math>\mathbb{K}^n,</math> a directional derivative operator <math>\vartheta</math> takes a function on that space, an <math>f</math> of type <math>(\mathbb{K}^n \to \mathbb{K}),</math> and maps it to a ground field value of type <math>\mathbb{K}.</math> This value is known as the ''derivative'' of <math>f</math> in the direction <math>\vartheta</math> [Che46, 76&ndash;77]. In the boolean case <math>\vartheta : (\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math> has the form of a proposition about propositions, in other words, a proposition of the next higher type.<br />
<br />
Next, by way of illustrating the propositions as types idea, consider a proposition of the form <math>X \Rightarrow (Y \Rightarrow Z).</math> One knows from propositional calculus that this is logically equivalent to a proposition of the form <math>(X \land Y) \Rightarrow Z.</math> But this equivalence should remind us of the functional isomorphism that exists between a construction of the type <math>X \to (Y \to Z)</math> and a construction of the type <math>(X \times Y) \to Z.</math> The propositions as types analogy permits us to take a functional type like this and, under the right conditions, replace the functional arrows &ldquo;<math>\to</math>&rdquo; and products &ldquo;<math>\times</math>&rdquo; with the respective logical arrows &ldquo;<math>\Rightarrow</math>&rdquo; and products &ldquo;<math>\land</math>&rdquo;. Accordingly, viewing the result as a proposition, we can employ axioms and theorems of propositional calculus to suggest appropriate isomorphisms among the categorical and functional constructions.<br />
<br />
Finally, examine the middle four rows of Table&nbsp;3. These display a series of isomorphic types that stretch from the categories that are labeled ''Vector Field'' to those that are labeled ''Derivation''. A ''vector field'', also known as an ''infinitesimal transformation'', associates a tangent vector at a point with each point of a space. In symbols, a vector field is a function of the form <math>\textstyle \xi : X \to \bigcup_{x \in X} \xi_x</math> that assigns to each point <math>x</math> of the space <math>X</math> a tangent vector to <math>X</math> at that point, namely, the tangent vector <math>\xi_x</math> [Che46, 82&ndash;83]. If <math>X</math> is of the type <math>\mathbb{K}^n,</math> then <math>\xi</math> is of the type <math>\mathbb{K}^n \to ((\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K}).</math> This has the pattern <math>X \to (Y \to Z),</math> with <math>X = \mathbb{K}^n,</math> <math>Y = (\mathbb{K}^n \to \mathbb{K}),</math> and <math>Z = \mathbb{K}.</math><br />
<br />
Applying the propositions as types analogy, one can follow this pattern through a series of metamorphoses from the type of a vector field to the type of a derivation, as traced out in Table&nbsp;4. Observe how the function <math>f : X \to \mathbb{K},</math> associated with the place of <math>Y</math> in the pattern, moves through its paces from the second to the first position. In this way, the vector field <math>\xi,</math> initially viewed as attaching each tangent vector <math>\xi_x</math> to the site <math>x</math> where it acts in <math>X,</math> now comes to be seen as acting on each scalar potential <math>f : X \to \mathbb{K}</math> like a generalized species of differentiation, producing another function <math>\xi f : X \to \mathbb{K}</math> of the same type.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 4.} ~~ \text{An Equivalence Based on the Propositions as Types Analogy}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Pattern}</math><br />
| <math>\text{Construct}</math><br />
| <math>\text{Instance}</math><br />
|-<br />
| <math>X \to (Y \to Z)</math><br />
| <math>\text{Vector Field}</math><br />
| <math>\mathbb{K}^n \to ((\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K})</math><br />
|-<br />
| <math>(X \times Y) \to Z</math><br />
| <math>\Uparrow</math><br />
| <math>(\mathbb{K}^n \times (\mathbb{K}^n \to \mathbb{K})) \to \mathbb{K}</math><br />
|-<br />
| <math>(Y \times X) \to Z</math><br />
| <math>\Downarrow</math><br />
| <math>((\mathbb{K}^n \to \mathbb{K}) \times \mathbb{K}^n) \to \mathbb{K}</math><br />
|-<br />
| <math>Y \to (X \to Z)</math><br />
| <math>\text{Derivation}</math><br />
| <math>(\mathbb{K}^n \to \mathbb{K}) \to (\mathbb{K}^n \to \mathbb{K})</math><br />
|}<br />
<br />
<br><br />
<br />
===Reality at the Threshold of Logic===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
But no science can rest entirely on measurement, and many scientific investigations are quite out of reach of that device. To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]<br />
|}<br />
<br />
Table 5 accumulates an array of notation that I hope will not be too distracting. Some of it is rarely needed, but has been filled in for the sake of completeness. Its purpose is simple, to give literal expression to the visual intuitions that come with venn diagrams, and to help build a bridge between our qualitative and quantitative outlooks on dynamic systems.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 5.} ~~ \text{A Bridge Over Troubled Waters}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| align="center" | <math>\text{Linear Space}</math><br />
| align="center" | <math>\text{Liminal Space}</math><br />
| align="center" | <math>\text{Logical Space}</math><br />
|-<br />
| <math>\begin{matrix}\mathcal{X} & = & \{ x_1, \ldots, x_n \}\end{matrix}</math><br />
| <math>\begin{matrix}\underline{\mathcal{X}} & = & \{ \underline{x}_1, \ldots, \underline{x}_n \}\end{matrix}</math><br />
| <math>\begin{matrix}\mathcal{A} & = & \{ a_1, \ldots, a_n \}\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X_i & = & \langle x_i \rangle<br />
\\<br />
& \cong & \mathbb{K}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}_i & = & \{ \texttt{(} \underline{x}_i \texttt{)}, \underline{x}_i \}<br />
\\<br />
& \cong & \mathbb{B}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A_i & = & \{ \texttt{(} a_i \texttt{)}, a_i \}<br />
\\<br />
& \cong & \mathbb{B}<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X<br />
\\<br />
= & \langle \mathcal{X} \rangle<br />
\\<br />
= & \langle x_1, \ldots, x_n \rangle<br />
\\<br />
= & X_1 \times \ldots \times X_n<br />
\\<br />
= & \prod_{i=1}^n X_i<br />
\\<br />
\cong & \mathbb{K}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}<br />
\\<br />
= & \langle \underline{\mathcal{X}} \rangle<br />
\\<br />
= & \langle \underline{x}_1, \ldots, \underline{x}_n \rangle<br />
\\<br />
= & \underline{X}_1 \times \ldots \times \underline{X}_n<br />
\\<br />
= & \prod_{i=1}^n \underline{X}_i<br />
\\<br />
\cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A<br />
\\<br />
= & \langle \mathcal{A} \rangle<br />
\\<br />
= & \langle a_1, \ldots, a_n \rangle<br />
\\<br />
= & A_1 \times \ldots \times A_n<br />
\\<br />
= & \prod_{i=1}^n A_i<br />
\\<br />
\cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X^* & = & (\ell : X \to \mathbb{K})<br />
\\<br />
& \cong & \mathbb{K}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}^* & = & (\ell : \underline{X} \to \mathbb{B})<br />
\\<br />
& \cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A^* & = & (\ell : A \to \mathbb{B})<br />
\\<br />
& \cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X^\uparrow & = & (X \to \mathbb{K})<br />
\\<br />
& \cong & (\mathbb{K}^n \to \mathbb{K})<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}^\uparrow & = & (\underline{X} \to \mathbb{B})<br />
\\<br />
& \cong & (\mathbb{B}^n \to \mathbb{B})<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A^\uparrow & = & (A \to \mathbb{B})<br />
\\<br />
& \cong & (\mathbb{B}^n \to \mathbb{B})<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X^\bullet<br />
\\<br />
= & [\mathcal{X}]<br />
\\<br />
= & [x_1, \ldots, x_n]<br />
\\<br />
= & (X, X^\uparrow)<br />
\\<br />
= & (X ~+\!\to \mathbb{K})<br />
\\<br />
= & (X, (X \to \mathbb{K}))<br />
\\<br />
\cong & (\mathbb{K}^n, (\mathbb{K}^n \to \mathbb{K}))<br />
\\<br />
= & (\mathbb{K}^n ~+\!\to \mathbb{K})<br />
\\<br />
= & [\mathbb{K}^n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}^\bullet<br />
\\<br />
= & [\underline{\mathcal{X}}]<br />
\\<br />
= & [\underline{x}_1, \ldots, \underline{x}_n]<br />
\\<br />
= & (\underline{X}, \underline{X}^\uparrow)<br />
\\<br />
= & (\underline{X} ~+\!\to \mathbb{B})<br />
\\<br />
= & (\underline{X}, (\underline{X} \to \mathbb{B}))<br />
\\<br />
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))<br />
\\<br />
= & (\mathbb{B}^n ~+\!\to \mathbb{B})<br />
\\<br />
= & [\mathbb{B}^n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A^\bullet<br />
\\<br />
= & [\mathcal{A}]<br />
\\<br />
= & [a_1, \ldots, a_n]<br />
\\<br />
= & (A, A^\uparrow)<br />
\\<br />
= & (A ~+\!\to \mathbb{B})<br />
\\<br />
= & (A, (A \to \mathbb{B}))<br />
\\<br />
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))<br />
\\<br />
= & (\mathbb{B}^n ~+\!\to \mathbb{B})<br />
\\<br />
= & [\mathbb{B}^n]<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
The left side of the Table collects mostly standard notation for an <math>n</math>-dimensional vector space over a field <math>\mathbb{K}.</math> The right side of the table repeats the first elements of a notation that I sketched above, to be used in further developments of propositional calculus. (I plan to use this notation in the logical analysis of neural network systems.) The middle column of the table is designed as a transitional step from the case of an arbitrary field <math>\mathbb{K},</math> with a special interest in the continuous line <math>\mathbb{R},</math> to the qualitative and discrete situations that are instanced and typified by <math>\mathbb{B}.</math><br />
<br />
I now proceed to explain these concepts in more detail. The most important ideas developed in Table&nbsp;5 are these:<br />
<br />
* The idea of a universe of discourse, which includes both a space of ''points'' and a space of ''maps'' on those points.<br />
<br />
* The idea of passing from a more complex universe to a simpler universe by a process of ''thresholding'' each dimension of variation down to a single bit of information.<br />
<br />
For the sake of concreteness, let us suppose that we start with a continuous <math>n</math>-dimensional vector space like <math>X = \langle x_1, \ldots, x_n \rangle \cong \mathbb{R}^n.</math> The coordinate system <math>\mathcal{X} = \{x_i\}</math> is a set of maps <math>x_i : \mathbb{R}^n \to \mathbb{R},</math> also known as the ''coordinate projections''. Given a "dataset" of points <math>\mathbf{x}</math> in <math>\mathbb{R}^n,</math> there are numerous ways of sensibly reducing the data down to one bit for each dimension. One strategy that is general enough for our present purposes is as follows. For each <math>i</math> we choose an <math>n</math>-ary relation <math>L_i</math> on <math>\mathbb{R}^n,</math> that is, a subset of <math>\mathbb{R}^n,</math> and then we define the <math>i^\mathrm{th}</math> threshold map, or ''limen'' <math>\underline{x}_i</math> as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\underline{x}_i : \mathbb{R}^n \to \mathbb{B}\ \text{such that:}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
\underline{x}_i(\mathbf{x}) = 1 & \text{if} & \mathbf{x} \in L_i,<br />
\\[4pt]<br />
\underline{x}_i(\mathbf{x}) = 0 & \text{if} & \mathbf{x} \not\in L_i.<br />
\end{matrix}</math><br />
|}<br />
<br />
In other notations that are sometimes used, the operator <math>\chi (\ldots)</math> or the corner brackets <math>\lceil\ldots\rceil</math> can be used to denote a ''characteristic function'', that is, a mapping from statements to their truth values in <math>\mathbb{B}.</math> Finally, it is not uncommon to use the name of the relation itself as a predicate that maps <math>n</math>-tuples into truth values. Thus we have the following notational variants of the above definition:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
\underline{x}_i (\mathbf{x}) & = & \chi (\mathbf{x} \in L_i) & = & \lceil \mathbf{x} \in L_i \rceil & = & L_i (\mathbf{x}).<br />
\end{matrix}</math><br />
|}<br />
<br />
Notice that, as defined here, there need be no actual relation between the <math>n</math>-dimensional subsets <math>\{L_i\}</math> and the coordinate axes corresponding to <math>\{x_i\},</math> aside from the circumstance that the two sets have the same cardinality. In concrete cases, though, one usually has some reason for associating these "volumes" with these "lines", for instance, <math>L_i</math> is bounded by some hyperplane that intersects the <math>i^\text{th}</math> axis at a unique threshold value <math>r_i \in \mathbb{R}.</math> Often, the hyperplane is chosen normal to the axis. In recognition of this motive, let us make the following convention. When the set <math>L_i</math> has points on the <math>i^\text{th}</math> axis, that is, points of the form <math>(0, \ldots, 0, r_i, 0, \ldots, 0)</math> where only the <math>x_i</math> coordinate is possibly non-zero, we may pick any one of these coordinate values as a parametric index of the relation. In this case we say that the indexing is ''real'', otherwise the indexing is ''imaginary''. For a knowledge based system <math>X,</math> this should serve once again to mark the distinction between ''acquaintance'' and ''opinion''.<br />
<br />
States of knowledge about the location of a system or about the distribution of a population of systems in a state space <math>X = \mathbb{R}^n</math> can now be expressed by taking the set <math>\underline{\mathcal{X}} = \{\underline{x}_i\}</math> as a basis of logical features. In picturesque terms, one may think of the underscore and the subscript as combining to form a subtextual spelling for the <math>i^\text{th}</math> threshold map. This can help to remind us that the ''threshold operator'' <math>(\underline{~})_i</math> acts on <math>\mathbf{x}</math> by setting up a kind of a &ldquo;hurdle&rdquo; for it. In this interpretation the coordinate proposition <math>\underline{x}_i</math> asserts that the representative point <math>\mathbf{x}</math> resides ''above'' the <math>i^\mathrm{th}</math> threshold.<br />
<br />
Primitive assertions of the form <math>\underline{x}_i (\mathbf{x})</math> may then be negated and joined by means of propositional connectives in the usual ways to provide information about the state <math>\mathbf{x}</math> of a contemplated system or a statistical ensemble of systems. Parentheses <math>\texttt{(} \ldots \texttt{)}</math> may be used to indicate logical negation. Eventually one discovers the usefulness of the <math>k</math>-ary ''just one false'' operators of the form <math>\texttt{(} a_1 \texttt{,} \ldots \texttt{,} a_k \texttt{)},</math> as treated in earlier reports. This much tackle generates a space of points (cells, interpretations), <math>\underline{X} \cong \mathbb{B}^n,</math> and a space of functions (regions, propositions), <math>\underline{X}^\uparrow \cong (\mathbb{B}^n \to \mathbb{B}).</math> Together these form a new universe of discourse <math>\underline{X}^\bullet</math> of the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})),</math> which we may abbreviate as <math>\mathbb{B}^n\ +\!\to \mathbb{B}</math> or most succinctly as <math>[\mathbb{B}^n].</math><br />
<br />
The square brackets have been chosen to recall the rectangular frame of a venn diagram. In thinking about a universe of discourse it is a good idea to keep this picture in mind, graphically illustrating the links among the elementary cells <math>\underline{\mathbf{x}},</math> the defining features <math>\underline{x}_i,</math> and the potential shadings <math>f : \underline{X} \to \mathbb{B}</math> all at the same time, not to mention the arbitrariness of the way we choose to inscribe our distinctions in the medium of a continuous space.<br />
<br />
Finally, let <math>X^*</math> denote the space of linear functions, <math>(\ell : X \to \mathbb{K}),</math> which has in the finite case the same dimensionality as <math>X,</math> and let the same notation be extended across the Table.<br />
<br />
We have just gone through a lot of work, apparently doing nothing more substantial than spinning a complex spell of notational devices through a labyrinth of baffled spaces and baffling maps. The reason for doing this was to bind together and to constitute the intuitive concept of a universe of discourse into a coherent categorical object, the kind of thing, once grasped, that can be turned over in the mind and considered in all its manifold changes and facets. The effort invested in these preliminary measures is intended to pay off later, when we need to consider the state transformations and the time evolution of neural network systems.<br />
<br />
===Tables of Propositional Forms===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope. It provides explicit techniques for manipulating the most basic ingredients of discourse.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7&ndash;8]<br />
|}<br />
<br />
To prepare for the next phase of discussion, Tables&nbsp;6 and 7 collect and summarize all of the propositional forms on one and two variables. These propositional forms are represented over bases of boolean variables as complete sets of boolean-valued functions. Adjacent to their names and specifications are listed what are roughly the simplest expressions in the ''[[Cactus_Language_&bull;_Overview|cactus language]]'', the particular syntax for propositional calculus that I use in formal and computational contexts. For the sake of orientation, the English paraphrases and the more common notations are listed in the last two columns. As simple and circumscribed as these low-dimensional universes may appear to be, a careful exploration of their differential extensions will involve us in complexities sufficient to demand our attention for some time to come.<br />
<br />
Propositional forms on one variable correspond to boolean functions <math>f : \mathbb{B}^1 \to \mathbb{B}.</math> In Table&nbsp;6 these functions are listed in a variant form of [[truth table]], one in which the axes of the usual arrangement are rotated through a right angle. Each function <math>f_i</math> is indexed by the string of values that it takes on the points of the universe <math>X^\bullet = [x] \cong \mathbb{B}^1.</math> The binary index generated in this way is converted to its decimal equivalent and these are used as conventional names for the <math>f_i,</math> as shown in the first column of the Table. In their own right the <math>2^1</math> points of the universe <math>X^\bullet</math> are coordinated as a space of type <math>\mathbb{B}^1,</math> this in light of the universe <math>X^\bullet</math> being a functional domain where the coordinate projection <math>x</math> takes on its values in <math>\mathbb{B}.</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 6.} ~~ \text{Propositional Forms on One Variable}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon</math><br />
| <math>1~0</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_0</math><br />
| <math>f_{00}</math><br />
| <math>0~0</math><br />
| <math>\texttt{(} ~ \texttt{)}</math><br />
| <math>\text{false}</math><br />
| <math>0</math><br />
|-<br />
| <math>f_1</math><br />
| <math>f_{01}</math><br />
| <math>0~1</math><br />
| <math>\texttt{(} x \texttt{)}</math><br />
| <math>\text{not}~ x</math><br />
| <math>\lnot x</math><br />
|-<br />
| <math>f_2</math><br />
| <math>f_{10}</math><br />
| <math>1~0</math><br />
| <math>x</math><br />
| <math>x</math><br />
| <math>x</math><br />
|-<br />
| <math>f_3</math><br />
| <math>f_{11}</math><br />
| <math>1~1</math><br />
| <math>\texttt{((} ~ \texttt{))}</math><br />
| <math>\text{true}</math><br />
| <math>1</math><br />
|}<br />
<br />
<br><br />
<br />
Propositional forms on two variables correspond to boolean functions <math>f : \mathbb{B}^2 \to \mathbb{B}.</math> In Table&nbsp;7 each function <math>f_i</math> is indexed by the values that it takes on the points of the universe <math>X^\bullet = [x, y] \cong \mathbb{B}^2.</math> Converting the binary index thus generated to a decimal equivalent, we obtain the functional nicknames that are listed in the first column. The <math>2^2</math> points of the universe <math>X^\bullet</math> are coordinated as a space of type <math>\mathbb{B}^2,</math> as indicated under the heading of the Table, where the coordinate projections <math>x</math> and <math>y</math> run through the various combinations of their values in <math>\mathbb{B}.</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 7-a.} ~~ \text{Propositional Forms on Two Variables}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}</math><br />
| style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon</math><br />
| <math>1~1~0~0</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon</math><br />
| <math>1~0~1~0</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[4pt]<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\\[4pt]<br />
f_{3}<br />
\\[4pt]<br />
f_{4}<br />
\\[4pt]<br />
f_{5}<br />
\\[4pt]<br />
f_{6}<br />
\\[4pt]<br />
f_{7}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0000}<br />
\\[4pt]<br />
f_{0001}<br />
\\[4pt]<br />
f_{0010}<br />
\\[4pt]<br />
f_{0011}<br />
\\[4pt]<br />
f_{0100}<br />
\\[4pt]<br />
f_{0101}<br />
\\[4pt]<br />
f_{0110}<br />
\\[4pt]<br />
f_{0111}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~0<br />
\\[4pt]<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~0~1~1<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
0~1~0~1<br />
\\[4pt]<br />
0~1~1~0<br />
\\[4pt]<br />
0~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{)} ~ y ~<br />
\\[4pt]<br />
\texttt{(} x \texttt{)}<br />
\\[4pt]<br />
~ x ~ \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{,} ~ y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x ~ y \texttt{)}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{false}<br />
\\[4pt]<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\[4pt]<br />
y ~\text{without}~ x<br />
\\[4pt]<br />
\text{not}~ x<br />
\\[4pt]<br />
x ~\text{without}~ y<br />
\\[4pt]<br />
\text{not}~ y<br />
\\[4pt]<br />
x ~\text{not equal to}~ y<br />
\\[4pt]<br />
\text{not both}~ x ~\text{and}~ y<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
\lnot x \land \lnot y<br />
\\[4pt]<br />
\lnot x \land y<br />
\\[4pt]<br />
\lnot x<br />
\\[4pt]<br />
x \land \lnot y<br />
\\[4pt]<br />
\lnot y<br />
\\[4pt]<br />
x \ne y<br />
\\[4pt]<br />
\lnot x \lor \lnot y<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{8}<br />
\\[4pt]<br />
f_{9}<br />
\\[4pt]<br />
f_{10}<br />
\\[4pt]<br />
f_{11}<br />
\\[4pt]<br />
f_{12}<br />
\\[4pt]<br />
f_{13}<br />
\\[4pt]<br />
f_{14}<br />
\\[4pt]<br />
f_{15}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1000}<br />
\\[4pt]<br />
f_{1001}<br />
\\[4pt]<br />
f_{1010}<br />
\\[4pt]<br />
f_{1011}<br />
\\[4pt]<br />
f_{1100}<br />
\\[4pt]<br />
f_{1101}<br />
\\[4pt]<br />
f_{1110}<br />
\\[4pt]<br />
f_{1111}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
1~0~0~0<br />
\\[4pt]<br />
1~0~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\\[4pt]<br />
1~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x ~ y<br />
\\[4pt]<br />
\texttt{((} x \texttt{,} ~ y \texttt{))}<br />
\\[4pt]<br />
y<br />
\\[4pt]<br />
\texttt{(} x ~ \texttt{(} y \texttt{))}<br />
\\[4pt]<br />
x<br />
\\[4pt]<br />
\texttt{((} x \texttt{)} ~ y \texttt{)}<br />
\\[4pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
\texttt{((} ~ \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x ~\text{and}~ y<br />
\\[4pt]<br />
x ~\text{equal to}~ y<br />
\\[4pt]<br />
y<br />
\\[4pt]<br />
\text{not}~ x ~\text{without}~ y<br />
\\[4pt]<br />
x<br />
\\[4pt]<br />
\text{not}~ y ~\text{without}~ x<br />
\\[4pt]<br />
x ~\text{or}~ y<br />
\\[4pt]<br />
\text{true}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x \land y<br />
\\[4pt]<br />
x = y<br />
\\[4pt]<br />
y<br />
\\[4pt]<br />
x \Rightarrow y<br />
\\[4pt]<br />
x<br />
\\[4pt]<br />
x \Leftarrow y<br />
\\[4pt]<br />
x \lor y<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 7-b.} ~~ \text{Propositional Forms on Two Variables}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}</math><br />
| style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon</math><br />
| <math>1~1~0~0</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon</math><br />
| <math>1~0~1~0</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_{0}</math><br />
| <math>f_{0000}</math><br />
| <math>0~0~0~0</math><br />
| <math>\texttt{(} ~ \texttt{)}</math><br />
| <math>\text{false}</math><br />
| <math>0</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\\[4pt]<br />
f_{4}<br />
\\[4pt]<br />
f_{8}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0001}<br />
\\[4pt]<br />
f_{0010}<br />
\\[4pt]<br />
f_{0100}<br />
\\[4pt]<br />
f_{1000}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
1~0~0~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{)} ~ y ~<br />
\\[4pt]<br />
~ x ~ \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
~ x ~~ y ~<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\[4pt]<br />
y ~\text{without}~ x<br />
\\[4pt]<br />
x ~\text{without}~ y<br />
\\[4pt]<br />
x ~\text{and}~ y<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot x \land \lnot y<br />
\\[4pt]<br />
\lnot x \land y<br />
\\[4pt]<br />
x \land \lnot y<br />
\\[4pt]<br />
x \land y<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{3}<br />
\\[4pt]<br />
f_{12}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0011}<br />
\\[4pt]<br />
f_{1100}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\[4pt]<br />
x<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{not}~ x<br />
\\[4pt]<br />
x<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot x<br />
\\[4pt]<br />
x<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[4pt]<br />
f_{9}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0110}<br />
\\[4pt]<br />
f_{1001}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[4pt]<br />
1~0~0~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{,} y \texttt{)}<br />
\\[4pt]<br />
\texttt{((} x \texttt{,} y \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x ~\text{not equal to}~ y<br />
\\[4pt]<br />
x ~\text{equal to}~ y<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x \ne y<br />
\\[4pt]<br />
x = y<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{5}<br />
\\[4pt]<br />
f_{10}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0101}<br />
\\[4pt]<br />
f_{1010}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\[4pt]<br />
y<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{not}~ y<br />
\\[4pt]<br />
y<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot y<br />
\\[4pt]<br />
y<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[4pt]<br />
f_{11}<br />
\\[4pt]<br />
f_{13}<br />
\\[4pt]<br />
f_{14}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0111}<br />
\\[4pt]<br />
f_{1011}<br />
\\[4pt]<br />
f_{1101}<br />
\\[4pt]<br />
f_{1110}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} ~ x ~~ y ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{(} ~ x ~ \texttt{(} y \texttt{))}<br />
\\[4pt]<br />
\texttt{((} x \texttt{)} ~ y ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{not both}~ x ~\text{and}~ y<br />
\\[4pt]<br />
\text{not}~ x ~\text{without}~ y<br />
\\[4pt]<br />
\text{not}~ y ~\text{without}~ x<br />
\\[4pt]<br />
x ~\text{or}~ y<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot x \lor \lnot y<br />
\\[4pt]<br />
x \Rightarrow y<br />
\\[4pt]<br />
x \Leftarrow y<br />
\\[4pt]<br />
x \lor y<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}</math><br />
| <math>f_{1111}</math><br />
| <math>1~1~1~1</math><br />
| <math>\texttt{((} ~ \texttt{))}</math><br />
| <math>\text{true}</math><br />
| <math>1</math><br />
|}<br />
<br />
<br><br />
<br />
==A Differential Extension of Propositional Calculus==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Fire over water:<br><br />
The image of the condition before transition.<br><br />
Thus the superior man is careful<br><br />
In the differentiation of things,<br><br />
So that each finds its place.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; ''I Ching'', Hexagram 64, [Wil, 249]<br />
|}<br />
<br />
This much preparation is enough to begin introducing my subject, if I excuse myself from giving full arguments for my definitional choices until some later stage. I am trying to develop a ''differential theory of qualitative equations'' that parallels the application of differential geometry to dynamic systems. The idea of a tangent vector is key to this work and a major goal is to find the right logical analogues of tangent spaces, bundles, and functors. The strategy is taken of looking for the simplest versions of these constructions that can be discovered within the realm of propositional calculus, so long as they serve to fill out the general theme.<br />
<br />
===Differential Propositions : Qualitative Analogues of Differential Equations===<br />
<br />
In order to define the differential extension of a universe of discourse <math>[\mathcal{A}],</math> the initial alphabet <math>\mathcal{A}</math> must be extended to include a collection of symbols for ''differential features'', or basic ''changes'' that are capable of occurring in <math>[\mathcal{A}].</math> Intuitively, these symbols may be construed as denoting primitive features of change, qualitative attributes of motion, or propositions about how things or points in the universe may be changing or moving with respect to the features that are noted in the initial alphabet.<br />
<br />
Therefore, let us define the corresponding ''differential alphabet'' or ''tangent alphabet'' as <math>\mathrm{d}\mathcal{A}\!</math> <math>=\!</math> <math>\{\mathrm{d}a_1, \ldots, \mathrm{d}a_n\},</math> in principle just an arbitrary alphabet of symbols, disjoint from the initial alphabet <math>\mathcal{A}\!</math> <math>=\!</math> <math>\{ a_1, \ldots, a_n \},\!</math> that is intended to be interpreted in the way just indicated. It only remains to be understood that the precise interpretation of the symbols in <math>\mathrm{d}\mathcal{A}\!</math> is often conceived to be changeable from point to point of the underlying space <math>A.\!</math> Indeed, for all we know, the state space <math>A\!</math> might well be the state space of a language interpreter, one that is concerned, among other things, with the idiomatic meanings of the dialect generated by <math>\mathcal{A}</math> and <math>\mathrm{d}\mathcal{A}.\!</math><br />
<br />
The ''tangent space'' to <math>A\!</math> at one of its points <math>x,\!</math> sometimes written <math>\mathrm{T}_x(A),</math> takes the form <math>\mathrm{d}A</math> <math>=\!</math> <math>\langle \mathrm{d}\mathcal{A} \rangle</math> <math>=\!</math> <math>\langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle.\!</math> Strictly speaking, the name ''cotangent space'' is probably more correct for this construction, but the fact that we take up spaces and their duals in pairs to form our universes of discourse allows our language to be pliable here.<br />
<br />
Proceeding as we did with the base space <math>A,\!</math> the tangent space <math>\mathrm{d}A</math> at a point of <math>A\!</math> can be analyzed as a product of distinct and independent factors:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\mathrm{d}A ~=~ \prod_{i=1}^n \mathrm{d}A_i ~=~ \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n.\!</math><br />
|}<br />
<br />
Here, <math>\mathrm{d}A_i\!</math> is a set of two differential propositions, <math>\mathrm{d}A_i = \{ (\mathrm{d}a_i), \mathrm{d}a_i \},\!</math> where <math>\texttt{(} \mathrm{d}a_i \texttt{)}\!</math> is a proposition with the logical value of <math>\text{not} ~ \mathrm{d}a_i.\!</math> Each component <math>\mathrm{d}A_i\!</math> has the type <math>\mathbb{B},\!</math> operating under the ordered correspondence <math>\{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \} \cong \{ 0, 1 \}.\!</math> However, clarity is often served by acknowledging this differential usage with a superficially distinct type <math>\mathbb{D},\!</math> whose intension may be indicated as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\mathbb{D} = \{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \} = \{ \text{same}, \text{different} \} = \{ \text{stay}, \text{change} \} = \{ \text{stop}, \text{step} \}.\!</math><br />
|}<br />
<br />
Viewed within a coordinate representation, spaces of type <math>\mathbb{B}^n\!</math> and <math>\mathbb{D}^n\!</math> may appear to be identical sets of binary vectors, but taking a view at this level of abstraction would be like ignoring the qualitative units and the diverse dimensions that distinguish position and momentum, or the different roles of quantity and impulse.<br />
<br />
===An Interlude on the Path===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
There would have been no beginnings: instead, speech would proceed from me, while I stood in its path &ndash; a slender gap &ndash; the point of its possible disappearance.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
A sense of the relation between <math>\mathbb{B}</math> and <math>\mathbb{D}</math> may be obtained by considering the ''path classifier'' (or the ''equivalence class of curves'') approach to tangent vectors. Consider a universe <math>[\mathcal{X}].\!</math> Given the boolean value system, a path in the space <math>X = \langle \mathcal{X} \rangle</math> is a map <math>q : \mathbb{B} \to X.</math> In this context the set of paths <math>(\mathbb{B} \to X)</math> is isomorphic to the cartesian square <math>X^2 = X \times X,</math> or the set of ordered pairs chosen from <math>X.\!</math><br />
<br />
We may analyze <math>X^2 = \{ (u, v) : u, v \in X \}</math> into two parts, specifically, the ordered pairs <math>(u, v)\!</math> that lie on and off the diagonal:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}X^2 & = & \{ (u, v) : u = v \} & \cup & \{ (u, v) : u \ne v \}.\end{matrix}</math><br />
|}<br />
<br />
This partition may also be expressed in the following symbolic form:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}X^2 & \cong & \operatorname{diag} (X) & + & 2 \binom{X}{2}.\end{matrix}</math><br />
|}<br />
<br />
The separate terms of this formula are defined as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}\operatorname{diag} (X) & = & \{ (x, x) : x \in X \}.\end{matrix}\!</math><br />
|}<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}\binom{X}{k} & = & X ~\text{choose}~ k & = & \{ k\text{-sets from}~ X \}.\end{matrix}\!</math><br />
|}<br />
<br />
Thus we have:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}\binom{X}{2} & = & \{ \{ u, v \} : u, v \in X \}.\end{matrix}</math><br />
|}<br />
<br />
We may now use the features in <math>\mathrm{d}\mathcal{X} = \{ \mathrm{d}x_i \} = \{ \mathrm{d}x_1, \ldots, \mathrm{d}x_n \}</math> to classify the paths of <math>(\mathbb{B} \to X)</math> by way of the pairs in <math>X^2.\!</math> If <math>X \cong \mathbb{B}^n,</math> then a path <math>q\!</math> in <math>X\!</math> has the following form:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
q : (\mathbb{B} \to \mathbb{B}^n) & \cong & \mathbb{B}^n \times \mathbb{B}^n & \cong & \mathbb{B}^{2n} & \cong & (\mathbb{B}^2)^n.<br />
\end{matrix}</math><br />
|}<br />
<br />
Intuitively, we want to map this <math>(\mathbb{B}^2)^n</math> onto <math>\mathbb{D}^n</math> by mapping each component <math>\mathbb{B}^2</math> onto a copy of <math>\mathbb{D}.</math> But in the presenting context <math>{}^{\backprime\backprime} \mathbb{D} {}^{\prime\prime}</math> is just a name associated with, or an incidental quality attributed to, coefficient values in <math>\mathbb{B}</math> when they are attached to features in <math>\mathrm{d}\mathcal{X}.</math><br />
<br />
Taking these intentions into account, define <math>\mathrm{d}x_i : X^2 \to \mathbb{B}</math> in the following manner:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcrcl}<br />
\mathrm{d}x_i(u, v)<br />
& = & \texttt{(} ~ x_i(u) & \texttt{,} & x_i(v) ~ \texttt{)}<br />
\\<br />
& = & x_i(u) & + & x_i(v)<br />
\\<br />
& = & x_i(v) & - & x_i(u).<br />
\end{array}</math><br />
|}<br />
<br />
In the above transcription, the operator bracket of the form <math>\texttt{(} \ldots \texttt{,} \ldots \texttt{)}\!</math> is a ''cactus lobe'', in general signifying that just one of the arguments listed is false. In the case of two arguments this is the same thing as saying that the arguments are not equal. The plus sign signifies boolean addition, in the sense of addition in <math>\mathrm{GF}(2),</math> and thus means the same thing in this context as the minus sign, in the sense of adding the additive inverse.<br />
<br />
The above definition of <math>\mathrm{d}x_i : X^2 \to \mathbb{B}</math> is equivalent to defining <math>\mathrm{d}x_i : (\mathbb{B} \to X) \to \mathbb{B}\!</math> in the following way:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcrcl}<br />
\mathrm{d}x_i (q)<br />
& = & \texttt{(} ~ x_i(q_0) & \texttt{,} & x_i(q_1) ~ \texttt{)}<br />
\\<br />
& = & x_i(q_0) & + & x_i(q_1)<br />
\\<br />
& = & x_i(q_1) & - & x_i(q_0).<br />
\end{array}</math><br />
|}<br />
<br />
In this definition <math>q_b = q(b),\!</math> for each <math>b\!</math> in <math>\mathbb{B}.</math> Thus, the proposition <math>\mathrm{d}x_i</math> is true of the path <math>q = (u, v)\!</math> exactly if the terms of <math>q,\!</math> the endpoints <math>u\!</math> and <math>v,\!</math> lie on different sides of the question <math>x_i.\!</math><br />
<br />
The language of features in <math>\langle \mathrm{d}\mathcal{X} \rangle,</math> indeed the whole calculus of propositions in <math>[\mathrm{d}\mathcal{X}],</math> may now be used to classify paths and sets of paths. In other words, the paths can be taken as models of the propositions <math>g : \mathrm{d}X \to \mathbb{B}.</math> For example, the paths corresponding to <math>\mathrm{diag}(X)</math> fall under the description <math>\texttt{(} \mathrm{d}x_1 \texttt{)} \cdots \texttt{(} \mathrm{d}x_n \texttt{)},\!</math> which says that nothing changes against the backdrop of the coordinate frame <math>\{ x_1, \ldots, x_n \}.\!</math><br />
<br />
Finally, a few words of explanation may be in order. If this concept of a path appears to be described in a roundabout fashion, it is because I am trying to avoid using any assumption of vector space properties for the space <math>X\!</math> that contains its range. In many ways the treatment is still unsatisfactory, but improvements will have to wait for the introduction of substitution operators acting on singular propositions.<br />
<br />
===The Extended Universe of Discourse===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
At the moment of speaking, I would like to have perceived a nameless voice, long preceding me, leaving me merely to enmesh myself in it, taking up its cadence, and to lodge myself, when no one was looking, in its interstices as if it had paused an instant, in suspense, to beckon to me.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
Next we define the ''extended alphabet'' or ''bundled alphabet'' <math>\mathrm{E}\mathcal{A}</math> as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lclcl}<br />
\mathrm{E}\mathcal{A}<br />
& = & \mathcal{A} \cup \mathrm{d}\mathcal{A}<br />
& = & \{ a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}.<br />
\end{array}</math><br />
|}<br />
<br />
This supplies enough material to construct the ''differential extension'' <math>\mathrm{E}A,</math> or the ''tangent bundle'' over the initial space <math>A,\!</math> in the following fashion:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcl}<br />
\mathrm{E}A<br />
& = & \langle \mathrm{E}\mathcal{A} \rangle<br />
\\[4pt]<br />
& = & \langle \mathcal{A} \cup \mathrm{d}\mathcal{A} \rangle<br />
\\[4pt]<br />
& = & \langle a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle,<br />
\end{array}</math><br />
|}<br />
<br />
and also:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcl}<br />
\mathrm{E}A<br />
& = & A \times \mathrm{d}A<br />
\\[4pt]<br />
& = & A_1 \times \ldots \times A_n \times \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n.<br />
\end{array}</math><br />
|}<br />
<br />
This gives <math>\mathrm{E}A</math> the type <math>\mathbb{B}^n \times \mathbb{D}^n.</math><br />
<br />
Finally, the tangent universe <math>\mathrm{E}A^\bullet = [\mathrm{E}\mathcal{A}]\!</math> is constituted from the totality of points and maps, or interpretations and propositions, that are based on the extended set of features <math>\mathrm{E}\mathcal{A},</math> and this fact is summed up in the following notation:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lclcl}<br />
\mathrm{E}A^\bullet<br />
& = & [\mathrm{E}\mathcal{A}]<br />
& = & [a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n].<br />
\end{array}</math><br />
|}<br />
<br />
This gives the tangent universe <math>\mathrm{E}A^\bullet\!</math> the type:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcl}<br />
(\mathbb{B}^n \times \mathbb{D}^n\ +\!\to \mathbb{B})<br />
& = & (\mathbb{B}^n \times \mathbb{D}^n, (\mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B})).<br />
\end{array}</math><br />
|}<br />
<br />
A proposition in the tangent universe <math>[\mathrm{E}\mathcal{A}]</math> is called a ''differential proposition'' and forms the analogue of a system of differential equations, constraints, or relations in ordinary calculus.<br />
<br />
With these constructions, the differential extension <math>\mathrm{E}A</math> and the space of differential propositions <math>(\mathrm{E}A \to \mathbb{B}),\!</math> we have arrived, in main outline, at one of the major subgoals of this study. Table&nbsp;8 summarizes the concepts that have been introduced for working with differentially extended universes of discourse.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 8.} ~~ \text{Differential Extension : Basic Notation}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Symbol}\!</math><br />
| <math>\text{Notation}\!</math><br />
| <math>\text{Description}\!</math><br />
| <math>\text{Type}\!</math><br />
|-<br />
| <math>\mathrm{d}\mathfrak{A}\!</math><br />
| <math>\{ {}^{\backprime\backprime} \mathrm{d}a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} \mathrm{d}a_n {}^{\prime\prime} \}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Alphabet of}<br />
\\[2pt]<br />
\text{differential symbols}<br />
\end{matrix}</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>\mathrm{d}\mathcal{A}\!</math><br />
| <math>\{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Basis of}<br />
\\[2pt]<br />
\text{differential features}<br />
\end{matrix}</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>\mathrm{d}A_i\!</math><br />
| <math>\{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \}\!</math><br />
| <math>\text{Differential dimension}~ i\!</math><br />
| <math>\mathbb{D}\!</math><br />
|-<br />
| <math>\mathrm{d}A\!</math><br />
|<br />
<math>\begin{matrix}<br />
\langle \mathrm{d}\mathcal{A} \rangle<br />
\\[2pt]<br />
\langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle<br />
\\[2pt]<br />
\{ (\mathrm{d}a_1, \ldots, \mathrm{d}a_n) \}<br />
\\[2pt]<br />
\mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n<br />
\\[2pt]<br />
\textstyle \prod_i \mathrm{d}A_i<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Tangent space at a point:}<br />
\\[2pt]<br />
\text{Set of changes, motions,}<br />
\\[2pt]<br />
\text{steps, tangent vectors}<br />
\\[2pt]<br />
\text{at a point}<br />
\end{matrix}</math><br />
| <math>\mathbb{D}^n\!</math><br />
|-<br />
| <math>\mathrm{d}A^*\!</math><br />
| <math>(\mathrm{hom} : \mathrm{d}A \to \mathbb{B})\!</math><br />
| <math>\text{Linear functions on}~ \mathrm{d}A\!</math><br />
| <math>(\mathbb{D}^n)^* \cong \mathbb{D}^n\!</math><br />
|-<br />
| <math>\mathrm{d}A^\uparrow\!</math><br />
| <math>(\mathrm{d}A \to \mathbb{B})\!</math><br />
| <math>\text{Boolean functions on}~ \mathrm{d}A\!</math><br />
| <math>\mathbb{D}^n \to \mathbb{B}\!</math><br />
|-<br />
| <math>\mathrm{d}A^\bullet\!</math><br />
|<br />
<math>\begin{matrix}<br />
[\mathrm{d}\mathcal{A}]<br />
\\[2pt]<br />
(\mathrm{d}A, \mathrm{d}A^\uparrow)<br />
\\[2pt]<br />
(\mathrm{d}A ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
(\mathrm{d}A, (\mathrm{d}A \to \mathbb{B}))<br />
\\[2pt]<br />
[\mathrm{d}a_1, \ldots, \mathrm{d}a_n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Tangent universe at a point of}~ A^\bullet,<br />
\\[2pt]<br />
\text{based on the tangent features}<br />
\\[2pt]<br />
\{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
(\mathbb{D}^n, (\mathbb{D}^n \to \mathbb{B}))<br />
\\[2pt]<br />
(\mathbb{D}^n ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
[\mathbb{D}^n]<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
The adjectives ''differential'' or ''tangent'' are systematically attached to every construct based on the differential alphabet <math>\mathrm{d}\mathfrak{A},</math> taken by itself. Strictly speaking, we probably ought to call <math>\mathrm{d}\mathcal{A}</math> the set of ''cotangent'' features derived from <math>\mathcal{A},</math> but the only time this distinction really seems to matter is when we need to distinguish the tangent vectors as maps of type <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math> from cotangent vectors as elements of type <math>\mathbb{D}^n.</math> In like fashion, having defined <math>\mathrm{E}\mathcal{A} = \mathcal{A} \cup \mathrm{d}\mathcal{A},</math> we can systematically attach the adjective ''extended'' or the substantive ''bundle'' to all of the constructs associated with this full complement of <math>{2n}\!</math> features.<br />
<br />
It eventually becomes necessary to extend the initial alphabet even further, to allow for the discussion of higher order differential expressions. Table&nbsp;9 provides a suggestion of how these further extensions can be carried out.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 9.} ~~ \text{Higher Order Differential Features}\!</math><br />
|<br />
<p><math>\begin{array}{lllll}<br />
\mathrm{d}^0 \mathcal{A} & = & \{ a_1, \ldots, a_n \} & = & \mathcal{A}<br />
\\<br />
\mathrm{d}^1 \mathcal{A} & = & \{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \} & = & \mathrm{d}\mathcal{A}<br />
\end{array}</math></p><br />
<br />
<p><math>\begin{array}{lll}<br />
\mathrm{d}^k \mathcal{A} & = & \{ \mathrm{d}^k a_1, \ldots, \mathrm{d}^k a_n \}<br />
\\<br />
\mathrm{d}^* \mathcal{A} & = & \{ \mathrm{d}^0 \mathcal{A}, \ldots, \mathrm{d}^k \mathcal{A}, \ldots \}<br />
\end{array}</math></p><br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}^0 \mathcal{A} & = & \mathrm{d}^0 \mathcal{A}<br />
\\<br />
\mathrm{E}^1 \mathcal{A} & = & \mathrm{d}^0 \mathcal{A} ~\cup~ \mathrm{d}^1 \mathcal{A}<br />
\\<br />
\mathrm{E}^k \mathcal{A} & = & \mathrm{d}^0 \mathcal{A} ~\cup~ \ldots ~\cup~ \mathrm{d}^k \mathcal{A}<br />
\\<br />
\mathrm{E}^\infty \mathcal{A} & = & \bigcup~ \mathrm{d}^* \mathcal{A}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
===Intentional Propositions===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Do you guess I have some intricate purpose?<br><br />
Well I have . . . . for the April rain has, and the mica on<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the side of a rock has.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 45]<br />
|}<br />
<br />
In order to analyze the behavior of a system at successive moments in time, while staying within the limitations of propositional logic, it is necessary to create independent alphabets of logical features for each moment of time that we contemplate using in our discussion. These moments have reference to typical instances and relative intervals, not actual or absolute times. For example, to discuss ''velocities'' (first order rates of change) we need to consider points of time in pairs. There are a number of natural ways of doing this. Given an initial alphabet, we could use its symbols as a lexical basis to generate successive alphabets of compound symbols, say, with temporal markers appended as suffixes.<br />
<br />
As a standard way of dealing with these situations, the following scheme of notation suggests a way of extending any alphabet of logical features through as many temporal moments as a particular order of analysis may demand. The lexical operators <math>\mathrm{p}^k</math> and <math>\mathrm{Q}^k</math> are convenient in many contexts where the accumulation of prime symbols and union symbols would otherwise be cumbersome.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 10.} ~~ \text{A Realm of Intentional Features}\!</math><br />
|<br />
<p><math>\begin{array}{lllll}<br />
\mathrm{p}^0 \mathcal{A} & = & \{ a_1, \ldots, a_n \} & = & \mathcal{A}<br />
\\<br />
\mathrm{p}^1 \mathcal{A} & = & \{ a_1^\prime, \ldots, a_n^\prime \} & = & \mathcal{A}^\prime<br />
\\<br />
\mathrm{p}^2 \mathcal{A} & = & \{ a_1^{\prime\prime}, \ldots, a_n^{\prime\prime} \} & = & \mathcal{A}^{\prime\prime}<br />
\\<br />
\cdots & & \cdots &<br />
\end{array}</math></p><br />
<br />
<p><math>\begin{array}{lll}<br />
\mathrm{p}^k \mathcal{A} & = & \{ \mathrm{p}^k a_1, \ldots, \mathrm{p}^k a_n \}<br />
\end{array}</math></p><br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{Q}^0 \mathcal{A} & = & \mathcal{A}<br />
\\<br />
\mathrm{Q}^1 \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}'<br />
\\<br />
\mathrm{Q}^2 \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}' \cup \mathcal{A}''<br />
\\<br />
\cdots & & \cdots<br />
\\<br />
\mathrm{Q}^k \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}' \cup \ldots \cup \mathrm{p}^k \mathcal{A}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
The resulting augmentations of our logical basis determine a series of discursive universes that may be called the ''intentional extension'' of propositional calculus. This extension follows a pattern analogous to the differential extension, which was developed in terms of the operators <math>\mathrm{d}^k</math> and <math>\mathrm{E}^k,</math> and there is a natural relation between these two extensions that bears further examination. In contexts displaying this pattern, where a sequence of domains stretches from an anchoring domain <math>X\!</math> through an indefinite number of higher reaches, a particular collection of domains based on <math>X\!</math> will be referred to as a ''realm'' of <math>X,\!</math> and when the succession exhibits a temporal aspect, as a ''reign'' of <math>X.\!</math><br />
<br />
For the purposes of this discussion, an ''intentional proposition'' is defined as a proposition in the universe of discourse <math>\mathrm{Q}X^\bullet = [\mathrm{Q}\mathcal{X}],</math> in other words, a map <math>q : \mathrm{Q}X \to \mathbb{B}.</math> The sense of this definition may be seen if we consider the following facts. First, the equivalence <math>\mathrm{Q}X = X \times X'</math> motivates the following chain of isomorphisms between spaces:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lllcl}<br />
(\mathrm{Q}X \to \mathbb{B})<br />
& \cong & (X & \times & ~X' \to \mathbb{B})<br />
\\[4pt]<br />
& \cong & (X & \to & (X' \to \mathbb{B}))<br />
\\[4pt]<br />
& \cong & (X' & \to & (X~ \to \mathbb{B})).<br />
\end{array}</math><br />
|}<br />
<br />
Viewed in this light, an intentional proposition <math>q\!</math> may be rephrased as a map <math>q : X \times X' \to \mathbb{B},</math> which judges the juxtaposition of states in <math>X\!</math> from one moment to the next. Alternatively, <math>q\!</math> may be parsed in two stages in two different ways, as <math>q : X \to (X' \to \mathbb{B})</math> and as <math>q : X' \to (X \to \mathbb{B}),</math> which associate to each point of <math>X\!</math> or <math>X'\!</math> a proposition about states in <math>X'\!</math> or <math>X,\!</math> respectively. In this way, an intentional proposition embodies a type of value system, in effect, a proposal that places a value on a collection of ends-in-view, or a project that evaluates a set of goals as regarded from each point of view in the state space of a system.<br />
<br />
In sum, the intentional proposition <math>q\!</math> indicates a method for the systematic selection of local goals. As a general form of description, a map of the type <math>q : \mathrm{Q}^i X \to \mathbb{B}\!</math> may be referred to as an "<math>i^\text{th}</math> order intentional proposition". Naturally, when we speak of intentional propositions without qualification, we usually mean first order intentions.<br />
<br />
Many different realms of discourse have the same structure as the extensions that have been indicated here. From a strictly logical point of view, each new layer of terms is composed of independent logical variables that are no different in kind from those that go before, and each further course of logical atoms is treated like so many individual, but otherwise indifferent bricks by the prototype computer program that I use as a propositional interpreter. Thus, the names that I use to single out the differential and the intentional extensions, and the lexical paradigms that I follow to construct them, are meant to suggest the interpretations that I have in mind, but they can only hint at the extra meanings that human communicators may pack into their terms and inflections.<br />
<br />
As applied here, the word ''intentional'' is drawn from common use and may have little bearing on its technical use in other, more properly philosophical, contexts. I am merely using the complex of intentional concepts &mdash; aims, ends, goals, objectives, purposes, and so on &mdash; metaphorically to flesh out and vividly to represent any situation where one needs to contemplate a system in multiple aspects of state and destination, that is, its being in certain states and at the same time acting as if headed through certain states. If confusion arises, more neutral words like ''conative'', ''contingent'', ''discretionary'', ''experimental'', ''kinetic'', ''progressive'', ''tentative'', or ''trial'' would probably serve as well.<br />
<br />
===Life on Easy Street===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Failing to fetch me at first keep encouraged,<br><br />
Missing me one place search another,<br><br />
I stop some where waiting for you<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]<br />
|}<br />
<br />
The finite character of the extended universe <math>[\mathrm{E}\mathcal{A}]</math> makes the problem of solving differential propositions relatively straightforward, at least, in principle. The solution set of the differential proposition <math>q : \mathrm{E}A \to \mathbb{B}</math> is the set of models <math>q^{-1}(1)\!</math> in <math>\mathrm{E}A.</math> Finding all the models of <math>q,\!</math> the extended interpretations in <math>\mathrm{E}A</math> that satisfy <math>q,\!</math> can be carried out by a finite search. Being in possession of complete algorithms for propositional calculus modeling, theorem checking, or theorem proving makes the analytic task fairly simple in principle, though the question of efficiency in the face of arbitrary complexity may always remain another matter entirely. While the fact that propositional satisfiability is NP-complete may be discouraging for the prospects of a single efficient algorithm that covers the whole space <math>[\mathrm{E}\mathcal{A}]</math> with equal facility, there appears to be much room for improvement in classifying special forms and in developing algorithms that are tailored to their practical processing.<br />
<br />
In view of these constraints and contingencies, our focus shifts to the tasks of approximation and interpretation that support intuition, especially in dealing with the natural kinds of differential propositions that arise in applications, and in the effort to understand, in succinct and adaptive forms, their dynamic implications. In the absence of direct insights, these tasks are partially carried out by forging analogies with the familiar situations and customary routines of ordinary calculus. But the indirect approach, going by way of specious analogy and intuitive habit, forces us to remain on guard against the circumstance that occurs when the word ''forging'' takes on its shadier nuance, indicting the constant risk of a counterfeit in the proportion.<br />
<br />
==Back to the Beginning : Exemplary Universes==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
I would have preferred to be enveloped in words, borne way beyond all possible beginnings.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
To anchor our understanding of differential logic, let us look at how the various concepts apply in the simplest possible concrete cases, where the initial dimension is only 1 or 2. In spite of the obvious simplicity of these cases, it is possible to observe how central difficulties of the subject begin to arise already at this stage.<br />
<br />
===A One-Dimensional Universe===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
There was never any more inception than there is now,<br><br />
Nor any more youth or age than there is now;<br><br />
And will never be any more perfection than there is now,<br><br />
Nor any more heaven or hell than there is now.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]<br />
|}<br />
<br />
Let <math>\mathcal{X} = \{ x_1 \} = \{ A \}</math> be an alphabet that represents one boolean variable or a single logical feature. In this example the capital letter <math>{}^{\backprime\backprime} A {}^{\prime\prime}\!</math> is used usual informally, to name a feature and not a space, in departure from our formerly stated formal conventions. At any rate, the basis element <math>A = x_1\!</math> may be interpreted as a simple proposition or a coordinate projection <math>A = x_1 : \mathbb{B} \xrightarrow{i} \mathbb{B}.</math> The space <math>X = \langle A \rangle = \{ \texttt{(} A \texttt{)}, A \}</math> of points (cells, vectors, interpretations) has cardinality <math>2^n = 2^1 = 2\!</math> and is isomorphic to <math>\mathbb{B} = \{ 0, 1 \}.</math> Moreover, <math>X\!</math> may be identified with the set of singular propositions <math>\{ x : \mathbb{B} \xrightarrow{s} \mathbb{B} \}.</math> The space of linear propositions <math>X^* = \{ \mathrm{hom} : \mathbb{B} \xrightarrow{\ell} \mathbb{B} \} = \{ 0, A \}</math> is algebraically dual to <math>X\!</math> and also has cardinality <math>2.\!</math> Here, <math>{}^{\backprime\backprime} 0 {}^{\prime\prime}\!</math> is interpreted as denoting the constant function <math>0 : \mathbb{B} \to \mathbb{B},</math> amounting to the linear proposition of rank <math>0,\!</math> while <math>A\!</math> is the linear proposition of rank <math>1.\!</math> Last but not least we have the positive propositions <math>\{ \mathrm{pos} : \mathbb{B} \xrightarrow{p} \mathbb{B} \} = \{ A, 1 \},\!</math> of rank <math>1\!</math> and <math>0,\!</math> respectively, where <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}\!</math> is understood as denoting the constant function <math>1 : \mathbb{B} \to \mathbb{B}.</math> In sum, there are <math>2^{2^n} = 2^{2^1} = 4</math> propositions altogether in the universe of discourse, comprising the set <math>X^\uparrow = \{ f : X \to \mathbb{B} \} = \{ 0, \texttt{(} A \texttt{)}, A, 1 \} \cong (\mathbb{B} \to \mathbb{B}).</math><br />
<br />
The first order differential extension of <math>\mathcal{X}</math> is <math>\mathrm{E}\mathcal{X} = \{ x_1, \mathrm{d}x_1 \} = \{ A, \mathrm{d}A \}.</math> If the feature <math>A\!</math> is understood as applying to some object or state, then the feature <math>\mathrm{d}A</math> may be interpreted as an attribute of the same object or state that says that it is changing ''significantly'' with respect to the property <math>A,\!</math> or that it has an ''escape velocity'' with respect to the state <math>A.\!</math> In practice, differential features acquire their logical meaning through a class of ''temporal inference rules''.<br />
<br />
For example, relative to a frame of observation that is left implicit for now, one is permitted to make the following sorts of inference: From the fact that <math>A\!</math> and <math>\mathrm{d}A</math> are true at a given moment one may infer that <math>\texttt{(} A \texttt{)}\!</math> will be true in the next moment of observation. Altogether in the present instance, there is the fourfold scheme of inference that is shown below:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:50%"<br />
|<br />
<math>\begin{matrix}<br />
\text{From} & \texttt{(} A \texttt{)}<br />
& \text{and} & \texttt{(} \mathrm{d}A \texttt{)}<br />
& \text{infer} & \texttt{(} A \texttt{)} & \text{next.}<br />
\\[8pt]<br />
\text{From} & \texttt{(} A \texttt{)}<br />
& \text{and} & \mathrm{d}A<br />
& \text{infer} & A & \text{next.}<br />
\\[8pt]<br />
\text{From} & A<br />
& \text{and} & \texttt{(} \mathrm{d}A \texttt{)}<br />
& \text{infer} & A & \text{next.}<br />
\\[8pt]<br />
\text{From} & A<br />
& \text{and} & \mathrm{d}A<br />
& \text{infer} & \texttt{(} A \texttt{)} & \text{next.}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
It might be thought that an independent time variable needs to be brought in at this point, but it is an insight of fundamental importance that the idea of process is logically prior to the notion of time. A time variable is a reference to a ''clock'' &mdash; a canonical, conventional process that is accepted or established as a standard of measurement, but in essence no different than any other process. This raises the question of how different subsystems in a more global process can be brought into comparison, and what it means for one process to serve the function of a local standard for others. But these inquiries only wrap up puzzles in further riddles, and are obviously too involved to be handled at our current level of approximation.<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
The clock indicates the moment . . . . but what does<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;eternity indicate?<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 79]<br />
|}<br />
<br />
Observe that the secular inference rules, used by themselves, involve a loss of information, since nothing in them can tell us whether the momenta <math>\{ \texttt{(} \mathrm{d}A \texttt{)}, \mathrm{d}A \}\!</math> are changed or unchanged in the next instance. In order to know this, one would have to determine <math>\mathrm{d}^2 A,\!</math> and so on, pursuing an infinite regress. Ultimately, in order to rest with a finitely determinate system, it is necessary to make an infinite assumption, for example, that <math>\mathrm{d}^k A = 0\!</math> for all <math>k\!</math> greater than some fixed value <math>M.\!</math> Another way to escape the regress is through the provision of a dynamic law, in typical form making higher order differentials dependent on lower degrees and estates.<br />
<br />
===Example 1. A Square Rigging===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Urge and urge and urge,<br><br />
Always the procreant urge of the world.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]<br />
|}<br />
<br />
By way of example, suppose that we are given the initial condition <math>A = \mathrm{d}A\!</math> and the law <math>\mathrm{d}^2 A = \texttt{(} A \texttt{)}.\!</math> Since the equation <math>A = \mathrm{d}A\!</math> is logically equivalent to the disjunction <math>A ~ \mathrm{d}A ~\text{or}~ \texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)},\!</math> we may infer two possible trajectories, as displayed in Table&nbsp;11. In either case the state <math>A ~ \texttt{(} \mathrm{d}A \texttt{)(} \mathrm{d}^2 A \texttt{)}\!</math> is a stable attractor or a terminal condition for both starting points.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 11.} ~~ \text{A Pair of Commodious Trajectories}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Time}\!</math><br />
| <math>\text{Trajectory 1}\!</math><br />
| <math>\text{Trajectory 2}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
2<br />
\\[4pt]<br />
3<br />
\\[4pt]<br />
4<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A & \mathrm{d}A & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
\texttt{(} A \texttt{)} & \mathrm{d}A & \mathrm{d}^2 A<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
{}^{\shortparallel} & {}^{\shortparallel} & {}^{\shortparallel}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{)} & \texttt{(} \mathrm{d}A \texttt{)} & \mathrm{d}^2 A<br />
\\[4pt]<br />
\texttt{(} A \texttt{)} & \mathrm{d}A & \mathrm{d}^2 A<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
{}^{\shortparallel} & {}^{\shortparallel} & {}^{\shortparallel}<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Because the initial space <math>X = \langle A \rangle\!</math> is one-dimensional, we can easily fit the second order extension <math>\mathrm{E}^2 X = \langle A, \mathrm{d}A, \mathrm{d}^2 A \rangle\!</math> within the compass of a single venn diagram, charting the couple of converging trajectories as shown in Figure&nbsp;12.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 12 -- The Anchor.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 12.} ~~ \text{The Anchor}\!</math><br />
|}<br />
<br />
If we eliminate from view the regions of <math>\mathrm{E}^2 X\!</math> that are ruled out by the dynamic law <math>\mathrm{d}^2 A = \texttt{(} A \texttt{)},\!</math> then what remains is the quotient structure that is shown in Figure&nbsp;13. This picture makes it easy to see that the dynamically allowable portion of the universe is partitioned between the properties <math>A\!</math> and <math>\mathrm{d}^2 A\!.</math> As it happens, this fact might have been expressed &ldquo;right off the bat&rdquo; by an equivalent formulation of the differential law, one that uses the exclusive disjunction to state the law as <math>\texttt{(} A \texttt{,} \mathrm{d}^2 A \texttt{)}\!.</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 13 -- The Tiller.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 13.} ~~ \text{The Tiller}\!</math><br />
|}<br />
<br />
What we have achieved in this example is to give a differential description of a simple dynamic process. In effect, we did this by embedding a directed graph, which can be taken to represent the state transitions of a finite automaton, in a dynamically allotted quotient structure that is created from a boolean lattice or an <math>n\!</math>-cube by nullifying all of the regions that the dynamics outlaws. With growth in the dimensions of our contemplated universes, it becomes essential, both for human comprehension and for computer implementation, that the dynamic structures of interest to us be represented not actually, by acquaintance, but virtually, by description. In our present study, we are using the language of propositional calculus to express the relevant descriptions, and to comprehend the structure that is implicit in the subsets of a <math>n\!</math>-cube without necessarily being forced to actualize all of its points.<br />
<br />
One of the reasons for engaging in this kind of extremely reduced, but explicitly controlled case study is to throw light on the general study of languages, formal and natural, in their full array of syntactic, semantic, and pragmatic aspects. Propositional calculus is one of the last points of departure where we can view these three aspects interacting in a non-trivial way without being immediately and totally overwhelmed by the complexity they generate. Often this complexity causes investigators of formal and natural languages to adopt the strategy of focusing on a single aspect and to abandon all hope of understanding the whole, whether it's the still living natural language or the dynamics of inquiry that lies crystallized in formal logic.<br />
<br />
From the perspective that I find most useful here, a language is a syntactic system that is designed or evolved in part to express a set of descriptions. When the explicit symbols of a language have extensions in its object world that are actually infinite, or when the implicit categories and generative devices of a linguistic theory have extensions in its subject matter that are potentially infinite, then the finite characters of terms, statements, arguments, grammars, logics, and rhetorics force an excess of intension to reside in all these symbols and functions, across the spectrum from the object language to the metalinguistic uses. In the aphorism from W. von Humboldt that Chomsky often cites, for example, in [Cho86, 30] and [Cho93, 49], language requires &ldquo;the infinite use of finite means&rdquo;. This is necessarily true when the extensions are infinite, when the referential symbols and grammatical categories of a language possess infinite sets of models and instances. But it also voices a practical truth when the extensions, though finite at every stage, tend to grow at exponential rates.<br />
<br />
This consequence of dealing with extensions that are &ldquo;practically infinite&rdquo; becomes crucial when one tries to build neural network systems that learn, since the learning competence of any intelligent system is limited to the objects and domains that it is able to represent. If we want to design systems that operate intelligently with the full deck of propositions dealt by intact universes of discourse, then we must supply them with succinct representations and efficient transformations in this domain. Furthermore, in the project of constructing inquiry driven systems, we find ourselves forced to contemplate the level of generality that is embodied in propositions, because the dynamic evolution of these systems is driven by the measurable discrepancies that occur among their expectations, intentions, and observations, and because each of these subsystems or components of knowledge constitutes a propositional modality that can take on the fully generic character of an empirical summary or an axiomatic theory.<br />
<br />
A compression scheme by any other name is a symbolic representation, and this is what the differential extension of propositional calculus, through all of its many universes of discourse, is intended to supply. Why is this particular program of mental calisthenics worth carrying out in general? By providing a uniform logical medium for describing dynamic systems we can make the task of understanding complex systems much easier, both in looking for invariant representations of individual cases and in finding points of comparison among diverse structures that would otherwise appear as isolated systems. All of this goes to facilitate the search for compact knowledge and to adapt what is learned from individual cases to the general realm.<br />
<br />
===Back to the Feature===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
I guess it must be the flag of my disposition, out of hopeful<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;green stuff woven.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 31]<br />
|}<br />
<br />
Let us assume that the sense intended for differential features is well enough established in the intuition, for now, that we may continue with outlining the structure of the differential extension <math>[\mathrm{E}\mathcal{X}] = [A, \mathrm{d}A].\!</math> Over the extended alphabet <math>\mathrm{E}\mathcal{X} = \{ x_1, \mathrm{d}x_1 \} = \{ A, \mathrm{d}A \}\!</math> of cardinality <math>2^n = 2\!</math> we generate the set of points <math>\mathrm{E}X\!</math> of cardinality <math>2^{2n} = 4\!</math> that bears the following chain of equivalent descriptions:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}X & = & \langle A, \mathrm{d}A \rangle<br />
\\[4pt]<br />
& = & \{ \texttt{(} A \texttt{)}, A \} ~\times~ \{ \texttt{(} \mathrm{d}A \texttt{)}, \mathrm{d}A \}<br />
\\[4pt]<br />
& = &<br />
\{<br />
\texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)},~<br />
\texttt{(} A \texttt{)} \mathrm{d}A,~<br />
A \texttt{(} \mathrm{d}A \texttt{)},~<br />
A ~ \mathrm{d}A<br />
\}.<br />
\end{array}</math><br />
|}<br />
<br />
The space <math>\mathrm{E}X\!</math> may be assigned the mnemonic type <math>\mathbb{B} \times \mathbb{D},\!</math> which is really no different than <math>\mathbb{B} \times \mathbb{B} = \mathbb{B}^2.\!</math> An individual element of <math>\mathrm{E}X\!</math> may be regarded as a ''disposition at a point'' or a ''situated direction'', in effect, a singular mode of change occurring at a single point in the universe of discourse. In applications, the modality of this change can be interpreted in various ways, for example, as an expectation, an intention, or an observation with respect to the behavior of a system.<br />
<br />
To complete the construction of the extended universe of discourse <math>\mathrm{E}X^\bullet = [x_1, \mathrm{d}x_1] = [A, \mathrm{d}A]\!</math> one must add the set of differential propositions <math>\mathrm{E}X^\uparrow = \{ g : \mathrm{E}X \to \mathbb{B} \} \cong (\mathbb{B} \times \mathbb{D} \to \mathbb{B})\!</math> to the set of dispositions in <math>\mathrm{E}X.\!</math> There are <math>2^{2^{2n}} = 16\!</math> propositions in <math>\mathrm{E}X^\uparrow,\!</math> as detailed in Table&nbsp;14.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 14.} ~~ \text{Differential Propositions}\!</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>A\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>\mathrm{d}A\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>g_{0}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{1}<br />
\\[4pt]<br />
g_{2}<br />
\\[4pt]<br />
g_{4}<br />
\\[4pt]<br />
g_{8}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
1~0~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
\texttt{(} A \texttt{)} ~ \mathrm{d}A ~<br />
\\[4pt]<br />
~ A ~ \texttt{(} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
~ A ~~ \mathrm{d}A ~<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{neither}~ A ~\text{nor}~ \mathrm{d}A<br />
\\[4pt]<br />
\mathrm{d}A ~\text{and not}~ A<br />
\\[4pt]<br />
A ~\text{and not}~ \mathrm{d}A<br />
\\[4pt]<br />
A ~\text{and}~ \mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot A \land \lnot \mathrm{d}A<br />
\\[4pt]<br />
\lnot A \land \mathrm{d}A<br />
\\[4pt]<br />
A \land \lnot \mathrm{d}A<br />
\\[4pt]<br />
A \land \mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
g_{3}<br />
\\[4pt]<br />
g_{12}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{)}<br />
\\[4pt]<br />
A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ A<br />
\\[4pt]<br />
A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot A<br />
\\[4pt]<br />
A<br />
\end{matrix}\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{6}<br />
\\[4pt]<br />
g_{9}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[4pt]<br />
1~0~0~1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{,} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
\texttt{((} A \texttt{,} \mathrm{d}A \texttt{))}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
A ~\text{not equal to}~ \mathrm{d}A<br />
\\[4pt]<br />
A ~\text{equal to}~ \mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
A \ne \mathrm{d}A<br />
\\[4pt]<br />
A = \mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{5}<br />
\\[4pt]<br />
g_{10}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
\mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ \mathrm{d}A<br />
\\[4pt]<br />
\mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot \mathrm{d}A<br />
\\[4pt]<br />
\mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{7}<br />
\\[4pt]<br />
g_{11}<br />
\\[4pt]<br />
g_{13}<br />
\\[4pt]<br />
g_{14}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} ~ A ~~ \mathrm{d}A ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{(} ~ A ~ \texttt{(} \mathrm{d}A \texttt{))}<br />
\\[4pt]<br />
\texttt{((} A \texttt{)} ~ \mathrm{d}A ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{((} A \texttt{)(} \mathrm{d}A \texttt{))}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not both}~ A ~\text{and}~ \mathrm{d}A<br />
\\[4pt]<br />
\text{not}~ A ~\text{without}~ \mathrm{d}A<br />
\\[4pt]<br />
\text{not}~ \mathrm{d}A ~\text{without}~ A<br />
\\[4pt]<br />
A ~\text{or}~ \mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot A \lor \lnot \mathrm{d}A<br />
\\[4pt]<br />
A \Rightarrow \mathrm{d}A<br />
\\[4pt]<br />
A \Leftarrow \mathrm{d}A<br />
\\[4pt]<br />
A \lor \mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
| <math>f_{3}\!</math><br />
| <math>g_{15}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
Aside from changing the names of variables and shuffling the order of rows, this Table follows the format that was used previously for boolean functions of two variables. The rows are grouped to reflect natural similarity classes among the propositions. In a future discussion, these classes will be given additional explanation and motivation as the orbits of a certain transformation group acting on the set of 16 propositions. Notice that four of the propositions, in their logical expressions, resemble those given in the table for <math>X^\uparrow.\!</math> Thus the first set of propositions <math>\{ f_i \}\!</math> is automatically embedded in the present set <math>\{ g_j \}\!</math> and the corresponding inclusions are indicated at the far left margin of the Table.<br />
<br />
===Tacit Extensions===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
I would really like to have slipped imperceptibly into this lecture, as into all the others I shall be delivering, perhaps over the years ahead.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
Strictly speaking, however, there is a subtle distinction in type between the function <math>f_i : X \to \mathbb{B}</math> and the corresponding function <math>g_j : \mathrm{E}X \to \mathbb{B},</math> even though they share the same logical expression. Naturally, we want to maintain the logical equivalence of expressions that represent the same proposition while appreciating the full diversity of that proposition's functional and typical representatives. Both perspectives, and all the levels of abstraction extending through them, have their reasons, as will develop in time.<br />
<br />
Because this special circumstance points up an important general theme, it is a good idea to discuss it more carefully. Whenever there arises a situation like this, where one alphabet <math>\mathcal{X}</math> is a subset of another alphabet <math>\mathcal{Y},</math> then we say that any proposition <math>f : \langle \mathcal{X} \rangle \to \mathbb{B}</math> has a ''tacit extension'' to a proposition <math>\boldsymbol\varepsilon f : \langle \mathcal{Y} \rangle \to \mathbb{B},\!</math> and that the space <math>(\langle \mathcal{X} \rangle \to \mathbb{B})</math> has an ''automatic embedding'' within the space <math>(\langle \mathcal{Y} \rangle \to \mathbb{B}).</math> The extension is defined in such a way that <math>\boldsymbol\varepsilon f\!</math> puts the same constraint on the variables of <math>\mathcal{X}</math> that are contained in <math>\mathcal{Y}</math> as the proposition <math>f\!</math> initially did, while it puts no constraint on the variables of <math>\mathcal{Y}</math> outside of <math>\mathcal{X},</math> in effect, conjoining the two constraints.<br />
<br />
If the variables in question are indexed as <math>\mathcal{X} = \{ x_1, \ldots, x_n \}</math> and <math>\mathcal{Y} = \{ x_1, \ldots, x_n, \ldots, x_{n+k} \},</math> then the definition of the tacit extension from <math>\mathcal{X}</math> to <math>\mathcal{Y}</math> may be expressed in the form of an equation:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\boldsymbol\varepsilon f(x_1, \ldots, x_n, \ldots, x_{n+k}) ~=~ f(x_1, \ldots, x_n).\!</math><br />
|}<br />
<br />
On formal occasions, such as the present context of definition, the tacit extension from <math>\mathcal{X}</math> to <math>\mathcal{Y}</math> is explicitly symbolized by the operator <math>\boldsymbol\varepsilon : (\langle \mathcal{X} \rangle \to \mathbb{B}) \to (\langle \mathcal{Y} \rangle \to \mathbb{B}),</math> where the appropriate alphabets <math>\mathcal{X}</math> and <math>\mathcal{Y}</math> are understood from context, but normally one may leave the "<math>\boldsymbol\varepsilon\!</math>" silent.<br />
<br />
Let's explore what this means for the present Example. Here, <math>\mathcal{X} = \{ A \}</math> and <math>\mathcal{Y} = \mathrm{E}\mathcal{X} = \{ A, \mathrm{d}A \}.</math> For each of the propositions <math>f_i\!</math> over <math>X\!,</math> specifically, those whose expression <math>e_i\!</math> lies in the collection <math>\{ 0, \texttt{(} A \texttt{)}, A, 1 \},\!</math> the tacit extension <math>\boldsymbol\varepsilon f\!</math> of <math>f\!</math> to <math>\mathrm{E}X</math> can be phrased as a logical conjunction of two factors, <math>f_i = e_i \cdot \tau ~ ,\!</math> where <math>\tau\!</math> is a logical tautology that uses all the variables of <math>\mathcal{Y} - \mathcal{X}.</math> Working in these terms, the tacit extensions <math>\boldsymbol\varepsilon f\!</math> of <math>f\!</math> to <math>\mathrm{E}X</math> may be explicated as shown in Table&nbsp;15.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:60%"<br />
|+ style="height:30px" | <math>\text{Table 15.} ~~ \text{Tacit Extension of}~ [A] ~\text{to}~ [A, \mathrm{d}A]\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0<br />
& = & 0 & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))} & = & & 0<br />
\\[8pt]<br />
\texttt{(} A \texttt{)}<br />
& = & \texttt{(} A \texttt{)} & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))}<br />
& = & \texttt{(} A \texttt{)} \, \mathrm{d}A ~ & + & \texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)}<br />
\\[8pt]<br />
A<br />
& = & ~A~ & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))}<br />
& = & ~A~ ~\mathrm{d}A~ & + & ~A~ \texttt{(} \mathrm{d}A \texttt{)}<br />
\\[8pt]<br />
1<br />
& = & 1 & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))} & = & & 1<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
In its effect on the singular propositions over <math>X,\!</math> this analysis has an interesting interpretation. The tacit extension takes us from thinking about a particular state, like <math>A\!</math> or <math>\texttt{(} A \texttt{)},\!</math> to considering the collection of outcomes, the outgoing changes or the singular dispositions, that spring from that state.<br />
<br />
===Example 2. Drives and Their Vicissitudes===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
I open my scuttle at night and see the far-sprinkled systems,<br><br />
And all I see, multiplied as high as I can cipher, edge but<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the rim of the farther systems.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 81]<br />
|}<br />
<br />
Before we leave the one-feature case let's look at a more substantial example, one that illustrates a general class of curves that can be charted through the extended feature spaces and that provides an opportunity to discuss a number of important themes concerning their structure and dynamics.<br />
<br />
Again, let <math>\mathcal{X} = \{ x_1 \} = \{ A \}.\!</math> In the discussion that follows we will consider a class of trajectories having the property that <math>\mathrm{d}^k A = 0\!</math> for all <math>k\!</math> greater than some fixed <math>m\!</math> and we may indulge in the use of some picturesque terms that describe salient classes of such curves. Given the finite order condition, there is a highest order non-zero difference <math>\mathrm{d}^m A\!</math> exhibited at each point in the course of any determinate trajectory that one may wish to consider. With respect to any point of the corresponding orbit or curve let us call this highest order differential feature <math>\mathrm{d}^m A\!</math> the ''drive'' at that point. Curves of constant drive <math>\mathrm{d}^m A\!</math> are then referred to as ''<math>m^\text{th}\!</math>-gear curves''.<br />
<br />
* '''Scholium.''' The fact that a difference calculus can be developed for boolean functions is well known [Fuji], [Koh, &sect; 8-4] and was probably familiar to Boole, who was an expert in difference equations before he turned to logic. And of course there is the strange but true story of how the Turin machines of the 1840s prefigured the Turing machines of the 1940s [Men, 225-297]. At the very outset of general purpose, mechanized computing we find that the motive power driving the Analytical Engine of Babbage, the kernel of an idea behind all of his wheels, was exactly his notion that difference operations, suitably trained, can serve as universal joints for any conceivable computation [M&M], [Mel, ch. 4].<br />
<br />
Given this language, the Example we take up here can be described as the family of <math>4^\text{th}\!</math>-gear curves through <math>\mathrm{E}^4 X\!</math> <math>=\!</math> <math>\langle A, ~\mathrm{d}A, ~\mathrm{d}^2\!A, ~\mathrm{d}^3\!A, ~\mathrm{d}^4\!A \rangle.</math> These are the trajectories generated subject to the dynamic law <math>\mathrm{d}^4 A = 1,\!</math> where it is understood in such a statement that all higher order differences are equal to <math>0.\!</math> Since <math>\mathrm{d}^4 A\!</math> and all higher <math>\mathrm{d}^k A\!</math> are fixed, the temporal or transitional conditions (initial, mediate, terminal &mdash; transient or stable states) vary only with respect to their projections as points of <math>\mathrm{E}^3 X = \langle A, ~\mathrm{d}A, ~\mathrm{d}^2\!A, ~\mathrm{d}^3\!A \rangle.</math> Thus, there is just enough space in a planar venn diagram to plot all of these orbits and to show how they partition the points of <math>\mathrm{E}^3 X.\!</math> It turns out that there are exactly two possible orbits, of eight points each, as illustrated in Figure&nbsp;16.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 16 -- A Couple of Fourth Gear Orbits.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 16.} ~~ \text{A Couple of Fourth Gear Orbits}\!</math><br />
|}<br />
<br />
With a little thought it is possible to devise an indexing scheme for the general run of dynamic states that allows for comparing universes of discourse that weigh in on different scales of observation. With this end in sight, let us index the states <math>q \in \mathrm{E}^m X\!</math> with the dyadic rationals (or the binary fractions) in the half-open interval <math>[0, 2).\!</math> Formally and canonically, a state <math>q_r\!</math> is indexed by a fraction <math>r = \tfrac{s}{t}\!</math> whose denominator is the power of two <math>t = 2^m\!</math> and whose numerator is a binary numeral formed from the coefficients of state in a manner to be described next. The ''differential coefficients'' of the state <math>q\!</math> are just the values <math>\mathrm{d}^k\!A(q)</math> for <math>k = 0 ~\text{to}~ m,\!</math> where <math>\mathrm{d}^0\!A</math> is defined as being identical to <math>A.\!</math> To form the binary index <math>d_0.d_1 \ldots d_m\!</math> of the state <math>q\!</math> the coefficient <math>\mathrm{d}^k\!A(q)</math> is read off as the binary digit <math>d_k\!</math> associated with the place value <math>2^{-k}.\!</math> Expressed by way of algebraic formulas, the rational index <math>r\!</math> of the state <math>q\!</math> can be given by the following equivalent formulations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
r(q)<br />
& = &<br />
\displaystyle\sum_k d_k \cdot 2^{-k}<br />
& = &<br />
\displaystyle\sum_k \text{d}^k A(q) \cdot 2^{-k}<br />
\\[8pt]<br />
=<br />
\\[8pt]<br />
\displaystyle\frac{s(q)}{t}<br />
& = &<br />
\displaystyle\frac{\sum_k d_k \cdot 2^{(m-k)}}{2^m}<br />
& = &<br />
\displaystyle\frac{\sum_k \text{d}^k A(q) \cdot 2^{(m-k)}}{2^m}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Applied to the example of <math>4^\text{th}\!</math>-gear curves, this scheme results in the data of Tables&nbsp;17-a and 17-b, which exhibit one period for each orbit. The states in each orbit are listed as ordered pairs <math>(p_i, q_j),\!</math> where <math>p_i\!</math> may be read as a temporal parameter that indicates the present time of the state and where <math>j\!</math> is the decimal equivalent of the binary numeral <math>s.\!</math> Informally and more casually, the Tables exhibit the states <math>q_s\!</math> as subscripted with the numerators of their rational indices, taking for granted the constant denominators of <math>2^m\! = 2^4 = 16.\!</math> In this set-up the temporal successions of states can be reckoned as given by a kind of ''parallel round-up rule''. That is, if <math>(d_k, d_{k+1})\!</math> is any pair of adjacent digits in the state index <math>r,\!</math> then the value of <math>d_k\!</math> in the next state is <math>{d_k}' = d_k + d_{k+1}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:55%"<br />
|+ style="height:30px" | <math>\text{Table 17-a.} ~~ \text{A Couple of Orbits in Fourth Gear : Orbit 1}\!</math><br />
|- style="background:ghostwhite"<br />
| <math>\text{Time}\!</math><br />
| <math>\text{State}\!</math><br />
| <math>A\!</math><br />
| <math>\mathrm{d}A\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| <math>p_i\!</math><br />
| <math>q_j\!</math><br />
| <math>\mathrm{d}^0\!A</math><br />
| <math>\mathrm{d}^1\!A</math><br />
| <math>\mathrm{d}^2\!A</math><br />
| <math>\mathrm{d}^3\!A</math><br />
| <math>\mathrm{d}^4\!A</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
p_0<br />
\\[4pt]<br />
p_1<br />
\\[4pt]<br />
p_2<br />
\\[4pt]<br />
p_3<br />
\\[4pt]<br />
p_4<br />
\\[4pt]<br />
p_5<br />
\\[4pt]<br />
p_6<br />
\\[4pt]<br />
p_7<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
q_{01}<br />
\\[4pt]<br />
q_{03}<br />
\\[4pt]<br />
q_{05}<br />
\\[4pt]<br />
q_{15}<br />
\\[4pt]<br />
q_{17}<br />
\\[4pt]<br />
q_{19}<br />
\\[4pt]<br />
q_{21}<br />
\\[4pt]<br />
q_{31}<br />
\end{matrix}\!</math><br />
| colspan="5" |<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="text-align:center; width:100%"<br />
|<br />
<math>\begin{matrix}<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
1.<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:55%"<br />
|+ style="height:30px" | <math>\text{Table 17-b.} ~~ \text{A Couple of Orbits in Fourth Gear : Orbit 2}\!</math><br />
|- style="background:ghostwhite"<br />
| <math>\text{Time}\!</math><br />
| <math>\text{State}\!</math><br />
| <math>A\!</math><br />
| <math>\mathrm{d}A\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| <math>p_i\!</math><br />
| <math>q_j\!</math><br />
| <math>\mathrm{d}^0\!A</math><br />
| <math>\mathrm{d}^1\!A</math><br />
| <math>\mathrm{d}^2\!A</math><br />
| <math>\mathrm{d}^3\!A</math><br />
| <math>\mathrm{d}^4\!A</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
p_0<br />
\\[4pt]<br />
p_1<br />
\\[4pt]<br />
p_2<br />
\\[4pt]<br />
p_3<br />
\\[4pt]<br />
p_4<br />
\\[4pt]<br />
p_5<br />
\\[4pt]<br />
p_6<br />
\\[4pt]<br />
p_7<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
q_{25}<br />
\\[4pt]<br />
q_{11}<br />
\\[4pt]<br />
q_{29}<br />
\\[4pt]<br />
q_{07}<br />
\\[4pt]<br />
q_{09}<br />
\\[4pt]<br />
q_{27}<br />
\\[4pt]<br />
q_{13}<br />
\\[4pt]<br />
q_{23}<br />
\end{matrix}\!</math><br />
| colspan="5" |<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="text-align:center; width:100%"<br />
|<br />
<math>\begin{matrix}<br />
1.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
==Transformations of Discourse==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
It is understandable that an engineer should be completely absorbed in his speciality, instead of pouring himself out into the freedom and vastness of the world of thought, even though his machines are being sent off to the ends of the earth; for he no more needs to be capable of applying to his own personal soul what is daring and new in the soul of his subject than a machine is in fact capable of applying to itself the differential calculus on which it is based. The same thing cannot, however, be said about mathematics; for here we have the new method of thought, pure intellect, the very well-spring of the times, the ''fons et origo'' of an unfathomable transformation.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 39]<br />
|}<br />
<br />
In this section we take up the general study of logical transformations, or maps that relate one universe of discourse to another. In many ways, and especially as applied to the subject of intelligent dynamic systems, my argument develops the antithesis of the statement just quoted. Along the way, if incidental to my ends, I hope this essay can pose a fittingly irenic epitaph to the frankly ironic epigraph inscribed at its head.<br />
<br />
My goal in this section is to answer a single question: What is a propositional tangent functor? In other words, my aim is to develop a clear conception of what manner of thing would pass in the logical realm for a genuine analogue of the tangent functor, an object conceived to generalize as far as possible in the abstract terms of category theory the ordinary notions of functional differentiation and the all too familiar operations of taking derivatives.<br />
<br />
As a first step I discuss the kinds of transformations that we already know as ''extensions'' and ''projections'', and I use these special cases to illustrate several different styles of logical and visual representation that will figure heavily in the sequel.<br />
<br />
===Foreshadowing Transformations : Extensions and Projections of Discourse===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
And, despite the care which she took to look behind her at every moment, she failed to see a shadow which followed her like her own shadow, which stopped when she stopped, which started again when she did, and which made no more noise than a well-conducted shadow should.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 126]<br />
|}<br />
<br />
Many times in our discussion we have occasion to place one universe of discourse in the context of a larger universe of discourse. An embedding of the general type <math>[\mathcal{X}] \to [\mathcal{Y}]\!</math> is implied any time that we make use of one alphabet <math>[\mathcal{X}]\!</math> that happens to be included in another alphabet <math>[\mathcal{Y}].\!</math> When we are discussing differential issues we usually have in mind that the extended alphabet <math>[\mathcal{Y}]\!</math> has a special construction or a specific lexical relation with respect to the initial alphabet <math>[\mathcal{X}],\!</math> one that is marked by characteristic types of accents, indices, or inflected forms.<br />
<br />
====Extension from 1 to 2 Dimensions====<br />
<br />
Figure 18-a lays out the ''angular form'' of venn diagram for universes of 1 and 2 dimensions, indicating the embedding map of type <math>\mathbb{B}^1 \to \mathbb{B}^2\!</math> and detailing the coordinates that are associated with individual cells. Because all points, cells, or logical interpretations are represented as connected geometric areas, we can say that these pictures provide us with an ''areal view'' of each universe of discourse.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-a -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-a.} ~~ \text{Extension from 1 to 2 Dimensions : Areal}\!</math><br />
|}<br />
<br />
Figure 18-b shows the differential extension from <math>X^\bullet = [x]\!</math> to <math>\mathrm{E}X^\bullet = [x, \mathrm{d}x]\!</math> in a ''bundle of boxes'' form of venn diagram. As awkward as it may seem at first, this type of picture is often the most natural and the most easily available representation when we want to conceptualize the localized information or momentary knowledge of an intelligent dynamic system. It gives a ready picture of a ''proposition at a point'', in the present instance, of a proposition about changing states which is itself associated with a particular dynamic state of a system. It is easy to see how this application might be extended to conceive of more general types of instantaneous knowledge that are possessed by a system.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-b -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-b.} ~~ \text{Extension from 1 to 2 Dimensions : Bundle}\!</math><br />
|}<br />
<br />
Figure 18-c shows the same extension in a ''compact'' style of venn diagram, where the differential features at each position are represented by arrows extending from that position that cross or do not cross, as the case may be, the corresponding feature boundaries.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-c -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-c.} ~~ \text{Extension from 1 to 2 Dimensions : Compact}\!</math><br />
|}<br />
<br />
Figure 18-d compresses the picture of the differential extension even further, yielding a directed graph or ''digraph'' form of representation. (Notice that my definition of a digraph allows for loops or ''slings'' at individual points, in addition to arcs or ''arrows'' between the points.)<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-d -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-d.} ~~ \text{Extension from 1 to 2 Dimensions : Digraph}\!</math><br />
|}<br />
<br />
====Extension from 2 to 4 Dimensions====<br />
<br />
Figure 19-a lays out the ''areal view'' or the ''angular form'' of venn diagram for universes of 2 and 4 dimensions, indicating the embedding map of type <math>\mathbb{B}^2 \to \mathbb{B}^4.\!</math> In many ways these pictures are the best kind there is, giving full canvass to an ideal vista. Their style allows the clearest, the fairest, and the plainest view that we can form of a universe of discourse, affording equal representation to all dispositions and maintaining a balance with respect to ordinary and differential features. If only we could extend this view! Unluckily, an obvious difficulty beclouds this prospect, and that is how precipitately we run into the limits of our plane and visual intuitions. Even within the scope of the spare few dimensions that we have scanned up to this point subtle discrepancies have crept in already. The circumstances that bind us and the frameworks that block us, the flat distortion of the planar projection and the inevitable ineffability that precludes us from wrapping its rhomb figure into rings around a torus, all of these factors disguise the underlying but true connectivity of the universe of discourse.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-a -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-a.} ~~ \text{Extension from 2 to 4 Dimensions : Areal}\!</math><br />
|}<br />
<br />
Figure 19-b shows the differential extension from <math>U^\bullet = [u, v]\!</math> to <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v]\!</math> in the ''bundle of boxes'' form of venn diagram.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-b -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-b.} ~~ \text{Extension from 2 to 4 Dimensions : Bundle}\!</math><br />
|}<br />
<br />
As dimensions increase, this factorization of the extended universe along the lines that are marked out by the bundle picture begins to look more and more like a practical necessity. But whenever we use a propositional model to address a real situation in the context of nature we need to remain aware that this articulation into factors, affecting our description, may be wholly artificial in nature and cleave to nothing, no joint in nature, nor any juncture in time to be in or out of joint.<br />
<br />
Figure 19-c illustrates the extension from 2 to 4 dimensions in the ''compact'' style of venn diagram. Here, just the changes with respect to the center cell are shown.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-c -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-c.} ~~ \text{Extension from 2 to 4 Dimensions : Compact}\!</math><br />
|}<br />
<br />
Figure 19-d gives the ''digraph'' form of representation for the differential extension <math>U^\bullet \to \mathrm{E}U^\bullet,\!</math> where the 4 nodes marked with a circle <math>{}^{\bigcirc}\!</math> are the cells <math>uv,\, u \texttt{(} v \texttt{)},\, \texttt{(} u \texttt{)} v,\, \texttt{(} u \texttt{)(} v \texttt{)},\!</math> respectively, and where a 2-headed arc counts as 2 arcs of the differential digraph.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-d -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-d.} ~~ \text{Extension from 2 to 4 Dimensions : Digraph}\!</math><br />
|}<br />
<br />
===Thematization of Functions : And a Declaration of Independence for Variables===<br />
<br />
{| width="100%"<br />
| align="left" |<br />
''And as imagination bodies forth''<br><br />
''The forms of things unknown, the poet's pen''<br><br />
''Turns them to shapes, and gives to airy nothing''<br><br />
''A local habitation and a name.''<br />
| align="right" valign="bottom" | A Midsummer Night's Dream, 5.1.18<br />
|}<br />
<br />
In the representation of propositions as functions it is possible to notice different degrees of explicitness in the way their functional character is symbolized. To indicate what I mean by this, the next series of Figures illustrates a set of graphic conventions that will be put to frequent use in the remainder of this discussion, both to mark the relevant distinctions and to help us convert between related expressions at different levels of explicitness in their functionality.<br />
<br />
====Thematization : Venn Diagrams====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
The known universe has one complete lover and that is the greatest poet. He consumes an eternal passion and is indifferent which chance happens and which possible contingency of fortune or misfortune and persuades daily and hourly his delicious pay.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 11&ndash;12]<br />
|}<br />
<br />
Figure 20-i traces the first couple of steps in this order of ''thematic'' progression, that will gradually run the gamut through a complete series of degrees of functional explicitness in the expression of logical propositions. The first venn diagram represents a situation where the function is indicated by a shaded figure and a logical expression. At this stage one may be thinking of the proposition only as expressed by a formula in a particular language and its content only as a subset of the universe of discourse, as when considering the proposition <math>u\!\cdot\!v</math> in the universe <math>[u, v].\!</math> The second venn diagram depicts a situation in which two significant steps have been taken. First, one has taken the trouble to give the proposition <math>u\!\cdot\!v</math> a distinctive functional name <math>{}^{\backprime\backprime} J {}^{\prime\prime}.\!</math> Second, one has come to think explicitly about the target domain that contains the functional values of <math>J,\!</math> as when writing <math>J : \langle u, v \rangle \to \mathbb{B}.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 20-i -- Thematization of Conjunction (Stage 1).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 20-i.} ~~ \text{Thematization of Conjunction (Stage 1)}\!</math><br />
|}<br />
<br />
In Figure 20-ii the proposition <math>J\!</math> is viewed explicitly as a transformation from one universe of discourse to another.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 20-ii -- Thematization of Conjunction (Stage 2).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 20-ii.} ~~ \text{Thematization of Conjunction (Stage 2)}\!</math><br />
|}<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
o-------------------------------o o-------------------------------o<br />
| | | |<br />
| o-----o o-----o | | o-----o o-----o |<br />
| / \ / \ | | / \ / \ |<br />
| / o \ | | / o \ |<br />
| / /`\ \ | | / /`\ \ |<br />
| o o```o o | | o o```o o |<br />
| | u |```| v | | | | u |```| v | |<br />
| o o```o o | | o o```o o |<br />
| \ \`/ / | | \ \`/ / |<br />
| \ o / | | \ o / |<br />
| \ / \ / | | \ / \ / |<br />
| o-----o o-----o | | o-----o o-----o |<br />
| | | |<br />
o-------------------------------o o-------------------------------o<br />
\ / \ /<br />
\ / \ /<br />
\ / \ J /<br />
\ / \ /<br />
\ / \ /<br />
o----------\---------/----------o o----------\---------/----------o<br />
| \ / | | \ / |<br />
| \ / | | \ / |<br />
| o-----@-----o | | o-----@-----o |<br />
| /`````````````\ | | /`````````````\ |<br />
| /```````````````\ | | /```````````````\ |<br />
| /`````````````````\ | | /`````````````````\ |<br />
| o```````````````````o | | o```````````````````o |<br />
| |```````````````````| | | |```````````````````| |<br />
| |```````` J ````````| | | |```````` x ````````| |<br />
| |```````````````````| | | |```````````````````| |<br />
| o```````````````````o | | o```````````````````o |<br />
| \`````````````````/ | | \`````````````````/ |<br />
| \```````````````/ | | \```````````````/ |<br />
| \`````````````/ | | \`````````````/ |<br />
| o-----------o | | o-----------o |<br />
| | | |<br />
| | | |<br />
o-------------------------------o o-------------------------------o<br />
J = u v x = J<u, v><br />
<br />
Figure 20-ii. Thematization of Conjunction (Stage 2)<br />
</pre><br />
|}<br />
<br />
In the first venn diagram the name that is assigned to a composite proposition, function, or region in the source universe is delegated to a simple feature in the target universe. This can result in a single character or term exceeding the responsibilities it can carry off well. Allowing the name of a function <math>J : \langle u, v \rangle \to \mathbb{B}\!</math> to serve as the name of its dependent variable <math>J : \mathbb{B}\!</math> does not mean that one has to confuse a function with any of its values, but it does put one at risk for a number of obvious problems, and we should not be surprised, on numerous and limiting occasions, when quibbling arises from the attempts of a too original syntax to serve these two masters.<br />
<br />
The second venn diagram circumvents these difficulties by introducing a new variable name for each basic feature of the target universe, as when writing <math>J : \langle u, v \rangle \to \langle x \rangle,\!</math> and thereby assigns a concrete type <math>\langle x \rangle</math> to the abstract codomain <math>\mathbb{B}.\!</math> To make this induction of variables more formal one can append subscripts, as in <math>x_J,\!</math> to indicate the origin or derivation of the new characters. Or we may use a lexical modifier to convert function names into variable names, for example, associating the function name <math>J\!</math> with the variable name <math>\check{J}.\!</math> Thus we may think of <math>x = x_J = \check{J}\!</math> as the ''cache variable'' corresponding to the function <math>J\!</math> or the symbol <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> considered as a contingent variable.<br />
<br />
In Figure 20-iii we arrive at a stage where the functional equations <math>J = u\!\cdot\!v</math> and <math>x = u\!\cdot\!v</math> are regarded as propositions in their own right, reigning in and ruling over the 3-feature universes of discourse <math>[u, v, J]~\!</math> and <math>[u, v, x],\!</math> respectively. Subject to the cautions already noted, the function name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> can be reinterpreted as the name of a feature <math>\check{J}</math> and the equation <math>J = u\!\cdot\!v</math> can be read as the logical equivalence <math>\texttt{((} J, u ~ v \texttt{))}.\!</math> To give it a generic name let us call this newly expressed, collateral proposition the ''thematization'' or the ''thematic extension'' of the original proposition <math>J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 20-iii -- Thematization of Conjunction (Stage 3).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 20-iii.} ~~ \text{Thematization of Conjunction (Stage 3)}\!</math><br />
|}<br />
<br />
The first venn diagram represents the thematization of the conjunction <math>J\!</math> with shading in the appropriate regions of the universe <math>[u, v, J].\!</math> Also, it illustrates a quick way of constructing a thematic extension. First, draw a line, in practice or the imagination, that bisects every cell of the original universe, placing half of each cell under the aegis of the thematized proposition and the other half under its antithesis. Next, on the scene where the theme applies leave the shade wherever it lies, and off the stage, where it plays otherwise, stagger the pattern in a harlequin guise.<br />
<br />
In the final venn diagram of this sequence the thematic progression comes full circle and completes one round of its development. The ambiguities that were occasioned by the changing role of the name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> are resolved by introducing a new variable name <math>{}^{\backprime\backprime} x {}^{\prime\prime}</math> to take the place of <math>\check{J},\!</math> and the region that represents this fresh featured <math>x\!</math> is circumscribed in a more conventional symmetry of form and placement. Just as we once gave the name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> to the proposition <math>u\!\cdot\!v,</math> we now give the name <math>{}^{\backprime\backprime} \iota {}^{\prime\prime}</math> to its thematization <math>\texttt{((} x, u ~ v \texttt{))}.\!</math> Already, again, at this culminating stage of reflection, we begin to think of the newly named proposition as a distinctive individual, a particular function <math>\iota : \langle u, v, x \rangle \to \mathbb{B}.\!</math><br />
<br />
From now on, the terms ''thematic extension'' and ''thematization'' will be used to describe both the process and degree of explication that progresses through this series of pictures, both the operation of increasingly explicit symbolization and the dimension of variation that is swept out by it. To speak of this change in general, that takes us in our current example from <math>J\!</math> to <math>\iota,\!</math> we introduce a class of operators symbolized by the Greek letter <math>\theta,\!</math> writing <math>\iota = \theta J\!</math> in the present instance. The operator <math>\theta,\!</math> in the present situation bearing the type <math>\theta : [u, v] \to [u, v, x],\!</math> provides us with a convenient way of recapitulating and summarizing the complete cycle of thematic developments.<br />
<br />
Figure 21 shows how the thematic extension operator <math>\theta\!</math> acts on two further examples, the disjunction <math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math> and the equality <math>\texttt{((} u, v \texttt{))}.\!</math> Referring to the disjunction as <math>f(u, v)\!</math> and the equality as <math>f(u, v),\!</math> we may express the thematic extensions as <math>\varphi = \theta f\!</math> and <math>\gamma = \theta g.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 21 -- Thematization of Disjunction and Equality.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 21.} ~~ \text{Thematization of Disjunction and Equality}\!</math><br />
|}<br />
<br />
====Thematization : Truth Tables====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
That which distorts honest shapes or which creates unearthly beings or places or contingencies is a nuisance and a revolt.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 19]<br />
|}<br />
<br />
Tables 22 through 25 outline a method for computing the thematic extensions of propositions in terms of their coordinate values.<br />
<br />
A preliminary step, as illustrated in Table&nbsp;22, is to write out the truth table representations of the propositional forms whose thematic extensions one wants to compute, in the present instance, the functions <math>f(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math> and <math>g(u, v) = \texttt{((} u, v \texttt{))}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:50%"<br />
|+ style="height:30px" | <math>\text{Table 22.} ~~ \text{Disjunction}~ f ~\text{and Equality}~ g\!</math><br />
|- style="height:35px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black" | <math>f\!</math><br />
| <math>g\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Next, each propositional form is individually represented in the fashion shown in Tables&nbsp;23-i and 23-ii, using <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} g {}^{\prime\prime}\!</math> as function names and creating new variables <math>x\!</math> and <math>y\!</math> to hold the associated functional values. This pair of Tables outlines the first stage in the transition from the <math>2\!</math>-dimensional universes of <math>f\!</math> and <math>g\!</math> to the <math>3\!</math>-dimensional universes of <math>\theta f\!</math> and <math>\theta g.\!</math> The top halves of the Tables replicate the truth table patterns for <math>f\!</math> and <math>g\!</math> in the form <math>f : [u, v] \to [x]\!</math> and <math>g : [u, v] \to [y].\!</math> The bottom halves of the tables print the negatives of these pictures, as it were, and paste the truth tables for <math>\texttt{(} f \texttt{)}\!</math> and <math>\texttt{(} g \texttt{)}\!</math> under the copies for <math>f\!</math> and <math>g.\!</math> At this stage, the columns for <math>\theta f\!</math> and <math>\theta g\!</math> are appended almost as afterthoughts, amounting to indicator functions for the sets of ordered triples that make up the functions <math>f\!</math> and <math>g.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 23-i and 23-ii.} ~~ \text{Thematics of Disjunction and Equality (1)}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 23-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black" | <math>f\!</math><br />
| <math>x\!</math><br />
| <math>\varphi\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |<br />
<math>\begin{matrix}\to\\\to\\\to\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" | &nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 23-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black" | <math>g\!</math><br />
| <math>y\!</math><br />
| <math>\gamma\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |<br />
<math>\begin{matrix}\to\\\to\\\to\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" | &nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
All the data are now in place to give the truth tables for <math>\theta f\!</math> and <math>\theta g.\!</math> All that remains to be done is to permute the rows and change the roles of <math>x\!</math> and <math>y\!</math> from dependent to independent variables. In Tables&nbsp;24-i and 24-ii the rows are arranged in such a way as to put the 3-tuples <math>(u, v, x)\!</math> and <math>(u, v, y)\!</math> in binary numerical order, suitable for viewing as the arguments of the maps <math>\theta f = \varphi : [u, v, x] \to \mathbb{B}\!</math> and <math>\theta g = \gamma : [u, v, y] \to \mathbb{B}.\!</math> Moreover, the structure of the tables is altered slightly, allowing the now vestigial functions <math>\theta f\!</math> and <math>\theta g\!</math> to be passed over without further attention and shifting the heavy vertical bars a notch to the right. In effect, this clinches the fact that the thematic variables <math>x := \check{f}\!</math> and <math>y := \check{g}\!</math> are now treated as independent variables.<br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 24-i and 24-ii.} ~~ \text{Thematics of Disjunction and Equality (2)}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 24-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>f\!</math><br />
| <math>x\!</math><br />
| style="border-left:1px solid black" | <math>\varphi\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <font size="+2">&nbsp;<br></font><math>\begin{matrix}\to\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 24-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>g\!</math><br />
| <math>y\!</math><br />
| style="border-left:1px solid black" | <math>\gamma\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | &nbsp;<br><math>\begin{matrix}\to\\\to\end{matrix}\!</math><br>&nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
An optional reshuffling of the rows brings additional features of the thematic extensions to light. Leaving the columns in place for the sake of comparison, Tables&nbsp;25-i and 25-ii sort the rows in a different order, in effect treating <math>x\!</math> and <math>y\!</math> as the primary variables in their respective 3-tuples. Regarding the thematic extensions in the form <math>\varphi : [x, u, v] \to \mathbb{B}\!</math> and <math>\gamma : [y, u, v] \to \mathbb{B}\!</math> makes it easier to see in this tabular setting a property that was graphically obvious in the venn diagrams above. Specifically, when the thematic variable <math>\check{F}\!</math> is true then <math>\theta F\!</math> exhibits the pattern of the original <math>F,\!</math> and when <math>\check{F}\!</math> is false then <math>\theta F\!</math> exhibits the pattern of its negation <math>\texttt{(} F \texttt{)}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 25-i and 25-ii.} ~~ \text{Thematics of Disjunction and Equality (3)}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 25-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>f\!</math><br />
| <math>x\!</math><br />
| style="border-left:1px solid black" | <math>\varphi\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>{\to}\!</math><br><font size="+2">&nbsp;<br>&nbsp;<br>&nbsp;<br></font><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <font size="+2">&nbsp;<br></font><math>\begin{matrix}\to\\\to\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 25-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>g\!</math><br />
| <math>y\!</math><br />
| style="border-left:1px solid black" | <math>\gamma\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | &nbsp;<br><math>\begin{matrix}\to\\\to\end{matrix}\!</math><br>&nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
Finally, Tables&nbsp;26-i and 26-ii compare the tacit extensions <math>\boldsymbol\varepsilon : [u, v] \to [u, v, x]\!</math> and <math>\boldsymbol\varepsilon : [u, v] \to [u, v, y]\!</math> with the thematic extensions of the same types, as applied to the propositions <math>f\!</math> and <math>g,\!</math> respectively.<br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 26-i and 26-ii.} ~~ \text{Tacit Extension and Thematization}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 26-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>x\!</math><br />
| style="border-left:1px solid black" | <math>\boldsymbol\varepsilon f\!</math><br />
| <math>\theta f\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 26-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>y\!</math><br />
| style="border-left:1px solid black" | <math>\boldsymbol\varepsilon g\!</math><br />
| <math>\theta g\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
Table 27 summarizes the thematic extensions of all propositions on two variables. Column&nbsp;4 lists the equations of form <math>\texttt{((} \check{f_i}, f_i (u, v) \texttt{))}\!</math> and Column&nbsp;5 simplifies these equations into the form of algebraic expressions. As always, <math>{}^{\backprime\backprime} + {}^{\prime\prime}\!</math> refers to exclusive disjunction and each <math>{}^{\backprime\backprime} \check{f} {}^{\prime\prime}\!</math> appearing in the last two Columns refers to the corresponding variable name <math>{}^{\backprime\backprime} \check{f_i} {}^{\prime\prime}.~\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 27.} ~~ \text{Thematization of Bivariate Propositions}\!</math><br />
|- style="height:30px; background:ghostwhite"<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>{f}\!</math><br />
| <math>\theta f\!</math><br />
| <math>\theta f\!</math><br />
|- style="background:ghostwhite"<br />
| align="right" | <math>u\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| align="right" | <math>v\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| align="left" | <math>\texttt{((} \check{f} \texttt{,~(~)~))}\!</math><br />
| align="left" | <math>\check{f} + 1\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\\[4pt]<br />
f_{4}<br />
\\[4pt]<br />
f_{8}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
1~0~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} u \texttt{)~} v \texttt{~}<br />
\\[4pt]<br />
\texttt{~} u \texttt{~(} v \texttt{)}<br />
\\[4pt]<br />
\texttt{~} u \texttt{~~} v \texttt{~}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(u)(v)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~(u)~v~~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~u~(v)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~u~~v~~))}<br />
\end{array}</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + u + v + uv<br />
\\[4pt]<br />
\check{f} + v + uv + 1<br />
\\[4pt]<br />
\check{f} + u + uv + 1<br />
\\[4pt]<br />
\check{f} + uv + 1<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{3}<br />
\\[4pt]<br />
f_{12}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{)}<br />
\\[4pt]<br />
\texttt{~} u \texttt{~}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(u)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~u~~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + u<br />
\\[4pt]<br />
\check{f} + u + 1<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[4pt]<br />
f_{9}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[4pt]<br />
1~0~0~1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{,} v \texttt{)}<br />
\\[4pt]<br />
\texttt{((} u \texttt{,} v \texttt{))}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~~(} u \texttt{,} v \texttt{)~~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~((} u \texttt{,} v \texttt{))~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + u + v + 1<br />
\\[4pt]<br />
\check{f} + u + v<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{5}<br />
\\[4pt]<br />
f_{10}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} v \texttt{)}<br />
\\[4pt]<br />
\texttt{~} v \texttt{~}<br />
\end{matrix}</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(} v \texttt{)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~} v \texttt{~~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + v<br />
\\[4pt]<br />
\check{f} + v + 1<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[4pt]<br />
f_{11}<br />
\\[4pt]<br />
f_{13}<br />
\\[4pt]<br />
f_{14}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(~} u \texttt{~~} v \texttt{~)}<br />
\\[4pt]<br />
\texttt{(~} u \texttt{~(} v \texttt{))}<br />
\\[4pt]<br />
\texttt{((} u \texttt{)~} v \texttt{~)}<br />
\\[4pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(~} u \texttt{~~} v \texttt{~)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~(~} u \texttt{~(} v \texttt{))~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~((} u \texttt{)~} v \texttt{~)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~((} u \texttt{)(} v \texttt{))~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + uv<br />
\\[4pt]<br />
\check{f} + u + uv<br />
\\[4pt]<br />
\check{f} + v + uv<br />
\\[4pt]<br />
\check{f} + u + v + uv + 1<br />
\end{array}\!</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| align="left" | <math>\texttt{((} \check{f} \texttt{,~((~))~))}\!</math><br />
| align="left" | <math>\check{f}\!</math><br />
|}<br />
<br />
<br><br />
<br />
In order to show what all of the thematic extensions from two dimensions to three dimensions look like in terms of coordinates, Tables&nbsp;28 and 29 present ordinary truth tables for the functions <math>f_i : \mathbb{B}^2 \to \mathbb{B}\!</math> and for the corresponding thematizations <math>\theta f_i = \varphi_i : \mathbb{B}^3 \to \mathbb{B}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 28.} ~~ \text{Propositions on Two Variables}\!</math><br />
|- style="height:35px; background:ghostwhite"<br />
| style="width:5%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:5%; border-bottom:1px solid black; border-left:1px solid black" | <math>f_{0}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{1}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{2}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{3}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{4}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{5}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{6}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{7}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{8}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{9}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{10}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{11}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{12}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{13}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{14}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{15}\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 29.} ~~ \text{Thematic Extensions of Bivariate Propositions}\!</math><br />
|- style="height:35px; background:ghostwhite"<br />
| style="width:5%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\check{f}\!</math><br />
| style="width:5%; border-bottom:1px solid black; border-left:1px solid black" | <math>\varphi_{0}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{1}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{2}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{3}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{4}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{5}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{6}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{7}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{8}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{9}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{10}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{11}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{12}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{13}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{14}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{15}\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
|-<br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Propositional Transformations===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
If only the word &lsquo;artificial&rsquo; were associated with the idea of ''art'', or expert skill gained through voluntary apprenticeship (instead of suggesting the factitious and unreal), we might say that ''logical'' refers to artificial thought.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 56&ndash;57]<br />
|}<br />
<br />
In this section we develop a comprehensive set of concepts for dealing with transformations between universes of discourse. In this most general setting the source and target universes of a transformation are allowed to be different, but may be the same. When we apply these concepts to dynamic systems we focus on the important special case of transformations that map a universe into itself, regarding them as the state transitions of a discrete dynamical process and placing them among the myriad ways that a universe of discourse might change, and by that change turn into itself.<br />
<br />
====Alias and Alibi Transformations====<br />
<br />
There are customarily two modes of understanding a transformation, at least, when we try to interpret its relevance to something in reality. A transformation always refers to a changing prospect, to say it in a unified but equivocal way, but this can be taken to mean either a subjective change in the interpreting observer's point of view or an objective change in the systematic subject of discussion. In practice these variant uses of the transformation concept are distinguished in the following terms:<br />
<br />
# A ''perspectival'' or ''alias'' transformation refers to a shift in perspective or a change in language that takes place in the observer's frame of reference.<br />
# A ''transitional'' or ''alibi'' transformation refers to a change of position or an alteration of state that occurs in the object system as it falls under study.<br />
<br />
(For a recent discussion of the alias vs. alibi issue, as it relates to linear transformations in vector spaces and to other issues of an algebraic nature, see [MaB, 256, 582-4].)<br />
<br />
Naturally, when we are concerned with the dynamical properties of a system, the transitional aspect of transformation is the factor that comes to the fore, and this involves us in contemplating all of the ways of changing a universe into itself while remaining under the rule of established dynamical laws. In the prospective application to dynamic systems, and to neural networks viewed in this light, our interest lies chiefly with the transformations of a state space into itself that constitute the state transitions of a discrete dynamic process. Nevertheless, many important properties of these transformations, and some constructions that we need to see most clearly, are independent of the transitional interpretation and are likely to be confounded with irrelevant features if presented first and only in that association.<br />
<br />
In addition, and in partial contrast, intelligent systems are exactly that species of dynamic agents that have the capacity to have a point of view, and we cannot do justice to their peculiar properties without examining their ability to form and transform their own frames of reference in exposure to the elements of their own experience. In this setting, the perspectival aspect of transformation is the facet that shines most brightly, perhaps too often leaving us fascinated with mere glimmerings of its actual potential. It needs to be emphasized that nothing of the ordinary sort needs be moved in carrying out a transformation under the alias interpretation, that it may only involve a change in the forms of address, an amendment of the terms which are customed to approach and fashioned to describe the very same things in the very same world. But again, working within a discipline of realistic computation, we know how formidably complex and resource-consuming such transformations of perspective can be to implement in practice, much less to endow in the self-governed form of a nascently intelligent dynamical system.<br />
<br />
====Transformations of General Type====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
''Es ist passiert'', &ldquo;it just sort of happened&rdquo;, people said there when other people in other places thought heaven knows what had occurred. It was a peculiar phrase, not known in this sense to the Germans and with no equivalent in other languages, the very breath of it transforming facts and the bludgeonings of fate into something light as eiderdown, as thought itself.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 34]<br />
|}<br />
<br />
Consider the situation illustrated in Figure&nbsp;30, where the alphabets <math>\mathcal{U} = \{ u, v \}\!</math> and <math>\mathcal{X} = \{ x, y, z \}\!</math> are used to label basic features in two different logical universes, <math>U^\bullet = [u, v]\!</math> and <math>X^\bullet = [x, y, z].\!</math><br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
o-------------------------------------------------------o<br />
| U |<br />
| |<br />
| o-----------o o-----------o |<br />
| / \ / \ |<br />
| / o \ |<br />
| / / \ \ |<br />
| / / \ \ |<br />
| o o o o |<br />
| | | | | |<br />
| | u | | v | |<br />
| | | | | |<br />
| o o o o |<br />
| \ \ / / |<br />
| \ \ / / |<br />
| \ o / |<br />
| \ / \ / |<br />
| o-----------o o-----------o |<br />
| |<br />
| |<br />
o---------------------------o---------------------------o<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
o-------------------------o o-------------------------o o-------------------------o<br />
| U | | U | | U |<br />
| o---o o---o | | o---o o---o | | o---o o---o |<br />
| / \ / \ | | / \ / \ | | / \ / \ |<br />
| / o \ | | / o \ | | / o \ |<br />
| / / \ \ | | / / \ \ | | / / \ \ |<br />
| o o o o | | o o o o | | o o o o |<br />
| | u | | v | | | | u | | v | | | | u | | v | |<br />
| o o o o | | o o o o | | o o o o |<br />
| \ \ / / | | \ \ / / | | \ \ / / |<br />
| \ o / | | \ o / | | \ o / |<br />
| \ / \ / | | \ / \ / | | \ / \ / |<br />
| o---o o---o | | o---o o---o | | o---o o---o |<br />
| | | | | |<br />
o-------------------------o o-------------------------o o-------------------------o<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ g | \ f / | h /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ o----------|-----------\-----/-----------|----------o /<br />
\ | X | \ / | | /<br />
\ | | \ / | | /<br />
\ | | o-----o-----o | | /<br />
\| | / \ | |/<br />
\ | / \ | /<br />
|\ | / \ | /|<br />
| \ | / \ | / |<br />
| \ | / \ | / |<br />
| \ | o x o | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \| | | |/ |<br />
| o--o--------o o--------o--o |<br />
| / \ \ / / \ |<br />
| / \ \ / / \ |<br />
| / \ o / \ |<br />
| / \ / \ / \ |<br />
| / \ / \ / \ |<br />
| o o--o-----o--o o |<br />
| | | | | |<br />
| | | | | |<br />
| | | | | |<br />
| | y | | z | |<br />
| | | | | |<br />
| | | | | |<br />
| o o o o |<br />
| \ \ / / |<br />
| \ \ / / |<br />
| \ o / |<br />
| \ / \ / |<br />
| \ / \ / |<br />
| o-----------o o-----------o |<br />
| |<br />
| |<br />
o---------------------------------------------------o<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ p , q /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
o<br />
<br />
Figure 30. Generic Frame of a Logical Transformation<br />
</pre><br />
|}<br />
<br />
Enter the picture, as we usually do, in the middle of things, with features like <math>x, y , z\!</math> that present themselves to be simple enough in their own right and that form a satisfactory, if temporary foundation to provide a basis for discussion. In this universe and on these terms we find expression for various propositions and questions of principal interest to ourselves, as indicated by the maps <math>p, q : X \to \mathbb{B}.\!</math> Then we discover that the simple features <math>\{ x, y, z \}\!</math> are really more complex than we thought at first, and it becomes useful to regard them as functions <math>\{ f, g, h \}\!</math> of other features <math>\{ u, v \}\!</math> that we place in a preface to our original discourse, or suppose as topics of a preliminary universe of discourse <math>U^\bullet = [u, v].\!</math> It may happen that these late-blooming but pre-ambling features are found to lie closer, in a sense that may be our job to determine, to the central nature of the situation of interest, in which case they earn our regard as being more fundamental, but these functions and features are only required to supply a critical stance on the universe of discourse or an alternate perspective on the nature of things in order to be preserved as useful.<br />
<br />
A particular transformation <math>F : [u, v] \to [x, y, z]\!</math> may be expressed by a system of equations, as shown below. Here, <math>F\!</math> is defined by its component maps <math>F = (F_1, F_2, F_3) = (f, g, h),\!</math> where each component map in <math>\{ f, g, h \}\!</math> is a proposition of type <math>\mathbb{B}^n \to \mathbb{B}^1.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:50%"<br />
|<br />
<math>\begin{matrix}<br />
x & = & f(u, v)<br />
\\[10pt]<br />
y & = & g(u, v)<br />
\\[10pt]<br />
z & = & h(u, v)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Regarded as a logical statement, this system of equations expresses a relation between a collection of freely chosen propositions <math>\{ f, g, h \}\!</math> in one universe of discourse and the special collection of simple propositions <math>\{ x, y, z \}\!</math> on which is founded another universe of discourse. Growing familiarity with a particular transformation of discourse, and the desire to achieve a ready understanding of its implications, requires that we be able to convert this information about generals and simples into information about all the main subtypes of propositions, including the linear and singular propositions.<br />
<br />
===Analytic Expansions : Operators and Functors===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Consider what effects that might ''conceivably'' have practical bearings you ''conceive'' the objects of your ''conception'' to have. Then, your ''conception'' of those effects is the whole of your ''conception'' of the object.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; C.S. Peirce, &ldquo;The Maxim of Pragmatism&rdquo;, CP 5.438<br />
|}<br />
<br />
Given the barest idea of a logical transformation, as suggested by the sketch in Figure&nbsp;30, and having conceptualized the universe of discourse, with all of its points and propositions, as a beginning object of discussion, we are ready to enter the next phase of our investigation.<br />
<br />
====Operators on Propositions and Transformations====<br />
<br />
The next step is naturally inclined toward objects of the next higher order, namely, with operators that take in argument lists of logical transformations and that give back specified types of logical transformations as their results. For our present aims, we do not need to consider the most general class of such operators, nor any one of them for its own sake. Rather, we are interested in the special sorts of operators that arise in the study and analysis of logical transformations. Figuratively speaking, these operators serve as instruments for the live tomography (and hopefully not the vivisection) of the forms of change under view. Beyond that, they open up ways to implement the changes of view that we need to grasp all the variations on a transformational theme, or to appreciate enough of its significant features to &ldquo;get the drift&rdquo; of the change occurring, to form a passing acquaintance or a synthetic comprehension of its general character and disposition.<br />
<br />
The simplest type of operator is one that takes a single transformation as an argument and returns a single transformation as a result, and most of the operators explicitly considered in our discussion will be of this kind. Figure&nbsp;31 illustrates the typical situation.<br />
<br />
{| align="center" border="0" cellpadding="20"<br />
|<br />
<pre><br />
o---------------------------------------o<br />
| |<br />
| |<br />
| U% F X% |<br />
| o------------------>o |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| !W! | | !W! |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| v v |<br />
| o------------------>o |<br />
| !W!U% !W!F !W!X% |<br />
| |<br />
| |<br />
o---------------------------------------o<br />
Figure 31. Operator Diagram (1)<br />
</pre><br />
|}<br />
<br />
In this Figure <math>{}^{\backprime\backprime} \mathsf{W} {}^{\prime\prime}\!</math> stands for a generic operator <math>\mathsf{W},\!</math> in this case one that takes a logical transformation <math>F\!</math> of type <math>(U^\bullet \to X^\bullet)\!</math> into a logical transformation <math>\mathsf{W}F\!</math> of type <math>(\mathsf{W}U^\bullet \to \mathsf{W}X^\bullet).\!</math> Thus, the operator <math>\mathsf{W}\!</math> must be viewed as making assignments for both families of objects we have previously considered, that is, for universes of discourse like <math>{U^\bullet}\!</math> and <math>{X^\bullet}\!</math> and for logical transformations like <math>F.\!</math><br />
<br />
'''Note.''' Strictly speaking, an operator like <math>\mathsf{W}\!</math> works between two whole categories of universes and transformations, which we call the ''source'' and the ''target'' categories of <math>\mathsf{W}.\!</math> Given this setting, <math>\mathsf{W}\!</math> specifies for each universe <math>U^\bullet\!</math> in its source category a definite universe <math>\mathsf{W}U^\bullet\!</math> in its target category, and to each transformation <math>F\!</math> in its source category it assigns a unique transformation <math>\mathsf{W}F\!</math> in its target category. Naturally, this only works if <math>\mathsf{W}\!</math> takes the source <math>U^\bullet</math> and the target <math>X^\bullet</math> of the map <math>F\!</math> over to the source <math>\mathsf{W}U^\bullet\!</math> and the target <math>\mathsf{W}X^\bullet\!</math> of the map <math>\mathsf{W}F.\!</math> With luck or care enough, we can avoid ever having to put anything like that in words again, letting diagrams do the work. In the situations of present concern we are usually focused on a single transformation <math>F,\!</math> and thus we can take it for granted that the assignment of universes under <math>\mathsf{W}\!</math> is defined appropriately at the source and target ends of <math>F.\!</math> It is not always the case, though, that we need to use the particular names (like <math>{}^{\backprime\backprime} \mathsf{W}U^\bullet {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \mathsf{W}X^\bullet {}^{\prime\prime}\!</math>) that <math>\mathsf{W}\!</math> assigns by default to its operative image universes. In most contexts we will usually have a prior acquaintance with these universes under other names and it is necessary only that we can tell from the information associated with an operator <math>\mathsf{W}\!</math> what universes they are.<br />
<br />
In Figure&nbsp;31 the maps <math>F\!</math> and <math>\mathsf{W}F\!</math> are displayed horizontally, the way one normally orients functional arrows in a written text, and <math>\mathsf{W}\!</math> rolls the map <math>F\!</math> downward into the images that are associated with <math>\mathsf{W}F.\!</math> In Figure&nbsp;32 the same information is redrawn so that the maps <math>F\!</math> and <math>\mathsf{W}F\!</math> flow down the page, and <math>\mathsf{W}\!</math> unfurls the map <math>F\!</math> rightward into domains that are the eminent purview of <math>\mathsf{W}F.\!</math><br />
<br />
{| align="center" border="0" cellpadding="20"<br />
|<br />
<pre><br />
o---------------------------------------o<br />
| |<br />
| |<br />
| U% !W! !W!U% |<br />
| o------------------>o |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| F | | !W!F |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| v v |<br />
| o------------------>o |<br />
| X% !W! !W!X% |<br />
| |<br />
| |<br />
o---------------------------------------o<br />
Figure 32. Operator Diagram (2)<br />
</pre><br />
|}<br />
<br />
The latter arrangement, as exhibited in Figure&nbsp;32, is more congruent with the thinking about operators that we shall do in the rest of this discussion, since all logical transformations from here on out will be pictured vertically, after the fashion of Figure&nbsp;30.<br />
<br />
====Differential Analysis of Propositions and Transformations====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" | The resultant metaphysical problem now is this: ''Does the man go round the squirrel or not?''<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 43]<br />
|}<br />
<br />
The approach to the differential analysis of logical propositions and transformations of discourse to be pursued here is carried out in terms of particular operators <math>\mathsf{W}\!</math> that act on propositions <math>F\!</math> or on transformations <math>F\!</math> to yield the corresponding operator maps <math>\mathsf{W}F.\!</math> The operator results then become the subject of a series of further stages of analysis, which take them apart into their propositional components, rendering them as a set of purely logical constituents. After this is done, all the parts are then re-integrated to reconstruct the original object in the light of a more complete understanding, at least in ways that enable one to appreciate certain aspects of it with fresh insight.<br />
<br />
* '''Remark on Strategy.''' At this point we run into a set of conceptual difficulties that force us to make a strategic choice in how we proceed. Part of the problem can be remedied by extending our discussion of tacit extensions to the transformational context. But the troubles that remain are much more obstinate and lead us to try two different types of solution. The approach that we develop first makes use of a variant type of extension operator, the ''trope extension'', to be defined below. This method is more conservative and requires less preparation, but has features which make it seem unsatisfactory in the long run. A more radical approach, but one with a better hope of long term success, makes use of the notion of ''contingency spaces''. These are an even more generous type of extended universe than the kind we currently use, but are defined subject to certain internal constraints. The extra work needed to set up this method forces us to put it off to a later stage. However, as a compromise, and to prepare the ground for the next pass, we call attention to the various conceptual difficulties as they arise along the way and try to give an honest estimate of how well our first approach deals with them.<br />
<br />
We now describe in general terms the particular operators that are instrumental to this form of analysis. The main series of operators all have the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathsf{W}<br />
& : &<br />
( U^\bullet \to X^\bullet )<br />
& \to &<br />
( \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet )<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
If we assume that the source universe <math>U^\bullet</math> and the target universe <math>X^\bullet</math> have finite dimensions <math>n\!</math> and <math>k,\!</math> respectively, then each operator <math>\mathsf{W}\!</math> is encompassed by the same abstract type:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathsf{W}<br />
& : &<br />
( [\mathbb{B}^n] \to [\mathbb{B}^k] )<br />
& \to &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k \times \mathbb{D}^k] )<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Since the range features of the operator result <math>\mathsf{W}F : [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k \times \mathbb{D}^k]</math> can be sorted by their ordinary versus differential qualities and the component maps can be examined independently, the complete operator <math>\mathsf{W}\!</math> can be separated accordingly into two components, in the form <math>\mathsf{W} = (\boldsymbol\varepsilon, \mathrm{W}).\!</math> Given a fixed context of source and target universes, <math>\boldsymbol\varepsilon\!</math> is always the same type of operator, a multiple component version of the tacit extension operators that were described earlier. In this context <math>\boldsymbol\varepsilon\!</math> has the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{array}{lccccc}<br />
\text{Concrete type}<br />
& \boldsymbol\varepsilon<br />
& : &<br />
( U^\bullet \to X^\bullet )<br />
& \to &<br />
( \mathrm{E}U^\bullet \to X^\bullet )<br />
\\[10pt]<br />
\text{Abstract type}<br />
& \boldsymbol\varepsilon<br />
& : &<br />
( [\mathbb{B}^n] \to [\mathbb{B}^k] )<br />
& \to &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k] )<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
On the other hand, the operator <math>\mathrm{W}\!</math> is specific to each <math>\mathsf{W}.\!</math> In this context <math>\mathrm{W}\!</math> always has the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{array}{lccccc}<br />
\text{Concrete type}<br />
& W<br />
& : &<br />
( U^\bullet \to X^\bullet )<br />
& \to &<br />
( \mathrm{E}U^\bullet \to \mathrm{d}X^\bullet )<br />
\\[10pt]<br />
\text{Abstract type}<br />
& W<br />
& : &<br />
( [\mathbb{B}^n] \to [\mathbb{B}^k] )<br />
& \to &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{D}^k] )<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
In the types just assigned to <math>\boldsymbol\varepsilon\!</math> and <math>\mathrm{W}\!</math> and by implication to their results <math>\boldsymbol\varepsilon F\!</math> and <math>\mathrm{W}F,\!</math> we have listed the most restrictive ranges defined for them rather than the more expansive target spaces that subsume these ranges. When there is need to recognize both, we may use type indications like the following:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\boldsymbol\varepsilon F<br />
& : &<br />
( \mathrm{E}U^\bullet \to X^\bullet \subseteq \mathrm{E}X^\bullet )<br />
& \cong &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k] \subseteq [\mathbb{B}^k \times \mathbb{D}^k] )<br />
\\[10pt]<br />
WF<br />
& : &<br />
( \mathrm{E}U^\bullet \to \mathrm{d}X^\bullet \subseteq \mathrm{E}X^\bullet )<br />
& \cong &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{D}^k] \subseteq [\mathbb{B}^k \times \mathbb{D}^k] )<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Hopefully, though, a general appreciation of these subsumptions will prevent us from having to make such declarations more often than absolutely necessary.<br />
<br />
In giving names to these operators we try to preserve as much of the traditional nomenclature and as many of the classical associations as possible. The chief difficulty in doing this is occasioned by the distinction between the &ldquo;sans&nbsp;serif&rdquo; operators <math>\mathsf{W}\!</math> and their &ldquo;serified&rdquo; components <math>\mathrm{W},\!</math> which forces us to find two distinct but parallel sets of terminology. Here is a plan to that purpose. First, the component operators <math>\mathrm{W}\!</math> are named by analogy with the corresponding operators in the classical difference calculus. Next, the complete operators <math>\mathsf{W} = (\boldsymbol\varepsilon, \mathrm{W})</math> are assigned titles according to their roles in a geometric or trigonometric allegory, if only to ensure that the tangent functor, that belongs to this family and whose exposition we are still working toward, comes out fit with its customary name. Finally, the operator results <math>\mathsf{W}F\!</math> and <math>\mathrm{W}F\!</math> can be fixed in our frame of reference by tethering the operative adjective for <math>\mathsf{W}\!</math> or <math>\mathrm{W}\!</math> to the anchoring epithet &ldquo;map&rdquo;, in conformity with an already standard practice.<br />
<br />
=====The Secant Operator : '''E'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Mr. Peirce, after pointing out that our beliefs are really rules for action, said that, to develop a thought's meaning, we need only determine what conduct it is fitted to produce: that conduct is for us its sole significance.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 46]<br />
|}<br />
<br />
Figures&nbsp;33-i and 33-ii depict two stages in the form of analysis that will be applied to transformations throughout the remainder of this study. From now on our interest is staked on an operator denoted <math>{}^{\backprime\backprime} \mathsf{E} {}^{\prime\prime},\!</math> which receives the principal investment of analytic attention, and on the constituent parts of <math>\mathsf{E},\!</math> which derive their shares of significance as developed by the analysis. In the sequel, we refer to <math>\mathsf{E}\!</math> as the ''secant operator'', taking it for granted that a context has been chosen that defines its type. The secant operator has the component description <math>\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}),\!</math> and its active ingredient <math>\mathrm{E}\!</math> is known as the ''enlargement operator''. (Here, we name <math>\mathrm{E}\!</math> after the literal ancestor of the shift operator in the calculus of finite differences, defined so that <math>\mathrm{E}f(x) = f(x+1)\!</math> for any suitable function <math>f,\!</math> though of course the logical analogue that we take up here must have a rather different definition.)<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
U% $E$ $E$U% $E$U% $E$U%<br />
o------------------>o============o============o<br />
| | | |<br />
| | | |<br />
| | | |<br />
| | | |<br />
F | | $E$F = | $d$^0.F + | $r$^0.F<br />
| | | |<br />
| | | |<br />
| | | |<br />
v v v v<br />
o------------------>o============o============o<br />
X% $E$ $E$X% $E$X% $E$X%<br />
<br />
Figure 33-i. Analytic Diagram (1)<br />
</pre><br />
|}<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
U% $E$ $E$U% $E$U% $E$U% $E$U%<br />
o------------------>o============o============o============o<br />
| | | | |<br />
| | | | |<br />
| | | | |<br />
| | | | |<br />
F | | $E$F = | $d$^0.F + | $d$^1.F + | $r$^1.F<br />
| | | | |<br />
| | | | |<br />
| | | | |<br />
v v v v v<br />
o------------------>o============o============o============o<br />
X% $E$ $E$X% $E$X% $E$X% $E$X%<br />
<br />
Figure 33-ii. Analytic Diagram (2)<br />
</pre><br />
|}<br />
<br />
In its action on universes <math>\mathsf{E}\!</math> yields the same result as <math>\mathrm{E},\!</math> a fact that can be expressed in equational form by writing <math>\mathsf{E}U^\bullet = \mathrm{E}U^\bullet\!</math> for any universe <math>U^\bullet.\!</math> Notice that the extended universes across the top and bottom of the diagram are indicated to be strictly identical, rather than requiring a corresponding decomposition for them. In a certain sense, the functional parts of <math>\mathsf{E}F\!</math> are partitioned into separate contexts that have to be re-integrated again, but the best image to use is that of making transparent copies of each universe and then overlapping their functional contents once more at the conclusion of the analysis, as suggested by the graphic conventions that are used at the top of Figure&nbsp;30.<br />
<br />
Acting on a transformation <math>F\!</math> from universe <math>U^\bullet\!</math> to universe <math>X^\bullet,\!</math> the operator <math>\mathsf{E}\!</math> determines a transformation <math>\mathsf{E}F\!</math> from <math>\mathsf{E}U^\bullet\!</math> to <math>\mathsf{E}X^\bullet.\!</math> The map <math>\mathsf{E}F\!</math> forms the main body of evidence to be investigated in performing a differential analysis of <math>F.\!</math> Because we shall frequently be focusing on small pieces of this map for considerable lengths of time, and consequently lose sight of the &ldquo;big picture&rdquo;, it is critically important to emphasize that the map <math>\mathsf{E}F\!</math> is a transformation that determines a relation from one extended universe into another. This means that we should not be satisfied with our understanding of a transformation <math>F\!</math> until we can lay out the full &ldquo;parts diagram&rdquo; of <math>\mathsf{E}F\!</math> along the lines of the generic frame in Figure&nbsp;30.<br />
<br />
Working within the confines of propositional calculus, it is possible to give an elementary definition of <math>\mathsf{E}F\!</math> by means of a system of propositional equations, as we now describe.<br />
<br />
Given a transformation<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>F = (F_1, \ldots, F_k) : \mathbb{B}^n \to \mathbb{B}^k\!</math><br />
|}<br />
<br />
of concrete type<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>F : [u_1, \ldots, u_n] \to [x_1, \ldots, x_k],\!</math><br />
|}<br />
<br />
the transformation<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\mathsf{E}F = (F_1, \ldots, F_k, \mathrm{E}F_1, \ldots, \mathrm{E}F_k) : \mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B}^k \times \mathbb{D}^k\!</math><br />
|}<br />
<br />
of concrete type<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\mathsf{E}F : [u_1, \dots, u_n, \mathrm{d}u_1, \dots, \mathrm{d}u_n] \to [x_1, \ldots, x_k, \mathrm{d}x_1, \ldots, \mathrm{d}x_k]\!</math><br />
|}<br />
<br />
is defined by means of the following system of logical equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\\[16pt]<br />
\mathrm{d}x_1<br />
& = & \mathrm{E}F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1 + \mathrm{d}u_1, \ldots, u_n + \mathrm{d}u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
\mathrm{d}x_k<br />
& = & \mathrm{E}F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1 + \mathrm{d}u_1, \ldots, u_n + \mathrm{d}u_n)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
It is important to note that this system of equations can be read as a conjunction of equational propositions, in effect, as a single proposition in the universe of discourse generated by all the named variables. Specifically, this is the universe of discourse over <math>2(n+k)\!</math> variables denoted by:<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
\mathrm{E}[\mathcal{U} \cup \mathcal{X}]<br />
& = &<br />
[u_1, \ldots, u_n, ~ x_1, \ldots, x_k, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n, ~ \mathrm{d}x_1, \ldots, \mathrm{d}x_k].<br />
\end{matrix}</math><br />
|}<br />
<br />
In this light, it should be clear that the system of equations defining <math>\mathsf{E}F\!</math> embodies, in a higher rank and differentially extended version, an analogy with the process of thematization that we treated earlier for propositions of type <math>F : \mathbb{B}^n \to \mathbb{B}.\!</math><br />
<br />
The entire collection of constraints that is represented in the above system of equations may be abbreviated by writing <math>\mathsf{E}F = (\boldsymbol\varepsilon F, \mathrm{E}F),\!</math> for any map <math>F.\!</math> This is tantamount to regarding <math>\mathsf{E}\!</math> as a complex operator, <math>\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}),\!</math> with a form of application that distributes each component of the operator to work on each component of the operand, as follows:<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
\mathsf{E}F<br />
& = &<br />
(\boldsymbol\varepsilon, \mathrm{E})F<br />
& = &<br />
(\boldsymbol\varepsilon F, \mathrm{E}F)<br />
& = &<br />
(\boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k, ~ \mathrm{E}F_1, \ldots, \mathrm{E}F_k).<br />
\end{matrix}</math><br />
|}<br />
<br />
Quite a lot of &ldquo;thematic infrastructure&rdquo; or interpretive information is being swept under the rug in the use of such abbreviations. When confusion arises about the meaning of such constructions, one always has recourse to the defining system of equations, in its totality a purely propositional expression. This means that the parenthesized argument lists, that were used in this context to build an image of multi-component transformations, should not be expected to determine a well-defined product in themselves but only to serve as reminders of the prior thematic decisions (choices of variable names, etc.) that have to be made in order to determine one. Accordingly, the argument list notation can be regarded as a kind of ''thematic frame'', an interpretive storage device that preserves the proper associations of concrete logical features between the extended universes at the source and target of <math>\mathsf{E}F.\!</math><br />
<br />
The generic notations <math>\mathsf{d}^0\!F, \mathsf{d}^1\!F, \ldots, \mathsf{d}^m\!F\!</math> in Figure&nbsp;33 refer to the increasing orders of differentials that are extracted in the course of analyzing <math>F.\!</math> When the analysis is halted at a partial stage of development, notations like <math>\mathsf{r}^0\!F, \mathsf{r}^1\!F, \ldots, \mathsf{r}^m\!F\!</math> may be used to summarize the contributions to <math>\mathsf{E}F\!</math> that remain to be analyzed. The Figure illustrates a convention that makes <math>\mathsf{r}^m\!F,\!</math> in effect, the sum of all differentials of order strictly greater than <math>m.\!</math><br />
<br />
We next discuss the operators that figure into this form of analysis, describing their effects on transformations. In simplified or specialized contexts these operators tend to take on a variety of different names and notations, some of whose number we introduce along the way.<br />
<br />
=====The Radius Operator : '''e'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
And the tangible fact at the root of all our thought-distinctions, however subtle, is that there is no one of them so fine as to consist in anything but a possible difference of practice.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 46]<br />
|}<br />
<br />
The operator identified as <math>\mathrm{d}^0\!</math> in the analytic diagram (Figure&nbsp;33) has the sole purpose of creating a proxy for <math>F\!</math> in the appropriately extended context. Construed in terms of its broadest components, <math>\mathrm{d}^0\!</math> is equivalent to the doubly tacit extension operator <math>(\boldsymbol\varepsilon, \boldsymbol\varepsilon),\!</math> in recognition of which let us redub it as <math>{}^{\backprime\backprime} \mathsf{e} {}^{\prime\prime}.\!</math> Pursuing a geometric analogy, we may refer to <math>\mathsf{e} =(\boldsymbol\varepsilon, \boldsymbol\varepsilon) = \mathrm{d}^0\!</math> as the ''radius operator''. The operation intended by all of these forms is defined by the following equation:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathsf{e}F<br />
& = &<br />
(\boldsymbol\varepsilon, \boldsymbol\varepsilon)F<br />
\\[4pt]<br />
& = &<br />
(\boldsymbol\varepsilon F, ~ \boldsymbol\varepsilon F)<br />
\\[4pt]<br />
& = &<br />
(\boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k, ~ \boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k).<br />
\end{array}</math><br />
|}<br />
<br />
which is tantamount to the system of equations below.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\\[16pt]<br />
\mathrm{d}x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
\mathrm{d}x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
=====The Phantom of the Operators : '''&eta;'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
I was wondering what the reason could be, when I myself raised my head and everything within me seemed drawn towards the Unseen, ''which was playing the most perfect music''!<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 81]<br />
|}<br />
<br />
We now describe an operator whose persistent but elusive action behind the scenes, whose slightly twisted and ambivalent character, and whose fugitive disposition, caught somewhere in flight between the arrantly negative and the positive but errant intent, has cost us some painstaking trouble to detect. In the end we shall place it among the other extensions and projections, as a shade among shadows, of muted tones and motley hue, that adumbrates its own thematic frame and paradoxically lights the way toward a whole new spectrum of values.<br />
<br />
Given a transformation <math>F : [u_1, \ldots, u_n] \to [x_1, \dots, x_k],\!</math> we often have call to consider a family of related transformations, all having the form:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>F^\dagger : [u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n] \to [\mathrm{d}x_1, \dots, \mathrm{d}x_k].\!</math><br />
|}<br />
<br />
The operator <math>\eta</math> is introduced to deal with the simplest one of these maps:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\eta F : [u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n] \to [\mathrm{d}x_1, \ldots \mathrm{d}x_k],\!</math><br />
|}<br />
<br />
which is defined by the following equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
\mathrm{d}x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
In effect, the operator <math>\eta\!</math> is nothing but the stand-alone version of a procedure that is otherwise invoked subordinate to the work of the radius operator <math>\mathsf{e}.\!</math> Operating independently, <math>\eta\!</math> achieves precisely the same results that the second <math>\boldsymbol\varepsilon\!</math> in <math>(\boldsymbol\varepsilon, \boldsymbol\varepsilon)\!</math> accomplishes by working within the context of its ordered pair thematic frame. From this point on, because the use of <math>\boldsymbol\varepsilon\!</math> and <math>\eta\!</math> in this setting combines the aims of both the tacit and the thematic extensions, and because <math>\eta\!</math> reflects in regard to <math>\boldsymbol\varepsilon\!</math> little more than the application of a differential twist, a mere turn of phrase, we refer to <math>\eta\!</math> as the ''trope extension'' operator.<br />
<br />
=====The Chord Operator : '''D'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
What difference would it practically make to any one if this notion rather than that notion were true? If no practical difference whatever can be traced, then the alternatives mean practically the same thing, and all dispute is idle.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 45]<br />
|}<br />
<br />
Next we discuss an operator that is always immanent in this form of analysis, and remains implicitly present in the entire proceeding. It may appear once as a record: a relic or revenant that reprises the reminders of an earlier stage of development. Or it may appear always as a resource: a reserve or redoubt that caches in advance an echo of what remains to be played out, cleared up, and requited in full at a future stage. And all of this remains true whether or not we recall the key at any time, and whether or not the subtending theme is recited explicitly at any stage of play.<br />
<br />
This is the operator that is referred to as <math>\mathsf{r}^0\!</math> in the initial stage of analysis (Figure&nbsp;33-i) and that is expanded as <math>\mathsf{d}^1 + \mathsf{r}^1\!</math> in the subsequent step (Figure&nbsp;33-ii). In congruence, but not quite harmony with our allusions of analogy that are not quite geometry, we call this the ''chord operator'' and denote it <math>\mathsf{D}.\!</math> In the more casual terms that are here introduced, <math>\mathsf{D}</math> is defined as the remainder of <math>\mathsf{E}\!</math> and <math>\mathsf{e}\!</math> and it assigns a due measure to each undertone of accord or discord that is struck between the note of enterprise <math>\mathsf{E}\!</math> and the bar of exigency <math>\mathsf{e}.\!</math><br />
<br />
The tension between these counterposed notions, in balance transient but regular in stridence, may be refracted along familiar lines, though never by any such fraction resolved. In this style we write <math>\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}),\!</math> calling <math>\mathrm{D}\!</math> the ''difference operator'' and noting that it plays a role in this realm of mutable and diverse discourse that is analogous to the part taken by the discrete difference operator in the ordinary difference calculus. Finally, we should note that the chord <math>\mathsf{D}\!</math> is not one that need be lost at any stage of development. At the <math>m^\text{th}\!</math> stage of play it can always be reconstituted in the following form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathsf{D}<br />
& = & \mathsf{E} - \mathsf{e}<br />
\\[6pt]<br />
& = & \mathsf{r}^0<br />
\\[6pt]<br />
& = & \mathsf{d}^1 + \mathsf{r}^1<br />
\\[6pt]<br />
& = & \mathsf{d}^1 + \ldots + \mathsf{d}^m + \mathsf{r}^m<br />
\\[6pt]<br />
& = & \displaystyle \sum_{i=1}^m \mathsf{d}^i + \mathsf{r}^m<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====The Tangent Operator : '''T'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
They take part in scenes of whose significance they have no inkling. They are merely tangent to curves of history the beginnings and ends and forms of which pass wholly beyond their ken. So we are tangent to the wider life of things.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 300]<br />
|}<br />
<br />
The operator tagged as <math>\mathsf{d}^1\!</math> in the analytic diagram (Figure&nbsp;33) is called the ''tangent operator'' and is usually denoted in this text as <math>\mathsf{d}\!</math> or <math>\mathsf{T}.\!</math> Because it has the properties required to qualify as a functor, namely, preserving the identity element of the composition operation and the articulated form of every composition of transformations, it also earns the title of a ''tangent functor''. According to the custom adopted here, we dissect it as <math>\mathsf{T} = \mathsf{d} = (\boldsymbol\varepsilon, \mathrm{d}),\!</math> where <math>\mathrm{d}\!</math> is the operator that yields the first order differential <math>\mathrm{d}F\!</math> when applied to a transformation <math>F,\!</math> and whose name is legion.<br />
<br />
Figure&nbsp;34 illustrates a stage of analysis where we ignore everything but the tangent functor <math>\mathsf{T}\!</math> and attend to it chiefly as it bears on the first order differential <math>\mathrm{d}F\!</math> in the analytic expansion of <math>F.\!</math> In this situation we often refer to the extended universes <math>\mathrm{E}U^\bullet\!</math> and <math>\mathrm{E}X^\bullet\!</math> under the equivalent designations <math>\mathsf{T}U^\bullet\!</math> and <math>\mathsf{T}X^\bullet,\!</math> respectively. The purpose of the tangent functor <math>\mathsf{T}\!</math> is to extract the tangent map <math>\mathsf{T}F\!</math> at each point of <math>U^\bullet,\!</math> and the tangent map <math>\mathsf{T}F = (\boldsymbol\varepsilon, \mathrm{d})F\!</math> tells us not only what the transformation <math>F\!</math> is doing at each point of the universe <math>U^\bullet\!</math> but also what <math>F\!</math> is doing to states in the neighborhood of that point, approximately, linearly, and relatively speaking.<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
U% $T$ $T$U% $T$U%<br />
o------------------>o============o<br />
| | |<br />
| | |<br />
| | |<br />
| | |<br />
F | | $T$F = | <!e!, d> F<br />
| | |<br />
| | |<br />
| | |<br />
v v v<br />
o------------------>o============o<br />
X% $T$ $T$X% $T$X%<br />
<br />
Figure 34. Tangent Functor Diagram<br />
</pre><br />
|}<br />
<br />
* '''NB.''' There is one aspect of the preceding construction that remains especially problematic. Why did we define the operators <math>\mathrm{W}\!</math> in <math>\{ \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}\!</math> so that the ranges of their resulting maps all fall within the realms of differential quality, even fabricating a variant of the tacit extension operator to have that character? Clearly, not all of the operator maps <math>\mathrm{W}F\!</math> have equally good reasons for placing their values in differential stocks. The reason for it appears to be that, without doing this, we cannot justify the comparison and combination of their functional values in the various analytic steps. By default, only those values in the same functional component can be brought into algebraic modes of interaction. Up till now the only mechanism provided for their broader association has been a purely logical one, their common placement in a target universe of discourse, but the task of converting this logical circumstance into algebraic forms of application has not yet been taken up.<br />
<br />
===Transformations of Type '''B'''<sup>2</sup> &rarr; '''B'''<sup>1</sup>===<br />
<br />
To study the effects of these analytic operators in the simplest possible setting, let us revert to a still more primitive case. Consider the singular proposition <math>J(u, v)= u\!\cdot\!v,\!</math> regarded either as the functional product of the maps <math>u\!</math> and <math>v\!</math> or as the logical conjunction of the features <math>u\!</math> and <math>v,\!</math> a map whose fiber of truth <math>J^{-1}(1)\!</math> picks out the single cell of that logical description in the universe of discourse <math>U^\bullet.\!</math> Thus <math>J,\!</math> or <math>u\!\cdot\!v,\!</math> may be treated as another name for the point whose coordinates are <math>(1, 1)\!</math> in <math>U^\bullet.\!</math><br />
<br />
====Analytic Expansion of Conjunction====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
<p>In her sufferings she read a great deal and discovered that she had lost something, the possession of which she had previously not been much aware of: a&nbsp;soul.</p><br />
<br />
<p>What is that? It is easily defined negatively: it is simply what curls up and hides when there is any mention of algebraic series.</p><br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 118]<br />
|}<br />
<br />
Figure&nbsp;35 pictures the form of conjunction <math>J : \mathbb{B}^2 \to \mathbb{B}\!</math> as a transformation from the <math>2\!</math>-dimensional universe <math>[u, v]\!</math> to the <math>1\!</math>-dimensional universe <math>[x].\!</math> This is a subtle but significant change of viewpoint on the proposition, attaching an arbitrary but concrete quality to its functional value. Using the language introduced earlier, we can express this change by saying that the proposition <math>J : \langle u, v \rangle \to \mathbb{B}\!</math> is being recast into the thematized role of a transformation <math>J : [u, v] \to [x],\!</math> where the new variable <math>x\!</math> takes the part of a thematic variable <math>\check{J}.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 35 -- A Conjunction Viewed as a Transformation.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 35.} ~~ \text{Conjunction as Transformation}\!</math><br />
|}<br />
<br />
=====Tacit Extension of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
I teach straying from me, yet who can stray from me?<br><br />
I follow you whoever you are from the present hour;<br><br />
My words itch at your ears till you understand them.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 83]<br />
|}<br />
<br />
Earlier we defined the tacit extension operators <math>\boldsymbol\varepsilon : X^\bullet \to Y^\bullet\!</math> as maps embedding each proposition of a given universe <math>X^\bullet~\!</math> in a more generously given universe <math>Y^\bullet \supset X^\bullet.\!</math> Of immediate interest are the tacit extensions <math>\boldsymbol\varepsilon : U^\bullet \to \mathrm{E}U^\bullet,\!</math> that locate each proposition of <math>U^\bullet\!</math> in the enlarged context of <math>\mathrm{E}U^\bullet.\!</math> In its application to the propositional conjunction <math>J = u\!\cdot\!v</math> in <math>[u, v],\!</math> the tacit extension operator <math>\boldsymbol\varepsilon\!</math> yields the proposition <math>\boldsymbol\varepsilon J\!</math> in <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v].\!</math> The extended proposition <math>\boldsymbol\varepsilon J\!</math> may be computed according to the scheme in Table&nbsp;36, in effect doing nothing more that conjoining a tautology of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> to <math>J\!</math> in <math>U^\bullet.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 36.} ~~ \text{Computation of}~ \boldsymbol\varepsilon J\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\boldsymbol\varepsilon J & = & J {}_{^\langle} u, v {}_{^\rangle}<br />
\\[4pt]<br />
& = & u \cdot v<br />
\\[4pt]<br />
& = & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{ } \mathrm{d}v \texttt{ }<br />
& + & u \cdot v \cdot \texttt{ } \mathrm{d}u \texttt{ } \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \cdot v \cdot \texttt{ } \mathrm{d}u \texttt{ } \cdot \texttt{ } \mathrm{d}v \texttt{ }<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{array}{*{4}{l}}<br />
\boldsymbol\varepsilon J<br />
& = && u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & u \cdot v \cdot \texttt{~} \mathrm{d}u \texttt{~} \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & u \cdot v \cdot \texttt{~} \mathrm{d}u \texttt{~} \cdot \texttt{~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
The lower portion of the Table contains the dispositional features of <math>\boldsymbol\varepsilon J\!</math> arranged in such a way that the variety of ordinary features spreads across the rows and the variety of differential features runs through the columns. This organization serves to facilitate pattern matching in the remainder of our computations. Again, the tacit extension is usually so trivial a concern that we do not always bother to make an explicit note of it, taking it for granted that any function <math>F\!</math> being employed in a differential context is equivalent to <math>\boldsymbol\varepsilon F\!</math> for a suitable <math>\boldsymbol\varepsilon.\!</math><br />
<br />
Figures&nbsp;37-a through 37-d present several pictures of the proposition <math>J\!</math> and its tacit extension <math>\boldsymbol\varepsilon J.\!</math> Notice in these Figures how <math>\boldsymbol\varepsilon J\!</math> in <math>\mathrm{E}U^\bullet\!</math> visibly extends <math>J\!</math> in <math>U^\bullet\!</math> by annexing to the indicated cells of <math>J\!</math> all the arcs that exit from or flow out of them. In effect, this extension attaches to these cells all the dispositions that spring from them, in other words, it attributes to these cells all the conceivable changes that are their issue.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-a -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-a.} ~~ \text{Tacit Extension of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-b -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-b.} ~~ \text{Tacit Extension of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-c -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-c.} ~~ \text{Tacit Extension of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-d -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-d.} ~~ \text{Tacit Extension of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
The computational scheme shown in Table&nbsp;36 treated <math>J\!</math> as a proposition in <math>U^\bullet\!</math> and formed <math>\boldsymbol\varepsilon J\!</math> as a proposition in <math>\mathrm{E}U^\bullet.\!</math> When <math>J\!</math> is regarded as a mapping <math>J : U^\bullet \to X^\bullet\!</math> then <math>\boldsymbol\varepsilon J\!</math> must be obtained as a mapping <math>\boldsymbol\varepsilon J : \mathrm{E}U^\bullet \to X^\bullet.\!</math> By default, the tacit extension of the map <math>J : [u, v] \to [x]\!</math> is naturally taken to be a particular map,<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\boldsymbol\varepsilon J : [u, v, \mathrm{d}u, \mathrm{d}v] \to [x] \subseteq [x, \mathrm{d}x],\!</math><br />
|}<br />
<br />
namely, the one that looks like <math>J\!</math> when painted in the frame of the extended source universe and that takes the same thematic variable in the extended target universe as the one that <math>J\!</math> already takes.<br />
<br />
But the choice of a particular thematic variable, for example <math>x\!</math> for <math>\check{J},\!</math> is a shade more arbitrary than the choice of original variable names <math>\{ u, v \},\!</math> so the map we are calling the ''trope extension'',<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\eta J : [u, v, \mathrm{d}u, \mathrm{d}v] \to [\mathrm{d}x] \subseteq [x, \mathrm{d}x],\!</math><br />
|}<br />
<br />
since it looks just the same as <math>\boldsymbol\varepsilon J\!</math> in the way its fibers paint the source domain, belongs just as fully to the family of tacit extensions, generically considered.<br />
<br />
These considerations have the practical consequence that all of our computations and illustrations of <math>\boldsymbol\varepsilon J\!</math> perform the double duty of capturing <math>\eta J\!</math> as well. In other words, we are saved the work of carrying out calculations and drawing figures for the trope extension <math>\eta J,\!</math> because it would be identical to the work already done for <math>\boldsymbol\varepsilon J.\!</math> Since the computations given for <math>\boldsymbol\varepsilon J\!</math> are expressed solely in terms of the variables <math>\{ u, v, \mathrm{d}u, \mathrm{d}v \},\!</math> they work equally well for finding <math>\eta J.\!</math> Further, since each of the above Figures shows only how the level sets of <math>\boldsymbol\varepsilon J\!</math> partition the extended source universe <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v],\!</math> all of them serve equally well as portraits of <math>\eta J.\!</math><br />
<br />
=====Enlargement Map of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
No one could have established the existence of any details that might not just as well have existed in earlier times too; but all the relations between things had shifted slightly. Ideas that had once been of lean account grew fat.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 62]<br />
|}<br />
<br />
The enlargement map <math>\mathrm{E}J\!</math> is computed from the proposition <math>J\!</math> by making a particular class of formal substitutions for its variables, in this case <math>u + \mathrm{d}u\!</math> for <math>u\!</math> and <math>v + \mathrm{d}v\!</math> for <math>v,\!</math> and afterwards expanding the result in whatever way is found convenient.<br />
<br />
Table&nbsp;38 shows a typical scheme of computation, following a systematic method of exploiting boolean expansions over selected variables and ultimately developing <math>\mathrm{E}J\!</math> over the cells of <math>[u, v].\!</math> The critical step of this procedure uses the facts that <math>\texttt{(} 0, x \texttt{)} = 0 + x = x\!</math> and <math>\texttt{(} 1, x \texttt{)} = 1 + x = \texttt{(} x \texttt{)}\!</math> for any boolean variable <math>x.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 38.} ~~ \text{Computation of}~ \mathrm{E}J ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{E}J & = & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}<br />
\\[4pt]<br />
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
& = & \texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot J_{(1 + \mathrm{d}u, 1 + \mathrm{d}v)}<br />
& + & \texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot J_{(1 + \mathrm{d}u, \mathrm{d}v)}<br />
& + & \texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot J_{(\mathrm{d}u, 1 + \mathrm{d}v)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot J_{(\mathrm{d}u, \mathrm{d}v)}<br />
\\[4pt]<br />
& = &<br />
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot J_{(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})}<br />
& + &<br />
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot J_{(\texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v)}<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot J_{(\mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)})}<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot J_{(\mathrm{d}u, \mathrm{d}v)}<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{E}J<br />
& = &<br />
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&& + &<br />
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{ } \mathrm{d}v \texttt{ }<br />
\\[4pt]<br />
&&&&& + &<br />
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot \texttt{ } \mathrm{d}u \texttt{ } \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&&&&&& + &<br />
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \texttt{ } \mathrm{d}u \texttt{ }~\texttt{ } \mathrm{d}v \texttt{ }<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;39 exhibits another method that happens to work quickly in this particular case, using distributive laws to multiply things out in an algebraic manner, arranging the notations of feature and fluxion according to a scale of simple character and degree. Proceeding this way leads through an intermediate step which, in chiming the changes of ordinary calculus, should take on a familiar ring. Consequential properties of exclusive disjunction then carry us on to the concluding line.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 39.} ~~ \text{Computation of}~ \mathrm{E}J ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{c}}<br />
\mathrm{E}J<br />
& = & (u + \mathrm{d}u) \cdot (v + \mathrm{d}v)<br />
\\[6pt]<br />
& = & u \cdot v<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = &<br />
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{ } \mathrm{d}v \texttt{ }<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot \texttt{ } \mathrm{d}u \texttt{ } \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \texttt{ } \mathrm{d}u \texttt{ }~\texttt{ } \mathrm{d}v \texttt{ }<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;40-a through 40-d present several views of the enlarged proposition <math>\mathrm{E}J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-a -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-a.} ~~ \text{Enlargement of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-b -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-b.} ~~ \text{Enlargement of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-c -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-c.} ~~ \text{Enlargement of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-d -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-d.} ~~ \text{Enlargement of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
An intuitive reading of the proposition <math>\mathrm{E}J\!</math> becomes available at this point. Recall that propositions in the extended universe <math>\mathrm{E}U^\bullet\!</math> express the ''dispositions'' of a system and the constraints that are placed on them. In other words, a differential proposition in <math>\mathrm{E}U^\bullet\!</math> can be read as referring to various changes that a system might undergo in and from its various states. In particular, we can understand <math>\mathrm{E}J\!</math> as a statement that tells us what changes need to be made with regard to each state in the universe of discourse in order to reach the ''truth'' of <math>J,\!</math> that is, the region of the universe where <math>J\!</math> is true. This interpretation is visibly clear in the Figures above and appeals to the imagination in a satisfying way but it has the added benefit of giving fresh meaning to the original name of the shift operator <math>\mathrm{E}.\!</math> Namely, <math>\mathrm{E}J\!</math> can be read as a proposition that ''enlarges'' on the meaning of <math>J,\!</math> in the sense of explaining its practical bearings and clarifying what it means in terms of actions and effects &mdash; the available options for differential action and the consequential effects that result from each choice.<br />
<br />
Read this way, the enlargement <math>\mathrm{E}J\!</math> has strong ties to the normal use of <math>J,\!</math> no matter whether it is understood as a proposition or a function, namely, to act as a figurative device for indicating the models of <math>J,\!</math> in effect, pointing to the interpretive elements in its fiber of truth <math>J^{-1}(1).\!</math> It is this kind of &ldquo;use&rdquo; that is often contrasted with the &ldquo;mention&rdquo; of a proposition, and thereby hangs a tale.<br />
<br />
=====Digression : Reflection on Use and Mention=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Reflection is turning a topic over in various aspects and in various lights so that nothing significant about it shall be overlooked &mdash; almost as one might turn a stone over to see what its hidden side is like or what is covered by it.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 57]<br />
|}<br />
<br />
The contrast drawn in logic between the ''use'' and the ''mention'' of a proposition corresponds to the difference that we observe in functional terms between using <math>{}^{\backprime\backprime} J \, {}^{\prime\prime}\!</math> to indicate the region <math>J^{-1}(1)\!</math> and using <math>{}^{\backprime\backprime} J \, {}^{\prime\prime}\!</math> to indicate the function <math>J.\!</math> You may think that one of these uses ought to be proscribed, and logicians are quick to prescribe against their confusion. But there seems to be no likelihood in practice that their interactions can be avoided. If the name <math>{}^{\backprime\backprime} J \, {}^{\prime\prime}\!</math> is used as a sign of the function <math>J,\!</math> and if the function <math>J\!</math> has its use in signifying something else, as would constantly be the case when some future theory of signs has given a functional meaning to every sign whatsoever, then is not <math>J,\!</math> by transitivity a sign of the thing itself? There are, of course, two answers to this question. Not every act of signifying or referring need be transitive. Not every warrant or guarantee or certificate is automatically transferable, indeed, not many. Not every feature of a feature is a feature of the featuree. Otherwise, if a buffalo is white, and white is a color, then a buffalo would ''be'' a color.<br />
<br />
The logical or pragmatic distinction between use and mention is cogent and necessary, and so is the analogous functional distinction between determining a value and determining what determines that value, but so are the normal techniques that we use to make these distinctions apply flexibly in practice. The way that the hue and cry about use and mention is raised in logical discussions, you might be led to think that this single dimension of choices embraces the only kinds of use worth mentioning and the only kinds of mention worth using. It will constitute the expeditionary and taxonomic tasks of that future theory of signs to explore and to classify the many other constellations and dimensions of use and mention that are yet to be opened up by the generative potential of full-fledged sign relations.<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
The well-known capacity that thoughts have &mdash; as doctors have discovered &mdash; for dissolving and dispersing those hard lumps of deep, ingrowing, morbidly entangled conflict that arise out of gloomy regions of the self probably rests on nothing other than their social and worldly nature, which links the individual being with other people and things; but unfortunately what gives them their power of healing seems to be the same as what diminishes the quality of personal experience in them.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 130]<br />
|}<br />
<br />
=====Difference Map of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
&ldquo;It doesn't matter what one does,&rdquo; the Man Without Qualities said to himself, shrugging his shoulders. &ldquo;In a tangle of forces like this it doesn't make a scrap of difference.&rdquo; He turned away like a man who has learned renunciation, almost indeed like a sick man who shrinks from any intensity of contact. And then, striding through his adjacent dressing-room, he passed a punching-ball that hung there; he gave it a blow far swifter and harder than is usual in moods of resignation or states of weakness.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 8]<br />
|}<br />
<br />
With the tacit extension map <math>\boldsymbol\varepsilon J\!</math> and the enlargement map <math>\mathrm{E}J\!</math> well in place, the difference map <math>\mathrm{D}J\!</math> can be computed along the lines displayed in Table&nbsp;41, ending up with an expansion of <math>\mathrm{D}J\!</math> over the cells of <math>[u, v].\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 41.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & \mathrm{E}J<br />
& + & \boldsymbol\varepsilon J<br />
\\[6pt]<br />
& = & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}<br />
& + & J_{(u, v)}<br />
\\[6pt]<br />
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \cdot v<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = &<br />
u \cdot v \cdot \qquad 0<br />
\\[6pt]<br />
& + &<br />
u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v<br />
& + &<br />
u \cdot \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v<br />
\\[6pt]<br />
& + &<br />
u \cdot v \cdot \texttt{~} \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
&&& + &<br />
\texttt{(} u \texttt{)} \cdot v \cdot \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
& + &<br />
u \cdot v \cdot \texttt{~} \mathrm{d}u \;\cdot\; \mathrm{d}v \texttt{~}<br />
&&&&& + &<br />
\texttt{(} u \texttt{)} \cdot \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = &<br />
u \cdot v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
u \cdot \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v<br />
& + &<br />
\texttt{(} u \texttt{)} \cdot v \cdot \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)} \cdot \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Alternatively, the difference map <math>\mathrm{D}J\!</math> can be expanded over the cells of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> to arrive at the formulation shown in Table&nbsp;42. The same development would be obtained from the previous Table by collecting terms in an alternate manner, along the rows rather than the columns in the middle portion of the Table.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 42.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & \boldsymbol\varepsilon J<br />
& + & \mathrm{E}J<br />
\\[6pt]<br />
& = & J_{(u, v)}<br />
& + & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}<br />
\\[6pt]<br />
& = & u \cdot v<br />
& + & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
& = & 0<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}J<br />
& = & 0<br />
& + & u \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Even more simply, the same result is reached by matching up the propositional coefficients of <math>\boldsymbol\varepsilon J</math> and <math>\mathrm{E}J\!</math> along the cells of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> and adding the pairs under boolean addition, that is, &ldquo;mod 2&rdquo;, where 1&nbsp;+&nbsp;1&nbsp;=&nbsp;0, as shown in Table&nbsp;43.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 43.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 3)}\!</math><br />
|<br />
<math>\begin{array}{*{5}{l}}<br />
\mathrm{D}J & = & \boldsymbol\varepsilon J & + & \mathrm{E}J<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\boldsymbol\varepsilon J<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ u \,\cdot\, v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~ u \;\cdot\; v \;\cdot\; \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u ~ \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} ~ v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot\, \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & ~~ 0 ~~ \,\cdot\, ~ \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~~ u ~ \,\cdot\, ~~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ ~ v ~~ \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
The difference map <math>\mathrm{D}J\!</math> can also be given a ''dispositional'' interpretation. First, recall that <math>\boldsymbol\varepsilon J\!</math> exhibits the dispositions to change from anywhere in <math>J\!</math> to anywhere at all in the universe of discourse and <math>\mathrm{E}J\!</math> exhibits the dispositions to change from anywhere in the universe to anywhere in <math>J.\!</math> Next, observe that each of these classes of dispositions may be divided in accordance with the case of <math>J\!</math> versus <math>\texttt{(} J \texttt{)}\!</math> that applies to their points of departure and destination, as shown below. Then, since the dispositions corresponding to <math>\boldsymbol\varepsilon J</math> and <math>\mathrm{E}J\!</math> have in common the dispositions to preserve <math>J,\!</math> their symmetric difference <math>\texttt{(} \boldsymbol\varepsilon J, \mathrm{E}J \texttt{)}\!</math> is made up of all the remaining dispositions, which are in fact disposed to cross the boundary of <math>J\!</math> in one direction or the other. In other words, we may conclude that <math>\mathrm{D}J\!</math> expresses the collective disposition to make a definite change with respect to <math>J,\!</math> no matter what value it holds in the current state of affairs.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
\boldsymbol\varepsilon J<br />
& = & \{ \text{Dispositions from}~ J ~\text{to}~ J \}<br />
& + & \{ \text{Dispositions from}~ J ~\text{to}~ \texttt{(} J \texttt{)} \}<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = & \{ \text{Dispositions from}~ J ~\text{to}~ J \}<br />
& + & \{ \text{Dispositions from}~ \texttt{(} J \texttt{)} ~\text{to}~ J \}<br />
\\[6pt]<br />
\mathrm{D}J<br />
& = & \{ \text{Dispositions from}~ J ~\text{to}~ \texttt{(} J \texttt{)} \}<br />
& + & \{ \text{Dispositions from}~ \texttt{(} J \texttt{)} ~\text{to}~ J \}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;44-a through 44-d illustrate the difference proposition <math>\mathrm{D}J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-a -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-a.} ~~ \text{Difference Map of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-b -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-b.} ~~ \text{Difference Map of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-c -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-c.} ~~ \text{Difference Map of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-d -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-d.} ~~ \text{Difference Map of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
=====Differential of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
By deploying discourse throughout a calendar, and by giving a date to each of its elements, one does not obtain a definitive hierarchy of precessions and originalities; this hierarchy is never more than relative to the systems of discourse that it sets out to evaluate.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Archaeology of Knowledge'', [Fou, 143]<br />
|}<br />
<br />
Finally, at long last, the differential proposition <math>\mathrm{d}J\!</math> can be gleaned from the difference proposition <math>\mathrm{D}J\!</math> by ranging over the cells of <math>[u, v]\!</math> and picking out the linear proposition of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> that is &ldquo;closest&rdquo; to the portion of <math>\mathrm{D}J\!</math> that touches on each point. The idea of distance that would give this definition unequivocal sense has been referred to in cautionary quotes, the kind we use to distance ourselves from taking a final position. There are obvious notions of approximation that suggest themselves, but finding one that can be justified as ultimately correct is not as straightforward as it seems.<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
He had drifted into the very heart of the world. From him to the distant beloved was as far as to the next tree.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 144]<br />
|}<br />
<br />
Let us venture a guess as to where these developments might be heading. From the present vantage point it appears that the ultimate answer to the quandary of distances and the question of a fitting measure may be that, rather than having the constitution of an analytic series depend on our familiar notions of approach, proximity, and approximation, it will be found preferable, and perhaps unavoidable, to turn the tables and let the orders of approximation be defined in terms of our favored and operative notions of formal analysis. Only the aftermath of this conversion, if it does converge, could be hoped to prove whether this hortatory form of analysis and the cohort idea of an analytic form &mdash; the limitary concept of a self-corrective process and the coefficient concept of a completable product &mdash; are truly (in practical reality) the more inceptive and persistent of principles and really (for all practical purposes) the more effective and regulative of ideas.<br />
<br />
Awaiting that determination, I proceed with what seems like the obvious course, and compute <math>\mathrm{d}J\!</math> according to the pattern in Table&nbsp;45.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 45.} ~~ \text{Computation of}~ \mathrm{d}J\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}J<br />
& = &<br />
u\!\cdot\!v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
u \, \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \, \mathrm{d}v<br />
& + &<br />
\texttt{(} u \texttt{)} \, v \cdot \mathrm{d}u \, \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \!\cdot\! \mathrm{d}v \texttt{~}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}J<br />
& = &<br />
u\!\cdot\!v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + &<br />
u \, \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + &<br />
\texttt{(} u \texttt{)} \, v \cdot \mathrm{d}u<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;46-a through 46-d illustrate the proposition <math>{\mathrm{d}J},\!</math> rounded out in our usual array of prospects. This proposition of <math>\mathrm{E}U^\bullet\!</math> is what we refer to as the (first order) differential of <math>J,\!</math> and normally regard as ''the'' differential proposition corresponding to <math>J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-a -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-a.} ~~ \text{Differential of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-b -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-b.} ~~ \text{Differential of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-c -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-c.} ~~ \text{Differential of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-d -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-d.} ~~ \text{Differential of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
=====Remainder of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
<p>I bequeath myself to the dirt to grow from the grass I love,<br><br />
If you want me again look for me under your bootsoles.</p><br />
<br />
<p>You will hardly know who I am or what I mean,<br><br />
But I shall be good health to you nevertheless,<br><br />
And filter and fibre your blood.</p><br />
<br />
<p>Failing to fetch me at first keep encouraged,<br><br />
Missing me one place search another,<br><br />
I stop some where waiting for you</p><br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]<br />
|}<br />
<br />
<br><br />
<br />
Let us recapitulate the story so far. We have in effect been carrying out a decomposition of the enlarged proposition <math>\mathrm{E}J\!</math> in a series of stages. First, we considered the equation <math>\mathrm{E}J = \boldsymbol\varepsilon J + \mathrm{D}J,\!</math> which was involved in the definition of <math>\mathrm{D}J\!</math> as the difference <math>\mathrm{E}J - \boldsymbol\varepsilon J.\!</math> Next, we contemplated the equation <math>\mathrm{D}J = \mathrm{d}J + \mathrm{r}J,\!</math> which expresses <math>\mathrm{D}J\!</math> in terms of two components, the differential <math>\mathrm{d}J\!</math> that was just extracted and the residual component <math>\mathrm{r}J = \mathrm{D}J - \mathrm{d}J.~\!</math> This remaining proposition <math>\mathrm{r}J\!</math> can be computed as shown in Table&nbsp;47.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 47.} ~~ \text{Computation of}~ \mathrm{r}J\!</math><br />
|<br />
<math>\begin{array}{*{5}{l}}<br />
\mathrm{r}J & = & \mathrm{D}J & + & \mathrm{d}J<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}J<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{r}J ~<br />
& = & u \!\cdot\! v \cdot ~ \mathrm{d}u \cdot \mathrm{d}v ~ ~ ~ ~ ~<br />
& + & u \texttt{(} v \texttt{)} \cdot \, \mathrm{d}u \cdot \mathrm{d}v \,<br />
& + & \texttt{(} u \texttt{)} v \cdot \, \mathrm{d}u \cdot \mathrm{d}v \,<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \, \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
As it happens, the remainder <math>\mathrm{r}J\!</math> falls under the description of a second order differential <math>\mathrm{r}J = \mathrm{d}^2 J.\!</math> This means that the expansion of <math>\mathrm{E}J\!</math> in the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|<br />
<math>\begin{array}{*{7}{l}}<br />
\mathrm{E}J<br />
& = & \boldsymbol\varepsilon J<br />
& + & \mathrm{D}J<br />
\\[6pt]<br />
& = & \boldsymbol\varepsilon J<br />
& + & \mathrm{d}J<br />
& + & \mathrm{r}J<br />
\\[6pt]<br />
& = & \mathrm{d}^0 J<br />
& + & \mathrm{d}^1 J<br />
& + & \mathrm{d}^2 J<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
which is nothing other than the propositional analogue of a Taylor series, is a decomposition that terminates in a finite number of steps.<br />
<br />
Figures&nbsp;48-a through 48-d illustrate the proposition <math>\mathrm{r}J = \mathrm{d}^2 J,\!</math> which forms the remainder map of <math>J\!</math> and also, in this instance, the second order differential of <math>J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-a -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-a.} ~~ \text{Remainder of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-b -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-b.} ~~ \text{Remainder of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-c -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-c.} ~~ \text{Remainder of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-d -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-d.} ~~ \text{Remainder of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
=====Summary of Conjunction=====<br />
<br />
To establish a convenient reference point for further discussion, Table&nbsp;49 summarizes the operator actions that have been computed for the form of conjunction, as exemplified by the proposition <math>J.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 49.} ~~ \text{Computation Summary for}~ J~\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon J<br />
& = & u \!\cdot\! v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}J<br />
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}J<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{r}J<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Analytic Series : Coordinate Method====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
And if he is told that something ''is'' the way it is, then he thinks: Well, it could probably just as easily be some other way. So the sense of possibility might be defined outright as the capacity to think how everything could &ldquo;just as easily&rdquo; be, and to attach no more importance to what is than to what is not.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 12]<br />
|}<br />
<br />
Table&nbsp;50 exhibits a truth table method for computing the analytic series (or the differential expansion) of a proposition in terms of coordinates.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:70%"<br />
|+ style="height:30px" | <math>\text{Table 50.} ~~ \text{Computation of an Analytic Series in Terms of Coordinates}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:8%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:8%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:8%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}u\!</math><br />
| style="width:8%; border-bottom:1px solid black" | <math>\mathrm{d}v\!</math><br />
| style="width:8%; border-bottom:1px solid black; border-left:1px solid black" | <math>u'\!</math><br />
| style="width:8%; border-bottom:1px solid black" | <math>v'\!</math><br />
| style="width:10%; border-bottom:1px solid black; border-left:4px double black" | <math>\boldsymbol\varepsilon J\!</math><br />
| style="width:10%; border-bottom:1px solid black" | <math>\mathrm{E}J\!</math><br />
| style="width:12%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{D}J\!</math><br />
| style="width:10%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}J\!</math><br />
| style="width:10%; border-bottom:1px solid black" | <math>\mathrm{d}^2\!J\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
The first six columns of the Table, taken as a whole, represent the variables of a construct called the ''contingent universe'' <math>[u, v, \mathrm{d}u, \mathrm{d}v, u', v'],\!</math> or the bundle of ''contingency spaces'' <math>[\mathrm{d}u, \mathrm{d}v, u', v']\!</math> over the universe <math>[u, v].\!</math> Their placement to the left of the double bar indicates that all of them amount to independent variables, but there is a co-dependency among them, as described by the following equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
u' & = & u + \mathrm{d}u & = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\\[8pt]<br />
v' & = & v + \mathrm{d}v & = & \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
These relations correspond to the formal substitutions that are made in defining <math>\mathrm{E}J\!</math> and <math>\mathrm{D}J.\!</math> For now, the whole rigamarole of contingency spaces can be regarded as a technical device for achieving the effect of these substitutions, adapted to a setting where functional compositions and other symbolic manipulations are difficult to contemplate and execute.<br />
<br />
The five columns to the right of the double bar in Table&nbsp;50 contain the values of the dependent variables <math>\{ \boldsymbol\varepsilon J, ~\mathrm{E}J, ~\mathrm{D}J, ~\mathrm{d}J, ~\mathrm{d}^2\!J \}.\!</math> These are normally interpreted as values of functions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B}\!</math> or as values of propositions in the extended universe <math>[u, v, \mathrm{d}u, \mathrm{d}v]\!</math> but the dependencies prevailing in the contingent universe make it possible to regard these same final values as arising via functions on alternative lists of arguments, for example, the set <math>\{ u, v, u', v' \}.\!</math><br />
<br />
The column for <math>\boldsymbol\varepsilon J\!</math> is computed as <math>J(u, v) = uv\!</math> and together with the columns for <math>u\!</math> and <math>v\!</math> illustrates how we &ldquo;share structure&rdquo; in the Table by listing only the first entries of each constant block.<br />
<br />
The column for <math>\mathrm{E}J\!</math> is computed by means of the following chain of identities, where the contingent variables <math>u'\!</math> and <math>v'\!</math> are defined as <math>u' = u + \mathrm{d}u\!</math> and <math>v' = v + \mathrm{d}v.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathrm{E}J(u, v, \mathrm{d}u, \mathrm{d}v)<br />
& = &<br />
J(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& = &<br />
J(u', v')<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
This makes it easy to determine <math>\mathrm{E}J\!</math> by inspection, computing the conjunction <math>J(u', v') = u'v'\!</math> from the columns headed <math>u'\!</math> and <math>v'.\!</math> Since each of these forms expresses the same proposition <math>\mathrm{E}J\!</math> in <math>\mathrm{E}U^\bullet,\!</math> the dependence on <math>\mathrm{d}u\!</math> and <math>\mathrm{d}v\!</math> is still present but merely left implicit in the final variant <math>J(u', v').\!</math><br />
<br />
* '''Note.''' On occasion, it is tempting to use the further notation <math>J'(u, v) = J(u', v'),\!</math> especially to suggest a transformation that acts on whole propositions, for example, taking the proposition <math>J\!</math> into the proposition <math>J' = \mathrm{E}J.\!</math> The prime <math>( {}^{\prime} )\!</math> then signifies an action that is mediated by a field of choices, namely, the values that are picked out for the contingent variables in sweeping through the initial universe. But this heaps an unwieldy lot of construed intentions on a rather slight character and puts too high a premium on the constant correctness of its interpretation. In practice, therefore, it is best to avoid this usage.<br />
<br />
Given the values of <math>\boldsymbol\varepsilon J\!</math> and <math>\mathrm{E}J,\!</math> the columns for the remaining functions can be filled in quickly. The difference map is computed according to the relation <math>\mathrm{D}J = \boldsymbol\varepsilon J + \mathrm{E}J.\!</math> The first order differential <math>\mathrm{d}J\!</math> is found by looking in each block of constant argument pairs <math>u, v\!</math> and choosing the linear function of <math>\mathrm{d}u, \mathrm{d}v\!</math> that best approximates <math>\mathrm{D}J\!</math> in that block. Finally, the remainder is computed as <math>\mathrm{r}J = \mathrm{D}J + \mathrm{d}J,\!</math> in this case yielding the second order differential <math>\mathrm{d}^2\!J.\!</math><br />
<br />
====Analytic Series : Recap====<br />
<br />
Let us now summarize the results of Table&nbsp;50 by writing down for each column and for each block of constant argument pairs <math>u, v\!</math> a reasonably canonical symbolic expression for the function of <math>\mathrm{d}u, \mathrm{d}v\!</math> that appears there. The synopsis formed in this way is presented in Table&nbsp;51. As one has a right to expect, it confirms the results that were obtained previously by operating solely in terms of the formal calculus.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:70%"<br />
|+ style="height:30px" | <math>\text{Table 51.} ~~ \text{Computation of an Analytic Series in Symbolic Terms}\!</math><br />
|- style="height:35px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black" | <math>J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{E}J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{D}J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{d}J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{d}^2\!J\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}u \!\;\cdot\;\! \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}u \!\;\cdot\;\! \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
\mathrm{d}u<br />
\\[4pt]<br />
\mathrm{d}v<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;52 and 53 provide a quick overview of the analysis performed so far, giving the successive decompositions of <math>\mathrm{E}J = J + \mathrm{D}J\!</math> and <math>\mathrm{D}J = \mathrm{d}J + \mathrm{r}J\!</math> in two different styles of diagram.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 52 -- Decomposition of EJ.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 52.} ~~ \text{Decomposition of}~ \mathrm{E}J\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 53 -- Decomposition of DJ.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 53.} ~~ \text{Decomposition of}~ \mathrm{D}J\!</math><br />
|}<br />
<br />
====Terminological Interlude====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Lastly, my attention was especially attracted, not so much to the scene, as to the mirrors that produced it. These mirrors were broken in parts. Yes, they were marked and scratched; they had been &ldquo;starred&rdquo;, in spite of their solidity &hellip;<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 230]<br />
|}<br />
<br />
At this point several issues of terminology have accrued enough substance to intrude on our discussion. The remarks of this Subsection are intended to accomplish two goals. First, we call attention to significant aspects of the previous series of Figures, translating into literal terms what they depict in iconic forms, and we re-stress the most important structural elements they indicate. Next, we prepare the way for taking on more complex examples of transformations, those whose target universes have more than one dimension.<br />
<br />
In talking about the actions of operators it is important to keep in mind the distinctions between the operators per&nbsp;se, their operands, and their results. Furthermore, in working with composite forms of operators <math>\mathrm{W} = (\mathrm{W}_1, \ldots, \mathrm{W}_n),\!</math> transformations <math>\mathrm{F} = (\mathrm{F}_1, \ldots, \mathrm{F}_n),\!</math> and target domains <math>X^\bullet = [x_1, \ldots, x_n],\!</math> we need to preserve a clear distinction between the compound entity of each given type and any one of its separate components. It is curious, given the usefulness of the concepts ''operator'' and ''operand'', that we seem to lack a generic term, formed on the same root, for the corresponding result of an operation. Following the obvious paradigm would lead to words like ''opus'', ''opera'', and ''operant'', but these words are too affected with clang associations to work well at present, though they might be adapted in time. One current usage gets around this problem by using the substantive ''map'' as a systematic epithet to express the result of each operator's action. We will follow this practice as far as possible, for example, using the phrase ''tangent map'' to denote the end product of the tangent functor acting on its operand map.<br />
<br />
* '''Scholium.''' See [JGH, 6-9] for a good account of tangent functors and tangent maps in ordinary analysis and for examples of their use in mechanics. This work as a whole is a model of clarity in applying functorial principles to problems in physical dynamics.<br />
<br />
Whenever we focus on isolated propositions, on single components of composite operators, or on the portions of transformations that have <math>1\!</math>-dimensional ranges, we are free to shift between the native form of a proposition <math>J : U \to \mathbb{B}\!</math> and the thematized form of a mapping <math>J : U^\bullet \to [x]\!</math> without much trouble. In these cases we are able to tolerate a higher degree of ambiguity about the precise nature of an operator's input and output domains than we otherwise might. For example, in the preceding treatment of the example <math>J,\!</math> and for each operator <math>\mathrm{W}\!</math> in the set <math>\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \},\!</math> both the operand <math>J\!</math> and the result <math>\mathrm{W}J\!</math> could be viewed in either one of two ways. On one hand we may treat them as propositions <math>J : U \to \mathbb{B}\!</math> and <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B},\!</math> ignoring the distinction between the range <math>[x] \cong \mathbb{B}\!</math> of <math>\boldsymbol\varepsilon J\!</math> and the range <math>[\mathrm{d}x] \cong \mathbb{D}\!</math> of the other types of <math>\mathrm{W}J.\!</math> This is what we usually do when we content ourselves with simply coloring in regions of venn diagrams. On the other hand we may view these entities as maps <math>J : U^\bullet \to [x] = X^\bullet\!</math> and <math>\boldsymbol\varepsilon J : \mathrm{E}U^\bullet \to [x] \subseteq \mathrm{E}X^\bullet\!</math> or <math>\mathrm{W}J : \mathrm{E}U^\bullet \to [\mathrm{d}x] \subseteq \mathrm{E}X^\bullet,\!</math> in which case the qualitative characters of the output features are not ignored.<br />
<br />
At the beginning of this Section we recast the natural form of a proposition <math>J : U \to \mathbb{B}\!</math> into the thematic role of a transformation <math>J : U^\bullet \to [x],\!</math> where <math>x\!</math> was a variable recruited to express the newly independent <math>\check{J}.\!</math> However, in our computations and representations of operator actions we immediately lapsed back to viewing the results as native elements of the extended universe <math>\mathrm{E}U^\bullet,\!</math> in other words, as propositions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B},\!</math> where <math>\mathrm{W}\!</math> ranged over the set <math>\{ \boldsymbol\varepsilon, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}.\!</math> That is as it should be. We have worked hard to devise a language that gives us these advantages &mdash; the flexibility to exchange terms and types of equal information value and the capacity to reflect as quickly and as wittingly as a controlled reflex on the fibers of our propositions, independently of whether they express amusements, beliefs, or conjectures.<br />
<br />
As we take on target spaces of increasing dimension, however, these types of confusions (and confusions of types) become less and less permissible. For this reason, Tables&nbsp;54 and 55 present a rather detailed summary of the notation and the terminology we are using, as applied to the case <math>J = uv.\!</math> The rationale of these Tables is not so much to train more elephant guns on this poor drosophila of a concrete example but to invest our paradigm with enough solidity to bear the weight of abstraction to come.<br />
<br />
Table&nbsp;54 provides basic notation and descriptive information for the objects and operators used in this Example, giving the generic type (or broadest defined type) for each entity. Here, the sans&nbsp;serif operators <math>\mathsf{W} \in \{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{d}, \mathsf{r} \}\!</math> and their components <math>\mathrm{W} \in \{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}\!</math> both have the same broad type <math>\mathsf{W}, \mathrm{W} : (U^\bullet \to X^\bullet) \to (\mathrm{E}U^\bullet \to \mathrm{E}X^\bullet),\!</math> as appropriate to operators that map transformations <math>J : U^\bullet \to X^\bullet\!</math> to extended transformations <math>\mathsf{W}J, \mathrm{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 54.} ~~ \text{Cast of Characters : Expansive Subtypes of Objects and Operators}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| align="center" | <math>\text{Symbol}\!</math><br />
| align="center" | <math>\text{Notation}\!</math><br />
| align="center" | <math>\text{Description}\!</math><br />
| align="center" | <math>\text{Type}\!</math><br />
|-<br />
| align="center" | <math>U^\bullet\!</math><br />
| <math>= [u, v]\!</math><br />
| <math>\text{Source universe}\!</math><br />
| <math>[\mathbb{B}^2]\!</math><br />
|-<br />
| align="center" | <math>X^\bullet~\!</math><br />
| <math>= [x]\!</math><br />
| <math>\text{Target universe}\!</math><br />
| <math>[\mathbb{B}^1]~\!</math><br />
|-<br />
| align="center" | <math>\mathrm{E}U^\bullet\!</math><br />
| <math>= [u, v, \mathrm{d}u, \mathrm{d}v]\!</math><br />
| <math>\text{Extended source universe}\!</math><br />
| <math>[\mathbb{B}^2 \!\times\! \mathbb{D}^2]</math><br />
|-<br />
| align="center" | <math>\mathrm{E}X^\bullet\!</math><br />
| <math>= [x, \mathrm{d}x]~\!</math><br />
| <math>\text{Extended target universe}\!</math><br />
| <math>[\mathbb{B}^1 \!\times\! \mathbb{D}^1]</math><br />
|-<br />
| align="center" | <math>J\!</math><br />
| <math>J : U \!\to\! \mathbb{B}\!</math><br />
| <math>\text{Proposition}\!</math><br />
| <math>(\mathbb{B}^2 \!\to\! \mathbb{B}) \in [\mathbb{B}^2]\!</math><br />
|-<br />
| align="center" | <math>J\!</math><br />
| <math>J : U^\bullet \!\to\! X^\bullet\!</math><br />
| <math>\text{Transformation or Map}\!</math><br />
| <math>[\mathbb{B}^2] \!\to\! [\mathbb{B}^1]\!</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\boldsymbol\varepsilon<br />
\\<br />
\eta<br />
\\<br />
\mathrm{E}<br />
\\<br />
\mathrm{D}<br />
\\<br />
\mathrm{d}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{W} : U^\bullet \!\to\! \mathrm{E}U^\bullet,<br />
\\<br />
\mathrm{W} : X^\bullet \!\to\! \mathrm{E}X^\bullet,<br />
\\<br />
\mathrm{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\\<br />
\text{for each}~ \mathrm{W} ~\text{in the set:}<br />
\\<br />
\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{ll}<br />
\text{Tacit extension operator} & \boldsymbol\varepsilon<br />
\\<br />
\text{Trope extension operator} & \eta<br />
\\<br />
\text{Enlargement operator} & \mathrm{E}<br />
\\<br />
\text{Difference operator} & \mathrm{D}<br />
\\<br />
\text{Differential operator} & \mathrm{d}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^2] \!\to\! [\mathbb{B}^2 \!\times\! \mathbb{D}^2]},<br />
\\<br />
{[\mathbb{B}^1] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]},<br />
\\\\<br />
([\mathbb{B}^2] \!\to\! [\mathbb{B}^1]) \!\to\!<br />
\\<br />
([\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1])<br />
\end{array}</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\mathsf{e}<br />
\\<br />
\mathsf{E}<br />
\\<br />
\mathsf{D}<br />
\\<br />
\mathsf{T}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{W} : U^\bullet \!\to\! \mathsf{T}U^\bullet = \mathrm{E}U^\bullet,<br />
\\<br />
\mathsf{W} : X^\bullet \!\to\! \mathsf{T}X^\bullet = \mathrm{E}X^\bullet,<br />
\\<br />
\mathsf{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathsf{T}U^\bullet \!\to\! \mathsf{T}X^\bullet)<br />
\\<br />
\text{for each}~ \mathsf{W} ~\text{in the set:}<br />
\\<br />
\{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{T} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{lll}<br />
\text{Radius operator} & \mathsf{e} & = (\boldsymbol\varepsilon, \eta)<br />
\\<br />
\text{Secant operator} & \mathsf{E} & = (\boldsymbol\varepsilon, \mathrm{E})<br />
\\<br />
\text{Chord operator} & \mathsf{D} & = (\boldsymbol\varepsilon, \mathrm{D})<br />
\\<br />
\text{Tangent functor} & \mathsf{T} & = (\boldsymbol\varepsilon, \mathrm{d})<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^2] \!\to\! [\mathbb{B}^2 \!\times\! \mathbb{D}^2]},<br />
\\<br />
{[\mathbb{B}^1] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]},<br />
\\\\<br />
([\mathbb{B}^2] \!\to\! [\mathbb{B}^1]) \!\to\!<br />
\\<br />
([\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1])<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;55 supplies a more detailed outline of terminology for operators and their results. Here, we list the restrictive subtype (or narrowest defined subtype) that applies to each entity and we indicate across the span of the Table the whole spectrum of alternative types that color the interpretation of each symbol. For example, all the component operator maps <math>\mathrm{W}J\!</math> have <math>1\!</math>-dimensional ranges, either <math>\mathbb{B}^1\!</math> or <math>\mathbb{D}^1,\!</math> and so they can be viewed either as propositions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B}\!</math> or as logical transformations <math>\mathrm{W}J : \mathrm{E}U^\bullet \to X^\bullet.\!</math> As a rule, the plan of the Table allows us to name each entry by detaching the underlined adjective at the left of its row and prefixing it to the generic noun at the top of its column. In one case, however, it is customary to depart from this scheme. Because the phrase ''differential proposition'', applied to the result <math>\mathrm{d}J : \mathrm{E}U \to \mathbb{D},\!</math> does not distinguish it from the general run of differential propositions <math>\mathrm{G}: \mathrm{E}U \to \mathbb{B},\!</math> it is usual to single out <math>\mathrm{d}J\!</math> as the ''tangent proposition'' of <math>J.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 55.} ~~ \text{Synopsis of Terminology : Restrictive and Alternative Subtypes}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| &nbsp;<br />
| align="center" | <math>\text{Operator}\!</math><br />
| align="center" | <math>\text{Proposition}\!</math><br />
| align="center" | <math>\text{Map}\!</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tacit}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\boldsymbol\varepsilon : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\boldsymbol\varepsilon : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{B} \\<br />
\boldsymbol\varepsilon J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{B}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x] \\<br />
\boldsymbol\varepsilon J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Trope}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\eta : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\eta : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\eta J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\eta J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Enlargement}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{E} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{E}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{E}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Difference}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{D} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{D}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{D}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Differential}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{d} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{d} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{d}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}\!</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Remainder}}\\\text{operator}\end{matrix}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{r} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{r} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{r}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{r}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Radius}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e} = (\boldsymbol\varepsilon, \eta) \\<br />
\mathsf{e} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{e}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Secant}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}) \\<br />
\mathsf{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{E}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Chord}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}) \\<br />
\mathsf{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{D}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tangent}}\\\text{functor}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T} = (\boldsymbol\varepsilon, \mathrm{d}) \\<br />
\mathsf{T} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{T}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====End of Perfunctory Chatter : Time to Roll the Clip!====<br />
<br />
Two steps remain to finish the analysis of <math>J\!</math> that we began so long ago. First, we need to paste our accumulated heap of flat pictures into the frames of transformations, filling out the shapes of the operator maps <math>\mathsf{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet.~\!</math> This scheme is executed in two styles, using the ''areal views'' in Figures&nbsp;56-a and the ''box views'' in Figures&nbsp;56-b. Finally, in Figures&nbsp;57-1 to 57-4 we put all the pieces together to construct the full operator diagrams for <math>\mathsf{W} : J \to \mathsf{W}J.\!</math> There is a considerable amount of redundancy among the following three series of Figures but that will hopefully provide a fuller picture of the operations under review, enabling these snapshots to serve as successive frames in the animation of logic they are meant to become.<br />
<br />
=====Operator Maps : Areal Views=====<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a1 -- Radius Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a1.} ~~ \text{Radius Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a2 -- Secant Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a2.} ~~ \text{Secant Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a3 -- Chord Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a3.} ~~ \text{Chord Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a4 -- Tangent Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a4.} ~~ \text{Tangent Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
=====Operator Maps : Box Views=====<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b1 -- Radius Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b1.} ~~ \text{Radius Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b2 -- Secant Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b2.} ~~ \text{Secant Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b3 -- Chord Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b3.} ~~ \text{Chord Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b4 -- Tangent Map of J ISW.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b4.} ~~ \text{Tangent Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
=====Operator Diagrams for the Conjunction J = uv=====<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-1 -- Radius Operator Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-1.} ~~ \text{Radius Operator Diagram for the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-2 -- Secant Operator Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-2.} ~~ \text{Secant Operator Diagram for the Conjunction}~ J = uv~\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-3 -- Chord Operator Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-3.} ~~ \text{Chord Operator Diagram for the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-4 -- Tangent Functor Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-4.} ~~ \text{Tangent Functor Diagram for the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
===Taking Aim at Higher Dimensional Targets===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
The past and present wilt . . . . I have filled them and<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;emptied them,<br><br />
And proceed to fill my next fold of the future.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 87]<br />
|}<br />
<br />
In the next Section we consider a transformation <math>F\!</math> of concrete type <math>F : [u, v] \to [x, y]\!</math> and abstract type <math>F : [\mathbb{B}^2] \to [\mathbb{B}^2].\!</math> From the standpoint of propositional calculus we naturally approach the task of understanding such a transformation by parsing it into component maps with <math>1\!</math>-dimensional ranges, as follows:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{ccccccl}<br />
F & = & (F_1, F_2) & = & (f, g) & : & [u, v] \to [x, y],<br />
\\[6pt]<br />
&& F_1 & = & f & : & [u, v] \to [x],<br />
\\[6pt]<br />
&& F_2 & = & g & : & [u, v] \to [y].<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Then we tackle the separate components, now viewed as propositions <math>F_i : U \to \mathbb{B},\!</math> one at a time. At the completion of this analytic phase, we return to the task of synthesizing these partial and transient impressions into an agile form of integrity, a solidly coordinated and deeply integrated comprehension of the ongoing transformation. (Very often, of course, in tangling with refractory cases, we never get as far as the beginning again.)<br />
<br />
Let us now refer to the dimension of the target space or codomain as the ''toll'' (or ''tole'') of a transformation, as distinguished from the dimension of the range or image that is customarily called the ''rank''. When we keep to transformations with a toll of <math>1,\!</math> as <math>J : [u, v] \to [x],\!</math> we tend to get lazy about distinguishing a logical transformation from its component propositions. However, if we deal with transformations of a higher toll, this form of indolence can no longer be tolerated.<br />
<br />
Well, perhaps we can carry it a little further. After all, the operator result <math>\mathrm{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet\!</math> is a map of toll <math>2,\!</math> and cannot be unfolded in one piece as a proposition. But when a map has rank <math>1,\!</math> like <math>\boldsymbol\varepsilon J : \mathrm{E}U \to X \subseteq \mathrm{E}X\!</math> or <math>\mathrm{d}J : \mathrm{E}U \to \mathrm{d}X \subseteq \mathrm{E}X,\!</math> we naturally choose to concentrate on the <math>1\!</math>-dimensional range of the operator result <math>\mathrm{W}J,\!</math> ignoring the final difference in quality between the spaces <math>X\!</math> and <math>\mathrm{d}X,\!</math> and view <math>\mathrm{W}J\!</math> as a proposition about <math>\mathrm{E}U.\!</math><br />
<br />
In this way, an initial ambivalence about the role of the operand <math>J\!</math> conveys a double duty to the result <math>\mathrm{W}J.\!</math> The pivot that is formed by our focus of attention is essential to the linkage that transfers this double moment, as the whole process takes its bearing and wheels around the precise measure of a narrow bead that we can draw on the range of <math>\mathrm{W}J.\!</math> This is the escapement that it takes to get away with what may otherwise seem to be a simple duplicity, and this is the tolerance that is needed to counterbalance a certain arrogance of equivocation, by all of which machinations we make ourselves free to indicate the operator results <math>\mathrm{W}J\!</math> as propositions or as transformations, indifferently.<br />
<br />
But that's it, and no further. Neglect of these distinctions in range and target universes of higher dimensions is bound to cause a hopeless confusion. To guard against these adverse prospects, Tables&nbsp;58 and 59 lay the groundwork for discussing a typical map <math>F : [\mathbb{B}^2] \to [\mathbb{B}^2],\!</math> and begin to pave the way to some extent for discussing any transformation of the form <math>F : [\mathbb{B}^n] \to [\mathbb{B}^k].\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 58.} ~~ \text{Cast of Characters : Expansive Subtypes of Objects and Operators}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| align="center" | <math>\text{Symbol}\!</math><br />
| align="center" | <math>\text{Notation}\!</math><br />
| align="center" | <math>\text{Description}\!</math><br />
| align="center" | <math>\text{Type}\!</math><br />
|-<br />
| align="center" | <math>U^\bullet\!</math><br />
| <math>= [u, v]\!</math><br />
| <math>\text{Source universe}\!</math><br />
| <math>[\mathbb{B}^n]\!</math><br />
|-<br />
| align="center" | <math>X^\bullet~\!</math><br />
| <math>\begin{array}{l}<br />
= [x, y] \\<br />
= [f, g]<br />
\end{array}</math><br />
| <math>\text{Target universe}\!</math><br />
| <math>[\mathbb{B}^k]\!</math><br />
|-<br />
| align="center" | <math>\mathrm{E}U^\bullet\!</math><br />
| <math>= [u, v, \mathrm{d}u, \mathrm{d}v]\!</math><br />
| <math>\text{Extended source universe}\!</math><br />
| <math>[\mathbb{B}^n \!\times\! \mathbb{D}^n]\!</math><br />
|-<br />
| align="center" | <math>\mathrm{E}X^\bullet\!</math><br />
| <math>\begin{array}{l}<br />
= [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
= [f, g, \mathrm{d}f, \mathrm{d}g]<br />
\end{array}</math><br />
| <math>\text{Extended target universe}\!</math><br />
| <math>[\mathbb{B}^k \!\times\! \mathbb{D}^k]\!</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
f \\ g<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{ll}<br />
f : U \!\to\! [x] \cong \mathbb{B} \\<br />
g : U \!\to\! [y] \cong \mathbb{B}<br />
\end{array}</math><br />
| <math>\text{Proposition}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathbb{B}^n \!\to\! \mathbb{B} \\<br />
\in (\mathbb{B}^n, \mathbb{B}^n \!\to\! \mathbb{B}) = [\mathbb{B}^n]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>F\!</math><br />
| <math>F = (f, g) : U^\bullet \!\to\! X^\bullet\!</math><br />
| <math>\text{Transformation of Map}\!</math><br />
| <math>[\mathbb{B}^n] \!\to\! [\mathbb{B}^k]</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\boldsymbol\varepsilon<br />
\\<br />
\eta<br />
\\<br />
\mathrm{E}<br />
\\<br />
\mathrm{D}<br />
\\<br />
\mathrm{d}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{W} : U^\bullet \!\to\! \mathrm{E}U^\bullet,<br />
\\<br />
\mathrm{W} : X^\bullet \!\to\! \mathrm{E}X^\bullet,<br />
\\<br />
\mathrm{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\\<br />
\text{for each}~ \mathrm{W} ~\text{in the set:}<br />
\\<br />
\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{ll}<br />
\text{Tacit extension operator} & \boldsymbol\varepsilon<br />
\\<br />
\text{Trope extension operator} & \eta<br />
\\<br />
\text{Enlargement operator} & \mathrm{E}<br />
\\<br />
\text{Difference operator} & \mathrm{D}<br />
\\<br />
\text{Differential operator} & \mathrm{d}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^n] \!\to\! [\mathbb{B}^n \!\times\! \mathbb{D}^n]},<br />
\\<br />
{[\mathbb{B}^k] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]},<br />
\\\\<br />
([\mathbb{B}^n] \!\to\! [\mathbb{B}^k]) \!\to\!<br />
\\<br />
([\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k])<br />
\end{array}</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\mathsf{e}<br />
\\<br />
\mathsf{E}<br />
\\<br />
\mathsf{D}<br />
\\<br />
\mathsf{T}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{W} : U^\bullet \!\to\! \mathsf{T}U^\bullet = \mathrm{E}U^\bullet,<br />
\\<br />
\mathsf{W} : X^\bullet \!\to\! \mathsf{T}X^\bullet = \mathrm{E}X^\bullet,<br />
\\<br />
\mathsf{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathsf{T}U^\bullet \!\to\! \mathsf{T}X^\bullet)<br />
\\<br />
\text{for each}~ \mathsf{W} ~\text{in the set:}<br />
\\<br />
\{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{T} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{lll}<br />
\text{Radius operator} & \mathsf{e} & = (\boldsymbol\varepsilon, \eta)<br />
\\<br />
\text{Secant operator} & \mathsf{E} & = (\boldsymbol\varepsilon, \mathrm{E})<br />
\\<br />
\text{Chord operator} & \mathsf{D} & = (\boldsymbol\varepsilon, \mathrm{D})<br />
\\<br />
\text{Tangent functor} & \mathsf{T} & = (\boldsymbol\varepsilon, \mathrm{d})<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^n] \!\to\! [\mathbb{B}^n \!\times\! \mathbb{D}^n]},<br />
\\<br />
{[\mathbb{B}^k] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]},<br />
\\\\<br />
([\mathbb{B}^n] \!\to\! [\mathbb{B}^k]) \!\to\!<br />
\\<br />
([\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k])<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 59.} ~~ \text{Synopsis of Terminology : Restrictive and Alternative Subtypes}~\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| &nbsp;<br />
| align="center" | <math>\begin{matrix}\text{Operator}\\\text{or}\\\text{Operand}\end{matrix}</math><br />
| align="center" | <math>\begin{matrix}\text{Proposition}\\\text{or}\\\text{Component}\end{matrix}</math><br />
| align="center" | <math>\begin{matrix}\text{Transformation}\\\text{or}\\\text{Map}\end{matrix}</math><br />
|-<br />
| align="center" | <math>\underline{\text{Operand}}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
F = (F_1, F_2) \\<br />
F = (f, g) : U \!\to\! X<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
F_i : \langle u, v \rangle \!\to\! \mathbb{B} \\<br />
F_i : \mathbb{B}^n \!\to\! \mathbb{B}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
F : [u, v] \!\to\! [x, y] \\<br />
F : [\mathbb{B}^n] \!\to\! [\mathbb{B}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tacit}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\boldsymbol\varepsilon : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\boldsymbol\varepsilon : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{B} \\<br />
\boldsymbol\varepsilon F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{B}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y] \\<br />
\boldsymbol\varepsilon F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Trope}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\eta : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\eta : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\eta F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\eta F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Enlargement}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{E} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{E}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{E}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Difference}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{D} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{D}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{D}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Differential}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{d} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{d} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{d}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}\!</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Remainder}}\\\text{operator}\end{matrix}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{r} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{r} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{r}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{r}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Radius}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e} = (\boldsymbol\varepsilon, \eta) \\<br />
\mathsf{e} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{e}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Secant}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}) \\<br />
\mathsf{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{E}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Chord}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}) \\<br />
\mathsf{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{D}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tangent}}\\\text{functor}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T} = (\boldsymbol\varepsilon, \mathrm{d}) \\<br />
\mathsf{T} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{T}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
===Transformations of Type '''B'''<sup>2</sup> &rarr; '''B'''<sup>2</sup>===<br />
<br />
To take up a slightly more complex example, but one that remains simple enough to pursue through a complete series of developments, consider the transformation from <math>U^\bullet = [u, v]\!</math> to <math>X^\bullet = [x, y]\!</math> that is defined by the following system of equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
x<br />
& = & f(u, v)<br />
& = & \texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[8pt]<br />
y<br />
& = & g(u, v)<br />
& = & \texttt{((} u \texttt{,} v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
The component notation <math>F = (F_1, F_2) = (f, g) : U^\bullet \to X^\bullet\!</math> allows us to give a name and a type to this transformation and permits defining it by the compact description that follows:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
(x, y)<br />
& = & F(u, v)<br />
& = & (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Logical Transformations====<br />
<br />
The information that defines the logical transformation <math>F\!</math> can be represented in the form of a truth table, as shown in Table&nbsp;60. To cut down on subscripts in this example we continue to use plain letter equivalents for all components of spaces and maps.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:60%"<br />
|+ style="height:30px" | <math>\text{Table 60.} ~~ \text{A Propositional Transformation}\!</math><br />
|- style="height:40px; background:ghostwhite; width:100%"<br />
| style="width:25%" | <math>u\!</math><br />
| style="width:25%" | <math>v\!</math><br />
| style="width:25%; border-left:1px solid black" | <math>f\!</math><br />
| style="width:25%" | <math>g\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="border-top:1px solid black" | <math>u\!</math><br />
| style="border-top:1px solid black" | <math>v\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;61 shows how we might paint a picture of the transformation <math>F\!</math> in the manner of Figure&nbsp;30.<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 61 -- Propositional Transformation.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 61.} ~~ \text{A Propositional Transformation}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;62 extracts the gist of Figure&nbsp;61, exhibiting a style of diagram that is adequate for most purposes.<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 62 -- Propositional Transformation (Short Form).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 62.} ~~ \text{A Propositional Transformation (Short Form)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Local Transformations====<br />
<br />
Figure&nbsp;63 gives a more complete picture of the transformation <math>F,\!</math> showing how the points of <math>U^\bullet\!</math> are transformed into points of <math>X^\bullet.\!</math> The bold lines crossing from one universe to the other trace the action that <math>F\!</math> induces on points, in other words, they show how the transformation acts as a mapping from points to points and chart its effects on the elements that are variously called cells, points, positions, or singular propositions.<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 63 -- Transformation of Positions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 63.} ~~ \text{A Transformation of Positions}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;64 shows how the action of <math>F\!</math> on cells or points can be computed in terms of coordinates.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 64.} ~~ \text{A Transformation of Positions}\!</math><br />
|- style="height:40px; background:ghostwhite; width:100%"<br />
| style="width:8%" | <math>u\!</math><br />
| style="width:8%" | <math>v\!</math><br />
| style="width:12%; border-left:1px solid black" | <math>x\!</math><br />
| style="width:12%" | <math>y\!</math><br />
| style="width:10%; border-left:1px solid black" | <math>x~y\!</math><br />
| style="width:10%" | <math>x \texttt{(} y \texttt{)}\!</math><br />
| style="width:10%" | <math>\texttt{(} x \texttt{)} y\!</math><br />
| style="width:10%" | <math>\texttt{(} x \texttt{)(} y \texttt{)}\!</math><br />
| style="width:20%; border-left:1px solid black" | <math>X^\bullet = [x, y]\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\uparrow<br />
\\[4pt]<br />
F =<br />
\\[4pt]<br />
(f, g)<br />
\\[4pt]<br />
\uparrow<br />
\end{matrix}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="border-top:1px solid black" | <math>u\!</math><br />
| style="border-top:1px solid black" | <math>v\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
| style="border-top:1px solid black" | <math>\texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>u~v\!</math><br />
| style="border-top:1px solid black" | <math>\texttt{(} u \texttt{,} v \texttt{)}\!</math><br />
| style="border-top:1px solid black" | <math>\texttt{(} u \texttt{)(} v \texttt{)}\!</math><br />
| style="border-top:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>U^\bullet = [u, v]\!</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;65 extends this scheme from single cells to arbitrary regions, showing how we might compute the action of a logical transformation on arbitrary propositions in the universe of discourse. The effect of a point-transformation on arbitrary propositions, or any other structures erected on points, is referred to as the ''induced action'' of the transformation on the structures in question.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 65-a.} ~~ \text{An Induced Transformation on Propositions}\!</math><br />
|- style="height:50px; background:ghostwhite"<br />
| style="width:20%" | <math>X^\bullet~\!</math><br />
| style="width:20%; border-right:none" | <math>\longleftarrow\!</math><br />
| style="width:20%; border-left:none; border-right:none" | <math>F = (f, g)\!</math><br />
| style="width:20%; border-left:none" | <math>\longleftarrow\!</math><br />
| style="width:20%" | <math>U^\bullet~\!</math><br />
|- style="background:ghostwhite"<br />
| rowspan="2" | <math>f_i (x, y)\!</math><br />
| align="right" | <math>\begin{matrix}u = \\ v =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~0~0\\1~0~1~0\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= u \\ = v\end{matrix}</math><br />
| rowspan="2" style="width:20%" | <math>f_j (u, v)\!</math><br />
|- style="background:ghostwhite"<br />
| align="right" | <math>\begin{matrix}x = \\ y =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~1~0\\1~0~0~1\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= f(u, v) \\ = g(u, v)\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{2}<br />
\\[2pt]<br />
f_{3}<br />
\\[2pt]<br />
f_{4}<br />
\\[2pt]<br />
f_{5}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{7}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)~ ~}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{~ ~(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{,~} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{~~} y \texttt{)}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~0<br />
\\[2pt]<br />
0~0~0~0<br />
\\[2pt]<br />
0~0~0~1<br />
\\[2pt]<br />
0~0~0~1<br />
\\[2pt]<br />
0~1~1~0<br />
\\[2pt]<br />
0~1~1~0<br />
\\[2pt]<br />
0~1~1~1<br />
\\[2pt]<br />
0~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{,~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{,~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{~~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{~~} v \texttt{)}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[2pt]<br />
f_{0}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{7}<br />
\\[2pt]<br />
f_{7}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{8}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{10}<br />
\\[2pt]<br />
f_{11}<br />
\\[2pt]<br />
f_{12}<br />
\\[2pt]<br />
f_{13}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{15}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[2pt]<br />
\texttt{~ ~ ~ ~} y \texttt{~~}<br />
\\[2pt]<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[2pt]<br />
\texttt{~~} x \texttt{~ ~ ~ ~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
1~0~0~0<br />
\\[2pt]<br />
1~0~0~0<br />
\\[2pt]<br />
1~0~0~1<br />
\\[2pt]<br />
1~0~0~1<br />
\\[2pt]<br />
1~1~1~0<br />
\\[2pt]<br />
1~1~1~0<br />
\\[2pt]<br />
1~1~1~1<br />
\\[2pt]<br />
1~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~~} u \texttt{~~} v \texttt{~~}<br />
\\[2pt]<br />
\texttt{~~} u \texttt{~~} v \texttt{~~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{8}<br />
\\[2pt]<br />
f_{8}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{15}<br />
\\[2pt]<br />
f_{15}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 65-b.} ~~ \text{An Induced Transformation on Propositions}\!</math><br />
|- style="height:50px; background:ghostwhite"<br />
| style="width:20%" | <math>X^\bullet~\!</math><br />
| style="width:20%; border-right:none" | <math>\longleftarrow\!</math><br />
| style="width:20%; border-left:none; border-right:none" | <math>F = (f, g)\!</math><br />
| style="width:20%; border-left:none" | <math>\longleftarrow\!</math><br />
| style="width:20%" | <math>U^\bullet~\!</math><br />
|- style="background:ghostwhite"<br />
| rowspan="2" | <math>f_i (x, y)\!</math><br />
| align="right" | <math>\begin{matrix}u = \\ v =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~0~0\\1~0~1~0\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= u \\ = v\end{matrix}</math><br />
| rowspan="2" style="width:20%" | <math>f_j (u, v)\!</math><br />
|- style="background:ghostwhite"<br />
| align="right" | <math>\begin{matrix}x = \\ y =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~1~0\\1~0~0~1\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= f(u, v) \\ = g(u, v)\end{matrix}</math><br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>f_{0}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[2pt]<br />
f_{2}<br />
\\[2pt]<br />
f_{4}<br />
\\[2pt]<br />
f_{8}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~0<br />
\\[2pt]<br />
0~0~0~1<br />
\\[2pt]<br />
0~1~1~0<br />
\\[2pt]<br />
1~0~0~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{,~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{~} u \texttt{~~} v \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{8}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{3}<br />
\\[2pt]<br />
f_{12}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[2pt]<br />
1~1~1~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} u \texttt{)(} v \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[2pt]<br />
f_{14}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[2pt]<br />
f_{9}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[2pt]<br />
1~0~0~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{~~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{~} u \texttt{~~} v \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[2pt]<br />
f_{8}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{5}<br />
\\[2pt]<br />
f_{10}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[2pt]<br />
1~0~0~1<br />
\end{matrix}~\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} u \texttt{,~} v \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[2pt]<br />
f_{9}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[2pt]<br />
f_{11}<br />
\\[2pt]<br />
f_{13}<br />
\\[2pt]<br />
f_{14}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[2pt]<br />
1~0~0~1<br />
\\[2pt]<br />
1~1~1~0<br />
\\[2pt]<br />
1~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} u \texttt{~~} v \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{15}<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>f_{15}\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Difference Operators and Tangent Functors====<br />
<br />
Given the alphabets <math>\mathcal{U} = \{ u, v \}\!</math> and <math>\mathcal{X} = \{ x, y \},\!</math> along with the corresponding universes of discourse <math>U^\bullet, X^\bullet \cong [\mathbb{B}^2],\!</math> how many logical transformations of the general form <math>G = (G_1, G_2) : U^\bullet \to X^\bullet\!</math> are there? Since <math>G_1\!</math> and <math>G_2\!</math> can be any propositions of the type <math>\mathbb{B}^2 \to \mathbb{B},\!</math> there are <math>2^4 = 16\!</math> choices for each of the maps <math>G_1\!</math> and <math>G_2\!</math> and thus there are <math>2^4 \cdot 2^4 = 2^8 = 256\!</math> different mappings altogether of the form <math>G : U^\bullet \to X^\bullet.\!</math> The set of functions of a given type is denoted by placing its type indicator in parentheses, in the present instance writing <math>(U^\bullet \to X^\bullet) = \{ G : U^\bullet \to X^\bullet \},\!</math> and so the cardinality of the ''function space'' <math>(U^\bullet \to X^\bullet)\!</math> is summed up by writing <math>|(U^\bullet \to X^\bullet)| = |(\mathbb{B}^2 \to \mathbb{B}^2)| = 4^4 = 256.\!</math><br />
<br />
Given a transformation <math>G = (G_1, G_2) : U^\bullet \to X^\bullet\!</math> of this type, we proceed to define a pair of further transformations, related to <math>G,\!</math> that operate between the extended universes, <math>\mathrm{E}U^\bullet\!</math> and <math>\mathrm{E}X^\bullet,\!</math> of its source and target domains.<br />
<br />
First, the ''enlargement map'' (or ''secant transformation'') <math>\mathrm{E}G = (\mathrm{E}G_1, \mathrm{E}G_2) : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet\!</math> is defined by the following set of component equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}G_i<br />
& = & G_i (u + \mathrm{d}u, v + \mathrm{d}v)<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Next, the ''difference map'' (or ''chordal transformation'') <math>\mathrm{D}G = (\mathrm{D}G_1, \mathrm{D}G_2) : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet~\!</math> is defined in component-wise fashion as the boolean sum of the initial proposition <math>G_i\!</math> and the enlarged proposition <math>\mathrm{E}G_i,\!</math> for <math>i = 1, 2,\!</math> according to the following set of equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
\mathrm{D}G_i<br />
& = & G_i (u, v)<br />
& + & \mathrm{E}G_i (u, v, \mathrm{d}u, \mathrm{d}v)<br />
\\[8pt]<br />
& = & G_i (u, v)<br />
& + & G_i (u + \mathrm{d}u, v + \mathrm{d}v)<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Maintaining a strict analogy with ordinary difference calculus would perhaps have us write <math>\mathrm{D}G_i = \mathrm{E}G_i - G_i,\!</math> but the sum and difference operations are the same thing in boolean arithmetic. It is more often natural in the logical context to consider an initial proposition <math>q,\!</math> then to compute the enlargement <math>\mathrm{E}q,\!</math> and finally to determine the difference <math>\mathrm{D}q = q + \mathrm{E}q,\!</math> so we let the variant order of terms reflect this sequence of considerations.<br />
<br />
Viewed in this light the difference operator <math>\mathrm{D}\!</math> is imagined to be a function of very wide scope and polymorphic application, one that is able to realize the association between each transformation <math>G\!</math> and its difference map <math>\mathrm{D}G,\!</math> for example, taking the function space <math>(U^\bullet \to X^\bullet)\!</math> into <math>(\mathrm{E}U^\bullet \to \mathrm{E}X^\bullet).\!</math> When we consider the variety of interpretations permitted to propositions over the contexts in which we put them to use, it should be clear that an operator of this scope is not at all a trivial matter to define in general and that it may take some trouble to work out. For the moment we content ourselves with returning to particular cases.<br />
<br />
Acting on the logical transformation <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;),\!</math> the operators <math>\mathrm{E}\!</math> and <math>\mathrm{D}\!</math> yield the enlarged map <math>\mathrm{E}F = (\mathrm{E}f, \mathrm{E}g)\!</math> and the difference map <math>\mathrm{D}F = (\mathrm{D}f, \mathrm{D}g),\!</math> respectively, whose components are given as follows.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}f<br />
& = & \texttt{((} u + \mathrm{d}u \texttt{)(} v + \mathrm{d}v \texttt{))}<br />
\\[8pt]<br />
\mathrm{E}g<br />
& = & \texttt{((} u + \mathrm{d}u \texttt{,~} v + \mathrm{d}v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
\mathrm{D}f<br />
& = & \texttt{((} u \texttt{)(} v \texttt{))}<br />
& + & \texttt{((} u + \mathrm{d}u \texttt{)(} v + \mathrm{d}v \texttt{))}<br />
\\[8pt]<br />
\mathrm{D}g<br />
& = & \texttt{((} u \texttt{,~} v \texttt{))}<br />
& + & \texttt{((} u + \mathrm{d}u \texttt{,~} v + \mathrm{d}v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
But these initial formulas are purely definitional, and help us little in understanding either the purpose of the operators or the meaning of their results. Working symbolically, let us apply the same method to the separate components <math>f\!</math> and <math>g\!</math> that we earlier used on <math>J.\!</math> This work is recorded in Appendix&nbsp;3 and a summary of the results is presented in Tables&nbsp;66-i and 66-ii.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 66-i.} ~~ \text{Computation Summary for}~ f(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f<br />
& = & u \!\cdot\! v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 1<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}f<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \cdot \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{d}f<br />
& = & u \!\cdot\! v \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}f<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 66-ii.} ~~ \text{Computation Summary for}~ g(u, v) = \texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon g<br />
& = & u \!\cdot\! v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}g<br />
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}g<br />
& = & u \!\cdot\! v \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;67 shows how to compute the analytic series for <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math> in terms of coordinates, and Table&nbsp;68 recaps these results in symbolic terms, agreeing with earlier derivations.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 67.} ~~ \text{Computation of an Analytic Series in Terms of Coordinates}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:6%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}u\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>\mathrm{d}v\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>u'\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>v'\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:4px double black" | <math>f\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>g\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{E}f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{E}g}\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{D}f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{D}g}\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{d}f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{d}g}\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{d}^2\!f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{d}^2\!g}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 68.} ~~ \text{Computation of an Analytic Series in Symbolic Terms}\!</math><br />
|- style="height:40px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black" | <math>f\!</math><br />
| <math>g\!</math><br />
| style="border-left:1px solid black" | <math>{\mathrm{D}f}\!</math><br />
| <math>{\mathrm{D}g}\!</math><br />
| style="border-left:1px solid black" | <math>{\mathrm{d}f}\!</math><br />
| <math>{\mathrm{d}g}\!</math><br />
| style="border-left:1px solid black" | <math>{\mathrm{d}^2\!f}\!</math><br />
| <math>{\mathrm{d}^2\!g}\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[4pt]<br />
\texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
\\[4pt]<br />
\texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
\\[4pt]<br />
\texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;69 gives a graphical picture of the difference map <math>\mathrm{D}F = (\mathrm{D}f, \mathrm{D}g)\!</math> for the transformation <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math> This represents the same information about <math>\mathrm{D}f~\!</math> and <math>\mathrm{D}g~\!</math> that was given in the corresponding rows of Tables&nbsp;66-i and 66-ii, for ease of reference repeated below.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[8pt]<br />
\mathrm{D}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 69 -- Difference Map (Short Form).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 69.} ~~ \text{Difference Map of}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;70-a shows a way of visualizing the tangent functor map <math>\mathrm{d}F = (\mathrm{d}f, \mathrm{d}g)\!</math> for the transformation <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math> This amounts to the same information about <math>\mathrm{d}f~\!</math> and <math>\mathrm{d}g~\!</math> that was given in Tables&nbsp;66-i and 66-ii, the corresponding rows of which are repeated below.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{d}f<br />
& = & u \!\cdot\! v \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[8pt]<br />
\mathrm{d}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 70-a -- Tangent Functor Diagram.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 70-a.} ~~ \text{Tangent Functor Diagram for}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math><br />
|}<br />
<br />
<br><br />
<br />
Figure 70-b shows another way to picture the action of the tangent functor on the logical transformation <math>F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 70-b -- Tangent Functor Diagram.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 70-b.} ~~ \text{Tangent Functor Ferris Wheel for}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math><br />
|}<br />
<br />
<br><br />
<br />
* '''Note.''' The original Figure&nbsp;70-b lost some of its labeling in a succession of platform metamorphoses over the years, so we have included an ASCII version below to indicate where the missing labels go.<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| /////\ /////\ | | /XXXX\ /XXXX\ | | /\\\\\ /\\\\\ |<br />
| ///////o//////\ | | /XXXXXXoXXXXXX\ | | /\\\\\\o\\\\\\\ |<br />
| //////// \//////\ | | /XXXXXX/ \XXXXXX\ | | /\\\\\\/ \\\\\\\\ |<br />
| o/////// \//////o | | oXXXXXX/ \XXXXXXo | | o\\\\\\/ \\\\\\\o |<br />
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |<br />
| |/du//| |//dv/| | | |XXXXX| |XXXXX| | | |\du\\| |\\dv\| |<br />
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |<br />
| o//////\ ///////o | | oXXXXXX\ /XXXXXXo | | o\\\\\\\ /\\\\\\o |<br />
| \//////\ //////// | | \XXXXXX\ /XXXXXX/ | | \\\\\\\\ /\\\\\\/ |<br />
| \//////o/////// | | \XXXXXXoXXXXXX/ | | \\\\\\\o\\\\\\/ |<br />
| \///// \///// | | \XXXX/ \XXXX/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ (u)(v) o-----------------------o dv' @ (u)(v) =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| / \ /////\ | | /\\\\\ /XXXX\ | | /\\\\\ /\\\\\ |<br />
| / o//////\ | | /\\\\\\oXXXXXX\ | | /\\\\\\o\\\\\\\ |<br />
| / //\//////\ | | /\\\\\\//\XXXXXX\ | | /\\\\\\/ \\\\\\\\ |<br />
| o ////\//////o | | o\\\\\\////\XXXXXXo | | o\\\\\\/ \\\\\\\o |<br />
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |<br />
| | du |/////|//dv/| | | |\\\\\|/////|XXXXX| | | |\du\\| |\\dv\| |<br />
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |<br />
| o \//////////o | | o\\\\\\\////XXXXXXo | | o\\\\\\\ /\\\\\\o |<br />
| \ \///////// | | \\\\\\\\//XXXXXX/ | | \\\\\\\\ /\\\\\\/ |<br />
| \ o/////// | | \\\\\\\oXXXXXX/ | | \\\\\\\o\\\\\\/ |<br />
| \ / \///// | | \\\\\/ \XXXX/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ (u) v o-----------------------o dv' @ (u) v =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| /////\ / \ | | /XXXX\ /\\\\\ | | /\\\\\ /\\\\\ |<br />
| ///////o \ | | /XXXXXXo\\\\\\\ | | /\\\\\\o\\\\\\\ |<br />
| /////////\ \ | | /XXXXXX//\\\\\\\\ | | /\\\\\\/ \\\\\\\\ |<br />
| o//////////\ o | | oXXXXXX////\\\\\\\o | | o\\\\\\/ \\\\\\\o |<br />
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |<br />
| |/du//|/////| dv | | | |XXXXX|/////|\\\\\| | | |\du\\| |\\dv\| |<br />
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |<br />
| o//////\//// o | | oXXXXXX\////\\\\\\o | | o\\\\\\\ /\\\\\\o |<br />
| \//////\// / | | \XXXXXX\//\\\\\\/ | | \\\\\\\\ /\\\\\\/ |<br />
| \//////o / | | \XXXXXXo\\\\\\/ | | \\\\\\\o\\\\\\/ |<br />
| \///// \ / | | \XXXX/ \\\\\/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ u (v) o-----------------------o dv' @ u (v) =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| / \ / \ | | /\\\\\ /\\\\\ | | /\\\\\ /\\\\\ |<br />
| / o \ | | /\\\\\\o\\\\\\\ | | /\\\\\\o\\\\\\\ |<br />
| / / \ \ | | /\\\\\\/ \\\\\\\\ | | /\\\\\\/ \\\\\\\\ |<br />
| o / \ o | | o\\\\\\/ \\\\\\\o | | o\\\\\\/ \\\\\\\o |<br />
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |<br />
| | du | | dv | | | |\\\\\| |\\\\\| | | |\du\\| |\\dv\| |<br />
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |<br />
| o \ / o | | o\\\\\\\ /\\\\\\o | | o\\\\\\\ /\\\\\\o |<br />
| \ \ / / | | \\\\\\\\ /\\\\\\/ | | \\\\\\\\ /\\\\\\/ |<br />
| \ o / | | \\\\\\\o\\\\\\/ | | \\\\\\\o\\\\\\/ |<br />
| \ / \ / | | \\\\\/ \\\\\/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ u v o-----------------------o dv' @ u v =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| U | |\U\\\\\\\\\\\\\\\\\\\\\| |\U\\\\\\\\\\\\\\\\\\\\\|<br />
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|<br />
| /////\ /////\ | |\\\\\/////\\/////\\\\\\| |\\\\\/ \\/ \\\\\\|<br />
| ///////o//////\ | |\\\\///////o//////\\\\\| |\\\\/ o \\\\\|<br />
| /////////\//////\ | |\\\////////X\//////\\\\| |\\\/ /\\ \\\\|<br />
| o//////////\//////o | |\\o///////XXX\//////o\\| |\\o /\\\\ o\\|<br />
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|<br />
| |//u//|/////|//v//| | |\\|//u//|XXXXX|//v//|\\| |\\| u |\\\\\| v |\\|<br />
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|<br />
| o//////\//////////o | |\\o//////\XXX///////o\\| |\\o \\\\/ o\\|<br />
| \//////\///////// | |\\\\//////\X////////\\\| |\\\\ \\/ /\\\|<br />
| \//////o/////// | |\\\\\//////o///////\\\\| |\\\\\ o /\\\\|<br />
| \///// \///// | |\\\\\\/////\\/////\\\\\| |\\\\\\ /\\ /\\\\\|<br />
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|<br />
| | |\\\\\\\\\\\\\\\\\\\\\\\| |\\\\\\\\\\\\\\\\\\\\\\\|<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= u' o-----------------------o v' =<br />
= | U' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/u'//|XXXXX|\\v'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
Figure 70-b. Tangent Functor Ferris Wheel for F<u, v> = <((u)(v)), ((u, v))><br />
</pre><br />
|}<br />
<br />
==Epilogue, Enchoiry, Exodus==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
It is time to explain myself . . . . let us stand up.<br />
| width="4%" | &nbsp;<br />
|- <br />
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 79]<br />
|}<br />
<br />
==Appendices==<br />
<br />
===Appendix 1. Propositional Forms and Differential Expansions===<br />
<br />
====Table A1. Propositional Forms on Two Variables====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A1.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="background:ghostwhite"<br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}</math><br />
| width="25%" | <math>\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{0}\\f_{1}\\f_{2}\\f_{3}\\f_{4}\\f_{5}\\f_{6}\\f_{7}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0000}\\f_{0001}\\f_{0010}\\f_{0011}\\f_{0100}\\f_{0101}\\f_{0110}\\f_{0111}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~0\\0~0~0~1\\0~0~1~0\\0~0~1~1\\0~1~0~0\\0~1~0~1\\0~1~1~0\\0~1~1~1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)~ ~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~ ~(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{,~} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{~~} y \texttt{)}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{false}<br />
\\<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\<br />
y ~\text{without}~ x<br />
\\<br />
\text{not}~ x<br />
\\<br />
x ~\text{without}~ y<br />
\\<br />
\text{not}~ y<br />
\\<br />
x ~\text{not equal to}~ y<br />
\\<br />
\text{not both}~ x ~\text{and}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\<br />
\lnot x \land \lnot y<br />
\\<br />
\lnot x \land y<br />
\\<br />
\lnot x<br />
\\<br />
x \land \lnot y<br />
\\<br />
\lnot y<br />
\\<br />
x \ne y<br />
\\<br />
\lnot x \lor \lnot y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{8}\\f_{9}\\f_{10}\\f_{11}\\f_{12}\\f_{13}\\f_{14}\\f_{15}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{1000}\\f_{1001}\\f_{1010}\\f_{1011}\\f_{1100}\\f_{1101}\\f_{1110}\\f_{1111}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1~0~0~0\\1~0~0~1\\1~0~1~0\\1~0~1~1\\1~1~0~0\\1~1~0~1\\1~1~1~0\\1~1~1~1<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~ ~ ~ ~} y \texttt{~~}<br />
\\<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{~~} x \texttt{~ ~ ~ ~}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{((~))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{and}~ y<br />
\\<br />
x ~\text{equal to}~ y<br />
\\<br />
y<br />
\\<br />
\text{not}~ x ~\text{without}~ y<br />
\\<br />
x<br />
\\<br />
\text{not}~ y ~\text{without}~ x<br />
\\<br />
x ~\text{or}~ y<br />
\\<br />
\text{true}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x \land y<br />
\\<br />
x = y<br />
\\<br />
y<br />
\\<br />
x \Rightarrow y<br />
\\<br />
x<br />
\\<br />
x \Leftarrow y<br />
\\<br />
x \lor y<br />
\\<br />
1<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A2. Propositional Forms on Two Variables====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A2.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="background:ghostwhite"<br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}</math><br />
| width="25%" | <math>\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>f_{0000}\!</math><br />
| <math>0~0~0~0</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0001}\\f_{0010}\\f_{0100}\\f_{1000}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~1\\0~0~1~0\\0~1~0~0\\1~0~0~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\<br />
y ~\text{without}~ x<br />
\\<br />
x ~\text{without}~ y<br />
\\<br />
x ~\text{and}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot x \land \lnot y<br />
\\<br />
\lnot x \land y<br />
\\<br />
x \land \lnot y<br />
\\<br />
x \land y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{3}\\f_{12}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0011}\\f_{1100}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~1~1\\1~1~0~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ x<br />
\\<br />
x<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot x<br />
\\<br />
x<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{6}\\f_{9}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0110}\\f_{1001}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~0\\1~0~0~1<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{not equal to}~ y<br />
\\<br />
x ~\text{equal to}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x \ne y<br />
\\<br />
x = y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{5}\\f_{10}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0101}\\f_{1010}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~0~1\\1~0~1~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ y<br />
\\<br />
y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot y<br />
\\<br />
y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{7}\\f_{11}\\f_{13}\\f_{14}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0111}\\f_{1011}\\f_{1101}\\f_{1110}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~1\\1~0~1~1\\1~1~0~1\\1~1~1~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not both}~ x ~\text{and}~ y<br />
\\<br />
\text{not}~ x ~\text{without}~ y<br />
\\<br />
\text{not}~ y ~\text{without}~ x<br />
\\<br />
x ~\text{or}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot x \lor \lnot y<br />
\\<br />
x \Rightarrow y<br />
\\<br />
x \Leftarrow y<br />
\\<br />
x \lor y<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>f_{1111}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A3. E''f'' Expanded Over Differential Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A3.} ~~ \mathrm{E}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{T}_{11}f\\\mathrm{E}f|_{\mathrm{d}x ~ \mathrm{d}y}\end{matrix}</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{T}_{10}f\\\mathrm{E}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}\end{matrix}</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{T}_{01}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}\end{matrix}</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{T}_{00}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{3}\\f_{12}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{6}\\f_{9}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{5}\\f_{10}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{7}\\f_{11}\\f_{13}\\f_{14}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
|- style="background:ghostwhite"<br />
| style="border-top:1px solid black" colspan="2" | <math>\text{Fixed Point Total}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>4\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>4\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>4\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>16\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A4. D''f'' Expanded Over Differential Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A4.} ~~ \mathrm{D}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\mathrm{D}f|_{\mathrm{d}x ~ \mathrm{d}y}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}~\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
y<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
y<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
x<br />
\\<br />
x<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
x<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
y<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
y<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <br />
<math>\begin{matrix}<br />
x<br />
\\<br />
x<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A5. E''f'' Expanded Over Ordinary Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A5.} ~~ \mathrm{E}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\mathrm{E}f|_{xy}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{E}f|_{x \texttt{(} y \texttt{)}}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{E}f|_{\texttt{(} x \texttt{)} y}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{E}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{3}\\f_{12}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{6}\\f_{9}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{5}\\f_{10}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{7}\\f_{11}\\f_{13}\\f_{14}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A6. D''f'' Expanded Over Ordinary Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A6.} ~~ \mathrm{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\mathrm{D}f|_{xy}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{x \texttt{(} y \texttt{)}}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} x \texttt{)} y}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\texttt{(} x \texttt{)}\\\texttt{~} x \texttt{~}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Appendix 2. Differential Forms===<br />
<br />
The actions of the difference operator <math>\mathrm{D}\!</math> and the tangent operator <math>\mathrm{d}\!</math> on the 16 bivariate propositions are shown in Tables&nbsp;A7 and A8.<br />
<br />
Table A7 expands the differential forms that result over a ''logical basis'':<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
|<br />
<math>\{~ \texttt{(}\mathrm{d}x\texttt{)(}\mathrm{d}y\texttt{)}, ~\mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}, ~\texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.\!</math><br />
|}<br />
<br />
This set consists of the singular propositions in the first order differential variables, indicating mutually exclusive and exhaustive ''cells'' of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the cell-basis, point-basis, or singular differential basis. In this setting it is frequently convenient to use the following abbreviations:<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
|<br />
<math>\partial x ~=~ \mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}\!</math> &nbsp; &nbsp; and &nbsp; &nbsp; <math>\partial y ~=~ \texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y.\!</math><br />
|}<br />
<br />
Table A8 expands the differential forms that result over an ''algebraic basis'':<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
| <math>\{~ 1, ~\mathrm{d}x, ~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.\!</math><br />
|}<br />
<br />
This set consists of the ''positive propositions'' in the first order differential variables, indicating overlapping positive regions of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the ''positive differential basis''.<br />
<br />
====Table A7. Differential Forms Expanded on a Logical Basis====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A7.} ~~ \text{Differential Forms Expanded on a Logical Basis}\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| &nbsp;<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" | <math>\mathrm{D}f~\!</math><br />
| <math>\mathrm{d}f~\!</math><br />
|-<br />
| <math>f_{0}\!</math><br />
| style="border-right:none" | <math>\texttt{(~)}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x) ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\\<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\\<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\\<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\partial x<br />
\\<br />
\partial x<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
\\<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\partial x & + & \partial y<br />
\\<br />
\partial x & + & \partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\partial y<br />
\\<br />
\partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x) ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\\<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\\<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\\<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| style="border-right:none" | <math>\texttt{((~))}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A8. Differential Forms Expanded on an Algebraic Basis====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A8.} ~~ \text{Differential Forms Expanded on an Algebraic Basis}\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| &nbsp;<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" | <math>\mathrm{D}f~\!</math><br />
| <math>\mathrm{d}f~\!</math><br />
|-<br />
| <math>f_{0}\!</math><br />
| style="border-right:none" | <math>\texttt{(~)}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x<br />
\\<br />
\mathrm{d}x<br />
\end{matrix}\!</math><br />
| <math>\begin{matrix}<br />
\mathrm{d}x<br />
\\<br />
\mathrm{d}x<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\\<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\\<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}y<br />
\\<br />
\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y<br />
\\<br />
\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| style="border-right:none" | <math>\texttt{((~))}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A9. Tangent Proposition as Pointwise Linear Approximation====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A9.} ~~ \text{Tangent Proposition}~ \mathrm{d}f = \text{Pointwise Linear Approximation to the Difference Map}~ \mathrm{D}f\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}f =<br />
\\[2pt]<br />
\partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}^2\!f =<br />
\\[2pt]<br />
\partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\mathrm{d}f|_{x \, y}</math><br />
| <math>\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math><br />
|-<br />
| style="border-right:none" | <math>f_0\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{5}\\f_{10}\end{matrix}\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\end{matrix}\!</math><br />
| <math>\begin{matrix}<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>f_{15}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A10. Taylor Series Expansion Df = d''f'' + d<sup>2</sup>''f''====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" |<br />
<math>\text{Table A10.} ~~ \text{Taylor Series Expansion}~ {\mathrm{D}f = \mathrm{d}f + \mathrm{d}^2\!f}\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{D}f<br />
\\<br />
= & \mathrm{d}f & + & \mathrm{d}^2\!f<br />
\\<br />
= & \partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y & + & \partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\mathrm{d}f|_{x \, y}</math><br />
| <math>\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math><br />
|-<br />
| style="border-right:none" | <math>f_0\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>f_{15}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A11. Partial Differentials and Relative Differentials====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A11.} ~~ \text{Partial Differentials and Relative Differentials}\!</math><br />
|- style="background:ghostwhite; height:50px"<br />
| &nbsp;<br />
| <math>f\!</math><br />
| <math>\frac{\partial f}{\partial x}\!</math><br />
| <math>\frac{\partial f}{\partial y}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}f =<br />
\\[2pt]<br />
\partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\left. \frac{\partial x}{\partial y} \right| f\!</math><br />
| <math>\left. \frac{\partial y}{\partial x} \right| f\!</math><br />
|-<br />
| <math>f_0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A12. Detail of Calculation for the Difference Map====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:4px double black; border-left:4px double black; border-right:4px double black; border-top:4px double black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A12.} ~~ \text{Detail of Calculation for}~ {\mathrm{E}f + f = \mathrm{D}f}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:6%" | &nbsp;<br />
| style="width:14%; border-left:1px solid black" | <math>f\!</math><br />
| style="width:20%; border-left:4px double black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\mathrm{d}x ~ \mathrm{d}y}<br />
\\[4pt]<br />
+ & f|_{\mathrm{d}x ~ \mathrm{d}y}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\mathrm{d}x ~ \mathrm{d}y}<br />
\end{array}</math><br />
| style="width:20%; border-left:1px solid black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}<br />
\\[4pt]<br />
+ & f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}<br />
\end{array}</math><br />
| style="width:20%; border-left:1px solid black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
+ & f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}<br />
\end{array}</math><br />
| style="width:20%; border-left:1px solid black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
+ & f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}<br />
\end{array}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{0}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:4px double black; border-left:4px double black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{1}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{)(} y \texttt{)~}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{2}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{)~} y \texttt{~~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{4}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~~} x \texttt{~(} y \texttt{)~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{8}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~~} x \texttt{~~} y \texttt{~~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{3}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{(} x \texttt{)}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{12}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>x\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{6}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{,~} y \texttt{)~}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{9}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} x \texttt{,~} y \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{5}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{(} y \texttt{)}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{10}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>y\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{7}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{~~} y \texttt{)~}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{11}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{~(} y \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{13}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} x \texttt{)~} y \texttt{)~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{14}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} x \texttt{)(} y \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{15}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:4px double black; border-left:4px double black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Appendix 3. Computational Details===<br />
<br />
====Operator Maps for the Logical Conjunction ''f''<sub>8</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{8}~\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{8}<br />
& = && f_{8}(u, v)<br />
\\[4pt]<br />
& = && uv<br />
\\[4pt]<br />
& = && uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & uv \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & uv \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{8}<br />
& = && uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & uv \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & uv \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & uv \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.2-i} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{E}f_{8}<br />
& = & f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{8}(\mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{8}(\mathrm{d}u, \mathrm{d}v)<br />
\\[4pt]<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[20pt]<br />
\mathrm{E}f_{8}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
\\[4pt]<br />
&&&&& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&&&&&& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.2-ii} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{c}}<br />
\mathrm{E}f_{8}<br />
& = & (u + \mathrm{d}u) \cdot (v + \mathrm{d}v)<br />
\\[6pt]<br />
& = & u \cdot v<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}f_{8}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{8}<br />
& = && \mathrm{E}f_{8}<br />
& + & \boldsymbol\varepsilon f_{8}<br />
\\[4pt]<br />
& = && f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{8}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & uv<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{~}<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}f_{8}<br />
& = & \boldsymbol\varepsilon f_{8}<br />
& + & \mathrm{E}f_{8}<br />
\\[6pt]<br />
& = & f_{8}(u, v)<br />
& + & f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[6pt]<br />
& = & uv<br />
& + & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
& = & 0<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}f_{8}<br />
& = & 0<br />
& + & u \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.3-iii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 3)}\!</math><br />
|<br />
<math>\begin{array}{c*{9}{l}}<br />
\mathrm{D}f_{8}<br />
& = & \boldsymbol\varepsilon f_{8} ~+~ \mathrm{E}f_{8}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{8}<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ u \,\cdot\, v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~ u \;\cdot\; v \;\cdot\; \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}f_{8}<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u ~ \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} ~ v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot\, \mathrm{d}u ~ \mathrm{d}v<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = & ~ ~ 0 ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~ ~ u ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ ~ ~ v ~~ \cdot ~ \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
=====Computation of d''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.4} ~~ \text{Computation of}~ \mathrm{d}f_{8}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{8}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.5} ~~ \text{Computation of}~ \mathrm{r}f_{8}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{8} & = & \mathrm{D}f_{8} ~+~ \mathrm{d}f_{8}<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[20pt]<br />
\mathrm{r}f_{8}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Conjunction=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.6} ~~ \text{Computation Summary for}~ f_{8}(u, v) = uv\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{8}<br />
& = & uv \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}f_{8}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{r}f_{8}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Operator Maps for the Logical Equality ''f''<sub>9</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{9}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{9}<br />
& = && f_{9}(u, v)<br />
\\[4pt]<br />
& = && \texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{9}(1, 1)<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{9}(1, 0)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{9}(0, 1)<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{9}(0, 0)<br />
\\[4pt]<br />
& = && u v & + & 0 & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{9}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.2} ~~ \text{Computation of}~ \mathrm{E}f_{9}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{E}f_{9}<br />
& = && f_{9}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! f_{9}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! f_{9}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{ } \mathrm{d}v \texttt{ })<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! f_{9}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! f_{9}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{ } \mathrm{d}v \texttt{ })<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{ (} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{) }<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{ (} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{) }<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{E}f_{9}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{9}<br />
& = && \mathrm{E}f_{9}<br />
& + & \boldsymbol\varepsilon f_{9}<br />
\\[4pt]<br />
& = && f_{9}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{9}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{9}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[20pt]<br />
\mathrm{D}f_{9}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}f_{9}<br />
& = & 0 \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & 1 \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & 1 \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & 0 \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of d''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.4} ~~ \text{Computation of}~ \mathrm{d}f_{9}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.5} ~~ \text{Computation of}~ \mathrm{r}f_{9}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{9} & = & \mathrm{D}f_{9} ~+~ \mathrm{d}f_{9}<br />
\\[20pt]<br />
\mathrm{D}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[20pt]<br />
\mathrm{r}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Equality=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.6} ~~ \text{Computation Summary for}~ f_{9}(u, v) = \texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{9}<br />
& = & uv \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}f_{9}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f_{9}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{9}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}f_{9}<br />
& = & uv \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Operator Maps for the Logical Implication ''f''<sub>11</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{11}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{11}<br />
& = && f_{11}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(} u \texttt{(} v \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{11}(1, 1)<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{11}(1, 0)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{11}(0, 1)<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{11}(0, 0)<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ }<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ }<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{11}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.2} ~~ \text{Computation of}~ \mathrm{E}f_{11}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{E}f_{11}<br />
& = && f_{11}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = &&<br />
\texttt{(}<br />
\\<br />
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\\<br />
&&& \texttt{(}<br />
\\<br />
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\<br />
&&& \texttt{))}<br />
\\[4pt]<br />
& = &&<br />
u v<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)((} \mathrm{d}v \texttt{)))}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u \texttt{((} \mathrm{d}v \texttt{)))}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[4pt]<br />
& = &&<br />
u v<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{E}f_{11}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{11}<br />
& = && \mathrm{E}f_{11}<br />
& + & \boldsymbol\varepsilon f_{11}<br />
\\[4pt]<br />
& = && f_{11}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{11}(u, v)<br />
\\[4pt]<br />
& = &&<br />
\texttt{(} \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\texttt{(} \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{(} v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & u v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & 0<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & 0<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = && u v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{c*{9}{l}}<br />
\mathrm{D}f_{11}<br />
& = & \boldsymbol\varepsilon f_{11} ~+~ \mathrm{E}f_{11}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{11}<br />
& = & u v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}f_{11}<br />
& = &<br />
u v<br />
\cdot<br />
\texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\cdot<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\cdot<br />
\texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\cdot<br />
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = &<br />
u v<br />
\cdot<br />
\texttt{~(} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{~}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\cdot<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\cdot<br />
\texttt{~} \mathrm{d}u ~ \mathrm{d}v \texttt{~}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\cdot<br />
\texttt{~} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of d''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.4} ~~ \text{Computation of}~ \mathrm{d}f_{11}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{11}<br />
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{11}<br />
& = & u v \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.5} ~~ \text{Computation of}~ \mathrm{r}f_{11}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{11} & = & \mathrm{D}f_{11} ~+~ \mathrm{d}f_{11}<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{11}<br />
& = & u v \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u<br />
\\[20pt]<br />
\mathrm{r}f_{11}<br />
& = & u v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Implication=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.6} ~~ \text{Computation Summary for}~ f_{11}(u, v) = \texttt{(} u \texttt{(} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{11}<br />
& = & u v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}f_{11}<br />
& = & u v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f_{11}<br />
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{11}<br />
& = & u v \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u<br />
\\[6pt]<br />
\mathrm{r}f_{11}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Operator Maps for the Logical Disjunction ''f''<sub>14</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{14}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{14}<br />
& = && f_{14}(u, v)<br />
\\[4pt]<br />
& = && \texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{14}(1, 1)<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{14}(1, 0)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{14}(0, 1)<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{14}(0, 0)<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ }<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ }<br />
& + & 0<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{14}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.2} ~~ \text{Computation of}~ \mathrm{E}f_{14}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{E}f_{14}<br />
& = && f_{14}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = &&<br />
\texttt{((}<br />
\\<br />
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\\<br />
&&& \texttt{)(}<br />
\\<br />
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\<br />
&&& \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! f_{14}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! f_{14}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{ } \mathrm{d}v \texttt{ })<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! f_{14}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! f_{14}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{ } \mathrm{d}v \texttt{ })<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{E}f_{14}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{14}<br />
& = && \mathrm{E}f_{14}<br />
& + & \boldsymbol\varepsilon f_{14}<br />
\\[4pt]<br />
& = && f_{14}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{14}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{))((} v \texttt{,} \mathrm{d}v \texttt{)))}<br />
& + & \texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{14}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
\\[4pt]<br />
&& + & 0<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
\\[4pt]<br />
&& + & uv \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
\\[20pt]<br />
\mathrm{D}f_{14}<br />
& = && uv \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}f_{14}<br />
& = & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & 0 \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of d''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.4} ~~ \text{Computation of}~ \mathrm{d}f_{14}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.5} ~~ \text{Computation of}~ \mathrm{r}f_{14}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{14} & = & \mathrm{D}f_{14} ~+~ \mathrm{d}f_{14}<br />
\\[20pt]<br />
\mathrm{D}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{d}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[20pt]<br />
\mathrm{r}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Disjunction=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.6} ~~ \text{Computation Summary for}~ f_{14}(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{14}<br />
& = & uv \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 1<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}f_{14}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f_{14}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{d}f_{14}<br />
& = & uv \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}f_{14}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
===Appendix 4. Source Materials===<br />
<br />
===Appendix 5. Various Definitions of the Tangent Vector===<br />
<br />
==References==<br />
<br />
===Works Cited===<br />
<br />
{| cellpadding=3<br />
| valign=top | [AuM]<br />
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|-<br />
| valign=top | [BiG]<br />
| Bishop, R.L., and Goldberg, S.I., ''Tensor Analysis on Manifolds'', Macmillan, 1968. Reprinted, Dover, New York, NY, 1980.<br />
|-<br />
| valign=top | [Boo]<br />
| Boole, G., ''An Investigation of The Laws of Thought'', Macmillan, 1854. Reprinted, Dover, New York, NY, 1958.<br />
|-<br />
| valign=top | [BoT]<br />
| Bott, R., and Tu, L.W., ''Differential Forms in Algebraic Topology'', Springer-Verlag, New York, NY, 1982.<br />
|-<br />
| valign=top | [dCa]<br />
| do Carmo, M.P., ''Riemannian Geometry''. Originally published in Portuguese, 1st editiom 1979, 2nd edition 1988. Translated by F. Flaherty, Birkhäuser, Boston, MA, 1992.<br />
|-<br />
| valign=top | [Che46]<br />
| Chevalley, C., ''Theory of Lie Groups'', Princeton University Press, Princeton, NJ, 1946.<br />
|-<br />
| valign=top | [Che56]<br />
| Chevalley, C., ''Fundamental Concepts of Algebra'', Academic Press, 1956.<br />
|-<br />
| valign=top | [Cho86]<br />
| Chomsky, N., ''Knowledge of Language : Its Nature, Origin, and Use'', Praeger, New York, NY, 1986.<br />
|-<br />
| valign=top | [Cho93]<br />
| Chomsky, N., ''Language and Thought'', Moyer Bell, Wakefield, RI, 1993.<br />
|-<br />
| valign=top | [DoM]<br />
| Doolin, B.F., and Martin, C.F., ''Introduction to Differential Geometry for Engineers'', Marcel Dekker, New York, NY, 1990.<br />
|-<br />
| valign=top | [Fuji]<br />
| Fujiwara, H., ''Logic Testing and Design for Testability'', MIT Press, Cambridge, MA, 1985.<br />
|-<br />
| valign=top | [Hic]<br />
| Hicks, N.J., ''Notes on Differential Geometry'', Van Nostrand, Princeton, NJ, 1965.<br />
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| valign=top | [Hir]<br />
| Hirsch, M.W., ''Differential Topology'', Springer-Verlag, New York, NY, 1976.<br />
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| valign=top | [How]<br />
| Howard, W.A., "The Formulae-as-Types Notion of Construction", Notes circulated from 1969. Reprinted in [SeH, 479-490].<br />
|-<br />
| valign=top | [JGH]<br />
| Jones, A., Gray, A., and Hutton, R., ''Manifolds and Mechanics'', Cambridge University Press, Cambridge, UK, 1987.<br />
|-<br />
| valign=top | [KoA]<br />
| Kosinski, A.A., ''Differential Manifolds'', Academic Press, San Diego, CA, 1993.<br />
|-<br />
| valign=top | [Koh]<br />
| Kohavi, Z., ''Switching and Finite Automata Theory'', 2nd edition, McGraw-Hill, New York, NY, 1978.<br />
|-<br />
| valign=top | [LaS]<br />
| Lambek, J., and Scott, P.J., ''Introduction to Higher Order Categorical Logic'', Cambridge University Press, Cambridge, UK, 1986.<br />
|-<br />
| valign=top | [La83]<br />
| Lang, S., ''Real Analysis'', 2nd edition, Addison-Wesley, Reading, MA, 1983.<br />
|-<br />
| valign=top | [La84]<br />
| Lang, S., ''Algebra'', 2nd edition, Addison-Wesley, Menlo Park, CA, 1984.<br />
|-<br />
| valign=top | [La85]<br />
| Lang, S., ''Differential Manifolds'', Springer-Verlag, New York, NY, 1985.<br />
|-<br />
| valign=top | [La93]<br />
| Lang, S., ''Real and Functional Analysis'', 3rd edition, Springer-Verlag, New York, NY, 1993.<br />
|-<br />
| valign=top | [Lie80]<br />
| Lie, S., "Sophus Lie's 1880 Transformation Group Paper", in ''Lie Groups : History, Frontiers, and Applications, Volume 1'', translated by M. Ackerman, comments by R. Hermann, Math Sci Press, Brookline, MA, 1975. Original paper 1880.<br />
|-<br />
| valign=top | [Lie84]<br />
| Lie, S., "Sophus Lie's 1884 Differential Invariant Paper", in ''Lie Groups : History, Frontiers, and Applications, Volume 3'', translated by M. Ackerman, comments by R. Hermann, Math Sci Press, Brookline, MA, 1976. Original paper 1884.<br />
|-<br />
| valign=top | [LoS]<br />
| Loomis, L.H., and Sternberg, S., ''Advanced Calculus'', Addison-Wesley, Reading, MA, 1968.<br />
|-<br />
| valign=top | [Mel]<br />
| Melzak, Z.A., ''Companion to Concrete Mathematics, Volume 2 : Mathematical Ideas, Modeling, and Applications'', John Wiley amd Sons, New York, NY, 1976.<br />
|-<br />
| valign=top | [Men]<br />
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|-<br />
| valign=top | [M&M]<br />
| Morrison, P., and Morrison, E. (eds.), ''Charles Babbage on the Principles and Development of the Calculator, and Other Seminal Writings by Charles Babbage and Others, With an Introduction by the Editors'', Dover, Mineola, NY, 1961.<br />
|-<br />
| valign=top | [P1]<br />
| Peirce, C.S., ''Collected Papers of Charles Sanders Peirce'', vols. 1–8, C. Hartshorne, P. Weiss, and A.W. Burks (eds.), Harvard University Press, Cambridge, MA, 1931–1960. Cited as CP [volume].[paragraph].<br />
|-<br />
| valign=top | [P2]<br />
| Peirce, C.S., "Qualitative Logic", in ''The New Elements of Mathematics, Volume 4'', C. Eisele (ed.), Mouton, The Hague, 1976. Cited as NE [volume], [page].<br />
|-<br />
| valign=top | [Rob]<br />
| Roberts, D.D., ''The Existential Graphs of Charles S. Peirce'', Mouton, The Hague, 1973.<br />
|-<br />
| valign=top | [SeH]<br />
| Seldin, J.P., and Hindley, J.R. (eds.), ''To H.B. Curry : Essays on Combinatory Logic, Lambda Calculus, and Formalism'', Academic Press, London, UK, 1980.<br />
|-<br />
| valign=top | [SpB]<br />
| Spencer-Brown, G., ''Laws of Form'', George Allen and Unwin, London, UK, 1969.<br />
|-<br />
| valign=top | [Sp65]<br />
| Spivak, M., ''Calculus on Manifolds : A Modern Approach to Classical Theorems of Advanced Calculus'', W.A. Benjamin, New York, NY, 1965.<br />
|-<br />
| valign=top | [Sp79]<br />
| Spivak, M., ''A Comprehensive Introduction to Differential Geometry'', vols. 1–2. 1st edition 1970. 2nd edition, Publish or Perish Inc., Houston, TX, 1979.<br />
|-<br />
| valign=top | [Sty]<br />
| Styazhkin, N.I., ''History of Mathematical Logic from Leibniz to Peano'', 1st published in Russian, Nauka, Moscow, 1964. MIT Press, Cambridge, MA, 1969.<br />
|-<br />
| valign=top | [Wie]<br />
| Wiener, N., ''Cybernetics : or Control and Communication in the Animal and the Machine'', 1st edition 1948. 2nd edition, MIT Press, Cambridge, MA, 1961.<br />
|}<br />
<br />
===Works Consulted===<br />
<br />
{| cellpadding=3<br />
| valign=top | [Ami]<br />
| Amit, D.J., ''Modeling Brain Function : The World of Attractor Neural Networks'', Cambridge University Press, Cambridge, UK, 1989.<br />
|-<br />
| valign=top | [Ed87]<br />
| Edelman, G.M., ''Neural Darwinism : The Theory of Neuronal Group Selection'', Basic Books, New York, NY, 1987.<br />
|-<br />
| valign=top | [Ed88]<br />
| Edelman, G.M., ''Topobiology : An Introduction to Molecular Embryology'', Basic Books, New York, NY, 1988.<br />
|-<br />
| valign=top | [Fla]<br />
| Flanders, H., ''Differential Forms with Applications to the Physical Sciences'', Academic Press, 1963. Reprinted, Dover, Mineola, NY, 1989. <br />
|-<br />
| valign=top | [Has]<br />
| Hassoun, M.H. (ed.), ''Associative Neural Memories : Theory and Implementation'', Oxford University Press, New York, NY, 1993.<br />
|-<br />
| valign=top | [KoB]<br />
| Kosko, B., ''Neural Networks and Fuzzy Systems : A Dynamical Systems Approach to Machine Intelligence'', Prentice-Hall, Englewood Cliffs, NJ, 1992.<br />
|-<br />
| valign=top | [MaB]<br />
| Mac Lane, S., and Birkhoff, G., ''Algebra'', 3rd edition, Chelsea, New York, NY, 1993.<br />
|-<br />
| valign=top | [Mac]<br />
| Mac Lane, S., ''Categories for the Working Mathematician'', Springer-Verlag, New York, NY, 1971.<br />
|-<br />
| valign=top | [McC]<br />
| McCulloch, W.S., ''Embodiments of Mind'', MIT Press, Cambridge, MA, 1965.<br />
|-<br />
| valign=top | [Mc1]<br />
| McCulloch, W.S., "A Heterarchy of Values Determined by the Topology of Nervous Nets", Bulletin of Mathematical Biophysics, vol. 7 (1945), pp. 89–93. Reprinted in [McC].<br />
|-<br />
| valign=top | [MiP]<br />
| Minsky, M.L., and Papert, S.A., ''Perceptrons : An Introduction to Computational Geometry'', MIT Press, Cambridge, MA, 1969. 2nd printing 1972. Expanded edition 1988.<br />
|-<br />
| valign=top | [Rum]<br />
| Rumelhart, D.E., Hinton, G.E., and McClelland, J.L., "A General Framework for Parallel Distributed Processing" = Chapter 2 in Rumelhart, McClelland, and the PDP Research Group, ''Parallel Distributed Processing, Explorations in the Microstructure of Cognition, Volume 1 : Foundations'', MIT Press, Cambridge, MA, 1986.<br />
|}<br />
<br />
===Incidental Works===<br />
<br />
{| cellpadding=3<br />
| valign=top | [Dew]<br />
| Dewey, John, ''How We Think'', D.C. Heath, Lexington, MA, 1910. Reprinted, Prometheus Books, Buffalo, NY, 1991.<br />
|-<br />
| valign=top | [Fou]<br />
| Foucault, Michel, ''The Archaeology of Knowledge and The Discourse on Language'', A.M. Sheridan-Smith and Rupert Swyer (trans.), Pantheon, New York, NY, 1972. Originally published as ''L´Archéologie du Savoir et L´ordre du discours'', Editions Gallimard, 1969 & 1971.<br />
|-<br />
| valign=top | [Hom]<br />
| Homer, ''The Odyssey'', with an English translation by A.T. Murray, Loeb Classical Library, Harvard University Press, Cambridge, MA, 1980. First printed 1919.<br />
|-<br />
| valign=top | [Jam]<br />
| James, William, ''Pragmatism : A New Name for Some Old Ways of Thinking'', Longmans, Green, and Company, New York, NY, 1907.<br />
|-<br />
| valign=top | [Ler]<br />
| Leroux, Gaston, ''The Phantom of the Opera'', foreword by P. Haining, Dorset Press, New York, NY, 1988. Originally published in French, 1911.<br />
|-<br />
| valign=top | [Mus]<br />
| Musil, Robert, ''The Man Without Qualities'', 3 volumes, translated with a foreword by Eithne Wilkins and Ernst Kaiser, Pan Books, London, UK, 1979. English edition first published by Secker and Warburg, 1954. Originally published in German, ''Der Mann ohne Eigenschaften'', 1930 & 1932.<br />
|-<br />
| valign=top | [PlaR]<br />
| Plato, ''The Republic'', with an English translation by Paul Shorey, Loeb Classical Library, Harvard University Press, Cambridge, MA, 1980. First printed 1930 & 1935.<br />
|-<br />
| valign=top | [PlaS]<br />
| Plato, ''The Sophist'', Loeb Classical Library, William Heinemann, London, 1921, 1987.<br />
|-<br />
| valign=top | [Qui]<br />
| Quine, W.V., ''Mathematical Logic'', 1st edition, 1940. Revised edition, 1951. Harvard University Press, Cambridge, MA, 1981.<br />
|-<br />
| valign=top | [SaD]<br />
| de Santillana, Giorgio, and von Dechend, Hertha, ''Hamlet's Mill : An Essay on Myth and the Frame of Time'', David R. Godine, Publisher, Boston, MA, 1977. 1st published 1969.<br />
|-<br />
| valign=top | [Sha]<br />
| Shakespeare, William, '' William Shakespeare : The Complete Works'', Compact Edition, S. Wells and G. Taylor (eds.), Oxford University Press, Oxford, UK, 1988.<br />
|-<br />
| valign=top | [Sh1]<br />
| Shakespeare, William, ''A Midsummer Night's Dream'', Washington Square Press, New York, NY, 1958.<br />
|-<br />
| valign=top | [Sh2]<br />
| Shakespeare, William, ''The Tragedy of Hamlet, Prince of Denmark'', In [Sha], pp. 654&ndash;690.<br />
|-<br />
| valign=top | [Sh3]<br />
| Shakespeare, William, ''Measure for Measure'', Washington Square Press, New York, NY, 1965.<br />
|-<br />
| valign=top | [Web]<br />
| ''Webster's Ninth New Collegiate Dictionary'', Merriam-Webster, Springfield, MA, 1983.<br />
|-<br />
| valign=top | [Whi]<br />
| Whitman, Walt, ''Leaves of Grass'', Vintage Books / The Library of America, New York, NY, 1992. Originally published in numerous editions, 1855&ndash;1892.<br />
|-<br />
| valign=top | [Wil]<br />
| Wilhelm, R., and Baynes, C.F. (trans.), ''The I Ching, or Book of Changes'', foreword by C.G. Jung, preface by H. Wilhelm, 3rd edition, Bollingen Series XIX, Princeton University Press, Princeton, NJ, 1967.<br />
|}<br />
<br />
==Document History==<br />
<br />
<pre><br />
Author: Jon Awbrey<br />
Created: 16 Dec 1993<br />
Relayed: 31 Oct 1994<br />
Revised: 03 Jun 2003<br />
Recoded: 03 Jun 2007<br />
</pre><br />
<br />
[[Category:Adaptive Systems]]<br />
[[Category:Artificial Intelligence]]<br />
[[Category:Boolean Algebra]]<br />
[[Category:Boolean Functions]]<br />
[[Category:Charles Sanders Peirce]]<br />
[[Category:Combinatorics]]<br />
[[Category:Computer Science]]<br />
[[Category:Cybernetics]]<br />
[[Category:Differential Logic]]<br />
[[Category:Discrete Systems]]<br />
[[Category:Dynamical Systems]]<br />
[[Category:Formal Languages]]<br />
[[Category:Formal Sciences]]<br />
[[Category:Formal Systems]]<br />
[[Category:Functional Logic]]<br />
[[Category:Graph Theory]]<br />
[[Category:Group Theory]]<br />
[[Category:Inquiry]]<br />
[[Category:Knowledge Representation]]<br />
[[Category:Linguistics]]<br />
[[Category:Logic]]<br />
[[Category:Logical Graphs]]<br />
[[Category:Mathematics]]<br />
[[Category:Mathematical Systems Theory]]<br />
[[Category:Philosophy]]<br />
[[Category:Science]]<br />
[[Category:Semiotics]]<br />
[[Category:Systems Science]]<br />
[[Category:Visualization]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Differential_Logic_and_Dynamic_Systems&diff=469885Differential Logic and Dynamic Systems2021-01-14T21:28:47Z<p>Jon Awbrey: /* Review and Transition */ update</p>
<hr />
<div>'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''<br />
<br />
{| align="center" cellpadding="10"<br />
| [[File:Tangent Functor Ferris Wheel.jpg]]<br />
|}<br />
<br />
{| style="height:36px; width:100%"<br />
| align="left" | ''Stand and unfold yourself.''<br />
| align="right" | Hamlet: Francsico&mdash;1.1.2<br />
|}<br />
<br />
This article develops a differential extension of propositional calculus and applies it to a context of problems arising in dynamic systems.&nbsp; The work pursued here is coordinated with a parallel application that focuses on neural network systems, but the dependencies are arranged to make the present article the main and the more self-contained work, to serve as a conceptual frame and a technical background for the network project.<br />
<br />
==Review and Transition==<br />
<br />
This note continues a previous discussion on the problem of dealing with change and diversity in logic-based intelligent systems. It is useful to begin by summarizing essential material from previous reports.<br />
<br />
Table 1 outlines a notation for propositional calculus based on two types of logical connectives, both of variable <math>k</math>-ary scope.<br />
<br />
* A bracketed list of propositional expressions in the form <math>\texttt{(} e_1, e_2, \ldots, e_{k-1}, e_k \texttt{)}</math> indicates that exactly one of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> is false.<br />
<br />
* A concatenation of propositional expressions in the form <math>e_1 ~ e_2 ~ \ldots ~ e_{k-1} ~ e_k</math> indicates that all of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> are true, in other words, that their [[logical conjunction]] is true.<br />
<br />
All other propositional connectives can be obtained in a very efficient style of representation through combinations of these two forms. Strictly speaking, the concatenation form is dispensable in light of the bracketed form, but it is convenient to maintain it as an abbreviation of more complicated bracket expressions.<br />
<br />
This treatment of propositional logic is derived from the work of C.S. Peirce [P1, P2], who gave this approach an extensive development in his graphical systems of predicate, relational, and modal logic [Rob]. More recently, these ideas were revived and supplemented in an alternative interpretation by George Spencer-Brown [SpB]. Both of these authors used other forms of enclosure where I use parentheses, but the structural topologies of expression and the functional varieties of interpretation are fundamentally the same.<br />
<br />
While working with expressions solely in propositional calculus, it is easiest to use plain parentheses for logical connectives. In contexts where parentheses are needed for other purposes &ldquo;teletype&rdquo; parentheses <math>\texttt{(} \ldots \texttt{)}</math> or barred parentheses <math>(\!| \ldots |\!)</math> may be used for logical operators.<br />
<br />
The briefest expression for logical truth is the empty word, usually denoted by <math>{}^{\backprime\backprime} \boldsymbol\varepsilon {}^{\prime\prime}</math> or <math>{}^{\backprime\backprime} \boldsymbol\lambda {}^{\prime\prime}</math> in formal languages, where it forms the identity element for concatenation. To make it visible in this text, it may be denoted by the equivalent expression <math>{}^{\backprime\backprime} \texttt{((} ~ \texttt{))} {}^{\prime\prime},</math> or, especially if operating in an algebraic context, by a simple <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}.</math> Also when working in an algebraic mode, the plus sign <math>{}^{\backprime\backprime} + {}^{\prime\prime}</math> may be used for [[exclusive disjunction]]. For example, we have the following paraphrases of algebraic expressions by bracket expressions:<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
|<br />
<math>\begin{matrix}<br />
x + y ~=~ \texttt{(} x, y \texttt{)}<br />
\\[6pt]<br />
x + y + z ~=~ \texttt{((} x, y \texttt{)}, z \texttt{)} ~=~ \texttt{(} x, \texttt{(} y, z \texttt{))}<br />
\end{matrix}</math><br />
|}<br />
<br />
It is important to note that the last expressions are not equivalent to the triple bracket <math>\texttt{(} x, y, z \texttt{)}.</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 1.} ~~ \text{Syntax and Semantics of a Calculus for Propositional Logic}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Expression}</math><br />
| <math>\text{Interpretation}</math><br />
| <math>\text{Other Notations}</math><br />
|-<br />
| &nbsp;<br />
| <math>\text{True}</math><br />
| <math>1</math><br />
|-<br />
| <math>\texttt{(} ~ \texttt{)}</math><br />
| <math>\text{False}</math><br />
| <math>0</math><br />
|-<br />
| <math>x</math><br />
| <math>x</math><br />
| <math>x</math><br />
|-<br />
| <math>\texttt{(} x \texttt{)}</math><br />
| <math>\text{Not}~ x</math><br />
|<br />
<math>\begin{matrix}<br />
x'<br />
\\<br />
\tilde{x}<br />
\\<br />
\lnot x<br />
\end{matrix}</math><br />
|-<br />
| <math>x~y~z</math><br />
| <math>x ~\text{and}~ y ~\text{and}~ z</math><br />
| <math>x \land y \land z</math><br />
|-<br />
| <math>\texttt{((} x \texttt{)(} y \texttt{)(} z \texttt{))}</math><br />
| <math>x ~\text{or}~ y ~\text{or}~ z</math><br />
| <math>x \lor y \lor z</math><br />
|-<br />
| <math>\texttt{(} x ~ \texttt{(} y \texttt{))}</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{implies}~ y<br />
\\<br />
\mathrm{If}~ x ~\text{then}~ y<br />
\end{matrix}</math><br />
| <math>x \Rightarrow y</math><br />
|-<br />
| <math>\texttt{(} x \texttt{,} y \texttt{)}</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{not equal to}~ y<br />
\\<br />
x ~\text{exclusive or}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x \ne y<br />
\\<br />
x + y<br />
\end{matrix}</math><br />
|-<br />
| <math>\texttt{((} x \texttt{,} y \texttt{))}</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{is equal to}~ y<br />
\\<br />
x ~\text{if and only if}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x = y<br />
\\<br />
x \Leftrightarrow y<br />
\end{matrix}</math><br />
|-<br />
| <math>\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Just one of}<br />
\\<br />
x, y, z<br />
\\<br />
\text{is false}.<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x'y~z~ & \lor<br />
\\<br />
x~y'z~ & \lor<br />
\\<br />
x~y~z' &<br />
\end{matrix}</math><br />
|-<br />
| <math>\texttt{((} x \texttt{),(} y \texttt{),(} z \texttt{))}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Just one of}<br />
\\<br />
x, y, z<br />
\\<br />
\text{is true}.<br />
\\<br />
&<br />
\\<br />
\text{Partition all}<br />
\\<br />
\text{into}~ x, y, z.<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x~y'z' & \lor<br />
\\<br />
x'y~z' & \lor<br />
\\<br />
x'y'z~ &<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,} y \texttt{),} z \texttt{)}<br />
\\<br />
&<br />
\\<br />
\texttt{(} x \texttt{,(} y \texttt{,} z \texttt{))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Oddly many of}<br />
\\<br />
x, y, z<br />
\\<br />
\text{are true}.<br />
\end{matrix}</math><br />
|<br />
<p><math>x + y + z</math></p><br />
<br><br />
<p><math>\begin{matrix}<br />
x~y~z~ & \lor<br />
\\<br />
x~y'z' & \lor<br />
\\<br />
x'y~z' & \lor<br />
\\<br />
x'y'z~ &<br />
\end{matrix}</math></p><br />
|-<br />
| <math>\texttt{(} w \texttt{,(} x \texttt{),(} y \texttt{),(} z \texttt{))}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Partition}~ w<br />
\\<br />
\text{into}~ x, y, z.<br />
\\<br />
&<br />
\\<br />
\text{Genus}~ w ~\text{comprises}<br />
\\<br />
\text{species}~ x, y, z.<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
w'x'y'z' & \lor<br />
\\<br />
w~x~y'z' & \lor<br />
\\<br />
w~x'y~z' & \lor<br />
\\<br />
w~x'y'z~ &<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
'''Note.''' The usage that one often sees, of a plus sign "<math>+</math>" to represent inclusive disjunction, and the reference to this operation as ''boolean addition'', is a misnomer on at least two counts. Boole used the plus sign to represent exclusive disjunction (at any rate, an operation of aggregation restricted in its logical interpretation to cases where the represented sets are disjoint (Boole, 32)), as any mathematician with a sensitivity to the ring and field properties of algebra would do:<br />
<br />
<blockquote><br />
The expression <math>x + y</math> seems indeed uninterpretable, unless it be assumed that the things represented by <math>x</math> and the things represented by <math>y</math> are entirely separate; that they embrace no individuals in common. (Boole, 66).<br />
</blockquote><br />
<br />
It was only later that Peirce and Jevons treated inclusive disjunction as a fundamental operation, but these authors, with a respect for the algebraic properties that were already associated with the plus sign, used a variety of other symbols for inclusive disjunction (Sty, 177, 189). It seems to have been Schröder who later reassigned the plus sign to inclusive disjunction (Sty, 208). Additional information, discussion, and references can be found in (Boole) and (Sty, 177&ndash;263). Aside from these historical points, which never really count against a current practice that has gained a life of its own, this usage does have a further disadvantage of cutting or confounding the lines of communication between algebra and logic. For this reason, it will be avoided here.<br />
<br />
==A Functional Conception of Propositional Calculus==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Out of the dimness opposite equals advance . . . .<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Always substance and increase,<br><br />
Always a knit of identity . . . . always distinction . . . .<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;always a breed of life.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]<br />
|}<br />
<br />
In the general case, we start with a set of logical features <math>\{a_1, \ldots, a_n\}</math> that represent properties of objects or propositions about the world. In concrete examples the features <math>\{a_i\!\}</math> commonly appear as capital letters from an ''alphabet'' like <math>\{A, B, C, \ldots\}</math> or as meaningful words from a linguistic ''vocabulary'' of codes. This language can be drawn from any sources, whether natural, technical, or artificial in character and interpretation. In the application to dynamic systems we tend to use the letters <math>\{x_1, \ldots, x_n\}</math> as our coordinate propositions, and to interpret them as denoting properties of a system's ''state'', that is, as propositions about its location in configuration space. Because I have to consider non-deterministic systems from the outset, I often use the word ''state'' in a loose sense, to denote the position or configuration component of a contemplated state vector, whether or not it ever gets a deterministic completion.<br />
<br />
The set of logical features <math>\{a_1, \ldots, a_n\}</math> provides a basis for generating an <math>n\!</math>-dimensional ''[[universe of discourse]]'' that I denote as <math>[a_1, \ldots, a_n].</math> It is useful to consider each universe of discourse as a unified categorical object that incorporates both the set of points <math>\langle a_1, \ldots, a_n \rangle</math> and the set of propositions <math>f : \langle a_1, \ldots, a_n \rangle \to \mathbb{B}</math> that are implicit with the ordinary picture of a venn diagram on <math>n\!</math> features. Thus, we may regard the universe of discourse <math>[a_1, \ldots, a_n]</math> as an ordered pair having the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}),</math> and we may abbreviate this last type designation as <math>\mathbb{B}^n\ +\!\to \mathbb{B},</math> or even more succinctly as <math>[\mathbb{B}^n].</math> (Used this way, the angle brackets <math>\langle\ldots\rangle</math> are referred to as ''generator brackets''.)<br />
<br />
Table 2 exhibits the scheme of notation I use to formalize the domain of propositional calculus, corresponding to the logical content of truth tables and venn diagrams. Although it overworks the square brackets a bit, I also use either one of the equivalent notations <math>[n]\!</math> or <math>\mathbf{n}</math> to denote the data type of a finite set on <math>n\!</math> elements.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 2.} ~~ \text{Propositional Calculus : Basic Notation}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Symbol}\!</math><br />
| <math>\text{Notation}\!</math><br />
| <math>\text{Description}\!</math><br />
| <math>\text{Type}\!</math><br />
|-<br />
| <math>\mathfrak{A}\!</math><br />
| <math>\{ {}^{\backprime\backprime} a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} a_n {}^{\prime\prime} \}\!</math><br />
| <math>\text{Alphabet}\!</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>\mathcal{A}\!</math><br />
| <math>\{ a_1, \ldots, a_n \}\!</math><br />
| <math>\text{Basis}\!</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>A_i\!</math><br />
| <math>\{ \texttt{(} a_i \texttt{)}, a_i \}\!</math><br />
| <math>\text{Dimension}~ i\!</math><br />
| <math>\mathbb{B}\!</math><br />
|-<br />
| <math>A\!</math><br />
|<br />
<math>\begin{matrix}<br />
\langle \mathcal{A} \rangle<br />
\\[2pt]<br />
\langle a_1, \ldots, a_n \rangle<br />
\\[2pt]<br />
\{ (a_1, \ldots, a_n) \}<br />
\\[2pt]<br />
A_1 \times \ldots \times A_n<br />
\\[2pt]<br />
\textstyle \prod_{i=1}^n A_i<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Set of cells},<br />
\\[2pt]<br />
\text{coordinate tuples},<br />
\\[2pt]<br />
\text{points, or vectors}<br />
\\[2pt]<br />
\text{in the universe}<br />
\\[2pt]<br />
\text{of discourse}<br />
\end{matrix}</math><br />
| <math>\mathbb{B}^n\!</math><br />
|-<br />
| <math>A^*\!</math><br />
| <math>(\mathrm{hom} : A \to \mathbb{B})\!</math><br />
| <math>\text{Linear functions}\!</math><br />
| <math>(\mathbb{B}^n)^* \cong \mathbb{B}^n\!</math><br />
|-<br />
| <math>A^\uparrow\!</math><br />
| <math>(A \to \mathbb{B})\!</math><br />
| <math>\text{Boolean functions}\!</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}\!</math><br />
|-<br />
| <math>A^\bullet\!</math><br />
|<br />
<math>\begin{matrix}<br />
[\mathcal{A}]<br />
\\[2pt]<br />
(A, A^\uparrow)<br />
\\[2pt]<br />
(A ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
(A, (A \to \mathbb{B}))<br />
\\[2pt]<br />
[a_1, \ldots, a_n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Universe of discourse}<br />
\\[2pt]<br />
\text{based on the features}<br />
\\[2pt]<br />
\{ a_1, \ldots, a_n \}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))<br />
\\[2pt]<br />
(\mathbb{B}^n ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
[\mathbb{B}^n]<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
===Qualitative Logic and Quantitative Analogy===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
''Logical'', however, is used in a third sense, which is at once more vital and more practical; to denote, namely, the systematic care, negative and positive, taken to safeguard reflection so that it may yield the best results under the given conditions.<br />
| width="4%" | &nbsp;<br />
|- <br />
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 56]<br />
|}<br />
<br />
These concepts and notations may now be explained in greater detail. In order to begin as simply as possible, let us distinguish two levels of analysis and set out initially on the easier path. On the first level of analysis we take spaces like <math>\mathbb{B},</math> <math>\mathbb{B}^n,</math> and <math>(\mathbb{B}^n \to \mathbb{B})</math> at face value and treat them as the primary objects of interest. On the second level of analysis we use these spaces as coordinate charts for talking about points and functions in more fundamental spaces.<br />
<br />
A pair of spaces, of types <math>\mathbb{B}^n</math> and <math>(\mathbb{B}^n \to \mathbb{B}),</math> give typical expression to everything that we commonly associate with the ordinary picture of a venn diagram. The dimension, <math>n,\!</math> counts the number of "circles" or simple closed curves that are inscribed in the universe of discourse, corresponding to its relevant logical features or basic propositions. Elements of type <math>\mathbb{B}^n</math> correspond to what are often called propositional ''interpretations'' in logic, that is, the different assignments of truth values to sentence letters. Relative to a given universe of discourse, these interpretations are visualized as its ''cells'', in other words, the smallest enclosed areas or undivided regions of the venn diagram. The functions <math>f : \mathbb{B}^n \to \mathbb{B}</math> correspond to the different ways of shading the venn diagram to indicate arbitrary propositions, regions, or sets. Regions included under a shading indicate the ''models'', and regions excluded represent the ''non-models'' of a proposition. To recognize and formalize the natural cohesion of these two layers of concepts into a single universe of discourse, we introduce the type notations <math>[\mathbb{B}^n] = \mathbb{B}^n\ +\!\to \mathbb{B}</math> to stand for the pair of types <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})).</math> The resulting "stereotype" serves to frame the universe of discourse as a unified categorical object, and makes it subject to prescribed sets of evaluations and transformations (categorical morphisms or ''arrows'') that affect the universe of discourse as an integrated whole.<br />
<br />
Most of the time we can serve the algebraic, geometric, and logical interests of our study without worrying about their occasional conflicts and incidental divergences. The conventions and definitions already set down will continue to cover most of the algebraic and functional aspects of our discussion, but to handle the logical and qualitative aspects we will need to add a few more. In general, abstract sets may be denoted by gothic, greek, or script capital variants of <math>A, B, C,\!</math> and so on, with their elements being denoted by a corresponding set of subscripted letters in plain lower case, for example, <math>\mathcal{A} = \{a_i\}.</math> Most of the time, a set such as <math>\mathcal{A} = \{a_i\}\!</math> will be employed as the ''alphabet'' of a [[formal language]]. These alphabet letters serve to name the logical features (properties or propositions) that generate a particular universe of discourse. When we want to discuss the particular features of a universe of discourse, beyond the abstract designation of a type like <math>(\mathbb{B}^n\ +\!\to \mathbb{B}),</math> then we may use the following notations. If <math>\mathcal{A} = \{a_1, \ldots, a_n\}</math> is an alphabet of logical features, then <math>A = \langle \mathcal{A} \rangle = \langle a_1, \ldots, a_n \rangle</math> is the set of interpretations, <math>A^\uparrow = (A \to \mathbb{B})</math> is the set of propositions, and <math>A^\bullet = [\mathcal{A}] = [a_1, \ldots, a_n]</math> is the combination of these interpretations and propositions into the universe of discourse that is based on the features <math>\{a_1, \ldots, a_n\}.</math><br />
<br />
As always, especially in concrete examples, these rules may be dropped whenever necessary, reverting to a free assortment of feature labels. However, when we need to talk about the logical aspects of a space that is already named as a vector space, it will be necessary to make special provisions. At any rate, these elaborations can be deferred until actually needed.<br />
<br />
===Philosophy of Notation : Formal Terms and Flexible Types===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Where number is irrelevant, regimented mathematical technique has hitherto tended to be lacking. Thus it is that the progress of natural science has depended so largely upon the discernment of measurable quantity of one sort or another.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]<br />
|}<br />
<br />
For much of our discussion propositions and boolean functions are treated as the same formal objects, or as different interpretations of the same formal calculus. This rule of interpretation has exceptions, though. There is a distinctively logical interest in the use of propositional calculus that is not exhausted by its functional interpretation. It is part of our task in this study to deal with these uniquely logical characteristics as they present themselves both in our subject matter and in our formal calculus. Just to provide a hint of what's at stake: In logic, as opposed to the more imaginative realms of mathematics, we consider it a good thing to always know what we are talking about. Where mathematics encourages tolerance for uninterpreted symbols as intermediate terms, logic exerts a keener effort to interpret directly each oblique carrier of meaning, no matter how slight, and to unfold the complicities of every indirection in the flow of information. Translated into functional terms, this means that we want to maintain a continual, immediate, and persistent sense of both the converse relation <math>f^{-1} \subseteq \mathbb{B} \times \mathbb{B}^n,</math> or what is the same thing, <math>f^{-1} : \mathbb{B} \to \mathcal{P}(\mathbb{B}^n),</math> and the ''fibers'' or inverse images <math>f^{-1}(0)\!</math> and <math>f^{-1}(1),\!</math> associated with each boolean function <math>f : \mathbb{B}^n \to \mathbb{B}</math> that we use. In practical terms, the desired implementation of a propositional interpreter should incorporate our intuitive recognition that the induced partition of the functional domain into level sets <math>f^{-1}(b),\!</math> for <math>b \in \mathbb{B},</math> is part and parcel of understanding the denotative uses of each propositional function <math>f.\!</math><br />
<br />
===Special Classes of Propositions===<br />
<br />
It is important to remember that the coordinate propositions <math>\{a_i\},\!</math> besides being projection maps <math>a_i : \mathbb{B}^n \to \mathbb{B},</math> are propositions on an equal footing with all others, even though employed as a basis in a particular moment. This set of <math>n\!</math> propositions may sometimes be referred to as the ''basic propositions'', the ''coordinate propositions'', or the ''simple propositions'' that found a universe of discourse. Either one of the equivalent notations, <math>\{a_i : \mathbb{B}^n \to \mathbb{B}\}</math> or <math>(\mathbb{B}^n \xrightarrow{i} \mathbb{B}),</math> may be used to indicate the adoption of the propositions <math>a_i\!</math> as a basis for describing a universe of discourse.<br />
<br />
Among the <math>2^{2^n}</math> propositions in <math>[a_1, \ldots, a_n]</math> are several families of <math>2^n\!</math> propositions each that take on special forms with respect to the basis <math>\{ a_1, \ldots, a_n \}.</math> Three of these families are especially prominent in the present context, the ''linear'', the ''positive'', and the ''singular'' propositions. Each family is naturally parameterized by the coordinate <math>n\!</math>-tuples in <math>\mathbb{B}^n</math> and falls into <math>n + 1\!</math> ranks, with a binomial coefficient <math>\tbinom{n}{k}</math> giving the number of propositions that have rank or weight <math>k.\!</math><br />
<br />
<ul><br />
<br />
<li><br />
<p>The ''linear propositions'', <math>\{ \ell : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{\ell} \mathbb{B}),\!</math> may be written as sums:</p><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\sum_{i=1}^n e_i ~=~ e_1 + \ldots + e_n<br />
~\text{where}~<br />
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 0 \end{matrix}\right\}<br />
~\text{for}~ i = 1 ~\text{to}~ n.\!</math><br />
|}<br />
</li><br />
<br />
<li><br />
<p>The ''positive propositions'', <math>\{ p : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{p} \mathbb{B}),\!</math> may be written as products:</p><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\prod_{i=1}^n e_i ~=~ e_1 \cdot \ldots \cdot e_n<br />
~\text{where}~<br />
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 1 \end{matrix}\right\}<br />
~\text{for}~ i = 1 ~\text{to}~ n.\!</math><br />
|}<br />
</li><br />
<br />
<li><br />
<p>The ''singular propositions'', <math>\{ \mathbf{x} : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{s} \mathbb{B}),\!</math> may be written as products:</p><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\prod_{i=1}^n e_i ~=~ e_1 \cdot \ldots \cdot e_n<br />
~\text{where}~<br />
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = \texttt{(} a_i \texttt{)} \end{matrix}\right\}<br />
~\text{for}~ i = 1 ~\text{to}~ n.\!</math><br />
|}<br />
</li><br />
<br />
</ul><br />
<br />
In each case the rank <math>k\!</math> ranges from <math>0\!</math> to <math>n\!</math> and counts the number of positive appearances of the coordinate propositions <math>a_1, \ldots, a_n\!</math> in the resulting expression. For example, for <math>{n = 3},\!</math> the linear proposition of rank <math>0\!</math> is <math>0,\!</math> the positive proposition of rank <math>0\!</math> is <math>1,\!</math> and the singular proposition of rank <math>0\!</math> is <math>\texttt{(} a_1 \texttt{)(} a_2 \texttt{)(} a_3\texttt{)}.\!</math><br />
<br />
The basic propositions <math>a_i : \mathbb{B}^n \to \mathbb{B}</math> are both linear and positive. So these two kinds of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions.<br />
<br />
Linear propositions and positive propositions are generated by taking boolean sums and products, respectively, over selected subsets of basic propositions, so both families of propositions are parameterized by the powerset <math>\mathcal{P}(\mathcal{I}),</math> that is, the set of all subsets <math>J\!</math> of the basic index set <math>\mathcal{I} = \{1, \ldots, n\}.\!</math><br />
<br />
Let us define <math>\mathcal{A}_J</math> as the subset of <math>\mathcal{A}</math> that is given by <math>\{a_i : i \in J\}.\!</math> Then we may comprehend the action of the linear and the positive propositions in the following terms:<br />
<br />
* The linear proposition <math>\ell_J : \mathbb{B}^n \to \mathbb{B}</math> evaluates each cell <math>\mathbf{x}</math> of <math>\mathbb{B}^n</math> by looking at the coefficients of <math>\mathbf{x}</math> with respect to the features that <math>\ell_J</math> "likes", namely those in <math>\mathcal{A}_J,</math> and then adds them up in <math>\mathbb{B}.</math> Thus, <math>\ell_J(\mathbf{x})</math> computes the parity of the number of features that <math>\mathbf{x}</math> has in <math>\mathcal{A}_J,</math> yielding one for odd and zero for even. Expressed in this idiom, <math>\ell_J(\mathbf{x}) = 1</math> says that <math>\mathbf{x}</math> seems ''odd'' (or ''oddly true'') to <math>\mathcal{A}_J,</math> whereas <math>\ell_J(\mathbf{x}) = 0</math> says that <math>\mathbf{x}</math> seems ''even'' (or ''evenly true'') to <math>\mathcal{A}_J,</math> so long as we recall that ''zero times'' is evenly often, too.<br />
<br />
* The positive proposition <math>p_J : \mathbb{B}^n \to \mathbb{B}</math> evaluates each cell <math>\mathbf{x}</math> of <math>\mathbb{B}^n</math> by looking at the coefficients of <math>\mathbf{x}</math> with regard to the features that <math>p_J\!</math> "likes", namely those in <math>\mathcal{A}_J,</math> and then takes their product in <math>\mathbb{B}.</math> Thus, <math>p_J(\mathbf{x})</math> assesses the unanimity of the multitude of features that <math>\mathbf{x}</math> has in <math>\mathcal{A}_J,</math> yielding one for all and aught for else. In these consensual or contractual terms, <math>p_J(\mathbf{x}) = 1</math> means that <math>\mathbf{x}</math> is ''AOK'' or congruent with all of the conditions of <math>\mathcal{A}_J,</math> while <math>p_J(\mathbf{x}) = 0</math> means that <math>\mathbf{x}</math> defaults or dissents from some condition of <math>\mathcal{A}_J.</math><br />
<br />
===Basis Relativity and Type Ambiguity===<br />
<br />
Finally, two things are important to keep in mind with regard to the simplicity, linearity, positivity, and singularity of propositions.<br />
<br />
First, all of these properties are relative to a particular basis. For example, a singular proposition with respect to a basis <math>\mathcal{A}</math> will not remain singular if <math>\mathcal{A}</math> is extended by a number of new and independent features. Even if we stick to the original set of pairwise options <math>\{a_i\} \cup \{ \texttt{(} a_i \texttt{)} \}\!</math> to select a new basis, the sets of linear and positive propositions are determined by the choice of simple propositions, and this determination is tantamount to the conventional choice of a cell as origin.<br />
<br />
Second, the singular propositions <math>\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B},</math> picking out as they do a single cell or a coordinate tuple <math>\mathbf{x}</math> of <math>\mathbb{B}^n,</math> become the carriers or the vehicles of a certain type-ambiguity that vacillates between the dual forms <math>\mathbb{B}^n</math> and <math>(\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B})</math> and infects the whole hierarchy of types built on them. In other words, the terms that signify the interpretations <math>\mathbf{x} : \mathbb{B}^n</math> and the singular propositions <math>\mathbf{x} : \mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B}</math> are fully equivalent in information, and this means that every token of the type <math>\mathbb{B}^n</math> can be reinterpreted as an appearance of the subtype <math>\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B}.</math> And vice versa, the two types can be exchanged with each other everywhere that they turn up. In practical terms, this allows the use of singular propositions as a way of denoting points, forming an alternative to coordinate tuples.<br />
<br />
For example, relative to the universe of discourse <math>[a_1, a_2, a_3]\!</math> the singular proposition <math>a_1 a_2 a_3 : \mathbb{B}^3 \xrightarrow{s} \mathbb{B}</math> could be explicitly retyped as <math>a_1 a_2 a_3 : \mathbb{B}^3</math> to indicate the point <math>(1, 1, 1)\!</math> but in most cases the proper interpretation could be gathered from context. Both notations remain dependent on a particular basis, but the code that is generated under the singular option has the advantage in its self-commenting features, in other words, it constantly reminds us of its basis in the process of denoting points. When the time comes to put a multiplicity of different bases into play, and to search for objects and properties that remain invariant under the transformations between them, this infinitesimal potential advantage may well evolve into an overwhelming practical necessity.<br />
<br />
===The Analogy Between Real and Boolean Types===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Measurement consists in correlating our subject matter with the series of real numbers; and such correlations are desirable because, once they are set up, all the well-worked theory of numerical mathematics lies ready at hand as a tool for our further reasoning.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]<br />
|}<br />
<br />
There are two further reasons why it useful to spend time on a careful treatment of types, and they both have to do with our being able to take full computational advantage of certain dimensions of flexibility in the types that apply to terms. First, the domains of differential geometry and logic programming are connected by analogies between real and boolean types of the same pattern. Second, the types involved in these patterns have important isomorphisms connecting them that apply on both the real and the boolean sides of the picture.<br />
<br />
Amazingly enough, these isomorphisms are themselves schematized by the axioms and theorems of propositional logic. This fact is known as the ''propositions as types'' analogy or the Curry&ndash;Howard isomorphism [How]. In another formulation it says that terms are to types as proofs are to propositions. See [LaS, 42&ndash;46] and [SeH] for a good discussion and further references. To anticipate the bearing of these issues on our immediate topic, Table&nbsp;3 sketches a partial overview of the Real to Boolean analogy that may serve to illustrate the paradigm in question.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 3.} ~~ \text{Analogy Between Real and Boolean Types}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Real Domain} ~ \mathbb{R}\!</math><br />
| <math>\longleftrightarrow\!</math><br />
| <math>\text{Boolean Domain} ~ \mathbb{B}\!</math><br />
|-<br />
| <math>\mathbb{R}^n\!</math><br />
| <math>\text{Basic Space}\!</math><br />
| <math>\mathbb{B}^n\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}\!</math><br />
| <math>\text{Function Space}\!</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R}\!</math><br />
| <math>\text{Tangent Vector}\!</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to ((\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R})\!</math><br />
| <math>\text{Vector Field}\!</math><br />
| <math>\mathbb{B}^n \to ((\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B})\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \times (\mathbb{R}^n \to \mathbb{R})) \to \mathbb{R}\!</math><br />
| '''<font size="4">"</font>'''<br />
| <math>(\mathbb{B}^n \times (\mathbb{B}^n \to \mathbb{B})) \to \mathbb{B}\!</math><br />
|-<br />
| <math>((\mathbb{R}^n \to \mathbb{R}) \times \mathbb{R}^n) \to \mathbb{R}\!</math><br />
| '''<font size="4">"</font>'''<br />
| <math>((\mathbb{B}^n \to \mathbb{B}) \times \mathbb{B}^n) \to \mathbb{B}\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^n \to \mathbb{R})~\!</math><br />
| <math>\text{Derivation}\!</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^n \to \mathbb{B})\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}^m\!</math><br />
| <math>\begin{matrix}\text{Basic}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}^m\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^m \to \mathbb{R})\!</math><br />
| <math>\begin{matrix}\text{Function}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^m \to \mathbb{B})\!</math><br />
|}<br />
<br />
<br><br />
<br />
The Table exhibits a sample of likely parallels between the real and boolean domains. The central column gives a selection of terminology that is borrowed from differential geometry and extended in its meaning to the logical side of the Table. These are the varieties of spaces that come up when we turn to analyzing the dynamics of processes that pursue their courses through the states of an arbitrary space <math>X.\!</math> Moreover, when it becomes necessary to approach situations of overwhelming dynamic complexity in a succession of qualitative reaches, then the methods of logic that are afforded by the boolean domains, with their declarative means of synthesis and deductive modes of analysis, supply a natural battery of tools for the task.<br />
<br />
It is usually expedient to take these spaces two at a time, in dual pairs of the form <math>X\!</math> and <math>(X \to \mathbb{K}).</math> In general, one creates pairs of type schemas by replacing any space <math>X\!</math> with its dual <math>(X \to \mathbb{K}),</math> for example, pairing the type <math>X \to Y</math> with the type <math>(X \to \mathbb{K}) \to (Y \to \mathbb{K}),</math> and <math>X \times Y</math> with <math>(X \to \mathbb{K}) \times (Y \to \mathbb{K}).</math> The word ''dual'' is used here in its broader sense to mean all of the functionals, not just the linear ones. Given any function <math>f : X \to \mathbb{K},</math> the ''converse'' or inverse relation corresponding to <math>f\!</math> is denoted <math>f^{-1},\!</math> and the subsets of <math>X\!</math> that are defined by <math>f^{-1}(k),\!</math> taken over <math>k\!</math> in <math>\mathbb{K},</math> are called the ''fibers'' or the ''level sets'' of the function <math>f.\!</math><br />
<br />
===Theory of Control and Control of Theory===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
You will hardly know who I am or what I mean,<br><br />
But I shall be good health to you nevertheless,<br><br />
And filter and fibre your blood.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]<br />
|}<br />
<br />
In the boolean context a function <math>f : X \to \mathbb{B}</math> is tantamount to a ''proposition'' about elements of <math>X,\!</math> and the elements of <math>X\!</math> constitute the ''interpretations'' of that proposition. The fiber <math>f^{-1}(1)\!</math> comprises the set of ''models'' of <math>f,\!</math> or examples of elements in <math>X\!</math> satisfying the proposition <math>f.\!</math> The fiber <math>f^{-1}(0)\!</math> collects the complementary set of ''anti-models'', or the exceptions to the proposition <math>f\!</math> that exist in <math>X.\!</math> Of course, the space of functions <math>(X \to \mathbb{B})\!</math> is isomorphic to the set of all subsets of <math>X,\!</math> called the ''power set'' of <math>X,\!</math> and often denoted <math>\mathcal{P}(X)</math> or <math>2^X.\!</math><br />
<br />
The operation of replacing <math>X\!</math> by <math>(X \to \mathbb{B})\!</math> in a type schema corresponds to a certain shift of attitude towards the space <math>X,\!</math> in which one passes from a focus on the ostensibly individual elements of <math>X\!</math> to a concern with the states of information and uncertainty that one possesses about objects and situations in <math>X.\!</math> The conceptual obstacles in the path of this transition can be smoothed over by using singular functions <math>(\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B})</math> as stepping stones. First of all, it's an easy step from an element <math>\mathbf{x}</math> of type <math>\mathbb{B}^n</math> to the equivalent information of a singular proposition <math>\mathbf{x} : X \xrightarrow{s} \mathbb{B}, </math> and then only a small jump of generalization remains to reach the type of an arbitrary proposition <math>f : X \to \mathbb{B},</math> perhaps understood to indicate a relaxed constraint on the singularity of points or a neighborhood circumscribing the original <math>\mathbf{x}.</math> This is frequently a useful transformation, communicating between the ''objective'' and the ''intentional'' perspectives, in spite perhaps of the open objection that this distinction is transient in the mean time and ultimately superficial.<br />
<br />
It is hoped that this measure of flexibility, allowing us to stretch a point into a proposition, can be useful in the examination of inquiry driven systems, where the differences between empirical, intentional, and theoretical propositions constitute the discrepancies and the distributions that drive experimental activity. I can give this model of inquiry a cybernetic cast by realizing that theory change and theory evolution, as well as the choice and the evaluation of experiments, are actions that are taken by a system or its agent in response to the differences that are detected between observational contents and theoretical coverage.<br />
<br />
All of the above notwithstanding, there are several points that distinguish these two tasks, namely, the ''theory of control'' and the ''control of theory'', features that are often obscured by too much precipitation in the quickness with which we understand their similarities. In the control of uncertainty through inquiry, some of the actuators that we need to be concerned with are axiom changers and theory modifiers, operators with the power to compile and to revise the theories that generate expectations and predictions, effectors that form and edit our grammars for the languages of observational data, and agencies that rework the proposed model to fit the actual sequences of events and the realized relationships of values that are observed in the environment. Moreover, when steps must be taken to carry out an experimental action, there must be something about the particular shape of our uncertainty that guides us in choosing what directions to explore, and this impression is more than likely influenced by previous accumulations of experience. Thus it must be anticipated that much of what goes into scientific progress, or any sustainable effort toward a goal of knowledge, is necessarily predicated on long term observation and modal expectations, not only on the more local or short term prediction and correction.<br />
<br />
===Propositions as Types and Higher Order Types===<br />
<br />
The types collected in Table&nbsp;3 (repeated below) serve to illustrate the themes of ''higher order propositional expressions'' and the ''propositions as types'' (PAT) analogy.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 3.} ~~ \text{Analogy Between Real and Boolean Types}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Real Domain} ~ \mathbb{R}\!</math><br />
| <math>\longleftrightarrow\!</math><br />
| <math>\text{Boolean Domain} ~ \mathbb{B}\!</math><br />
|-<br />
| <math>\mathbb{R}^n\!</math><br />
| <math>\text{Basic Space}\!</math><br />
| <math>\mathbb{B}^n\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}\!</math><br />
| <math>\text{Function Space}\!</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R}\!</math><br />
| <math>\text{Tangent Vector}\!</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to ((\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R})\!</math><br />
| <math>\text{Vector Field}\!</math><br />
| <math>\mathbb{B}^n \to ((\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B})\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \times (\mathbb{R}^n \to \mathbb{R})) \to \mathbb{R}\!</math><br />
| '''<font size="4">"</font>'''<br />
| <math>(\mathbb{B}^n \times (\mathbb{B}^n \to \mathbb{B})) \to \mathbb{B}\!</math><br />
|-<br />
| <math>((\mathbb{R}^n \to \mathbb{R}) \times \mathbb{R}^n) \to \mathbb{R}\!</math><br />
| '''<font size="4">"</font>'''<br />
| <math>((\mathbb{B}^n \to \mathbb{B}) \times \mathbb{B}^n) \to \mathbb{B}\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^n \to \mathbb{R})~\!</math><br />
| <math>\text{Derivation}\!</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^n \to \mathbb{B})\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}^m\!</math><br />
| <math>\begin{matrix}\text{Basic}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}^m\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^m \to \mathbb{R})\!</math><br />
| <math>\begin{matrix}\text{Function}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^m \to \mathbb{B})\!</math><br />
|}<br />
<br />
<br><br />
<br />
First, observe that the type of a ''tangent vector at a point'', also known as a ''directional derivative'' at that point, has the form <math>(\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K},</math> where <math>\mathbb{K}</math> is the chosen ground field, in the present case either <math>\mathbb{R}</math> or <math>\mathbb{B}.</math> At a point in a space of type <math>\mathbb{K}^n,</math> a directional derivative operator <math>\vartheta\!</math> takes a function on that space, an <math>f\!</math> of type <math>(\mathbb{K}^n \to \mathbb{K}),</math> and maps it to a ground field value of type <math>\mathbb{K}.</math> This value is known as the ''derivative'' of <math>f\!</math> in the direction <math>\vartheta\!</math> [Che46, 76&ndash;77]. In the boolean case <math>\vartheta : (\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math> has the form of a proposition about propositions, in other words, a proposition of the next higher type.<br />
<br />
Next, by way of illustrating the propositions as types idea, consider a proposition of the form <math>X \Rightarrow (Y \Rightarrow Z).</math> One knows from propositional calculus that this is logically equivalent to a proposition of the form <math>(X \land Y) \Rightarrow Z.</math> But this equivalence should remind us of the functional isomorphism that exists between a construction of the type <math>X \to (Y \to Z)</math> and a construction of the type <math>(X \times Y) \to Z.</math> The propositions as types analogy permits us to take a functional type like this and, under the right conditions, replace the functional arrows &ldquo;<math>\to~\!</math>&rdquo; and products &ldquo;<math>\times\!</math>&rdquo; with the respective logical arrows &ldquo;<math>\Rightarrow\!</math>&rdquo; and products &ldquo;<math>\land\!</math>&rdquo;. Accordingly, viewing the result as a proposition, we can employ axioms and theorems of propositional calculus to suggest appropriate isomorphisms among the categorical and functional constructions.<br />
<br />
Finally, examine the middle four rows of Table&nbsp;3. These display a series of isomorphic types that stretch from the categories that are labeled ''Vector Field'' to those that are labeled ''Derivation''. A ''vector field'', also known as an ''infinitesimal transformation'', associates a tangent vector at a point with each point of a space. In symbols, a vector field is a function of the form <math>\textstyle \xi : X \to \bigcup_{x \in X} \xi_x\!</math> that assigns to each point <math>x\!</math> of the space <math>X\!</math> a tangent vector to <math>X\!</math> at that point, namely, the tangent vector <math>\xi_x\!</math> [Che46, 82&ndash;83]. If <math>X\!</math> is of the type <math>\mathbb{K}^n,</math> then <math>\xi\!</math> is of the type <math>\mathbb{K}^n \to ((\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K}).</math> This has the pattern <math>X \to (Y \to Z),</math> with <math>X = \mathbb{K}^n,</math> <math>Y = (\mathbb{K}^n \to \mathbb{K}),</math> and <math>Z = \mathbb{K}.</math><br />
<br />
Applying the propositions as types analogy, one can follow this pattern through a series of metamorphoses from the type of a vector field to the type of a derivation, as traced out in Table&nbsp;4. Observe how the function <math>f : X \to \mathbb{K},</math> associated with the place of <math>Y\!</math> in the pattern, moves through its paces from the second to the first position. In this way, the vector field <math>\xi,\!</math> initially viewed as attaching each tangent vector <math>\xi_x\!</math> to the site <math>x\!</math> where it acts in <math>X,\!</math> now comes to be seen as acting on each scalar potential <math>f : X \to \mathbb{K}</math> like a generalized species of differentiation, producing another function <math>\xi f : X \to \mathbb{K}</math> of the same type.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 4.} ~~ \text{An Equivalence Based on the Propositions as Types Analogy}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Pattern}\!</math><br />
| <math>\text{Construct}\!</math><br />
| <math>\text{Instance}\!</math><br />
|-<br />
| <math>X \to (Y \to Z)\!</math><br />
| <math>\text{Vector Field}\!</math><br />
| <math>\mathbb{K}^n \to ((\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K})\!</math><br />
|-<br />
| <math>(X \times Y) \to Z\!</math><br />
| <math>\Uparrow\!</math><br />
| <math>(\mathbb{K}^n \times (\mathbb{K}^n \to \mathbb{K})) \to \mathbb{K}\!</math><br />
|-<br />
| <math>(Y \times X) \to Z\!</math><br />
| <math>\Downarrow\!</math><br />
| <math>((\mathbb{K}^n \to \mathbb{K}) \times \mathbb{K}^n) \to \mathbb{K}\!</math><br />
|-<br />
| <math>Y \to (X \to Z)\!</math><br />
| <math>\text{Derivation}\!</math><br />
| <math>(\mathbb{K}^n \to \mathbb{K}) \to (\mathbb{K}^n \to \mathbb{K})\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Reality at the Threshold of Logic===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
But no science can rest entirely on measurement, and many scientific investigations are quite out of reach of that device. To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]<br />
|}<br />
<br />
Table 5 accumulates an array of notation that I hope will not be too distracting. Some of it is rarely needed, but has been filled in for the sake of completeness. Its purpose is simple, to give literal expression to the visual intuitions that come with venn diagrams, and to help build a bridge between our qualitative and quantitative outlooks on dynamic systems.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 5.} ~~ \text{A Bridge Over Troubled Waters}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| align="center" | <math>\text{Linear Space}\!</math><br />
| align="center" | <math>\text{Liminal Space}\!</math><br />
| align="center" | <math>\text{Logical Space}\!</math><br />
|-<br />
| <math>\begin{matrix}\mathcal{X} & = & \{ x_1, \ldots, x_n \}\end{matrix}</math><br />
| <math>\begin{matrix}\underline{\mathcal{X}} & = & \{ \underline{x}_1, \ldots, \underline{x}_n \}\end{matrix}</math><br />
| <math>\begin{matrix}\mathcal{A} & = & \{ a_1, \ldots, a_n \}\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X_i & = & \langle x_i \rangle<br />
\\<br />
& \cong & \mathbb{K}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}_i & = & \{ \texttt{(} \underline{x}_i \texttt{)}, \underline{x}_i \}<br />
\\<br />
& \cong & \mathbb{B}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A_i & = & \{ \texttt{(} a_i \texttt{)}, a_i \}<br />
\\<br />
& \cong & \mathbb{B}<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X<br />
\\<br />
= & \langle \mathcal{X} \rangle<br />
\\<br />
= & \langle x_1, \ldots, x_n \rangle<br />
\\<br />
= & X_1 \times \ldots \times X_n<br />
\\<br />
= & \prod_{i=1}^n X_i<br />
\\<br />
\cong & \mathbb{K}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}<br />
\\<br />
= & \langle \underline{\mathcal{X}} \rangle<br />
\\<br />
= & \langle \underline{x}_1, \ldots, \underline{x}_n \rangle<br />
\\<br />
= & \underline{X}_1 \times \ldots \times \underline{X}_n<br />
\\<br />
= & \prod_{i=1}^n \underline{X}_i<br />
\\<br />
\cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A<br />
\\<br />
= & \langle \mathcal{A} \rangle<br />
\\<br />
= & \langle a_1, \ldots, a_n \rangle<br />
\\<br />
= & A_1 \times \ldots \times A_n<br />
\\<br />
= & \prod_{i=1}^n A_i<br />
\\<br />
\cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X^* & = & (\ell : X \to \mathbb{K})<br />
\\<br />
& \cong & \mathbb{K}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}^* & = & (\ell : \underline{X} \to \mathbb{B})<br />
\\<br />
& \cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A^* & = & (\ell : A \to \mathbb{B})<br />
\\<br />
& \cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X^\uparrow & = & (X \to \mathbb{K})<br />
\\<br />
& \cong & (\mathbb{K}^n \to \mathbb{K})<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}^\uparrow & = & (\underline{X} \to \mathbb{B})<br />
\\<br />
& \cong & (\mathbb{B}^n \to \mathbb{B})<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A^\uparrow & = & (A \to \mathbb{B})<br />
\\<br />
& \cong & (\mathbb{B}^n \to \mathbb{B})<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X^\bullet<br />
\\<br />
= & [\mathcal{X}]<br />
\\<br />
= & [x_1, \ldots, x_n]<br />
\\<br />
= & (X, X^\uparrow)<br />
\\<br />
= & (X ~+\!\to \mathbb{K})<br />
\\<br />
= & (X, (X \to \mathbb{K}))<br />
\\<br />
\cong & (\mathbb{K}^n, (\mathbb{K}^n \to \mathbb{K}))<br />
\\<br />
= & (\mathbb{K}^n ~+\!\to \mathbb{K})<br />
\\<br />
= & [\mathbb{K}^n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}^\bullet<br />
\\<br />
= & [\underline{\mathcal{X}}]<br />
\\<br />
= & [\underline{x}_1, \ldots, \underline{x}_n]<br />
\\<br />
= & (\underline{X}, \underline{X}^\uparrow)<br />
\\<br />
= & (\underline{X} ~+\!\to \mathbb{B})<br />
\\<br />
= & (\underline{X}, (\underline{X} \to \mathbb{B}))<br />
\\<br />
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))<br />
\\<br />
= & (\mathbb{B}^n ~+\!\to \mathbb{B})<br />
\\<br />
= & [\mathbb{B}^n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A^\bullet<br />
\\<br />
= & [\mathcal{A}]<br />
\\<br />
= & [a_1, \ldots, a_n]<br />
\\<br />
= & (A, A^\uparrow)<br />
\\<br />
= & (A ~+\!\to \mathbb{B})<br />
\\<br />
= & (A, (A \to \mathbb{B}))<br />
\\<br />
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))<br />
\\<br />
= & (\mathbb{B}^n ~+\!\to \mathbb{B})<br />
\\<br />
= & [\mathbb{B}^n]<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
The left side of the Table collects mostly standard notation for an <math>n\!</math>-dimensional vector space over a field <math>\mathbb{K}.</math> The right side of the table repeats the first elements of a notation that I sketched above, to be used in further developments of propositional calculus. (I plan to use this notation in the logical analysis of neural network systems.) The middle column of the table is designed as a transitional step from the case of an arbitrary field <math>\mathbb{K},</math> with a special interest in the continuous line <math>\mathbb{R},</math> to the qualitative and discrete situations that are instanced and typified by <math>\mathbb{B}.</math><br />
<br />
I now proceed to explain these concepts in more detail. The most important ideas developed in Table&nbsp;5 are these:<br />
<br />
* The idea of a universe of discourse, which includes both a space of ''points'' and a space of ''maps'' on those points.<br />
<br />
* The idea of passing from a more complex universe to a simpler universe by a process of ''thresholding'' each dimension of variation down to a single bit of information.<br />
<br />
For the sake of concreteness, let us suppose that we start with a continuous <math>n\!</math>-dimensional vector space like <math>X = \langle x_1, \ldots, x_n \rangle \cong \mathbb{R}^n.</math> The coordinate system <math>\mathcal{X} = \{x_i\}</math> is a set of maps <math>x_i : \mathbb{R}^n \to \mathbb{R},</math> also known as the ''coordinate projections''. Given a "dataset" of points <math>\mathbf{x}</math> in <math>\mathbb{R}^n,</math> there are numerous ways of sensibly reducing the data down to one bit for each dimension. One strategy that is general enough for our present purposes is as follows. For each <math>i\!</math> we choose an <math>n\!</math>-ary relation <math>L_i\!</math> on <math>\mathbb{R}^n,</math> that is, a subset of <math>\mathbb{R}^n,</math> and then we define the <math>i^\mathrm{th}\!</math> threshold map, or ''limen'' <math>\underline{x}_i</math> as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\underline{x}_i : \mathbb{R}^n \to \mathbb{B}\ \text{such that:}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
\underline{x}_i(\mathbf{x}) = 1 & \text{if} & \mathbf{x} \in L_i,<br />
\\[4pt]<br />
\underline{x}_i(\mathbf{x}) = 0 & \text{if} & \mathbf{x} \not\in L_i.<br />
\end{matrix}</math><br />
|}<br />
<br />
In other notations that are sometimes used, the operator <math>\chi (\ldots)</math> or the corner brackets <math>\lceil\ldots\rceil</math> can be used to denote a ''characteristic function'', that is, a mapping from statements to their truth values in <math>\mathbb{B}.</math> Finally, it is not uncommon to use the name of the relation itself as a predicate that maps <math>n\!</math>-tuples into truth values. Thus we have the following notational variants of the above definition:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
\underline{x}_i (\mathbf{x}) & = & \chi (\mathbf{x} \in L_i) & = & \lceil \mathbf{x} \in L_i \rceil & = & L_i (\mathbf{x}).<br />
\end{matrix}</math><br />
|}<br />
<br />
Notice that, as defined here, there need be no actual relation between the <math>n\!</math>-dimensional subsets <math>\{L_i\}\!</math> and the coordinate axes corresponding to <math>\{x_i\},\!</math> aside from the circumstance that the two sets have the same cardinality. In concrete cases, though, one usually has some reason for associating these "volumes" with these "lines", for instance, <math>L_i\!</math> is bounded by some hyperplane that intersects the <math>i^\text{th}\!</math> axis at a unique threshold value <math>r_i \in \mathbb{R}.\!</math> Often, the hyperplane is chosen normal to the axis. In recognition of this motive, let us make the following convention. When the set <math>L_i\!</math> has points on the <math>i^\text{th}\!</math> axis, that is, points of the form <math>(0, \ldots, 0, r_i, 0, \ldots, 0)</math> where only the <math>x_i\!</math> coordinate is possibly non-zero, we may pick any one of these coordinate values as a parametric index of the relation. In this case we say that the indexing is ''real'', otherwise the indexing is ''imaginary''. For a knowledge based system <math>X,\!</math> this should serve once again to mark the distinction between ''acquaintance'' and ''opinion''.<br />
<br />
States of knowledge about the location of a system or about the distribution of a population of systems in a state space <math>X = \mathbb{R}^n</math> can now be expressed by taking the set <math>\underline{\mathcal{X}} = \{\underline{x}_i\}</math> as a basis of logical features. In picturesque terms, one may think of the underscore and the subscript as combining to form a subtextual spelling for the <math>i^\text{th}\!</math> threshold map. This can help to remind us that the ''threshold operator'' <math>(\underline{~})_i</math> acts on <math>\mathbf{x}</math> by setting up a kind of a &ldquo;hurdle&rdquo; for it. In this interpretation the coordinate proposition <math>\underline{x}_i</math> asserts that the representative point <math>\mathbf{x}</math> resides ''above'' the <math>i^\mathrm{th}\!</math> threshold.<br />
<br />
Primitive assertions of the form <math>\underline{x}_i (\mathbf{x})</math> may then be negated and joined by means of propositional connectives in the usual ways to provide information about the state <math>\mathbf{x}</math> of a contemplated system or a statistical ensemble of systems. Parentheses <math>\texttt{(} \ldots \texttt{)}\!</math> may be used to indicate logical negation. Eventually one discovers the usefulness of the <math>k\!</math>-ary ''just one false'' operators of the form <math>\texttt{(} a_1 \texttt{,} \ldots \texttt{,} a_k \texttt{)},\!</math> as treated in earlier reports. This much tackle generates a space of points (cells, interpretations), <math>\underline{X} \cong \mathbb{B}^n,</math> and a space of functions (regions, propositions), <math>\underline{X}^\uparrow \cong (\mathbb{B}^n \to \mathbb{B}).</math> Together these form a new universe of discourse <math>\underline{X}^\bullet</math> of the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})),</math> which we may abbreviate as <math>\mathbb{B}^n\ +\!\to \mathbb{B}</math> or most succinctly as <math>[\mathbb{B}^n].</math><br />
<br />
The square brackets have been chosen to recall the rectangular frame of a venn diagram. In thinking about a universe of discourse it is a good idea to keep this picture in mind, graphically illustrating the links among the elementary cells <math>\underline{\mathbf{x}},</math> the defining features <math>\underline{x}_i,</math> and the potential shadings <math>f : \underline{X} \to \mathbb{B}</math> all at the same time, not to mention the arbitrariness of the way we choose to inscribe our distinctions in the medium of a continuous space.<br />
<br />
Finally, let <math>X^*\!</math> denote the space of linear functions, <math>(\ell : X \to \mathbb{K}),</math> which has in the finite case the same dimensionality as <math>X,\!</math> and let the same notation be extended across the Table.<br />
<br />
We have just gone through a lot of work, apparently doing nothing more substantial than spinning a complex spell of notational devices through a labyrinth of baffled spaces and baffling maps. The reason for doing this was to bind together and to constitute the intuitive concept of a universe of discourse into a coherent categorical object, the kind of thing, once grasped, that can be turned over in the mind and considered in all its manifold changes and facets. The effort invested in these preliminary measures is intended to pay off later, when we need to consider the state transformations and the time evolution of neural network systems.<br />
<br />
===Tables of Propositional Forms===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope. It provides explicit techniques for manipulating the most basic ingredients of discourse.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7&ndash;8]<br />
|}<br />
<br />
To prepare for the next phase of discussion, Tables&nbsp;6 and 7 collect and summarize all of the propositional forms on one and two variables. These propositional forms are represented over bases of boolean variables as complete sets of boolean-valued functions. Adjacent to their names and specifications are listed what are roughly the simplest expressions in the ''[[cactus language]]'', the particular syntax for propositional calculus that I use in formal and computational contexts. For the sake of orientation, the English paraphrases and the more common notations are listed in the last two columns. As simple and circumscribed as these low-dimensional universes may appear to be, a careful exploration of their differential extensions will involve us in complexities sufficient to demand our attention for some time to come.<br />
<br />
Propositional forms on one variable correspond to boolean functions <math>f : \mathbb{B}^1 \to \mathbb{B}.</math> In Table&nbsp;6 these functions are listed in a variant form of [[truth table]], one in which the axes of the usual arrangement are rotated through a right angle. Each function <math>f_i\!</math> is indexed by the string of values that it takes on the points of the universe <math>X^\bullet = [x] \cong \mathbb{B}^1.</math> The binary index generated in this way is converted to its decimal equivalent and these are used as conventional names for the <math>f_i,\!</math> as shown in the first column of the Table. In their own right the <math>2^1\!</math> points of the universe <math>X^\bullet</math> are coordinated as a space of type <math>\mathbb{B}^1,</math> this in light of the universe <math>X^\bullet</math> being a functional domain where the coordinate projection <math>x\!</math> takes on its values in <math>\mathbb{B}.</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 6.} ~~ \text{Propositional Forms on One Variable}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_0\!</math><br />
| <math>f_{00}\!</math><br />
| <math>0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>f_1\!</math><br />
| <math>f_{01}\!</math><br />
| <math>0~1\!</math><br />
| <math>\texttt{(} x \texttt{)}\!</math><br />
| <math>\text{not}~ x\!</math><br />
| <math>\lnot x\!</math><br />
|-<br />
| <math>f_2\!</math><br />
| <math>f_{10}\!</math><br />
| <math>1~0\!</math><br />
| <math>x\!</math><br />
| <math>x\!</math><br />
| <math>x\!</math><br />
|-<br />
| <math>f_3\!</math><br />
| <math>f_{11}\!</math><br />
| <math>1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
Propositional forms on two variables correspond to boolean functions <math>f : \mathbb{B}^2 \to \mathbb{B}.</math> In Table&nbsp;7 each function <math>f_i\!</math> is indexed by the values that it takes on the points of the universe <math>X^\bullet = [x, y] \cong \mathbb{B}^2.</math> Converting the binary index thus generated to a decimal equivalent, we obtain the functional nicknames that are listed in the first column. The <math>2^2\!</math> points of the universe <math>X^\bullet</math> are coordinated as a space of type <math>\mathbb{B}^2,</math> as indicated under the heading of the Table, where the coordinate projections <math>x\!</math> and <math>y\!</math> run through the various combinations of their values in <math>\mathbb{B}.</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 7-a.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}\!</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}\!</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}\!</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}\!</math><br />
| style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}\!</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}\!</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[4pt]<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\\[4pt]<br />
f_{3}<br />
\\[4pt]<br />
f_{4}<br />
\\[4pt]<br />
f_{5}<br />
\\[4pt]<br />
f_{6}<br />
\\[4pt]<br />
f_{7}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0000}<br />
\\[4pt]<br />
f_{0001}<br />
\\[4pt]<br />
f_{0010}<br />
\\[4pt]<br />
f_{0011}<br />
\\[4pt]<br />
f_{0100}<br />
\\[4pt]<br />
f_{0101}<br />
\\[4pt]<br />
f_{0110}<br />
\\[4pt]<br />
f_{0111}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~0<br />
\\[4pt]<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~0~1~1<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
0~1~0~1<br />
\\[4pt]<br />
0~1~1~0<br />
\\[4pt]<br />
0~1~1~1<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{)} ~ y ~<br />
\\[4pt]<br />
\texttt{(} x \texttt{)}<br />
\\[4pt]<br />
~ x ~ \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{,} ~ y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x ~ y \texttt{)}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{false}<br />
\\[4pt]<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\[4pt]<br />
y ~\text{without}~ x<br />
\\[4pt]<br />
\text{not}~ x<br />
\\[4pt]<br />
x ~\text{without}~ y<br />
\\[4pt]<br />
\text{not}~ y<br />
\\[4pt]<br />
x ~\text{not equal to}~ y<br />
\\[4pt]<br />
\text{not both}~ x ~\text{and}~ y<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
\lnot x \land \lnot y<br />
\\[4pt]<br />
\lnot x \land y<br />
\\[4pt]<br />
\lnot x<br />
\\[4pt]<br />
x \land \lnot y<br />
\\[4pt]<br />
\lnot y<br />
\\[4pt]<br />
x \ne y<br />
\\[4pt]<br />
\lnot x \lor \lnot y<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{8}<br />
\\[4pt]<br />
f_{9}<br />
\\[4pt]<br />
f_{10}<br />
\\[4pt]<br />
f_{11}<br />
\\[4pt]<br />
f_{12}<br />
\\[4pt]<br />
f_{13}<br />
\\[4pt]<br />
f_{14}<br />
\\[4pt]<br />
f_{15}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1000}<br />
\\[4pt]<br />
f_{1001}<br />
\\[4pt]<br />
f_{1010}<br />
\\[4pt]<br />
f_{1011}<br />
\\[4pt]<br />
f_{1100}<br />
\\[4pt]<br />
f_{1101}<br />
\\[4pt]<br />
f_{1110}<br />
\\[4pt]<br />
f_{1111}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
1~0~0~0<br />
\\[4pt]<br />
1~0~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\\[4pt]<br />
1~1~1~1<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x ~ y<br />
\\[4pt]<br />
\texttt{((} x \texttt{,} ~ y \texttt{))}<br />
\\[4pt]<br />
y<br />
\\[4pt]<br />
\texttt{(} x ~ \texttt{(} y \texttt{))}<br />
\\[4pt]<br />
x<br />
\\[4pt]<br />
\texttt{((} x \texttt{)} ~ y \texttt{)}<br />
\\[4pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
\texttt{((~))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x ~\text{and}~ y<br />
\\[4pt]<br />
x ~\text{equal to}~ y<br />
\\[4pt]<br />
y<br />
\\[4pt]<br />
\text{not}~ x ~\text{without}~ y<br />
\\[4pt]<br />
x<br />
\\[4pt]<br />
\text{not}~ y ~\text{without}~ x<br />
\\[4pt]<br />
x ~\text{or}~ y<br />
\\[4pt]<br />
\text{true}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x \land y<br />
\\[4pt]<br />
x = y<br />
\\[4pt]<br />
y<br />
\\[4pt]<br />
x \Rightarrow y<br />
\\[4pt]<br />
x<br />
\\[4pt]<br />
x \Leftarrow y<br />
\\[4pt]<br />
x \lor y<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 7-b.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}</math><br />
| style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>f_{0000}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\\[4pt]<br />
f_{4}<br />
\\[4pt]<br />
f_{8}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0001}<br />
\\[4pt]<br />
f_{0010}<br />
\\[4pt]<br />
f_{0100}<br />
\\[4pt]<br />
f_{1000}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
1~0~0~0<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{)} ~ y ~<br />
\\[4pt]<br />
~ x ~ \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
~ x ~~ y ~<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\[4pt]<br />
y ~\text{without}~ x<br />
\\[4pt]<br />
x ~\text{without}~ y<br />
\\[4pt]<br />
x ~\text{and}~ y<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot x \land \lnot y<br />
\\[4pt]<br />
\lnot x \land y<br />
\\[4pt]<br />
x \land \lnot y<br />
\\[4pt]<br />
x \land y<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{3}<br />
\\[4pt]<br />
f_{12}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0011}<br />
\\[4pt]<br />
f_{1100}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\[4pt]<br />
x<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{not}~ x<br />
\\[4pt]<br />
x<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot x<br />
\\[4pt]<br />
x<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[4pt]<br />
f_{9}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0110}<br />
\\[4pt]<br />
f_{1001}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[4pt]<br />
1~0~0~1<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{,} y \texttt{)}<br />
\\[4pt]<br />
\texttt{((} x \texttt{,} y \texttt{))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x ~\text{not equal to}~ y<br />
\\[4pt]<br />
x ~\text{equal to}~ y<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x \ne y<br />
\\[4pt]<br />
x = y<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{5}<br />
\\[4pt]<br />
f_{10}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0101}<br />
\\[4pt]<br />
f_{1010}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\[4pt]<br />
y<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{not}~ y<br />
\\[4pt]<br />
y<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot y<br />
\\[4pt]<br />
y<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[4pt]<br />
f_{11}<br />
\\[4pt]<br />
f_{13}<br />
\\[4pt]<br />
f_{14}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0111}<br />
\\[4pt]<br />
f_{1011}<br />
\\[4pt]<br />
f_{1101}<br />
\\[4pt]<br />
f_{1110}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} ~ x ~~ y ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{(} ~ x ~ \texttt{(} y \texttt{))}<br />
\\[4pt]<br />
\texttt{((} x \texttt{)} ~ y ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{not both}~ x ~\text{and}~ y<br />
\\[4pt]<br />
\text{not}~ x ~\text{without}~ y<br />
\\[4pt]<br />
\text{not}~ y ~\text{without}~ x<br />
\\[4pt]<br />
x ~\text{or}~ y<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot x \lor \lnot y<br />
\\[4pt]<br />
x \Rightarrow y<br />
\\[4pt]<br />
x \Leftarrow y<br />
\\[4pt]<br />
x \lor y<br />
\end{matrix}\!</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>f_{1111}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
==A Differential Extension of Propositional Calculus==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Fire over water:<br><br />
The image of the condition before transition.<br><br />
Thus the superior man is careful<br><br />
In the differentiation of things,<br><br />
So that each finds its place.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; ''I Ching'', Hexagram 64, [Wil, 249]<br />
|}<br />
<br />
This much preparation is enough to begin introducing my subject, if I excuse myself from giving full arguments for my definitional choices until some later stage. I am trying to develop a ''differential theory of qualitative equations'' that parallels the application of differential geometry to dynamic systems. The idea of a tangent vector is key to this work and a major goal is to find the right logical analogues of tangent spaces, bundles, and functors. The strategy is taken of looking for the simplest versions of these constructions that can be discovered within the realm of propositional calculus, so long as they serve to fill out the general theme.<br />
<br />
===Differential Propositions : Qualitative Analogues of Differential Equations===<br />
<br />
In order to define the differential extension of a universe of discourse <math>[\mathcal{A}],</math> the initial alphabet <math>\mathcal{A}</math> must be extended to include a collection of symbols for ''differential features'', or basic ''changes'' that are capable of occurring in <math>[\mathcal{A}].</math> Intuitively, these symbols may be construed as denoting primitive features of change, qualitative attributes of motion, or propositions about how things or points in the universe may be changing or moving with respect to the features that are noted in the initial alphabet.<br />
<br />
Therefore, let us define the corresponding ''differential alphabet'' or ''tangent alphabet'' as <math>\mathrm{d}\mathcal{A}\!</math> <math>=\!</math> <math>\{\mathrm{d}a_1, \ldots, \mathrm{d}a_n\},</math> in principle just an arbitrary alphabet of symbols, disjoint from the initial alphabet <math>\mathcal{A}\!</math> <math>=\!</math> <math>\{ a_1, \ldots, a_n \},\!</math> that is intended to be interpreted in the way just indicated. It only remains to be understood that the precise interpretation of the symbols in <math>\mathrm{d}\mathcal{A}\!</math> is often conceived to be changeable from point to point of the underlying space <math>A.\!</math> Indeed, for all we know, the state space <math>A\!</math> might well be the state space of a language interpreter, one that is concerned, among other things, with the idiomatic meanings of the dialect generated by <math>\mathcal{A}</math> and <math>\mathrm{d}\mathcal{A}.\!</math><br />
<br />
The ''tangent space'' to <math>A\!</math> at one of its points <math>x,\!</math> sometimes written <math>\mathrm{T}_x(A),</math> takes the form <math>\mathrm{d}A</math> <math>=\!</math> <math>\langle \mathrm{d}\mathcal{A} \rangle</math> <math>=\!</math> <math>\langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle.\!</math> Strictly speaking, the name ''cotangent space'' is probably more correct for this construction, but the fact that we take up spaces and their duals in pairs to form our universes of discourse allows our language to be pliable here.<br />
<br />
Proceeding as we did with the base space <math>A,\!</math> the tangent space <math>\mathrm{d}A</math> at a point of <math>A\!</math> can be analyzed as a product of distinct and independent factors:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\mathrm{d}A ~=~ \prod_{i=1}^n \mathrm{d}A_i ~=~ \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n.\!</math><br />
|}<br />
<br />
Here, <math>\mathrm{d}A_i\!</math> is a set of two differential propositions, <math>\mathrm{d}A_i = \{ (\mathrm{d}a_i), \mathrm{d}a_i \},\!</math> where <math>\texttt{(} \mathrm{d}a_i \texttt{)}\!</math> is a proposition with the logical value of <math>\text{not} ~ \mathrm{d}a_i.\!</math> Each component <math>\mathrm{d}A_i\!</math> has the type <math>\mathbb{B},\!</math> operating under the ordered correspondence <math>\{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \} \cong \{ 0, 1 \}.\!</math> However, clarity is often served by acknowledging this differential usage with a superficially distinct type <math>\mathbb{D},\!</math> whose intension may be indicated as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\mathbb{D} = \{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \} = \{ \text{same}, \text{different} \} = \{ \text{stay}, \text{change} \} = \{ \text{stop}, \text{step} \}.\!</math><br />
|}<br />
<br />
Viewed within a coordinate representation, spaces of type <math>\mathbb{B}^n\!</math> and <math>\mathbb{D}^n\!</math> may appear to be identical sets of binary vectors, but taking a view at this level of abstraction would be like ignoring the qualitative units and the diverse dimensions that distinguish position and momentum, or the different roles of quantity and impulse.<br />
<br />
===An Interlude on the Path===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
There would have been no beginnings: instead, speech would proceed from me, while I stood in its path &ndash; a slender gap &ndash; the point of its possible disappearance.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
A sense of the relation between <math>\mathbb{B}</math> and <math>\mathbb{D}</math> may be obtained by considering the ''path classifier'' (or the ''equivalence class of curves'') approach to tangent vectors. Consider a universe <math>[\mathcal{X}].\!</math> Given the boolean value system, a path in the space <math>X = \langle \mathcal{X} \rangle</math> is a map <math>q : \mathbb{B} \to X.</math> In this context the set of paths <math>(\mathbb{B} \to X)</math> is isomorphic to the cartesian square <math>X^2 = X \times X,</math> or the set of ordered pairs chosen from <math>X.\!</math><br />
<br />
We may analyze <math>X^2 = \{ (u, v) : u, v \in X \}</math> into two parts, specifically, the ordered pairs <math>(u, v)\!</math> that lie on and off the diagonal:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}X^2 & = & \{ (u, v) : u = v \} & \cup & \{ (u, v) : u \ne v \}.\end{matrix}</math><br />
|}<br />
<br />
This partition may also be expressed in the following symbolic form:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}X^2 & \cong & \operatorname{diag} (X) & + & 2 \binom{X}{2}.\end{matrix}</math><br />
|}<br />
<br />
The separate terms of this formula are defined as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}\operatorname{diag} (X) & = & \{ (x, x) : x \in X \}.\end{matrix}\!</math><br />
|}<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}\binom{X}{k} & = & X ~\text{choose}~ k & = & \{ k\text{-sets from}~ X \}.\end{matrix}\!</math><br />
|}<br />
<br />
Thus we have:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}\binom{X}{2} & = & \{ \{ u, v \} : u, v \in X \}.\end{matrix}</math><br />
|}<br />
<br />
We may now use the features in <math>\mathrm{d}\mathcal{X} = \{ \mathrm{d}x_i \} = \{ \mathrm{d}x_1, \ldots, \mathrm{d}x_n \}</math> to classify the paths of <math>(\mathbb{B} \to X)</math> by way of the pairs in <math>X^2.\!</math> If <math>X \cong \mathbb{B}^n,</math> then a path <math>q\!</math> in <math>X\!</math> has the following form:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
q : (\mathbb{B} \to \mathbb{B}^n) & \cong & \mathbb{B}^n \times \mathbb{B}^n & \cong & \mathbb{B}^{2n} & \cong & (\mathbb{B}^2)^n.<br />
\end{matrix}</math><br />
|}<br />
<br />
Intuitively, we want to map this <math>(\mathbb{B}^2)^n</math> onto <math>\mathbb{D}^n</math> by mapping each component <math>\mathbb{B}^2</math> onto a copy of <math>\mathbb{D}.</math> But in the presenting context <math>{}^{\backprime\backprime} \mathbb{D} {}^{\prime\prime}</math> is just a name associated with, or an incidental quality attributed to, coefficient values in <math>\mathbb{B}</math> when they are attached to features in <math>\mathrm{d}\mathcal{X}.</math><br />
<br />
Taking these intentions into account, define <math>\mathrm{d}x_i : X^2 \to \mathbb{B}</math> in the following manner:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcrcl}<br />
\mathrm{d}x_i(u, v)<br />
& = & \texttt{(} ~ x_i(u) & \texttt{,} & x_i(v) ~ \texttt{)}<br />
\\<br />
& = & x_i(u) & + & x_i(v)<br />
\\<br />
& = & x_i(v) & - & x_i(u).<br />
\end{array}</math><br />
|}<br />
<br />
In the above transcription, the operator bracket of the form <math>\texttt{(} \ldots \texttt{,} \ldots \texttt{)}\!</math> is a ''cactus lobe'', in general signifying that just one of the arguments listed is false. In the case of two arguments this is the same thing as saying that the arguments are not equal. The plus sign signifies boolean addition, in the sense of addition in <math>\mathrm{GF}(2),</math> and thus means the same thing in this context as the minus sign, in the sense of adding the additive inverse.<br />
<br />
The above definition of <math>\mathrm{d}x_i : X^2 \to \mathbb{B}</math> is equivalent to defining <math>\mathrm{d}x_i : (\mathbb{B} \to X) \to \mathbb{B}\!</math> in the following way:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcrcl}<br />
\mathrm{d}x_i (q)<br />
& = & \texttt{(} ~ x_i(q_0) & \texttt{,} & x_i(q_1) ~ \texttt{)}<br />
\\<br />
& = & x_i(q_0) & + & x_i(q_1)<br />
\\<br />
& = & x_i(q_1) & - & x_i(q_0).<br />
\end{array}</math><br />
|}<br />
<br />
In this definition <math>q_b = q(b),\!</math> for each <math>b\!</math> in <math>\mathbb{B}.</math> Thus, the proposition <math>\mathrm{d}x_i</math> is true of the path <math>q = (u, v)\!</math> exactly if the terms of <math>q,\!</math> the endpoints <math>u\!</math> and <math>v,\!</math> lie on different sides of the question <math>x_i.\!</math><br />
<br />
The language of features in <math>\langle \mathrm{d}\mathcal{X} \rangle,</math> indeed the whole calculus of propositions in <math>[\mathrm{d}\mathcal{X}],</math> may now be used to classify paths and sets of paths. In other words, the paths can be taken as models of the propositions <math>g : \mathrm{d}X \to \mathbb{B}.</math> For example, the paths corresponding to <math>\mathrm{diag}(X)</math> fall under the description <math>\texttt{(} \mathrm{d}x_1 \texttt{)} \cdots \texttt{(} \mathrm{d}x_n \texttt{)},\!</math> which says that nothing changes against the backdrop of the coordinate frame <math>\{ x_1, \ldots, x_n \}.\!</math><br />
<br />
Finally, a few words of explanation may be in order. If this concept of a path appears to be described in a roundabout fashion, it is because I am trying to avoid using any assumption of vector space properties for the space <math>X\!</math> that contains its range. In many ways the treatment is still unsatisfactory, but improvements will have to wait for the introduction of substitution operators acting on singular propositions.<br />
<br />
===The Extended Universe of Discourse===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
At the moment of speaking, I would like to have perceived a nameless voice, long preceding me, leaving me merely to enmesh myself in it, taking up its cadence, and to lodge myself, when no one was looking, in its interstices as if it had paused an instant, in suspense, to beckon to me.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
Next we define the ''extended alphabet'' or ''bundled alphabet'' <math>\mathrm{E}\mathcal{A}</math> as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lclcl}<br />
\mathrm{E}\mathcal{A}<br />
& = & \mathcal{A} \cup \mathrm{d}\mathcal{A}<br />
& = & \{ a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}.<br />
\end{array}</math><br />
|}<br />
<br />
This supplies enough material to construct the ''differential extension'' <math>\mathrm{E}A,</math> or the ''tangent bundle'' over the initial space <math>A,\!</math> in the following fashion:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcl}<br />
\mathrm{E}A<br />
& = & \langle \mathrm{E}\mathcal{A} \rangle<br />
\\[4pt]<br />
& = & \langle \mathcal{A} \cup \mathrm{d}\mathcal{A} \rangle<br />
\\[4pt]<br />
& = & \langle a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle,<br />
\end{array}</math><br />
|}<br />
<br />
and also:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcl}<br />
\mathrm{E}A<br />
& = & A \times \mathrm{d}A<br />
\\[4pt]<br />
& = & A_1 \times \ldots \times A_n \times \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n.<br />
\end{array}</math><br />
|}<br />
<br />
This gives <math>\mathrm{E}A</math> the type <math>\mathbb{B}^n \times \mathbb{D}^n.</math><br />
<br />
Finally, the tangent universe <math>\mathrm{E}A^\bullet = [\mathrm{E}\mathcal{A}]\!</math> is constituted from the totality of points and maps, or interpretations and propositions, that are based on the extended set of features <math>\mathrm{E}\mathcal{A},</math> and this fact is summed up in the following notation:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lclcl}<br />
\mathrm{E}A^\bullet<br />
& = & [\mathrm{E}\mathcal{A}]<br />
& = & [a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n].<br />
\end{array}</math><br />
|}<br />
<br />
This gives the tangent universe <math>\mathrm{E}A^\bullet\!</math> the type:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcl}<br />
(\mathbb{B}^n \times \mathbb{D}^n\ +\!\to \mathbb{B})<br />
& = & (\mathbb{B}^n \times \mathbb{D}^n, (\mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B})).<br />
\end{array}</math><br />
|}<br />
<br />
A proposition in the tangent universe <math>[\mathrm{E}\mathcal{A}]</math> is called a ''differential proposition'' and forms the analogue of a system of differential equations, constraints, or relations in ordinary calculus.<br />
<br />
With these constructions, the differential extension <math>\mathrm{E}A</math> and the space of differential propositions <math>(\mathrm{E}A \to \mathbb{B}),\!</math> we have arrived, in main outline, at one of the major subgoals of this study. Table&nbsp;8 summarizes the concepts that have been introduced for working with differentially extended universes of discourse.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 8.} ~~ \text{Differential Extension : Basic Notation}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Symbol}\!</math><br />
| <math>\text{Notation}\!</math><br />
| <math>\text{Description}\!</math><br />
| <math>\text{Type}\!</math><br />
|-<br />
| <math>\mathrm{d}\mathfrak{A}\!</math><br />
| <math>\{ {}^{\backprime\backprime} \mathrm{d}a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} \mathrm{d}a_n {}^{\prime\prime} \}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Alphabet of}<br />
\\[2pt]<br />
\text{differential symbols}<br />
\end{matrix}</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>\mathrm{d}\mathcal{A}\!</math><br />
| <math>\{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Basis of}<br />
\\[2pt]<br />
\text{differential features}<br />
\end{matrix}</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>\mathrm{d}A_i\!</math><br />
| <math>\{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \}\!</math><br />
| <math>\text{Differential dimension}~ i\!</math><br />
| <math>\mathbb{D}\!</math><br />
|-<br />
| <math>\mathrm{d}A\!</math><br />
|<br />
<math>\begin{matrix}<br />
\langle \mathrm{d}\mathcal{A} \rangle<br />
\\[2pt]<br />
\langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle<br />
\\[2pt]<br />
\{ (\mathrm{d}a_1, \ldots, \mathrm{d}a_n) \}<br />
\\[2pt]<br />
\mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n<br />
\\[2pt]<br />
\textstyle \prod_i \mathrm{d}A_i<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Tangent space at a point:}<br />
\\[2pt]<br />
\text{Set of changes, motions,}<br />
\\[2pt]<br />
\text{steps, tangent vectors}<br />
\\[2pt]<br />
\text{at a point}<br />
\end{matrix}</math><br />
| <math>\mathbb{D}^n\!</math><br />
|-<br />
| <math>\mathrm{d}A^*\!</math><br />
| <math>(\mathrm{hom} : \mathrm{d}A \to \mathbb{B})\!</math><br />
| <math>\text{Linear functions on}~ \mathrm{d}A\!</math><br />
| <math>(\mathbb{D}^n)^* \cong \mathbb{D}^n\!</math><br />
|-<br />
| <math>\mathrm{d}A^\uparrow\!</math><br />
| <math>(\mathrm{d}A \to \mathbb{B})\!</math><br />
| <math>\text{Boolean functions on}~ \mathrm{d}A\!</math><br />
| <math>\mathbb{D}^n \to \mathbb{B}\!</math><br />
|-<br />
| <math>\mathrm{d}A^\bullet\!</math><br />
|<br />
<math>\begin{matrix}<br />
[\mathrm{d}\mathcal{A}]<br />
\\[2pt]<br />
(\mathrm{d}A, \mathrm{d}A^\uparrow)<br />
\\[2pt]<br />
(\mathrm{d}A ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
(\mathrm{d}A, (\mathrm{d}A \to \mathbb{B}))<br />
\\[2pt]<br />
[\mathrm{d}a_1, \ldots, \mathrm{d}a_n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Tangent universe at a point of}~ A^\bullet,<br />
\\[2pt]<br />
\text{based on the tangent features}<br />
\\[2pt]<br />
\{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
(\mathbb{D}^n, (\mathbb{D}^n \to \mathbb{B}))<br />
\\[2pt]<br />
(\mathbb{D}^n ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
[\mathbb{D}^n]<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
The adjectives ''differential'' or ''tangent'' are systematically attached to every construct based on the differential alphabet <math>\mathrm{d}\mathfrak{A},</math> taken by itself. Strictly speaking, we probably ought to call <math>\mathrm{d}\mathcal{A}</math> the set of ''cotangent'' features derived from <math>\mathcal{A},</math> but the only time this distinction really seems to matter is when we need to distinguish the tangent vectors as maps of type <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math> from cotangent vectors as elements of type <math>\mathbb{D}^n.</math> In like fashion, having defined <math>\mathrm{E}\mathcal{A} = \mathcal{A} \cup \mathrm{d}\mathcal{A},</math> we can systematically attach the adjective ''extended'' or the substantive ''bundle'' to all of the constructs associated with this full complement of <math>{2n}\!</math> features.<br />
<br />
It eventually becomes necessary to extend the initial alphabet even further, to allow for the discussion of higher order differential expressions. Table&nbsp;9 provides a suggestion of how these further extensions can be carried out.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 9.} ~~ \text{Higher Order Differential Features}\!</math><br />
|<br />
<p><math>\begin{array}{lllll}<br />
\mathrm{d}^0 \mathcal{A} & = & \{ a_1, \ldots, a_n \} & = & \mathcal{A}<br />
\\<br />
\mathrm{d}^1 \mathcal{A} & = & \{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \} & = & \mathrm{d}\mathcal{A}<br />
\end{array}</math></p><br />
<br />
<p><math>\begin{array}{lll}<br />
\mathrm{d}^k \mathcal{A} & = & \{ \mathrm{d}^k a_1, \ldots, \mathrm{d}^k a_n \}<br />
\\<br />
\mathrm{d}^* \mathcal{A} & = & \{ \mathrm{d}^0 \mathcal{A}, \ldots, \mathrm{d}^k \mathcal{A}, \ldots \}<br />
\end{array}</math></p><br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}^0 \mathcal{A} & = & \mathrm{d}^0 \mathcal{A}<br />
\\<br />
\mathrm{E}^1 \mathcal{A} & = & \mathrm{d}^0 \mathcal{A} ~\cup~ \mathrm{d}^1 \mathcal{A}<br />
\\<br />
\mathrm{E}^k \mathcal{A} & = & \mathrm{d}^0 \mathcal{A} ~\cup~ \ldots ~\cup~ \mathrm{d}^k \mathcal{A}<br />
\\<br />
\mathrm{E}^\infty \mathcal{A} & = & \bigcup~ \mathrm{d}^* \mathcal{A}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
===Intentional Propositions===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Do you guess I have some intricate purpose?<br><br />
Well I have . . . . for the April rain has, and the mica on<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the side of a rock has.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 45]<br />
|}<br />
<br />
In order to analyze the behavior of a system at successive moments in time, while staying within the limitations of propositional logic, it is necessary to create independent alphabets of logical features for each moment of time that we contemplate using in our discussion. These moments have reference to typical instances and relative intervals, not actual or absolute times. For example, to discuss ''velocities'' (first order rates of change) we need to consider points of time in pairs. There are a number of natural ways of doing this. Given an initial alphabet, we could use its symbols as a lexical basis to generate successive alphabets of compound symbols, say, with temporal markers appended as suffixes.<br />
<br />
As a standard way of dealing with these situations, the following scheme of notation suggests a way of extending any alphabet of logical features through as many temporal moments as a particular order of analysis may demand. The lexical operators <math>\mathrm{p}^k</math> and <math>\mathrm{Q}^k</math> are convenient in many contexts where the accumulation of prime symbols and union symbols would otherwise be cumbersome.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 10.} ~~ \text{A Realm of Intentional Features}\!</math><br />
|<br />
<p><math>\begin{array}{lllll}<br />
\mathrm{p}^0 \mathcal{A} & = & \{ a_1, \ldots, a_n \} & = & \mathcal{A}<br />
\\<br />
\mathrm{p}^1 \mathcal{A} & = & \{ a_1^\prime, \ldots, a_n^\prime \} & = & \mathcal{A}^\prime<br />
\\<br />
\mathrm{p}^2 \mathcal{A} & = & \{ a_1^{\prime\prime}, \ldots, a_n^{\prime\prime} \} & = & \mathcal{A}^{\prime\prime}<br />
\\<br />
\cdots & & \cdots &<br />
\end{array}</math></p><br />
<br />
<p><math>\begin{array}{lll}<br />
\mathrm{p}^k \mathcal{A} & = & \{ \mathrm{p}^k a_1, \ldots, \mathrm{p}^k a_n \}<br />
\end{array}</math></p><br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{Q}^0 \mathcal{A} & = & \mathcal{A}<br />
\\<br />
\mathrm{Q}^1 \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}'<br />
\\<br />
\mathrm{Q}^2 \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}' \cup \mathcal{A}''<br />
\\<br />
\cdots & & \cdots<br />
\\<br />
\mathrm{Q}^k \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}' \cup \ldots \cup \mathrm{p}^k \mathcal{A}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
The resulting augmentations of our logical basis determine a series of discursive universes that may be called the ''intentional extension'' of propositional calculus. This extension follows a pattern analogous to the differential extension, which was developed in terms of the operators <math>\mathrm{d}^k</math> and <math>\mathrm{E}^k,</math> and there is a natural relation between these two extensions that bears further examination. In contexts displaying this pattern, where a sequence of domains stretches from an anchoring domain <math>X\!</math> through an indefinite number of higher reaches, a particular collection of domains based on <math>X\!</math> will be referred to as a ''realm'' of <math>X,\!</math> and when the succession exhibits a temporal aspect, as a ''reign'' of <math>X.\!</math><br />
<br />
For the purposes of this discussion, an ''intentional proposition'' is defined as a proposition in the universe of discourse <math>\mathrm{Q}X^\bullet = [\mathrm{Q}\mathcal{X}],</math> in other words, a map <math>q : \mathrm{Q}X \to \mathbb{B}.</math> The sense of this definition may be seen if we consider the following facts. First, the equivalence <math>\mathrm{Q}X = X \times X'</math> motivates the following chain of isomorphisms between spaces:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lllcl}<br />
(\mathrm{Q}X \to \mathbb{B})<br />
& \cong & (X & \times & ~X' \to \mathbb{B})<br />
\\[4pt]<br />
& \cong & (X & \to & (X' \to \mathbb{B}))<br />
\\[4pt]<br />
& \cong & (X' & \to & (X~ \to \mathbb{B})).<br />
\end{array}</math><br />
|}<br />
<br />
Viewed in this light, an intentional proposition <math>q\!</math> may be rephrased as a map <math>q : X \times X' \to \mathbb{B},</math> which judges the juxtaposition of states in <math>X\!</math> from one moment to the next. Alternatively, <math>q\!</math> may be parsed in two stages in two different ways, as <math>q : X \to (X' \to \mathbb{B})</math> and as <math>q : X' \to (X \to \mathbb{B}),</math> which associate to each point of <math>X\!</math> or <math>X'\!</math> a proposition about states in <math>X'\!</math> or <math>X,\!</math> respectively. In this way, an intentional proposition embodies a type of value system, in effect, a proposal that places a value on a collection of ends-in-view, or a project that evaluates a set of goals as regarded from each point of view in the state space of a system.<br />
<br />
In sum, the intentional proposition <math>q\!</math> indicates a method for the systematic selection of local goals. As a general form of description, a map of the type <math>q : \mathrm{Q}^i X \to \mathbb{B}\!</math> may be referred to as an "<math>i^\text{th}</math> order intentional proposition". Naturally, when we speak of intentional propositions without qualification, we usually mean first order intentions.<br />
<br />
Many different realms of discourse have the same structure as the extensions that have been indicated here. From a strictly logical point of view, each new layer of terms is composed of independent logical variables that are no different in kind from those that go before, and each further course of logical atoms is treated like so many individual, but otherwise indifferent bricks by the prototype computer program that I use as a propositional interpreter. Thus, the names that I use to single out the differential and the intentional extensions, and the lexical paradigms that I follow to construct them, are meant to suggest the interpretations that I have in mind, but they can only hint at the extra meanings that human communicators may pack into their terms and inflections.<br />
<br />
As applied here, the word ''intentional'' is drawn from common use and may have little bearing on its technical use in other, more properly philosophical, contexts. I am merely using the complex of intentional concepts &mdash; aims, ends, goals, objectives, purposes, and so on &mdash; metaphorically to flesh out and vividly to represent any situation where one needs to contemplate a system in multiple aspects of state and destination, that is, its being in certain states and at the same time acting as if headed through certain states. If confusion arises, more neutral words like ''conative'', ''contingent'', ''discretionary'', ''experimental'', ''kinetic'', ''progressive'', ''tentative'', or ''trial'' would probably serve as well.<br />
<br />
===Life on Easy Street===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Failing to fetch me at first keep encouraged,<br><br />
Missing me one place search another,<br><br />
I stop some where waiting for you<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]<br />
|}<br />
<br />
The finite character of the extended universe <math>[\mathrm{E}\mathcal{A}]</math> makes the problem of solving differential propositions relatively straightforward, at least, in principle. The solution set of the differential proposition <math>q : \mathrm{E}A \to \mathbb{B}</math> is the set of models <math>q^{-1}(1)\!</math> in <math>\mathrm{E}A.</math> Finding all the models of <math>q,\!</math> the extended interpretations in <math>\mathrm{E}A</math> that satisfy <math>q,\!</math> can be carried out by a finite search. Being in possession of complete algorithms for propositional calculus modeling, theorem checking, or theorem proving makes the analytic task fairly simple in principle, though the question of efficiency in the face of arbitrary complexity may always remain another matter entirely. While the fact that propositional satisfiability is NP-complete may be discouraging for the prospects of a single efficient algorithm that covers the whole space <math>[\mathrm{E}\mathcal{A}]</math> with equal facility, there appears to be much room for improvement in classifying special forms and in developing algorithms that are tailored to their practical processing.<br />
<br />
In view of these constraints and contingencies, our focus shifts to the tasks of approximation and interpretation that support intuition, especially in dealing with the natural kinds of differential propositions that arise in applications, and in the effort to understand, in succinct and adaptive forms, their dynamic implications. In the absence of direct insights, these tasks are partially carried out by forging analogies with the familiar situations and customary routines of ordinary calculus. But the indirect approach, going by way of specious analogy and intuitive habit, forces us to remain on guard against the circumstance that occurs when the word ''forging'' takes on its shadier nuance, indicting the constant risk of a counterfeit in the proportion.<br />
<br />
==Back to the Beginning : Exemplary Universes==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
I would have preferred to be enveloped in words, borne way beyond all possible beginnings.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
To anchor our understanding of differential logic, let us look at how the various concepts apply in the simplest possible concrete cases, where the initial dimension is only 1 or 2. In spite of the obvious simplicity of these cases, it is possible to observe how central difficulties of the subject begin to arise already at this stage.<br />
<br />
===A One-Dimensional Universe===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
There was never any more inception than there is now,<br><br />
Nor any more youth or age than there is now;<br><br />
And will never be any more perfection than there is now,<br><br />
Nor any more heaven or hell than there is now.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]<br />
|}<br />
<br />
Let <math>\mathcal{X} = \{ x_1 \} = \{ A \}</math> be an alphabet that represents one boolean variable or a single logical feature. In this example the capital letter <math>{}^{\backprime\backprime} A {}^{\prime\prime}\!</math> is used usual informally, to name a feature and not a space, in departure from our formerly stated formal conventions. At any rate, the basis element <math>A = x_1\!</math> may be interpreted as a simple proposition or a coordinate projection <math>A = x_1 : \mathbb{B} \xrightarrow{i} \mathbb{B}.</math> The space <math>X = \langle A \rangle = \{ \texttt{(} A \texttt{)}, A \}</math> of points (cells, vectors, interpretations) has cardinality <math>2^n = 2^1 = 2\!</math> and is isomorphic to <math>\mathbb{B} = \{ 0, 1 \}.</math> Moreover, <math>X\!</math> may be identified with the set of singular propositions <math>\{ x : \mathbb{B} \xrightarrow{s} \mathbb{B} \}.</math> The space of linear propositions <math>X^* = \{ \mathrm{hom} : \mathbb{B} \xrightarrow{\ell} \mathbb{B} \} = \{ 0, A \}</math> is algebraically dual to <math>X\!</math> and also has cardinality <math>2.\!</math> Here, <math>{}^{\backprime\backprime} 0 {}^{\prime\prime}\!</math> is interpreted as denoting the constant function <math>0 : \mathbb{B} \to \mathbb{B},</math> amounting to the linear proposition of rank <math>0,\!</math> while <math>A\!</math> is the linear proposition of rank <math>1.\!</math> Last but not least we have the positive propositions <math>\{ \mathrm{pos} : \mathbb{B} \xrightarrow{p} \mathbb{B} \} = \{ A, 1 \},\!</math> of rank <math>1\!</math> and <math>0,\!</math> respectively, where <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}\!</math> is understood as denoting the constant function <math>1 : \mathbb{B} \to \mathbb{B}.</math> In sum, there are <math>2^{2^n} = 2^{2^1} = 4</math> propositions altogether in the universe of discourse, comprising the set <math>X^\uparrow = \{ f : X \to \mathbb{B} \} = \{ 0, \texttt{(} A \texttt{)}, A, 1 \} \cong (\mathbb{B} \to \mathbb{B}).</math><br />
<br />
The first order differential extension of <math>\mathcal{X}</math> is <math>\mathrm{E}\mathcal{X} = \{ x_1, \mathrm{d}x_1 \} = \{ A, \mathrm{d}A \}.</math> If the feature <math>A\!</math> is understood as applying to some object or state, then the feature <math>\mathrm{d}A</math> may be interpreted as an attribute of the same object or state that says that it is changing ''significantly'' with respect to the property <math>A,\!</math> or that it has an ''escape velocity'' with respect to the state <math>A.\!</math> In practice, differential features acquire their logical meaning through a class of ''temporal inference rules''.<br />
<br />
For example, relative to a frame of observation that is left implicit for now, one is permitted to make the following sorts of inference: From the fact that <math>A\!</math> and <math>\mathrm{d}A</math> are true at a given moment one may infer that <math>\texttt{(} A \texttt{)}\!</math> will be true in the next moment of observation. Altogether in the present instance, there is the fourfold scheme of inference that is shown below:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:50%"<br />
|<br />
<math>\begin{matrix}<br />
\text{From} & \texttt{(} A \texttt{)}<br />
& \text{and} & \texttt{(} \mathrm{d}A \texttt{)}<br />
& \text{infer} & \texttt{(} A \texttt{)} & \text{next.}<br />
\\[8pt]<br />
\text{From} & \texttt{(} A \texttt{)}<br />
& \text{and} & \mathrm{d}A<br />
& \text{infer} & A & \text{next.}<br />
\\[8pt]<br />
\text{From} & A<br />
& \text{and} & \texttt{(} \mathrm{d}A \texttt{)}<br />
& \text{infer} & A & \text{next.}<br />
\\[8pt]<br />
\text{From} & A<br />
& \text{and} & \mathrm{d}A<br />
& \text{infer} & \texttt{(} A \texttt{)} & \text{next.}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
It might be thought that an independent time variable needs to be brought in at this point, but it is an insight of fundamental importance that the idea of process is logically prior to the notion of time. A time variable is a reference to a ''clock'' &mdash; a canonical, conventional process that is accepted or established as a standard of measurement, but in essence no different than any other process. This raises the question of how different subsystems in a more global process can be brought into comparison, and what it means for one process to serve the function of a local standard for others. But these inquiries only wrap up puzzles in further riddles, and are obviously too involved to be handled at our current level of approximation.<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
The clock indicates the moment . . . . but what does<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;eternity indicate?<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 79]<br />
|}<br />
<br />
Observe that the secular inference rules, used by themselves, involve a loss of information, since nothing in them can tell us whether the momenta <math>\{ \texttt{(} \mathrm{d}A \texttt{)}, \mathrm{d}A \}\!</math> are changed or unchanged in the next instance. In order to know this, one would have to determine <math>\mathrm{d}^2 A,\!</math> and so on, pursuing an infinite regress. Ultimately, in order to rest with a finitely determinate system, it is necessary to make an infinite assumption, for example, that <math>\mathrm{d}^k A = 0\!</math> for all <math>k\!</math> greater than some fixed value <math>M.\!</math> Another way to escape the regress is through the provision of a dynamic law, in typical form making higher order differentials dependent on lower degrees and estates.<br />
<br />
===Example 1. A Square Rigging===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Urge and urge and urge,<br><br />
Always the procreant urge of the world.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]<br />
|}<br />
<br />
By way of example, suppose that we are given the initial condition <math>A = \mathrm{d}A\!</math> and the law <math>\mathrm{d}^2 A = \texttt{(} A \texttt{)}.\!</math> Since the equation <math>A = \mathrm{d}A\!</math> is logically equivalent to the disjunction <math>A ~ \mathrm{d}A ~\text{or}~ \texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)},\!</math> we may infer two possible trajectories, as displayed in Table&nbsp;11. In either case the state <math>A ~ \texttt{(} \mathrm{d}A \texttt{)(} \mathrm{d}^2 A \texttt{)}\!</math> is a stable attractor or a terminal condition for both starting points.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 11.} ~~ \text{A Pair of Commodious Trajectories}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Time}\!</math><br />
| <math>\text{Trajectory 1}\!</math><br />
| <math>\text{Trajectory 2}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
2<br />
\\[4pt]<br />
3<br />
\\[4pt]<br />
4<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A & \mathrm{d}A & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
\texttt{(} A \texttt{)} & \mathrm{d}A & \mathrm{d}^2 A<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
{}^{\shortparallel} & {}^{\shortparallel} & {}^{\shortparallel}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{)} & \texttt{(} \mathrm{d}A \texttt{)} & \mathrm{d}^2 A<br />
\\[4pt]<br />
\texttt{(} A \texttt{)} & \mathrm{d}A & \mathrm{d}^2 A<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
{}^{\shortparallel} & {}^{\shortparallel} & {}^{\shortparallel}<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Because the initial space <math>X = \langle A \rangle\!</math> is one-dimensional, we can easily fit the second order extension <math>\mathrm{E}^2 X = \langle A, \mathrm{d}A, \mathrm{d}^2 A \rangle\!</math> within the compass of a single venn diagram, charting the couple of converging trajectories as shown in Figure&nbsp;12.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 12 -- The Anchor.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 12.} ~~ \text{The Anchor}\!</math><br />
|}<br />
<br />
If we eliminate from view the regions of <math>\mathrm{E}^2 X\!</math> that are ruled out by the dynamic law <math>\mathrm{d}^2 A = \texttt{(} A \texttt{)},\!</math> then what remains is the quotient structure that is shown in Figure&nbsp;13. This picture makes it easy to see that the dynamically allowable portion of the universe is partitioned between the properties <math>A\!</math> and <math>\mathrm{d}^2 A\!.</math> As it happens, this fact might have been expressed &ldquo;right off the bat&rdquo; by an equivalent formulation of the differential law, one that uses the exclusive disjunction to state the law as <math>\texttt{(} A \texttt{,} \mathrm{d}^2 A \texttt{)}\!.</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 13 -- The Tiller.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 13.} ~~ \text{The Tiller}\!</math><br />
|}<br />
<br />
What we have achieved in this example is to give a differential description of a simple dynamic process. In effect, we did this by embedding a directed graph, which can be taken to represent the state transitions of a finite automaton, in a dynamically allotted quotient structure that is created from a boolean lattice or an <math>n\!</math>-cube by nullifying all of the regions that the dynamics outlaws. With growth in the dimensions of our contemplated universes, it becomes essential, both for human comprehension and for computer implementation, that the dynamic structures of interest to us be represented not actually, by acquaintance, but virtually, by description. In our present study, we are using the language of propositional calculus to express the relevant descriptions, and to comprehend the structure that is implicit in the subsets of a <math>n\!</math>-cube without necessarily being forced to actualize all of its points.<br />
<br />
One of the reasons for engaging in this kind of extremely reduced, but explicitly controlled case study is to throw light on the general study of languages, formal and natural, in their full array of syntactic, semantic, and pragmatic aspects. Propositional calculus is one of the last points of departure where we can view these three aspects interacting in a non-trivial way without being immediately and totally overwhelmed by the complexity they generate. Often this complexity causes investigators of formal and natural languages to adopt the strategy of focusing on a single aspect and to abandon all hope of understanding the whole, whether it's the still living natural language or the dynamics of inquiry that lies crystallized in formal logic.<br />
<br />
From the perspective that I find most useful here, a language is a syntactic system that is designed or evolved in part to express a set of descriptions. When the explicit symbols of a language have extensions in its object world that are actually infinite, or when the implicit categories and generative devices of a linguistic theory have extensions in its subject matter that are potentially infinite, then the finite characters of terms, statements, arguments, grammars, logics, and rhetorics force an excess of intension to reside in all these symbols and functions, across the spectrum from the object language to the metalinguistic uses. In the aphorism from W. von Humboldt that Chomsky often cites, for example, in [Cho86, 30] and [Cho93, 49], language requires &ldquo;the infinite use of finite means&rdquo;. This is necessarily true when the extensions are infinite, when the referential symbols and grammatical categories of a language possess infinite sets of models and instances. But it also voices a practical truth when the extensions, though finite at every stage, tend to grow at exponential rates.<br />
<br />
This consequence of dealing with extensions that are &ldquo;practically infinite&rdquo; becomes crucial when one tries to build neural network systems that learn, since the learning competence of any intelligent system is limited to the objects and domains that it is able to represent. If we want to design systems that operate intelligently with the full deck of propositions dealt by intact universes of discourse, then we must supply them with succinct representations and efficient transformations in this domain. Furthermore, in the project of constructing inquiry driven systems, we find ourselves forced to contemplate the level of generality that is embodied in propositions, because the dynamic evolution of these systems is driven by the measurable discrepancies that occur among their expectations, intentions, and observations, and because each of these subsystems or components of knowledge constitutes a propositional modality that can take on the fully generic character of an empirical summary or an axiomatic theory.<br />
<br />
A compression scheme by any other name is a symbolic representation, and this is what the differential extension of propositional calculus, through all of its many universes of discourse, is intended to supply. Why is this particular program of mental calisthenics worth carrying out in general? By providing a uniform logical medium for describing dynamic systems we can make the task of understanding complex systems much easier, both in looking for invariant representations of individual cases and in finding points of comparison among diverse structures that would otherwise appear as isolated systems. All of this goes to facilitate the search for compact knowledge and to adapt what is learned from individual cases to the general realm.<br />
<br />
===Back to the Feature===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
I guess it must be the flag of my disposition, out of hopeful<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;green stuff woven.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 31]<br />
|}<br />
<br />
Let us assume that the sense intended for differential features is well enough established in the intuition, for now, that we may continue with outlining the structure of the differential extension <math>[\mathrm{E}\mathcal{X}] = [A, \mathrm{d}A].\!</math> Over the extended alphabet <math>\mathrm{E}\mathcal{X} = \{ x_1, \mathrm{d}x_1 \} = \{ A, \mathrm{d}A \}\!</math> of cardinality <math>2^n = 2\!</math> we generate the set of points <math>\mathrm{E}X\!</math> of cardinality <math>2^{2n} = 4\!</math> that bears the following chain of equivalent descriptions:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}X & = & \langle A, \mathrm{d}A \rangle<br />
\\[4pt]<br />
& = & \{ \texttt{(} A \texttt{)}, A \} ~\times~ \{ \texttt{(} \mathrm{d}A \texttt{)}, \mathrm{d}A \}<br />
\\[4pt]<br />
& = &<br />
\{<br />
\texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)},~<br />
\texttt{(} A \texttt{)} \mathrm{d}A,~<br />
A \texttt{(} \mathrm{d}A \texttt{)},~<br />
A ~ \mathrm{d}A<br />
\}.<br />
\end{array}</math><br />
|}<br />
<br />
The space <math>\mathrm{E}X\!</math> may be assigned the mnemonic type <math>\mathbb{B} \times \mathbb{D},\!</math> which is really no different than <math>\mathbb{B} \times \mathbb{B} = \mathbb{B}^2.\!</math> An individual element of <math>\mathrm{E}X\!</math> may be regarded as a ''disposition at a point'' or a ''situated direction'', in effect, a singular mode of change occurring at a single point in the universe of discourse. In applications, the modality of this change can be interpreted in various ways, for example, as an expectation, an intention, or an observation with respect to the behavior of a system.<br />
<br />
To complete the construction of the extended universe of discourse <math>\mathrm{E}X^\bullet = [x_1, \mathrm{d}x_1] = [A, \mathrm{d}A]\!</math> one must add the set of differential propositions <math>\mathrm{E}X^\uparrow = \{ g : \mathrm{E}X \to \mathbb{B} \} \cong (\mathbb{B} \times \mathbb{D} \to \mathbb{B})\!</math> to the set of dispositions in <math>\mathrm{E}X.\!</math> There are <math>2^{2^{2n}} = 16\!</math> propositions in <math>\mathrm{E}X^\uparrow,\!</math> as detailed in Table&nbsp;14.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 14.} ~~ \text{Differential Propositions}\!</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>A\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>\mathrm{d}A\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>g_{0}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{1}<br />
\\[4pt]<br />
g_{2}<br />
\\[4pt]<br />
g_{4}<br />
\\[4pt]<br />
g_{8}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
1~0~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
\texttt{(} A \texttt{)} ~ \mathrm{d}A ~<br />
\\[4pt]<br />
~ A ~ \texttt{(} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
~ A ~~ \mathrm{d}A ~<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{neither}~ A ~\text{nor}~ \mathrm{d}A<br />
\\[4pt]<br />
\mathrm{d}A ~\text{and not}~ A<br />
\\[4pt]<br />
A ~\text{and not}~ \mathrm{d}A<br />
\\[4pt]<br />
A ~\text{and}~ \mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot A \land \lnot \mathrm{d}A<br />
\\[4pt]<br />
\lnot A \land \mathrm{d}A<br />
\\[4pt]<br />
A \land \lnot \mathrm{d}A<br />
\\[4pt]<br />
A \land \mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
g_{3}<br />
\\[4pt]<br />
g_{12}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{)}<br />
\\[4pt]<br />
A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ A<br />
\\[4pt]<br />
A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot A<br />
\\[4pt]<br />
A<br />
\end{matrix}\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{6}<br />
\\[4pt]<br />
g_{9}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[4pt]<br />
1~0~0~1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{,} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
\texttt{((} A \texttt{,} \mathrm{d}A \texttt{))}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
A ~\text{not equal to}~ \mathrm{d}A<br />
\\[4pt]<br />
A ~\text{equal to}~ \mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
A \ne \mathrm{d}A<br />
\\[4pt]<br />
A = \mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{5}<br />
\\[4pt]<br />
g_{10}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
\mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ \mathrm{d}A<br />
\\[4pt]<br />
\mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot \mathrm{d}A<br />
\\[4pt]<br />
\mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{7}<br />
\\[4pt]<br />
g_{11}<br />
\\[4pt]<br />
g_{13}<br />
\\[4pt]<br />
g_{14}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} ~ A ~~ \mathrm{d}A ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{(} ~ A ~ \texttt{(} \mathrm{d}A \texttt{))}<br />
\\[4pt]<br />
\texttt{((} A \texttt{)} ~ \mathrm{d}A ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{((} A \texttt{)(} \mathrm{d}A \texttt{))}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not both}~ A ~\text{and}~ \mathrm{d}A<br />
\\[4pt]<br />
\text{not}~ A ~\text{without}~ \mathrm{d}A<br />
\\[4pt]<br />
\text{not}~ \mathrm{d}A ~\text{without}~ A<br />
\\[4pt]<br />
A ~\text{or}~ \mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot A \lor \lnot \mathrm{d}A<br />
\\[4pt]<br />
A \Rightarrow \mathrm{d}A<br />
\\[4pt]<br />
A \Leftarrow \mathrm{d}A<br />
\\[4pt]<br />
A \lor \mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
| <math>f_{3}\!</math><br />
| <math>g_{15}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
Aside from changing the names of variables and shuffling the order of rows, this Table follows the format that was used previously for boolean functions of two variables. The rows are grouped to reflect natural similarity classes among the propositions. In a future discussion, these classes will be given additional explanation and motivation as the orbits of a certain transformation group acting on the set of 16 propositions. Notice that four of the propositions, in their logical expressions, resemble those given in the table for <math>X^\uparrow.\!</math> Thus the first set of propositions <math>\{ f_i \}\!</math> is automatically embedded in the present set <math>\{ g_j \}\!</math> and the corresponding inclusions are indicated at the far left margin of the Table.<br />
<br />
===Tacit Extensions===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
I would really like to have slipped imperceptibly into this lecture, as into all the others I shall be delivering, perhaps over the years ahead.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
Strictly speaking, however, there is a subtle distinction in type between the function <math>f_i : X \to \mathbb{B}</math> and the corresponding function <math>g_j : \mathrm{E}X \to \mathbb{B},</math> even though they share the same logical expression. Naturally, we want to maintain the logical equivalence of expressions that represent the same proposition while appreciating the full diversity of that proposition's functional and typical representatives. Both perspectives, and all the levels of abstraction extending through them, have their reasons, as will develop in time.<br />
<br />
Because this special circumstance points up an important general theme, it is a good idea to discuss it more carefully. Whenever there arises a situation like this, where one alphabet <math>\mathcal{X}</math> is a subset of another alphabet <math>\mathcal{Y},</math> then we say that any proposition <math>f : \langle \mathcal{X} \rangle \to \mathbb{B}</math> has a ''tacit extension'' to a proposition <math>\boldsymbol\varepsilon f : \langle \mathcal{Y} \rangle \to \mathbb{B},\!</math> and that the space <math>(\langle \mathcal{X} \rangle \to \mathbb{B})</math> has an ''automatic embedding'' within the space <math>(\langle \mathcal{Y} \rangle \to \mathbb{B}).</math> The extension is defined in such a way that <math>\boldsymbol\varepsilon f\!</math> puts the same constraint on the variables of <math>\mathcal{X}</math> that are contained in <math>\mathcal{Y}</math> as the proposition <math>f\!</math> initially did, while it puts no constraint on the variables of <math>\mathcal{Y}</math> outside of <math>\mathcal{X},</math> in effect, conjoining the two constraints.<br />
<br />
If the variables in question are indexed as <math>\mathcal{X} = \{ x_1, \ldots, x_n \}</math> and <math>\mathcal{Y} = \{ x_1, \ldots, x_n, \ldots, x_{n+k} \},</math> then the definition of the tacit extension from <math>\mathcal{X}</math> to <math>\mathcal{Y}</math> may be expressed in the form of an equation:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\boldsymbol\varepsilon f(x_1, \ldots, x_n, \ldots, x_{n+k}) ~=~ f(x_1, \ldots, x_n).\!</math><br />
|}<br />
<br />
On formal occasions, such as the present context of definition, the tacit extension from <math>\mathcal{X}</math> to <math>\mathcal{Y}</math> is explicitly symbolized by the operator <math>\boldsymbol\varepsilon : (\langle \mathcal{X} \rangle \to \mathbb{B}) \to (\langle \mathcal{Y} \rangle \to \mathbb{B}),</math> where the appropriate alphabets <math>\mathcal{X}</math> and <math>\mathcal{Y}</math> are understood from context, but normally one may leave the "<math>\boldsymbol\varepsilon\!</math>" silent.<br />
<br />
Let's explore what this means for the present Example. Here, <math>\mathcal{X} = \{ A \}</math> and <math>\mathcal{Y} = \mathrm{E}\mathcal{X} = \{ A, \mathrm{d}A \}.</math> For each of the propositions <math>f_i\!</math> over <math>X\!,</math> specifically, those whose expression <math>e_i\!</math> lies in the collection <math>\{ 0, \texttt{(} A \texttt{)}, A, 1 \},\!</math> the tacit extension <math>\boldsymbol\varepsilon f\!</math> of <math>f\!</math> to <math>\mathrm{E}X</math> can be phrased as a logical conjunction of two factors, <math>f_i = e_i \cdot \tau ~ ,\!</math> where <math>\tau\!</math> is a logical tautology that uses all the variables of <math>\mathcal{Y} - \mathcal{X}.</math> Working in these terms, the tacit extensions <math>\boldsymbol\varepsilon f\!</math> of <math>f\!</math> to <math>\mathrm{E}X</math> may be explicated as shown in Table&nbsp;15.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:60%"<br />
|+ style="height:30px" | <math>\text{Table 15.} ~~ \text{Tacit Extension of}~ [A] ~\text{to}~ [A, \mathrm{d}A]\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0<br />
& = & 0 & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))} & = & & 0<br />
\\[8pt]<br />
\texttt{(} A \texttt{)}<br />
& = & \texttt{(} A \texttt{)} & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))}<br />
& = & \texttt{(} A \texttt{)} \, \mathrm{d}A ~ & + & \texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)}<br />
\\[8pt]<br />
A<br />
& = & ~A~ & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))}<br />
& = & ~A~ ~\mathrm{d}A~ & + & ~A~ \texttt{(} \mathrm{d}A \texttt{)}<br />
\\[8pt]<br />
1<br />
& = & 1 & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))} & = & & 1<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
In its effect on the singular propositions over <math>X,\!</math> this analysis has an interesting interpretation. The tacit extension takes us from thinking about a particular state, like <math>A\!</math> or <math>\texttt{(} A \texttt{)},\!</math> to considering the collection of outcomes, the outgoing changes or the singular dispositions, that spring from that state.<br />
<br />
===Example 2. Drives and Their Vicissitudes===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
I open my scuttle at night and see the far-sprinkled systems,<br><br />
And all I see, multiplied as high as I can cipher, edge but<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the rim of the farther systems.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 81]<br />
|}<br />
<br />
Before we leave the one-feature case let's look at a more substantial example, one that illustrates a general class of curves that can be charted through the extended feature spaces and that provides an opportunity to discuss a number of important themes concerning their structure and dynamics.<br />
<br />
Again, let <math>\mathcal{X} = \{ x_1 \} = \{ A \}.\!</math> In the discussion that follows we will consider a class of trajectories having the property that <math>\mathrm{d}^k A = 0\!</math> for all <math>k\!</math> greater than some fixed <math>m\!</math> and we may indulge in the use of some picturesque terms that describe salient classes of such curves. Given the finite order condition, there is a highest order non-zero difference <math>\mathrm{d}^m A\!</math> exhibited at each point in the course of any determinate trajectory that one may wish to consider. With respect to any point of the corresponding orbit or curve let us call this highest order differential feature <math>\mathrm{d}^m A\!</math> the ''drive'' at that point. Curves of constant drive <math>\mathrm{d}^m A\!</math> are then referred to as ''<math>m^\text{th}\!</math>-gear curves''.<br />
<br />
* '''Scholium.''' The fact that a difference calculus can be developed for boolean functions is well known [Fuji], [Koh, &sect; 8-4] and was probably familiar to Boole, who was an expert in difference equations before he turned to logic. And of course there is the strange but true story of how the Turin machines of the 1840s prefigured the Turing machines of the 1940s [Men, 225-297]. At the very outset of general purpose, mechanized computing we find that the motive power driving the Analytical Engine of Babbage, the kernel of an idea behind all of his wheels, was exactly his notion that difference operations, suitably trained, can serve as universal joints for any conceivable computation [M&M], [Mel, ch. 4].<br />
<br />
Given this language, the Example we take up here can be described as the family of <math>4^\text{th}\!</math>-gear curves through <math>\mathrm{E}^4 X\!</math> <math>=\!</math> <math>\langle A, ~\mathrm{d}A, ~\mathrm{d}^2\!A, ~\mathrm{d}^3\!A, ~\mathrm{d}^4\!A \rangle.</math> These are the trajectories generated subject to the dynamic law <math>\mathrm{d}^4 A = 1,\!</math> where it is understood in such a statement that all higher order differences are equal to <math>0.\!</math> Since <math>\mathrm{d}^4 A\!</math> and all higher <math>\mathrm{d}^k A\!</math> are fixed, the temporal or transitional conditions (initial, mediate, terminal &mdash; transient or stable states) vary only with respect to their projections as points of <math>\mathrm{E}^3 X = \langle A, ~\mathrm{d}A, ~\mathrm{d}^2\!A, ~\mathrm{d}^3\!A \rangle.</math> Thus, there is just enough space in a planar venn diagram to plot all of these orbits and to show how they partition the points of <math>\mathrm{E}^3 X.\!</math> It turns out that there are exactly two possible orbits, of eight points each, as illustrated in Figure&nbsp;16.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 16 -- A Couple of Fourth Gear Orbits.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 16.} ~~ \text{A Couple of Fourth Gear Orbits}\!</math><br />
|}<br />
<br />
With a little thought it is possible to devise an indexing scheme for the general run of dynamic states that allows for comparing universes of discourse that weigh in on different scales of observation. With this end in sight, let us index the states <math>q \in \mathrm{E}^m X\!</math> with the dyadic rationals (or the binary fractions) in the half-open interval <math>[0, 2).\!</math> Formally and canonically, a state <math>q_r\!</math> is indexed by a fraction <math>r = \tfrac{s}{t}\!</math> whose denominator is the power of two <math>t = 2^m\!</math> and whose numerator is a binary numeral formed from the coefficients of state in a manner to be described next. The ''differential coefficients'' of the state <math>q\!</math> are just the values <math>\mathrm{d}^k\!A(q)</math> for <math>k = 0 ~\text{to}~ m,\!</math> where <math>\mathrm{d}^0\!A</math> is defined as being identical to <math>A.\!</math> To form the binary index <math>d_0.d_1 \ldots d_m\!</math> of the state <math>q\!</math> the coefficient <math>\mathrm{d}^k\!A(q)</math> is read off as the binary digit <math>d_k\!</math> associated with the place value <math>2^{-k}.\!</math> Expressed by way of algebraic formulas, the rational index <math>r\!</math> of the state <math>q\!</math> can be given by the following equivalent formulations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
r(q)<br />
& = &<br />
\displaystyle\sum_k d_k \cdot 2^{-k}<br />
& = &<br />
\displaystyle\sum_k \text{d}^k A(q) \cdot 2^{-k}<br />
\\[8pt]<br />
=<br />
\\[8pt]<br />
\displaystyle\frac{s(q)}{t}<br />
& = &<br />
\displaystyle\frac{\sum_k d_k \cdot 2^{(m-k)}}{2^m}<br />
& = &<br />
\displaystyle\frac{\sum_k \text{d}^k A(q) \cdot 2^{(m-k)}}{2^m}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Applied to the example of <math>4^\text{th}\!</math>-gear curves, this scheme results in the data of Tables&nbsp;17-a and 17-b, which exhibit one period for each orbit. The states in each orbit are listed as ordered pairs <math>(p_i, q_j),\!</math> where <math>p_i\!</math> may be read as a temporal parameter that indicates the present time of the state and where <math>j\!</math> is the decimal equivalent of the binary numeral <math>s.\!</math> Informally and more casually, the Tables exhibit the states <math>q_s\!</math> as subscripted with the numerators of their rational indices, taking for granted the constant denominators of <math>2^m\! = 2^4 = 16.\!</math> In this set-up the temporal successions of states can be reckoned as given by a kind of ''parallel round-up rule''. That is, if <math>(d_k, d_{k+1})\!</math> is any pair of adjacent digits in the state index <math>r,\!</math> then the value of <math>d_k\!</math> in the next state is <math>{d_k}' = d_k + d_{k+1}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:55%"<br />
|+ style="height:30px" | <math>\text{Table 17-a.} ~~ \text{A Couple of Orbits in Fourth Gear : Orbit 1}\!</math><br />
|- style="background:ghostwhite"<br />
| <math>\text{Time}\!</math><br />
| <math>\text{State}\!</math><br />
| <math>A\!</math><br />
| <math>\mathrm{d}A\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| <math>p_i\!</math><br />
| <math>q_j\!</math><br />
| <math>\mathrm{d}^0\!A</math><br />
| <math>\mathrm{d}^1\!A</math><br />
| <math>\mathrm{d}^2\!A</math><br />
| <math>\mathrm{d}^3\!A</math><br />
| <math>\mathrm{d}^4\!A</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
p_0<br />
\\[4pt]<br />
p_1<br />
\\[4pt]<br />
p_2<br />
\\[4pt]<br />
p_3<br />
\\[4pt]<br />
p_4<br />
\\[4pt]<br />
p_5<br />
\\[4pt]<br />
p_6<br />
\\[4pt]<br />
p_7<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
q_{01}<br />
\\[4pt]<br />
q_{03}<br />
\\[4pt]<br />
q_{05}<br />
\\[4pt]<br />
q_{15}<br />
\\[4pt]<br />
q_{17}<br />
\\[4pt]<br />
q_{19}<br />
\\[4pt]<br />
q_{21}<br />
\\[4pt]<br />
q_{31}<br />
\end{matrix}\!</math><br />
| colspan="5" |<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="text-align:center; width:100%"<br />
|<br />
<math>\begin{matrix}<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
1.<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:55%"<br />
|+ style="height:30px" | <math>\text{Table 17-b.} ~~ \text{A Couple of Orbits in Fourth Gear : Orbit 2}\!</math><br />
|- style="background:ghostwhite"<br />
| <math>\text{Time}\!</math><br />
| <math>\text{State}\!</math><br />
| <math>A\!</math><br />
| <math>\mathrm{d}A\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| <math>p_i\!</math><br />
| <math>q_j\!</math><br />
| <math>\mathrm{d}^0\!A</math><br />
| <math>\mathrm{d}^1\!A</math><br />
| <math>\mathrm{d}^2\!A</math><br />
| <math>\mathrm{d}^3\!A</math><br />
| <math>\mathrm{d}^4\!A</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
p_0<br />
\\[4pt]<br />
p_1<br />
\\[4pt]<br />
p_2<br />
\\[4pt]<br />
p_3<br />
\\[4pt]<br />
p_4<br />
\\[4pt]<br />
p_5<br />
\\[4pt]<br />
p_6<br />
\\[4pt]<br />
p_7<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
q_{25}<br />
\\[4pt]<br />
q_{11}<br />
\\[4pt]<br />
q_{29}<br />
\\[4pt]<br />
q_{07}<br />
\\[4pt]<br />
q_{09}<br />
\\[4pt]<br />
q_{27}<br />
\\[4pt]<br />
q_{13}<br />
\\[4pt]<br />
q_{23}<br />
\end{matrix}\!</math><br />
| colspan="5" |<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="text-align:center; width:100%"<br />
|<br />
<math>\begin{matrix}<br />
1.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
==Transformations of Discourse==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
It is understandable that an engineer should be completely absorbed in his speciality, instead of pouring himself out into the freedom and vastness of the world of thought, even though his machines are being sent off to the ends of the earth; for he no more needs to be capable of applying to his own personal soul what is daring and new in the soul of his subject than a machine is in fact capable of applying to itself the differential calculus on which it is based. The same thing cannot, however, be said about mathematics; for here we have the new method of thought, pure intellect, the very well-spring of the times, the ''fons et origo'' of an unfathomable transformation.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 39]<br />
|}<br />
<br />
In this section we take up the general study of logical transformations, or maps that relate one universe of discourse to another. In many ways, and especially as applied to the subject of intelligent dynamic systems, my argument develops the antithesis of the statement just quoted. Along the way, if incidental to my ends, I hope this essay can pose a fittingly irenic epitaph to the frankly ironic epigraph inscribed at its head.<br />
<br />
My goal in this section is to answer a single question: What is a propositional tangent functor? In other words, my aim is to develop a clear conception of what manner of thing would pass in the logical realm for a genuine analogue of the tangent functor, an object conceived to generalize as far as possible in the abstract terms of category theory the ordinary notions of functional differentiation and the all too familiar operations of taking derivatives.<br />
<br />
As a first step I discuss the kinds of transformations that we already know as ''extensions'' and ''projections'', and I use these special cases to illustrate several different styles of logical and visual representation that will figure heavily in the sequel.<br />
<br />
===Foreshadowing Transformations : Extensions and Projections of Discourse===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
And, despite the care which she took to look behind her at every moment, she failed to see a shadow which followed her like her own shadow, which stopped when she stopped, which started again when she did, and which made no more noise than a well-conducted shadow should.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 126]<br />
|}<br />
<br />
Many times in our discussion we have occasion to place one universe of discourse in the context of a larger universe of discourse. An embedding of the general type <math>[\mathcal{X}] \to [\mathcal{Y}]\!</math> is implied any time that we make use of one alphabet <math>[\mathcal{X}]\!</math> that happens to be included in another alphabet <math>[\mathcal{Y}].\!</math> When we are discussing differential issues we usually have in mind that the extended alphabet <math>[\mathcal{Y}]\!</math> has a special construction or a specific lexical relation with respect to the initial alphabet <math>[\mathcal{X}],\!</math> one that is marked by characteristic types of accents, indices, or inflected forms.<br />
<br />
====Extension from 1 to 2 Dimensions====<br />
<br />
Figure 18-a lays out the ''angular form'' of venn diagram for universes of 1 and 2 dimensions, indicating the embedding map of type <math>\mathbb{B}^1 \to \mathbb{B}^2\!</math> and detailing the coordinates that are associated with individual cells. Because all points, cells, or logical interpretations are represented as connected geometric areas, we can say that these pictures provide us with an ''areal view'' of each universe of discourse.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-a -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-a.} ~~ \text{Extension from 1 to 2 Dimensions : Areal}\!</math><br />
|}<br />
<br />
Figure 18-b shows the differential extension from <math>X^\bullet = [x]\!</math> to <math>\mathrm{E}X^\bullet = [x, \mathrm{d}x]\!</math> in a ''bundle of boxes'' form of venn diagram. As awkward as it may seem at first, this type of picture is often the most natural and the most easily available representation when we want to conceptualize the localized information or momentary knowledge of an intelligent dynamic system. It gives a ready picture of a ''proposition at a point'', in the present instance, of a proposition about changing states which is itself associated with a particular dynamic state of a system. It is easy to see how this application might be extended to conceive of more general types of instantaneous knowledge that are possessed by a system.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-b -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-b.} ~~ \text{Extension from 1 to 2 Dimensions : Bundle}\!</math><br />
|}<br />
<br />
Figure 18-c shows the same extension in a ''compact'' style of venn diagram, where the differential features at each position are represented by arrows extending from that position that cross or do not cross, as the case may be, the corresponding feature boundaries.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-c -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-c.} ~~ \text{Extension from 1 to 2 Dimensions : Compact}\!</math><br />
|}<br />
<br />
Figure 18-d compresses the picture of the differential extension even further, yielding a directed graph or ''digraph'' form of representation. (Notice that my definition of a digraph allows for loops or ''slings'' at individual points, in addition to arcs or ''arrows'' between the points.)<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-d -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-d.} ~~ \text{Extension from 1 to 2 Dimensions : Digraph}\!</math><br />
|}<br />
<br />
====Extension from 2 to 4 Dimensions====<br />
<br />
Figure 19-a lays out the ''areal view'' or the ''angular form'' of venn diagram for universes of 2 and 4 dimensions, indicating the embedding map of type <math>\mathbb{B}^2 \to \mathbb{B}^4.\!</math> In many ways these pictures are the best kind there is, giving full canvass to an ideal vista. Their style allows the clearest, the fairest, and the plainest view that we can form of a universe of discourse, affording equal representation to all dispositions and maintaining a balance with respect to ordinary and differential features. If only we could extend this view! Unluckily, an obvious difficulty beclouds this prospect, and that is how precipitately we run into the limits of our plane and visual intuitions. Even within the scope of the spare few dimensions that we have scanned up to this point subtle discrepancies have crept in already. The circumstances that bind us and the frameworks that block us, the flat distortion of the planar projection and the inevitable ineffability that precludes us from wrapping its rhomb figure into rings around a torus, all of these factors disguise the underlying but true connectivity of the universe of discourse.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-a -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-a.} ~~ \text{Extension from 2 to 4 Dimensions : Areal}\!</math><br />
|}<br />
<br />
Figure 19-b shows the differential extension from <math>U^\bullet = [u, v]\!</math> to <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v]\!</math> in the ''bundle of boxes'' form of venn diagram.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-b -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-b.} ~~ \text{Extension from 2 to 4 Dimensions : Bundle}\!</math><br />
|}<br />
<br />
As dimensions increase, this factorization of the extended universe along the lines that are marked out by the bundle picture begins to look more and more like a practical necessity. But whenever we use a propositional model to address a real situation in the context of nature we need to remain aware that this articulation into factors, affecting our description, may be wholly artificial in nature and cleave to nothing, no joint in nature, nor any juncture in time to be in or out of joint.<br />
<br />
Figure 19-c illustrates the extension from 2 to 4 dimensions in the ''compact'' style of venn diagram. Here, just the changes with respect to the center cell are shown.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-c -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-c.} ~~ \text{Extension from 2 to 4 Dimensions : Compact}\!</math><br />
|}<br />
<br />
Figure 19-d gives the ''digraph'' form of representation for the differential extension <math>U^\bullet \to \mathrm{E}U^\bullet,\!</math> where the 4 nodes marked with a circle <math>{}^{\bigcirc}\!</math> are the cells <math>uv,\, u \texttt{(} v \texttt{)},\, \texttt{(} u \texttt{)} v,\, \texttt{(} u \texttt{)(} v \texttt{)},\!</math> respectively, and where a 2-headed arc counts as 2 arcs of the differential digraph.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-d -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-d.} ~~ \text{Extension from 2 to 4 Dimensions : Digraph}\!</math><br />
|}<br />
<br />
===Thematization of Functions : And a Declaration of Independence for Variables===<br />
<br />
{| width="100%"<br />
| align="left" |<br />
''And as imagination bodies forth''<br><br />
''The forms of things unknown, the poet's pen''<br><br />
''Turns them to shapes, and gives to airy nothing''<br><br />
''A local habitation and a name.''<br />
| align="right" valign="bottom" | A Midsummer Night's Dream, 5.1.18<br />
|}<br />
<br />
In the representation of propositions as functions it is possible to notice different degrees of explicitness in the way their functional character is symbolized. To indicate what I mean by this, the next series of Figures illustrates a set of graphic conventions that will be put to frequent use in the remainder of this discussion, both to mark the relevant distinctions and to help us convert between related expressions at different levels of explicitness in their functionality.<br />
<br />
====Thematization : Venn Diagrams====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
The known universe has one complete lover and that is the greatest poet. He consumes an eternal passion and is indifferent which chance happens and which possible contingency of fortune or misfortune and persuades daily and hourly his delicious pay.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 11&ndash;12]<br />
|}<br />
<br />
Figure 20-i traces the first couple of steps in this order of ''thematic'' progression, that will gradually run the gamut through a complete series of degrees of functional explicitness in the expression of logical propositions. The first venn diagram represents a situation where the function is indicated by a shaded figure and a logical expression. At this stage one may be thinking of the proposition only as expressed by a formula in a particular language and its content only as a subset of the universe of discourse, as when considering the proposition <math>u\!\cdot\!v</math> in the universe <math>[u, v].\!</math> The second venn diagram depicts a situation in which two significant steps have been taken. First, one has taken the trouble to give the proposition <math>u\!\cdot\!v</math> a distinctive functional name <math>{}^{\backprime\backprime} J {}^{\prime\prime}.\!</math> Second, one has come to think explicitly about the target domain that contains the functional values of <math>J,\!</math> as when writing <math>J : \langle u, v \rangle \to \mathbb{B}.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 20-i -- Thematization of Conjunction (Stage 1).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 20-i.} ~~ \text{Thematization of Conjunction (Stage 1)}\!</math><br />
|}<br />
<br />
In Figure 20-ii the proposition <math>J\!</math> is viewed explicitly as a transformation from one universe of discourse to another.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 20-ii -- Thematization of Conjunction (Stage 2).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 20-ii.} ~~ \text{Thematization of Conjunction (Stage 2)}\!</math><br />
|}<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
o-------------------------------o o-------------------------------o<br />
| | | |<br />
| o-----o o-----o | | o-----o o-----o |<br />
| / \ / \ | | / \ / \ |<br />
| / o \ | | / o \ |<br />
| / /`\ \ | | / /`\ \ |<br />
| o o```o o | | o o```o o |<br />
| | u |```| v | | | | u |```| v | |<br />
| o o```o o | | o o```o o |<br />
| \ \`/ / | | \ \`/ / |<br />
| \ o / | | \ o / |<br />
| \ / \ / | | \ / \ / |<br />
| o-----o o-----o | | o-----o o-----o |<br />
| | | |<br />
o-------------------------------o o-------------------------------o<br />
\ / \ /<br />
\ / \ /<br />
\ / \ J /<br />
\ / \ /<br />
\ / \ /<br />
o----------\---------/----------o o----------\---------/----------o<br />
| \ / | | \ / |<br />
| \ / | | \ / |<br />
| o-----@-----o | | o-----@-----o |<br />
| /`````````````\ | | /`````````````\ |<br />
| /```````````````\ | | /```````````````\ |<br />
| /`````````````````\ | | /`````````````````\ |<br />
| o```````````````````o | | o```````````````````o |<br />
| |```````````````````| | | |```````````````````| |<br />
| |```````` J ````````| | | |```````` x ````````| |<br />
| |```````````````````| | | |```````````````````| |<br />
| o```````````````````o | | o```````````````````o |<br />
| \`````````````````/ | | \`````````````````/ |<br />
| \```````````````/ | | \```````````````/ |<br />
| \`````````````/ | | \`````````````/ |<br />
| o-----------o | | o-----------o |<br />
| | | |<br />
| | | |<br />
o-------------------------------o o-------------------------------o<br />
J = u v x = J<u, v><br />
<br />
Figure 20-ii. Thematization of Conjunction (Stage 2)<br />
</pre><br />
|}<br />
<br />
In the first venn diagram the name that is assigned to a composite proposition, function, or region in the source universe is delegated to a simple feature in the target universe. This can result in a single character or term exceeding the responsibilities it can carry off well. Allowing the name of a function <math>J : \langle u, v \rangle \to \mathbb{B}\!</math> to serve as the name of its dependent variable <math>J : \mathbb{B}\!</math> does not mean that one has to confuse a function with any of its values, but it does put one at risk for a number of obvious problems, and we should not be surprised, on numerous and limiting occasions, when quibbling arises from the attempts of a too original syntax to serve these two masters.<br />
<br />
The second venn diagram circumvents these difficulties by introducing a new variable name for each basic feature of the target universe, as when writing <math>J : \langle u, v \rangle \to \langle x \rangle,\!</math> and thereby assigns a concrete type <math>\langle x \rangle</math> to the abstract codomain <math>\mathbb{B}.\!</math> To make this induction of variables more formal one can append subscripts, as in <math>x_J,\!</math> to indicate the origin or derivation of the new characters. Or we may use a lexical modifier to convert function names into variable names, for example, associating the function name <math>J\!</math> with the variable name <math>\check{J}.\!</math> Thus we may think of <math>x = x_J = \check{J}\!</math> as the ''cache variable'' corresponding to the function <math>J\!</math> or the symbol <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> considered as a contingent variable.<br />
<br />
In Figure 20-iii we arrive at a stage where the functional equations <math>J = u\!\cdot\!v</math> and <math>x = u\!\cdot\!v</math> are regarded as propositions in their own right, reigning in and ruling over the 3-feature universes of discourse <math>[u, v, J]~\!</math> and <math>[u, v, x],\!</math> respectively. Subject to the cautions already noted, the function name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> can be reinterpreted as the name of a feature <math>\check{J}</math> and the equation <math>J = u\!\cdot\!v</math> can be read as the logical equivalence <math>\texttt{((} J, u ~ v \texttt{))}.\!</math> To give it a generic name let us call this newly expressed, collateral proposition the ''thematization'' or the ''thematic extension'' of the original proposition <math>J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 20-iii -- Thematization of Conjunction (Stage 3).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 20-iii.} ~~ \text{Thematization of Conjunction (Stage 3)}\!</math><br />
|}<br />
<br />
The first venn diagram represents the thematization of the conjunction <math>J\!</math> with shading in the appropriate regions of the universe <math>[u, v, J].\!</math> Also, it illustrates a quick way of constructing a thematic extension. First, draw a line, in practice or the imagination, that bisects every cell of the original universe, placing half of each cell under the aegis of the thematized proposition and the other half under its antithesis. Next, on the scene where the theme applies leave the shade wherever it lies, and off the stage, where it plays otherwise, stagger the pattern in a harlequin guise.<br />
<br />
In the final venn diagram of this sequence the thematic progression comes full circle and completes one round of its development. The ambiguities that were occasioned by the changing role of the name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> are resolved by introducing a new variable name <math>{}^{\backprime\backprime} x {}^{\prime\prime}</math> to take the place of <math>\check{J},\!</math> and the region that represents this fresh featured <math>x\!</math> is circumscribed in a more conventional symmetry of form and placement. Just as we once gave the name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> to the proposition <math>u\!\cdot\!v,</math> we now give the name <math>{}^{\backprime\backprime} \iota {}^{\prime\prime}</math> to its thematization <math>\texttt{((} x, u ~ v \texttt{))}.\!</math> Already, again, at this culminating stage of reflection, we begin to think of the newly named proposition as a distinctive individual, a particular function <math>\iota : \langle u, v, x \rangle \to \mathbb{B}.\!</math><br />
<br />
From now on, the terms ''thematic extension'' and ''thematization'' will be used to describe both the process and degree of explication that progresses through this series of pictures, both the operation of increasingly explicit symbolization and the dimension of variation that is swept out by it. To speak of this change in general, that takes us in our current example from <math>J\!</math> to <math>\iota,\!</math> we introduce a class of operators symbolized by the Greek letter <math>\theta,\!</math> writing <math>\iota = \theta J\!</math> in the present instance. The operator <math>\theta,\!</math> in the present situation bearing the type <math>\theta : [u, v] \to [u, v, x],\!</math> provides us with a convenient way of recapitulating and summarizing the complete cycle of thematic developments.<br />
<br />
Figure 21 shows how the thematic extension operator <math>\theta\!</math> acts on two further examples, the disjunction <math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math> and the equality <math>\texttt{((} u, v \texttt{))}.\!</math> Referring to the disjunction as <math>f(u, v)\!</math> and the equality as <math>f(u, v),\!</math> we may express the thematic extensions as <math>\varphi = \theta f\!</math> and <math>\gamma = \theta g.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 21 -- Thematization of Disjunction and Equality.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 21.} ~~ \text{Thematization of Disjunction and Equality}\!</math><br />
|}<br />
<br />
====Thematization : Truth Tables====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
That which distorts honest shapes or which creates unearthly beings or places or contingencies is a nuisance and a revolt.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 19]<br />
|}<br />
<br />
Tables 22 through 25 outline a method for computing the thematic extensions of propositions in terms of their coordinate values.<br />
<br />
A preliminary step, as illustrated in Table&nbsp;22, is to write out the truth table representations of the propositional forms whose thematic extensions one wants to compute, in the present instance, the functions <math>f(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math> and <math>g(u, v) = \texttt{((} u, v \texttt{))}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:50%"<br />
|+ style="height:30px" | <math>\text{Table 22.} ~~ \text{Disjunction}~ f ~\text{and Equality}~ g\!</math><br />
|- style="height:35px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black" | <math>f\!</math><br />
| <math>g\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Next, each propositional form is individually represented in the fashion shown in Tables&nbsp;23-i and 23-ii, using <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} g {}^{\prime\prime}\!</math> as function names and creating new variables <math>x\!</math> and <math>y\!</math> to hold the associated functional values. This pair of Tables outlines the first stage in the transition from the <math>2\!</math>-dimensional universes of <math>f\!</math> and <math>g\!</math> to the <math>3\!</math>-dimensional universes of <math>\theta f\!</math> and <math>\theta g.\!</math> The top halves of the Tables replicate the truth table patterns for <math>f\!</math> and <math>g\!</math> in the form <math>f : [u, v] \to [x]\!</math> and <math>g : [u, v] \to [y].\!</math> The bottom halves of the tables print the negatives of these pictures, as it were, and paste the truth tables for <math>\texttt{(} f \texttt{)}\!</math> and <math>\texttt{(} g \texttt{)}\!</math> under the copies for <math>f\!</math> and <math>g.\!</math> At this stage, the columns for <math>\theta f\!</math> and <math>\theta g\!</math> are appended almost as afterthoughts, amounting to indicator functions for the sets of ordered triples that make up the functions <math>f\!</math> and <math>g.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 23-i and 23-ii.} ~~ \text{Thematics of Disjunction and Equality (1)}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 23-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black" | <math>f\!</math><br />
| <math>x\!</math><br />
| <math>\varphi\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |<br />
<math>\begin{matrix}\to\\\to\\\to\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" | &nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 23-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black" | <math>g\!</math><br />
| <math>y\!</math><br />
| <math>\gamma\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |<br />
<math>\begin{matrix}\to\\\to\\\to\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" | &nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
All the data are now in place to give the truth tables for <math>\theta f\!</math> and <math>\theta g.\!</math> All that remains to be done is to permute the rows and change the roles of <math>x\!</math> and <math>y\!</math> from dependent to independent variables. In Tables&nbsp;24-i and 24-ii the rows are arranged in such a way as to put the 3-tuples <math>(u, v, x)\!</math> and <math>(u, v, y)\!</math> in binary numerical order, suitable for viewing as the arguments of the maps <math>\theta f = \varphi : [u, v, x] \to \mathbb{B}\!</math> and <math>\theta g = \gamma : [u, v, y] \to \mathbb{B}.\!</math> Moreover, the structure of the tables is altered slightly, allowing the now vestigial functions <math>\theta f\!</math> and <math>\theta g\!</math> to be passed over without further attention and shifting the heavy vertical bars a notch to the right. In effect, this clinches the fact that the thematic variables <math>x := \check{f}\!</math> and <math>y := \check{g}\!</math> are now treated as independent variables.<br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 24-i and 24-ii.} ~~ \text{Thematics of Disjunction and Equality (2)}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 24-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>f\!</math><br />
| <math>x\!</math><br />
| style="border-left:1px solid black" | <math>\varphi\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <font size="+2">&nbsp;<br></font><math>\begin{matrix}\to\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 24-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>g\!</math><br />
| <math>y\!</math><br />
| style="border-left:1px solid black" | <math>\gamma\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | &nbsp;<br><math>\begin{matrix}\to\\\to\end{matrix}\!</math><br>&nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
An optional reshuffling of the rows brings additional features of the thematic extensions to light. Leaving the columns in place for the sake of comparison, Tables&nbsp;25-i and 25-ii sort the rows in a different order, in effect treating <math>x\!</math> and <math>y\!</math> as the primary variables in their respective 3-tuples. Regarding the thematic extensions in the form <math>\varphi : [x, u, v] \to \mathbb{B}\!</math> and <math>\gamma : [y, u, v] \to \mathbb{B}\!</math> makes it easier to see in this tabular setting a property that was graphically obvious in the venn diagrams above. Specifically, when the thematic variable <math>\check{F}\!</math> is true then <math>\theta F\!</math> exhibits the pattern of the original <math>F,\!</math> and when <math>\check{F}\!</math> is false then <math>\theta F\!</math> exhibits the pattern of its negation <math>\texttt{(} F \texttt{)}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 25-i and 25-ii.} ~~ \text{Thematics of Disjunction and Equality (3)}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 25-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>f\!</math><br />
| <math>x\!</math><br />
| style="border-left:1px solid black" | <math>\varphi\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>{\to}\!</math><br><font size="+2">&nbsp;<br>&nbsp;<br>&nbsp;<br></font><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <font size="+2">&nbsp;<br></font><math>\begin{matrix}\to\\\to\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 25-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>g\!</math><br />
| <math>y\!</math><br />
| style="border-left:1px solid black" | <math>\gamma\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | &nbsp;<br><math>\begin{matrix}\to\\\to\end{matrix}\!</math><br>&nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
Finally, Tables&nbsp;26-i and 26-ii compare the tacit extensions <math>\boldsymbol\varepsilon : [u, v] \to [u, v, x]\!</math> and <math>\boldsymbol\varepsilon : [u, v] \to [u, v, y]\!</math> with the thematic extensions of the same types, as applied to the propositions <math>f\!</math> and <math>g,\!</math> respectively.<br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 26-i and 26-ii.} ~~ \text{Tacit Extension and Thematization}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 26-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>x\!</math><br />
| style="border-left:1px solid black" | <math>\boldsymbol\varepsilon f\!</math><br />
| <math>\theta f\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 26-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>y\!</math><br />
| style="border-left:1px solid black" | <math>\boldsymbol\varepsilon g\!</math><br />
| <math>\theta g\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
Table 27 summarizes the thematic extensions of all propositions on two variables. Column&nbsp;4 lists the equations of form <math>\texttt{((} \check{f_i}, f_i (u, v) \texttt{))}\!</math> and Column&nbsp;5 simplifies these equations into the form of algebraic expressions. As always, <math>{}^{\backprime\backprime} + {}^{\prime\prime}\!</math> refers to exclusive disjunction and each <math>{}^{\backprime\backprime} \check{f} {}^{\prime\prime}\!</math> appearing in the last two Columns refers to the corresponding variable name <math>{}^{\backprime\backprime} \check{f_i} {}^{\prime\prime}.~\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 27.} ~~ \text{Thematization of Bivariate Propositions}\!</math><br />
|- style="height:30px; background:ghostwhite"<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>{f}\!</math><br />
| <math>\theta f\!</math><br />
| <math>\theta f\!</math><br />
|- style="background:ghostwhite"<br />
| align="right" | <math>u\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| align="right" | <math>v\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| align="left" | <math>\texttt{((} \check{f} \texttt{,~(~)~))}\!</math><br />
| align="left" | <math>\check{f} + 1\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\\[4pt]<br />
f_{4}<br />
\\[4pt]<br />
f_{8}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
1~0~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} u \texttt{)~} v \texttt{~}<br />
\\[4pt]<br />
\texttt{~} u \texttt{~(} v \texttt{)}<br />
\\[4pt]<br />
\texttt{~} u \texttt{~~} v \texttt{~}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(u)(v)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~(u)~v~~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~u~(v)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~u~~v~~))}<br />
\end{array}</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + u + v + uv<br />
\\[4pt]<br />
\check{f} + v + uv + 1<br />
\\[4pt]<br />
\check{f} + u + uv + 1<br />
\\[4pt]<br />
\check{f} + uv + 1<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{3}<br />
\\[4pt]<br />
f_{12}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{)}<br />
\\[4pt]<br />
\texttt{~} u \texttt{~}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(u)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~u~~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + u<br />
\\[4pt]<br />
\check{f} + u + 1<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[4pt]<br />
f_{9}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[4pt]<br />
1~0~0~1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{,} v \texttt{)}<br />
\\[4pt]<br />
\texttt{((} u \texttt{,} v \texttt{))}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~~(} u \texttt{,} v \texttt{)~~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~((} u \texttt{,} v \texttt{))~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + u + v + 1<br />
\\[4pt]<br />
\check{f} + u + v<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{5}<br />
\\[4pt]<br />
f_{10}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} v \texttt{)}<br />
\\[4pt]<br />
\texttt{~} v \texttt{~}<br />
\end{matrix}</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(} v \texttt{)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~} v \texttt{~~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + v<br />
\\[4pt]<br />
\check{f} + v + 1<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[4pt]<br />
f_{11}<br />
\\[4pt]<br />
f_{13}<br />
\\[4pt]<br />
f_{14}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(~} u \texttt{~~} v \texttt{~)}<br />
\\[4pt]<br />
\texttt{(~} u \texttt{~(} v \texttt{))}<br />
\\[4pt]<br />
\texttt{((} u \texttt{)~} v \texttt{~)}<br />
\\[4pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(~} u \texttt{~~} v \texttt{~)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~(~} u \texttt{~(} v \texttt{))~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~((} u \texttt{)~} v \texttt{~)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~((} u \texttt{)(} v \texttt{))~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + uv<br />
\\[4pt]<br />
\check{f} + u + uv<br />
\\[4pt]<br />
\check{f} + v + uv<br />
\\[4pt]<br />
\check{f} + u + v + uv + 1<br />
\end{array}\!</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| align="left" | <math>\texttt{((} \check{f} \texttt{,~((~))~))}\!</math><br />
| align="left" | <math>\check{f}\!</math><br />
|}<br />
<br />
<br><br />
<br />
In order to show what all of the thematic extensions from two dimensions to three dimensions look like in terms of coordinates, Tables&nbsp;28 and 29 present ordinary truth tables for the functions <math>f_i : \mathbb{B}^2 \to \mathbb{B}\!</math> and for the corresponding thematizations <math>\theta f_i = \varphi_i : \mathbb{B}^3 \to \mathbb{B}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 28.} ~~ \text{Propositions on Two Variables}\!</math><br />
|- style="height:35px; background:ghostwhite"<br />
| style="width:5%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:5%; border-bottom:1px solid black; border-left:1px solid black" | <math>f_{0}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{1}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{2}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{3}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{4}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{5}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{6}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{7}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{8}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{9}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{10}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{11}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{12}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{13}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{14}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{15}\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 29.} ~~ \text{Thematic Extensions of Bivariate Propositions}\!</math><br />
|- style="height:35px; background:ghostwhite"<br />
| style="width:5%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\check{f}\!</math><br />
| style="width:5%; border-bottom:1px solid black; border-left:1px solid black" | <math>\varphi_{0}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{1}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{2}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{3}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{4}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{5}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{6}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{7}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{8}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{9}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{10}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{11}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{12}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{13}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{14}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{15}\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
|-<br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Propositional Transformations===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
If only the word &lsquo;artificial&rsquo; were associated with the idea of ''art'', or expert skill gained through voluntary apprenticeship (instead of suggesting the factitious and unreal), we might say that ''logical'' refers to artificial thought.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 56&ndash;57]<br />
|}<br />
<br />
In this section we develop a comprehensive set of concepts for dealing with transformations between universes of discourse. In this most general setting the source and target universes of a transformation are allowed to be different, but may be the same. When we apply these concepts to dynamic systems we focus on the important special case of transformations that map a universe into itself, regarding them as the state transitions of a discrete dynamical process and placing them among the myriad ways that a universe of discourse might change, and by that change turn into itself.<br />
<br />
====Alias and Alibi Transformations====<br />
<br />
There are customarily two modes of understanding a transformation, at least, when we try to interpret its relevance to something in reality. A transformation always refers to a changing prospect, to say it in a unified but equivocal way, but this can be taken to mean either a subjective change in the interpreting observer's point of view or an objective change in the systematic subject of discussion. In practice these variant uses of the transformation concept are distinguished in the following terms:<br />
<br />
# A ''perspectival'' or ''alias'' transformation refers to a shift in perspective or a change in language that takes place in the observer's frame of reference.<br />
# A ''transitional'' or ''alibi'' transformation refers to a change of position or an alteration of state that occurs in the object system as it falls under study.<br />
<br />
(For a recent discussion of the alias vs. alibi issue, as it relates to linear transformations in vector spaces and to other issues of an algebraic nature, see [MaB, 256, 582-4].)<br />
<br />
Naturally, when we are concerned with the dynamical properties of a system, the transitional aspect of transformation is the factor that comes to the fore, and this involves us in contemplating all of the ways of changing a universe into itself while remaining under the rule of established dynamical laws. In the prospective application to dynamic systems, and to neural networks viewed in this light, our interest lies chiefly with the transformations of a state space into itself that constitute the state transitions of a discrete dynamic process. Nevertheless, many important properties of these transformations, and some constructions that we need to see most clearly, are independent of the transitional interpretation and are likely to be confounded with irrelevant features if presented first and only in that association.<br />
<br />
In addition, and in partial contrast, intelligent systems are exactly that species of dynamic agents that have the capacity to have a point of view, and we cannot do justice to their peculiar properties without examining their ability to form and transform their own frames of reference in exposure to the elements of their own experience. In this setting, the perspectival aspect of transformation is the facet that shines most brightly, perhaps too often leaving us fascinated with mere glimmerings of its actual potential. It needs to be emphasized that nothing of the ordinary sort needs be moved in carrying out a transformation under the alias interpretation, that it may only involve a change in the forms of address, an amendment of the terms which are customed to approach and fashioned to describe the very same things in the very same world. But again, working within a discipline of realistic computation, we know how formidably complex and resource-consuming such transformations of perspective can be to implement in practice, much less to endow in the self-governed form of a nascently intelligent dynamical system.<br />
<br />
====Transformations of General Type====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
''Es ist passiert'', &ldquo;it just sort of happened&rdquo;, people said there when other people in other places thought heaven knows what had occurred. It was a peculiar phrase, not known in this sense to the Germans and with no equivalent in other languages, the very breath of it transforming facts and the bludgeonings of fate into something light as eiderdown, as thought itself.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 34]<br />
|}<br />
<br />
Consider the situation illustrated in Figure&nbsp;30, where the alphabets <math>\mathcal{U} = \{ u, v \}\!</math> and <math>\mathcal{X} = \{ x, y, z \}\!</math> are used to label basic features in two different logical universes, <math>U^\bullet = [u, v]\!</math> and <math>X^\bullet = [x, y, z].\!</math><br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
o-------------------------------------------------------o<br />
| U |<br />
| |<br />
| o-----------o o-----------o |<br />
| / \ / \ |<br />
| / o \ |<br />
| / / \ \ |<br />
| / / \ \ |<br />
| o o o o |<br />
| | | | | |<br />
| | u | | v | |<br />
| | | | | |<br />
| o o o o |<br />
| \ \ / / |<br />
| \ \ / / |<br />
| \ o / |<br />
| \ / \ / |<br />
| o-----------o o-----------o |<br />
| |<br />
| |<br />
o---------------------------o---------------------------o<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
o-------------------------o o-------------------------o o-------------------------o<br />
| U | | U | | U |<br />
| o---o o---o | | o---o o---o | | o---o o---o |<br />
| / \ / \ | | / \ / \ | | / \ / \ |<br />
| / o \ | | / o \ | | / o \ |<br />
| / / \ \ | | / / \ \ | | / / \ \ |<br />
| o o o o | | o o o o | | o o o o |<br />
| | u | | v | | | | u | | v | | | | u | | v | |<br />
| o o o o | | o o o o | | o o o o |<br />
| \ \ / / | | \ \ / / | | \ \ / / |<br />
| \ o / | | \ o / | | \ o / |<br />
| \ / \ / | | \ / \ / | | \ / \ / |<br />
| o---o o---o | | o---o o---o | | o---o o---o |<br />
| | | | | |<br />
o-------------------------o o-------------------------o o-------------------------o<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ g | \ f / | h /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ o----------|-----------\-----/-----------|----------o /<br />
\ | X | \ / | | /<br />
\ | | \ / | | /<br />
\ | | o-----o-----o | | /<br />
\| | / \ | |/<br />
\ | / \ | /<br />
|\ | / \ | /|<br />
| \ | / \ | / |<br />
| \ | / \ | / |<br />
| \ | o x o | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \| | | |/ |<br />
| o--o--------o o--------o--o |<br />
| / \ \ / / \ |<br />
| / \ \ / / \ |<br />
| / \ o / \ |<br />
| / \ / \ / \ |<br />
| / \ / \ / \ |<br />
| o o--o-----o--o o |<br />
| | | | | |<br />
| | | | | |<br />
| | | | | |<br />
| | y | | z | |<br />
| | | | | |<br />
| | | | | |<br />
| o o o o |<br />
| \ \ / / |<br />
| \ \ / / |<br />
| \ o / |<br />
| \ / \ / |<br />
| \ / \ / |<br />
| o-----------o o-----------o |<br />
| |<br />
| |<br />
o---------------------------------------------------o<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ p , q /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
o<br />
<br />
Figure 30. Generic Frame of a Logical Transformation<br />
</pre><br />
|}<br />
<br />
Enter the picture, as we usually do, in the middle of things, with features like <math>x, y , z\!</math> that present themselves to be simple enough in their own right and that form a satisfactory, if temporary foundation to provide a basis for discussion. In this universe and on these terms we find expression for various propositions and questions of principal interest to ourselves, as indicated by the maps <math>p, q : X \to \mathbb{B}.\!</math> Then we discover that the simple features <math>\{ x, y, z \}\!</math> are really more complex than we thought at first, and it becomes useful to regard them as functions <math>\{ f, g, h \}\!</math> of other features <math>\{ u, v \}\!</math> that we place in a preface to our original discourse, or suppose as topics of a preliminary universe of discourse <math>U^\bullet = [u, v].\!</math> It may happen that these late-blooming but pre-ambling features are found to lie closer, in a sense that may be our job to determine, to the central nature of the situation of interest, in which case they earn our regard as being more fundamental, but these functions and features are only required to supply a critical stance on the universe of discourse or an alternate perspective on the nature of things in order to be preserved as useful.<br />
<br />
A particular transformation <math>F : [u, v] \to [x, y, z]\!</math> may be expressed by a system of equations, as shown below. Here, <math>F\!</math> is defined by its component maps <math>F = (F_1, F_2, F_3) = (f, g, h),\!</math> where each component map in <math>\{ f, g, h \}\!</math> is a proposition of type <math>\mathbb{B}^n \to \mathbb{B}^1.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:50%"<br />
|<br />
<math>\begin{matrix}<br />
x & = & f(u, v)<br />
\\[10pt]<br />
y & = & g(u, v)<br />
\\[10pt]<br />
z & = & h(u, v)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Regarded as a logical statement, this system of equations expresses a relation between a collection of freely chosen propositions <math>\{ f, g, h \}\!</math> in one universe of discourse and the special collection of simple propositions <math>\{ x, y, z \}\!</math> on which is founded another universe of discourse. Growing familiarity with a particular transformation of discourse, and the desire to achieve a ready understanding of its implications, requires that we be able to convert this information about generals and simples into information about all the main subtypes of propositions, including the linear and singular propositions.<br />
<br />
===Analytic Expansions : Operators and Functors===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Consider what effects that might ''conceivably'' have practical bearings you ''conceive'' the objects of your ''conception'' to have. Then, your ''conception'' of those effects is the whole of your ''conception'' of the object.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; C.S. Peirce, &ldquo;The Maxim of Pragmatism&rdquo;, CP 5.438<br />
|}<br />
<br />
Given the barest idea of a logical transformation, as suggested by the sketch in Figure&nbsp;30, and having conceptualized the universe of discourse, with all of its points and propositions, as a beginning object of discussion, we are ready to enter the next phase of our investigation.<br />
<br />
====Operators on Propositions and Transformations====<br />
<br />
The next step is naturally inclined toward objects of the next higher order, namely, with operators that take in argument lists of logical transformations and that give back specified types of logical transformations as their results. For our present aims, we do not need to consider the most general class of such operators, nor any one of them for its own sake. Rather, we are interested in the special sorts of operators that arise in the study and analysis of logical transformations. Figuratively speaking, these operators serve as instruments for the live tomography (and hopefully not the vivisection) of the forms of change under view. Beyond that, they open up ways to implement the changes of view that we need to grasp all the variations on a transformational theme, or to appreciate enough of its significant features to &ldquo;get the drift&rdquo; of the change occurring, to form a passing acquaintance or a synthetic comprehension of its general character and disposition.<br />
<br />
The simplest type of operator is one that takes a single transformation as an argument and returns a single transformation as a result, and most of the operators explicitly considered in our discussion will be of this kind. Figure&nbsp;31 illustrates the typical situation.<br />
<br />
{| align="center" border="0" cellpadding="20"<br />
|<br />
<pre><br />
o---------------------------------------o<br />
| |<br />
| |<br />
| U% F X% |<br />
| o------------------>o |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| !W! | | !W! |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| v v |<br />
| o------------------>o |<br />
| !W!U% !W!F !W!X% |<br />
| |<br />
| |<br />
o---------------------------------------o<br />
Figure 31. Operator Diagram (1)<br />
</pre><br />
|}<br />
<br />
In this Figure <math>{}^{\backprime\backprime} \mathsf{W} {}^{\prime\prime}\!</math> stands for a generic operator <math>\mathsf{W},\!</math> in this case one that takes a logical transformation <math>F\!</math> of type <math>(U^\bullet \to X^\bullet)\!</math> into a logical transformation <math>\mathsf{W}F\!</math> of type <math>(\mathsf{W}U^\bullet \to \mathsf{W}X^\bullet).\!</math> Thus, the operator <math>\mathsf{W}\!</math> must be viewed as making assignments for both families of objects we have previously considered, that is, for universes of discourse like <math>{U^\bullet}\!</math> and <math>{X^\bullet}\!</math> and for logical transformations like <math>F.\!</math><br />
<br />
'''Note.''' Strictly speaking, an operator like <math>\mathsf{W}\!</math> works between two whole categories of universes and transformations, which we call the ''source'' and the ''target'' categories of <math>\mathsf{W}.\!</math> Given this setting, <math>\mathsf{W}\!</math> specifies for each universe <math>U^\bullet\!</math> in its source category a definite universe <math>\mathsf{W}U^\bullet\!</math> in its target category, and to each transformation <math>F\!</math> in its source category it assigns a unique transformation <math>\mathsf{W}F\!</math> in its target category. Naturally, this only works if <math>\mathsf{W}\!</math> takes the source <math>U^\bullet</math> and the target <math>X^\bullet</math> of the map <math>F\!</math> over to the source <math>\mathsf{W}U^\bullet\!</math> and the target <math>\mathsf{W}X^\bullet\!</math> of the map <math>\mathsf{W}F.\!</math> With luck or care enough, we can avoid ever having to put anything like that in words again, letting diagrams do the work. In the situations of present concern we are usually focused on a single transformation <math>F,\!</math> and thus we can take it for granted that the assignment of universes under <math>\mathsf{W}\!</math> is defined appropriately at the source and target ends of <math>F.\!</math> It is not always the case, though, that we need to use the particular names (like <math>{}^{\backprime\backprime} \mathsf{W}U^\bullet {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \mathsf{W}X^\bullet {}^{\prime\prime}\!</math>) that <math>\mathsf{W}\!</math> assigns by default to its operative image universes. In most contexts we will usually have a prior acquaintance with these universes under other names and it is necessary only that we can tell from the information associated with an operator <math>\mathsf{W}\!</math> what universes they are.<br />
<br />
In Figure&nbsp;31 the maps <math>F\!</math> and <math>\mathsf{W}F\!</math> are displayed horizontally, the way one normally orients functional arrows in a written text, and <math>\mathsf{W}\!</math> rolls the map <math>F\!</math> downward into the images that are associated with <math>\mathsf{W}F.\!</math> In Figure&nbsp;32 the same information is redrawn so that the maps <math>F\!</math> and <math>\mathsf{W}F\!</math> flow down the page, and <math>\mathsf{W}\!</math> unfurls the map <math>F\!</math> rightward into domains that are the eminent purview of <math>\mathsf{W}F.\!</math><br />
<br />
{| align="center" border="0" cellpadding="20"<br />
|<br />
<pre><br />
o---------------------------------------o<br />
| |<br />
| |<br />
| U% !W! !W!U% |<br />
| o------------------>o |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| F | | !W!F |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| v v |<br />
| o------------------>o |<br />
| X% !W! !W!X% |<br />
| |<br />
| |<br />
o---------------------------------------o<br />
Figure 32. Operator Diagram (2)<br />
</pre><br />
|}<br />
<br />
The latter arrangement, as exhibited in Figure&nbsp;32, is more congruent with the thinking about operators that we shall do in the rest of this discussion, since all logical transformations from here on out will be pictured vertically, after the fashion of Figure&nbsp;30.<br />
<br />
====Differential Analysis of Propositions and Transformations====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" | The resultant metaphysical problem now is this: ''Does the man go round the squirrel or not?''<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 43]<br />
|}<br />
<br />
The approach to the differential analysis of logical propositions and transformations of discourse to be pursued here is carried out in terms of particular operators <math>\mathsf{W}\!</math> that act on propositions <math>F\!</math> or on transformations <math>F\!</math> to yield the corresponding operator maps <math>\mathsf{W}F.\!</math> The operator results then become the subject of a series of further stages of analysis, which take them apart into their propositional components, rendering them as a set of purely logical constituents. After this is done, all the parts are then re-integrated to reconstruct the original object in the light of a more complete understanding, at least in ways that enable one to appreciate certain aspects of it with fresh insight.<br />
<br />
* '''Remark on Strategy.''' At this point we run into a set of conceptual difficulties that force us to make a strategic choice in how we proceed. Part of the problem can be remedied by extending our discussion of tacit extensions to the transformational context. But the troubles that remain are much more obstinate and lead us to try two different types of solution. The approach that we develop first makes use of a variant type of extension operator, the ''trope extension'', to be defined below. This method is more conservative and requires less preparation, but has features which make it seem unsatisfactory in the long run. A more radical approach, but one with a better hope of long term success, makes use of the notion of ''contingency spaces''. These are an even more generous type of extended universe than the kind we currently use, but are defined subject to certain internal constraints. The extra work needed to set up this method forces us to put it off to a later stage. However, as a compromise, and to prepare the ground for the next pass, we call attention to the various conceptual difficulties as they arise along the way and try to give an honest estimate of how well our first approach deals with them.<br />
<br />
We now describe in general terms the particular operators that are instrumental to this form of analysis. The main series of operators all have the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathsf{W}<br />
& : &<br />
( U^\bullet \to X^\bullet )<br />
& \to &<br />
( \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet )<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
If we assume that the source universe <math>U^\bullet</math> and the target universe <math>X^\bullet</math> have finite dimensions <math>n\!</math> and <math>k,\!</math> respectively, then each operator <math>\mathsf{W}\!</math> is encompassed by the same abstract type:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathsf{W}<br />
& : &<br />
( [\mathbb{B}^n] \to [\mathbb{B}^k] )<br />
& \to &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k \times \mathbb{D}^k] )<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Since the range features of the operator result <math>\mathsf{W}F : [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k \times \mathbb{D}^k]</math> can be sorted by their ordinary versus differential qualities and the component maps can be examined independently, the complete operator <math>\mathsf{W}\!</math> can be separated accordingly into two components, in the form <math>\mathsf{W} = (\boldsymbol\varepsilon, \mathrm{W}).\!</math> Given a fixed context of source and target universes, <math>\boldsymbol\varepsilon\!</math> is always the same type of operator, a multiple component version of the tacit extension operators that were described earlier. In this context <math>\boldsymbol\varepsilon\!</math> has the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{array}{lccccc}<br />
\text{Concrete type}<br />
& \boldsymbol\varepsilon<br />
& : &<br />
( U^\bullet \to X^\bullet )<br />
& \to &<br />
( \mathrm{E}U^\bullet \to X^\bullet )<br />
\\[10pt]<br />
\text{Abstract type}<br />
& \boldsymbol\varepsilon<br />
& : &<br />
( [\mathbb{B}^n] \to [\mathbb{B}^k] )<br />
& \to &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k] )<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
On the other hand, the operator <math>\mathrm{W}\!</math> is specific to each <math>\mathsf{W}.\!</math> In this context <math>\mathrm{W}\!</math> always has the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{array}{lccccc}<br />
\text{Concrete type}<br />
& W<br />
& : &<br />
( U^\bullet \to X^\bullet )<br />
& \to &<br />
( \mathrm{E}U^\bullet \to \mathrm{d}X^\bullet )<br />
\\[10pt]<br />
\text{Abstract type}<br />
& W<br />
& : &<br />
( [\mathbb{B}^n] \to [\mathbb{B}^k] )<br />
& \to &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{D}^k] )<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
In the types just assigned to <math>\boldsymbol\varepsilon\!</math> and <math>\mathrm{W}\!</math> and by implication to their results <math>\boldsymbol\varepsilon F\!</math> and <math>\mathrm{W}F,\!</math> we have listed the most restrictive ranges defined for them rather than the more expansive target spaces that subsume these ranges. When there is need to recognize both, we may use type indications like the following:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\boldsymbol\varepsilon F<br />
& : &<br />
( \mathrm{E}U^\bullet \to X^\bullet \subseteq \mathrm{E}X^\bullet )<br />
& \cong &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k] \subseteq [\mathbb{B}^k \times \mathbb{D}^k] )<br />
\\[10pt]<br />
WF<br />
& : &<br />
( \mathrm{E}U^\bullet \to \mathrm{d}X^\bullet \subseteq \mathrm{E}X^\bullet )<br />
& \cong &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{D}^k] \subseteq [\mathbb{B}^k \times \mathbb{D}^k] )<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Hopefully, though, a general appreciation of these subsumptions will prevent us from having to make such declarations more often than absolutely necessary.<br />
<br />
In giving names to these operators we try to preserve as much of the traditional nomenclature and as many of the classical associations as possible. The chief difficulty in doing this is occasioned by the distinction between the &ldquo;sans&nbsp;serif&rdquo; operators <math>\mathsf{W}\!</math> and their &ldquo;serified&rdquo; components <math>\mathrm{W},\!</math> which forces us to find two distinct but parallel sets of terminology. Here is a plan to that purpose. First, the component operators <math>\mathrm{W}\!</math> are named by analogy with the corresponding operators in the classical difference calculus. Next, the complete operators <math>\mathsf{W} = (\boldsymbol\varepsilon, \mathrm{W})</math> are assigned titles according to their roles in a geometric or trigonometric allegory, if only to ensure that the tangent functor, that belongs to this family and whose exposition we are still working toward, comes out fit with its customary name. Finally, the operator results <math>\mathsf{W}F\!</math> and <math>\mathrm{W}F\!</math> can be fixed in our frame of reference by tethering the operative adjective for <math>\mathsf{W}\!</math> or <math>\mathrm{W}\!</math> to the anchoring epithet &ldquo;map&rdquo;, in conformity with an already standard practice.<br />
<br />
=====The Secant Operator : '''E'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Mr. Peirce, after pointing out that our beliefs are really rules for action, said that, to develop a thought's meaning, we need only determine what conduct it is fitted to produce: that conduct is for us its sole significance.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 46]<br />
|}<br />
<br />
Figures&nbsp;33-i and 33-ii depict two stages in the form of analysis that will be applied to transformations throughout the remainder of this study. From now on our interest is staked on an operator denoted <math>{}^{\backprime\backprime} \mathsf{E} {}^{\prime\prime},\!</math> which receives the principal investment of analytic attention, and on the constituent parts of <math>\mathsf{E},\!</math> which derive their shares of significance as developed by the analysis. In the sequel, we refer to <math>\mathsf{E}\!</math> as the ''secant operator'', taking it for granted that a context has been chosen that defines its type. The secant operator has the component description <math>\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}),\!</math> and its active ingredient <math>\mathrm{E}\!</math> is known as the ''enlargement operator''. (Here, we name <math>\mathrm{E}\!</math> after the literal ancestor of the shift operator in the calculus of finite differences, defined so that <math>\mathrm{E}f(x) = f(x+1)\!</math> for any suitable function <math>f,\!</math> though of course the logical analogue that we take up here must have a rather different definition.)<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
U% $E$ $E$U% $E$U% $E$U%<br />
o------------------>o============o============o<br />
| | | |<br />
| | | |<br />
| | | |<br />
| | | |<br />
F | | $E$F = | $d$^0.F + | $r$^0.F<br />
| | | |<br />
| | | |<br />
| | | |<br />
v v v v<br />
o------------------>o============o============o<br />
X% $E$ $E$X% $E$X% $E$X%<br />
<br />
Figure 33-i. Analytic Diagram (1)<br />
</pre><br />
|}<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
U% $E$ $E$U% $E$U% $E$U% $E$U%<br />
o------------------>o============o============o============o<br />
| | | | |<br />
| | | | |<br />
| | | | |<br />
| | | | |<br />
F | | $E$F = | $d$^0.F + | $d$^1.F + | $r$^1.F<br />
| | | | |<br />
| | | | |<br />
| | | | |<br />
v v v v v<br />
o------------------>o============o============o============o<br />
X% $E$ $E$X% $E$X% $E$X% $E$X%<br />
<br />
Figure 33-ii. Analytic Diagram (2)<br />
</pre><br />
|}<br />
<br />
In its action on universes <math>\mathsf{E}\!</math> yields the same result as <math>\mathrm{E},\!</math> a fact that can be expressed in equational form by writing <math>\mathsf{E}U^\bullet = \mathrm{E}U^\bullet\!</math> for any universe <math>U^\bullet.\!</math> Notice that the extended universes across the top and bottom of the diagram are indicated to be strictly identical, rather than requiring a corresponding decomposition for them. In a certain sense, the functional parts of <math>\mathsf{E}F\!</math> are partitioned into separate contexts that have to be re-integrated again, but the best image to use is that of making transparent copies of each universe and then overlapping their functional contents once more at the conclusion of the analysis, as suggested by the graphic conventions that are used at the top of Figure&nbsp;30.<br />
<br />
Acting on a transformation <math>F\!</math> from universe <math>U^\bullet\!</math> to universe <math>X^\bullet,\!</math> the operator <math>\mathsf{E}\!</math> determines a transformation <math>\mathsf{E}F\!</math> from <math>\mathsf{E}U^\bullet\!</math> to <math>\mathsf{E}X^\bullet.\!</math> The map <math>\mathsf{E}F\!</math> forms the main body of evidence to be investigated in performing a differential analysis of <math>F.\!</math> Because we shall frequently be focusing on small pieces of this map for considerable lengths of time, and consequently lose sight of the &ldquo;big picture&rdquo;, it is critically important to emphasize that the map <math>\mathsf{E}F\!</math> is a transformation that determines a relation from one extended universe into another. This means that we should not be satisfied with our understanding of a transformation <math>F\!</math> until we can lay out the full &ldquo;parts diagram&rdquo; of <math>\mathsf{E}F\!</math> along the lines of the generic frame in Figure&nbsp;30.<br />
<br />
Working within the confines of propositional calculus, it is possible to give an elementary definition of <math>\mathsf{E}F\!</math> by means of a system of propositional equations, as we now describe.<br />
<br />
Given a transformation<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>F = (F_1, \ldots, F_k) : \mathbb{B}^n \to \mathbb{B}^k\!</math><br />
|}<br />
<br />
of concrete type<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>F : [u_1, \ldots, u_n] \to [x_1, \ldots, x_k],\!</math><br />
|}<br />
<br />
the transformation<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\mathsf{E}F = (F_1, \ldots, F_k, \mathrm{E}F_1, \ldots, \mathrm{E}F_k) : \mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B}^k \times \mathbb{D}^k\!</math><br />
|}<br />
<br />
of concrete type<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\mathsf{E}F : [u_1, \dots, u_n, \mathrm{d}u_1, \dots, \mathrm{d}u_n] \to [x_1, \ldots, x_k, \mathrm{d}x_1, \ldots, \mathrm{d}x_k]\!</math><br />
|}<br />
<br />
is defined by means of the following system of logical equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\\[16pt]<br />
\mathrm{d}x_1<br />
& = & \mathrm{E}F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1 + \mathrm{d}u_1, \ldots, u_n + \mathrm{d}u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
\mathrm{d}x_k<br />
& = & \mathrm{E}F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1 + \mathrm{d}u_1, \ldots, u_n + \mathrm{d}u_n)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
It is important to note that this system of equations can be read as a conjunction of equational propositions, in effect, as a single proposition in the universe of discourse generated by all the named variables. Specifically, this is the universe of discourse over <math>2(n+k)\!</math> variables denoted by:<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
\mathrm{E}[\mathcal{U} \cup \mathcal{X}]<br />
& = &<br />
[u_1, \ldots, u_n, ~ x_1, \ldots, x_k, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n, ~ \mathrm{d}x_1, \ldots, \mathrm{d}x_k].<br />
\end{matrix}</math><br />
|}<br />
<br />
In this light, it should be clear that the system of equations defining <math>\mathsf{E}F\!</math> embodies, in a higher rank and differentially extended version, an analogy with the process of thematization that we treated earlier for propositions of type <math>F : \mathbb{B}^n \to \mathbb{B}.\!</math><br />
<br />
The entire collection of constraints that is represented in the above system of equations may be abbreviated by writing <math>\mathsf{E}F = (\boldsymbol\varepsilon F, \mathrm{E}F),\!</math> for any map <math>F.\!</math> This is tantamount to regarding <math>\mathsf{E}\!</math> as a complex operator, <math>\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}),\!</math> with a form of application that distributes each component of the operator to work on each component of the operand, as follows:<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
\mathsf{E}F<br />
& = &<br />
(\boldsymbol\varepsilon, \mathrm{E})F<br />
& = &<br />
(\boldsymbol\varepsilon F, \mathrm{E}F)<br />
& = &<br />
(\boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k, ~ \mathrm{E}F_1, \ldots, \mathrm{E}F_k).<br />
\end{matrix}</math><br />
|}<br />
<br />
Quite a lot of &ldquo;thematic infrastructure&rdquo; or interpretive information is being swept under the rug in the use of such abbreviations. When confusion arises about the meaning of such constructions, one always has recourse to the defining system of equations, in its totality a purely propositional expression. This means that the parenthesized argument lists, that were used in this context to build an image of multi-component transformations, should not be expected to determine a well-defined product in themselves but only to serve as reminders of the prior thematic decisions (choices of variable names, etc.) that have to be made in order to determine one. Accordingly, the argument list notation can be regarded as a kind of ''thematic frame'', an interpretive storage device that preserves the proper associations of concrete logical features between the extended universes at the source and target of <math>\mathsf{E}F.\!</math><br />
<br />
The generic notations <math>\mathsf{d}^0\!F, \mathsf{d}^1\!F, \ldots, \mathsf{d}^m\!F\!</math> in Figure&nbsp;33 refer to the increasing orders of differentials that are extracted in the course of analyzing <math>F.\!</math> When the analysis is halted at a partial stage of development, notations like <math>\mathsf{r}^0\!F, \mathsf{r}^1\!F, \ldots, \mathsf{r}^m\!F\!</math> may be used to summarize the contributions to <math>\mathsf{E}F\!</math> that remain to be analyzed. The Figure illustrates a convention that makes <math>\mathsf{r}^m\!F,\!</math> in effect, the sum of all differentials of order strictly greater than <math>m.\!</math><br />
<br />
We next discuss the operators that figure into this form of analysis, describing their effects on transformations. In simplified or specialized contexts these operators tend to take on a variety of different names and notations, some of whose number we introduce along the way.<br />
<br />
=====The Radius Operator : '''e'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
And the tangible fact at the root of all our thought-distinctions, however subtle, is that there is no one of them so fine as to consist in anything but a possible difference of practice.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 46]<br />
|}<br />
<br />
The operator identified as <math>\mathrm{d}^0\!</math> in the analytic diagram (Figure&nbsp;33) has the sole purpose of creating a proxy for <math>F\!</math> in the appropriately extended context. Construed in terms of its broadest components, <math>\mathrm{d}^0\!</math> is equivalent to the doubly tacit extension operator <math>(\boldsymbol\varepsilon, \boldsymbol\varepsilon),\!</math> in recognition of which let us redub it as <math>{}^{\backprime\backprime} \mathsf{e} {}^{\prime\prime}.\!</math> Pursuing a geometric analogy, we may refer to <math>\mathsf{e} =(\boldsymbol\varepsilon, \boldsymbol\varepsilon) = \mathrm{d}^0\!</math> as the ''radius operator''. The operation intended by all of these forms is defined by the following equation:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathsf{e}F<br />
& = &<br />
(\boldsymbol\varepsilon, \boldsymbol\varepsilon)F<br />
\\[4pt]<br />
& = &<br />
(\boldsymbol\varepsilon F, ~ \boldsymbol\varepsilon F)<br />
\\[4pt]<br />
& = &<br />
(\boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k, ~ \boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k).<br />
\end{array}</math><br />
|}<br />
<br />
which is tantamount to the system of equations below.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\\[16pt]<br />
\mathrm{d}x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
\mathrm{d}x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
=====The Phantom of the Operators : '''&eta;'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
I was wondering what the reason could be, when I myself raised my head and everything within me seemed drawn towards the Unseen, ''which was playing the most perfect music''!<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 81]<br />
|}<br />
<br />
We now describe an operator whose persistent but elusive action behind the scenes, whose slightly twisted and ambivalent character, and whose fugitive disposition, caught somewhere in flight between the arrantly negative and the positive but errant intent, has cost us some painstaking trouble to detect. In the end we shall place it among the other extensions and projections, as a shade among shadows, of muted tones and motley hue, that adumbrates its own thematic frame and paradoxically lights the way toward a whole new spectrum of values.<br />
<br />
Given a transformation <math>F : [u_1, \ldots, u_n] \to [x_1, \dots, x_k],\!</math> we often have call to consider a family of related transformations, all having the form:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>F^\dagger : [u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n] \to [\mathrm{d}x_1, \dots, \mathrm{d}x_k].\!</math><br />
|}<br />
<br />
The operator <math>\eta</math> is introduced to deal with the simplest one of these maps:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\eta F : [u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n] \to [\mathrm{d}x_1, \ldots \mathrm{d}x_k],\!</math><br />
|}<br />
<br />
which is defined by the following equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
\mathrm{d}x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
In effect, the operator <math>\eta\!</math> is nothing but the stand-alone version of a procedure that is otherwise invoked subordinate to the work of the radius operator <math>\mathsf{e}.\!</math> Operating independently, <math>\eta\!</math> achieves precisely the same results that the second <math>\boldsymbol\varepsilon\!</math> in <math>(\boldsymbol\varepsilon, \boldsymbol\varepsilon)\!</math> accomplishes by working within the context of its ordered pair thematic frame. From this point on, because the use of <math>\boldsymbol\varepsilon\!</math> and <math>\eta\!</math> in this setting combines the aims of both the tacit and the thematic extensions, and because <math>\eta\!</math> reflects in regard to <math>\boldsymbol\varepsilon\!</math> little more than the application of a differential twist, a mere turn of phrase, we refer to <math>\eta\!</math> as the ''trope extension'' operator.<br />
<br />
=====The Chord Operator : '''D'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
What difference would it practically make to any one if this notion rather than that notion were true? If no practical difference whatever can be traced, then the alternatives mean practically the same thing, and all dispute is idle.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 45]<br />
|}<br />
<br />
Next we discuss an operator that is always immanent in this form of analysis, and remains implicitly present in the entire proceeding. It may appear once as a record: a relic or revenant that reprises the reminders of an earlier stage of development. Or it may appear always as a resource: a reserve or redoubt that caches in advance an echo of what remains to be played out, cleared up, and requited in full at a future stage. And all of this remains true whether or not we recall the key at any time, and whether or not the subtending theme is recited explicitly at any stage of play.<br />
<br />
This is the operator that is referred to as <math>\mathsf{r}^0\!</math> in the initial stage of analysis (Figure&nbsp;33-i) and that is expanded as <math>\mathsf{d}^1 + \mathsf{r}^1\!</math> in the subsequent step (Figure&nbsp;33-ii). In congruence, but not quite harmony with our allusions of analogy that are not quite geometry, we call this the ''chord operator'' and denote it <math>\mathsf{D}.\!</math> In the more casual terms that are here introduced, <math>\mathsf{D}</math> is defined as the remainder of <math>\mathsf{E}\!</math> and <math>\mathsf{e}\!</math> and it assigns a due measure to each undertone of accord or discord that is struck between the note of enterprise <math>\mathsf{E}\!</math> and the bar of exigency <math>\mathsf{e}.\!</math><br />
<br />
The tension between these counterposed notions, in balance transient but regular in stridence, may be refracted along familiar lines, though never by any such fraction resolved. In this style we write <math>\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}),\!</math> calling <math>\mathrm{D}\!</math> the ''difference operator'' and noting that it plays a role in this realm of mutable and diverse discourse that is analogous to the part taken by the discrete difference operator in the ordinary difference calculus. Finally, we should note that the chord <math>\mathsf{D}\!</math> is not one that need be lost at any stage of development. At the <math>m^\text{th}\!</math> stage of play it can always be reconstituted in the following form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathsf{D}<br />
& = & \mathsf{E} - \mathsf{e}<br />
\\[6pt]<br />
& = & \mathsf{r}^0<br />
\\[6pt]<br />
& = & \mathsf{d}^1 + \mathsf{r}^1<br />
\\[6pt]<br />
& = & \mathsf{d}^1 + \ldots + \mathsf{d}^m + \mathsf{r}^m<br />
\\[6pt]<br />
& = & \displaystyle \sum_{i=1}^m \mathsf{d}^i + \mathsf{r}^m<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====The Tangent Operator : '''T'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
They take part in scenes of whose significance they have no inkling. They are merely tangent to curves of history the beginnings and ends and forms of which pass wholly beyond their ken. So we are tangent to the wider life of things.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 300]<br />
|}<br />
<br />
The operator tagged as <math>\mathsf{d}^1\!</math> in the analytic diagram (Figure&nbsp;33) is called the ''tangent operator'' and is usually denoted in this text as <math>\mathsf{d}\!</math> or <math>\mathsf{T}.\!</math> Because it has the properties required to qualify as a functor, namely, preserving the identity element of the composition operation and the articulated form of every composition of transformations, it also earns the title of a ''tangent functor''. According to the custom adopted here, we dissect it as <math>\mathsf{T} = \mathsf{d} = (\boldsymbol\varepsilon, \mathrm{d}),\!</math> where <math>\mathrm{d}\!</math> is the operator that yields the first order differential <math>\mathrm{d}F\!</math> when applied to a transformation <math>F,\!</math> and whose name is legion.<br />
<br />
Figure&nbsp;34 illustrates a stage of analysis where we ignore everything but the tangent functor <math>\mathsf{T}\!</math> and attend to it chiefly as it bears on the first order differential <math>\mathrm{d}F\!</math> in the analytic expansion of <math>F.\!</math> In this situation we often refer to the extended universes <math>\mathrm{E}U^\bullet\!</math> and <math>\mathrm{E}X^\bullet\!</math> under the equivalent designations <math>\mathsf{T}U^\bullet\!</math> and <math>\mathsf{T}X^\bullet,\!</math> respectively. The purpose of the tangent functor <math>\mathsf{T}\!</math> is to extract the tangent map <math>\mathsf{T}F\!</math> at each point of <math>U^\bullet,\!</math> and the tangent map <math>\mathsf{T}F = (\boldsymbol\varepsilon, \mathrm{d})F\!</math> tells us not only what the transformation <math>F\!</math> is doing at each point of the universe <math>U^\bullet\!</math> but also what <math>F\!</math> is doing to states in the neighborhood of that point, approximately, linearly, and relatively speaking.<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
U% $T$ $T$U% $T$U%<br />
o------------------>o============o<br />
| | |<br />
| | |<br />
| | |<br />
| | |<br />
F | | $T$F = | <!e!, d> F<br />
| | |<br />
| | |<br />
| | |<br />
v v v<br />
o------------------>o============o<br />
X% $T$ $T$X% $T$X%<br />
<br />
Figure 34. Tangent Functor Diagram<br />
</pre><br />
|}<br />
<br />
* '''NB.''' There is one aspect of the preceding construction that remains especially problematic. Why did we define the operators <math>\mathrm{W}\!</math> in <math>\{ \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}\!</math> so that the ranges of their resulting maps all fall within the realms of differential quality, even fabricating a variant of the tacit extension operator to have that character? Clearly, not all of the operator maps <math>\mathrm{W}F\!</math> have equally good reasons for placing their values in differential stocks. The reason for it appears to be that, without doing this, we cannot justify the comparison and combination of their functional values in the various analytic steps. By default, only those values in the same functional component can be brought into algebraic modes of interaction. Up till now the only mechanism provided for their broader association has been a purely logical one, their common placement in a target universe of discourse, but the task of converting this logical circumstance into algebraic forms of application has not yet been taken up.<br />
<br />
===Transformations of Type '''B'''<sup>2</sup> &rarr; '''B'''<sup>1</sup>===<br />
<br />
To study the effects of these analytic operators in the simplest possible setting, let us revert to a still more primitive case. Consider the singular proposition <math>J(u, v)= u\!\cdot\!v,\!</math> regarded either as the functional product of the maps <math>u\!</math> and <math>v\!</math> or as the logical conjunction of the features <math>u\!</math> and <math>v,\!</math> a map whose fiber of truth <math>J^{-1}(1)\!</math> picks out the single cell of that logical description in the universe of discourse <math>U^\bullet.\!</math> Thus <math>J,\!</math> or <math>u\!\cdot\!v,\!</math> may be treated as another name for the point whose coordinates are <math>(1, 1)\!</math> in <math>U^\bullet.\!</math><br />
<br />
====Analytic Expansion of Conjunction====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
<p>In her sufferings she read a great deal and discovered that she had lost something, the possession of which she had previously not been much aware of: a&nbsp;soul.</p><br />
<br />
<p>What is that? It is easily defined negatively: it is simply what curls up and hides when there is any mention of algebraic series.</p><br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 118]<br />
|}<br />
<br />
Figure&nbsp;35 pictures the form of conjunction <math>J : \mathbb{B}^2 \to \mathbb{B}\!</math> as a transformation from the <math>2\!</math>-dimensional universe <math>[u, v]\!</math> to the <math>1\!</math>-dimensional universe <math>[x].\!</math> This is a subtle but significant change of viewpoint on the proposition, attaching an arbitrary but concrete quality to its functional value. Using the language introduced earlier, we can express this change by saying that the proposition <math>J : \langle u, v \rangle \to \mathbb{B}\!</math> is being recast into the thematized role of a transformation <math>J : [u, v] \to [x],\!</math> where the new variable <math>x\!</math> takes the part of a thematic variable <math>\check{J}.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 35 -- A Conjunction Viewed as a Transformation.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 35.} ~~ \text{Conjunction as Transformation}\!</math><br />
|}<br />
<br />
=====Tacit Extension of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
I teach straying from me, yet who can stray from me?<br><br />
I follow you whoever you are from the present hour;<br><br />
My words itch at your ears till you understand them.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 83]<br />
|}<br />
<br />
Earlier we defined the tacit extension operators <math>\boldsymbol\varepsilon : X^\bullet \to Y^\bullet\!</math> as maps embedding each proposition of a given universe <math>X^\bullet~\!</math> in a more generously given universe <math>Y^\bullet \supset X^\bullet.\!</math> Of immediate interest are the tacit extensions <math>\boldsymbol\varepsilon : U^\bullet \to \mathrm{E}U^\bullet,\!</math> that locate each proposition of <math>U^\bullet\!</math> in the enlarged context of <math>\mathrm{E}U^\bullet.\!</math> In its application to the propositional conjunction <math>J = u\!\cdot\!v</math> in <math>[u, v],\!</math> the tacit extension operator <math>\boldsymbol\varepsilon\!</math> yields the proposition <math>\boldsymbol\varepsilon J\!</math> in <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v].\!</math> The extended proposition <math>\boldsymbol\varepsilon J\!</math> may be computed according to the scheme in Table&nbsp;36, in effect doing nothing more that conjoining a tautology of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> to <math>J\!</math> in <math>U^\bullet.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 36.} ~~ \text{Computation of}~ \boldsymbol\varepsilon J\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\boldsymbol\varepsilon J & = & J {}_{^\langle} u, v {}_{^\rangle}<br />
\\[4pt]<br />
& = & u \cdot v<br />
\\[4pt]<br />
& = & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{ } \mathrm{d}v \texttt{ }<br />
& + & u \cdot v \cdot \texttt{ } \mathrm{d}u \texttt{ } \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \cdot v \cdot \texttt{ } \mathrm{d}u \texttt{ } \cdot \texttt{ } \mathrm{d}v \texttt{ }<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{array}{*{4}{l}}<br />
\boldsymbol\varepsilon J<br />
& = && u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & u \cdot v \cdot \texttt{~} \mathrm{d}u \texttt{~} \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & u \cdot v \cdot \texttt{~} \mathrm{d}u \texttt{~} \cdot \texttt{~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
The lower portion of the Table contains the dispositional features of <math>\boldsymbol\varepsilon J\!</math> arranged in such a way that the variety of ordinary features spreads across the rows and the variety of differential features runs through the columns. This organization serves to facilitate pattern matching in the remainder of our computations. Again, the tacit extension is usually so trivial a concern that we do not always bother to make an explicit note of it, taking it for granted that any function <math>F\!</math> being employed in a differential context is equivalent to <math>\boldsymbol\varepsilon F\!</math> for a suitable <math>\boldsymbol\varepsilon.\!</math><br />
<br />
Figures&nbsp;37-a through 37-d present several pictures of the proposition <math>J\!</math> and its tacit extension <math>\boldsymbol\varepsilon J.\!</math> Notice in these Figures how <math>\boldsymbol\varepsilon J\!</math> in <math>\mathrm{E}U^\bullet\!</math> visibly extends <math>J\!</math> in <math>U^\bullet\!</math> by annexing to the indicated cells of <math>J\!</math> all the arcs that exit from or flow out of them. In effect, this extension attaches to these cells all the dispositions that spring from them, in other words, it attributes to these cells all the conceivable changes that are their issue.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-a -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-a.} ~~ \text{Tacit Extension of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-b -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-b.} ~~ \text{Tacit Extension of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-c -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-c.} ~~ \text{Tacit Extension of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-d -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-d.} ~~ \text{Tacit Extension of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
The computational scheme shown in Table&nbsp;36 treated <math>J\!</math> as a proposition in <math>U^\bullet\!</math> and formed <math>\boldsymbol\varepsilon J\!</math> as a proposition in <math>\mathrm{E}U^\bullet.\!</math> When <math>J\!</math> is regarded as a mapping <math>J : U^\bullet \to X^\bullet\!</math> then <math>\boldsymbol\varepsilon J\!</math> must be obtained as a mapping <math>\boldsymbol\varepsilon J : \mathrm{E}U^\bullet \to X^\bullet.\!</math> By default, the tacit extension of the map <math>J : [u, v] \to [x]\!</math> is naturally taken to be a particular map,<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\boldsymbol\varepsilon J : [u, v, \mathrm{d}u, \mathrm{d}v] \to [x] \subseteq [x, \mathrm{d}x],\!</math><br />
|}<br />
<br />
namely, the one that looks like <math>J\!</math> when painted in the frame of the extended source universe and that takes the same thematic variable in the extended target universe as the one that <math>J\!</math> already takes.<br />
<br />
But the choice of a particular thematic variable, for example <math>x\!</math> for <math>\check{J},\!</math> is a shade more arbitrary than the choice of original variable names <math>\{ u, v \},\!</math> so the map we are calling the ''trope extension'',<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\eta J : [u, v, \mathrm{d}u, \mathrm{d}v] \to [\mathrm{d}x] \subseteq [x, \mathrm{d}x],\!</math><br />
|}<br />
<br />
since it looks just the same as <math>\boldsymbol\varepsilon J\!</math> in the way its fibers paint the source domain, belongs just as fully to the family of tacit extensions, generically considered.<br />
<br />
These considerations have the practical consequence that all of our computations and illustrations of <math>\boldsymbol\varepsilon J\!</math> perform the double duty of capturing <math>\eta J\!</math> as well. In other words, we are saved the work of carrying out calculations and drawing figures for the trope extension <math>\eta J,\!</math> because it would be identical to the work already done for <math>\boldsymbol\varepsilon J.\!</math> Since the computations given for <math>\boldsymbol\varepsilon J\!</math> are expressed solely in terms of the variables <math>\{ u, v, \mathrm{d}u, \mathrm{d}v \},\!</math> they work equally well for finding <math>\eta J.\!</math> Further, since each of the above Figures shows only how the level sets of <math>\boldsymbol\varepsilon J\!</math> partition the extended source universe <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v],\!</math> all of them serve equally well as portraits of <math>\eta J.\!</math><br />
<br />
=====Enlargement Map of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
No one could have established the existence of any details that might not just as well have existed in earlier times too; but all the relations between things had shifted slightly. Ideas that had once been of lean account grew fat.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 62]<br />
|}<br />
<br />
The enlargement map <math>\mathrm{E}J\!</math> is computed from the proposition <math>J\!</math> by making a particular class of formal substitutions for its variables, in this case <math>u + \mathrm{d}u\!</math> for <math>u\!</math> and <math>v + \mathrm{d}v\!</math> for <math>v,\!</math> and afterwards expanding the result in whatever way is found convenient.<br />
<br />
Table&nbsp;38 shows a typical scheme of computation, following a systematic method of exploiting boolean expansions over selected variables and ultimately developing <math>\mathrm{E}J\!</math> over the cells of <math>[u, v].\!</math> The critical step of this procedure uses the facts that <math>\texttt{(} 0, x \texttt{)} = 0 + x = x\!</math> and <math>\texttt{(} 1, x \texttt{)} = 1 + x = \texttt{(} x \texttt{)}\!</math> for any boolean variable <math>x.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 38.} ~~ \text{Computation of}~ \mathrm{E}J ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{E}J & = & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}<br />
\\[4pt]<br />
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
& = & \texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot J_{(1 + \mathrm{d}u, 1 + \mathrm{d}v)}<br />
& + & \texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot J_{(1 + \mathrm{d}u, \mathrm{d}v)}<br />
& + & \texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot J_{(\mathrm{d}u, 1 + \mathrm{d}v)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot J_{(\mathrm{d}u, \mathrm{d}v)}<br />
\\[4pt]<br />
& = &<br />
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot J_{(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})}<br />
& + &<br />
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot J_{(\texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v)}<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot J_{(\mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)})}<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot J_{(\mathrm{d}u, \mathrm{d}v)}<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{E}J<br />
& = &<br />
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&& + &<br />
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{ } \mathrm{d}v \texttt{ }<br />
\\[4pt]<br />
&&&&& + &<br />
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot \texttt{ } \mathrm{d}u \texttt{ } \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&&&&&& + &<br />
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \texttt{ } \mathrm{d}u \texttt{ }~\texttt{ } \mathrm{d}v \texttt{ }<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;39 exhibits another method that happens to work quickly in this particular case, using distributive laws to multiply things out in an algebraic manner, arranging the notations of feature and fluxion according to a scale of simple character and degree. Proceeding this way leads through an intermediate step which, in chiming the changes of ordinary calculus, should take on a familiar ring. Consequential properties of exclusive disjunction then carry us on to the concluding line.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 39.} ~~ \text{Computation of}~ \mathrm{E}J ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{c}}<br />
\mathrm{E}J<br />
& = & (u + \mathrm{d}u) \cdot (v + \mathrm{d}v)<br />
\\[6pt]<br />
& = & u \cdot v<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = &<br />
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{ } \mathrm{d}v \texttt{ }<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot \texttt{ } \mathrm{d}u \texttt{ } \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \texttt{ } \mathrm{d}u \texttt{ }~\texttt{ } \mathrm{d}v \texttt{ }<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;40-a through 40-d present several views of the enlarged proposition <math>\mathrm{E}J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-a -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-a.} ~~ \text{Enlargement of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-b -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-b.} ~~ \text{Enlargement of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-c -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-c.} ~~ \text{Enlargement of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-d -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-d.} ~~ \text{Enlargement of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
An intuitive reading of the proposition <math>\mathrm{E}J\!</math> becomes available at this point. Recall that propositions in the extended universe <math>\mathrm{E}U^\bullet\!</math> express the ''dispositions'' of a system and the constraints that are placed on them. In other words, a differential proposition in <math>\mathrm{E}U^\bullet\!</math> can be read as referring to various changes that a system might undergo in and from its various states. In particular, we can understand <math>\mathrm{E}J\!</math> as a statement that tells us what changes need to be made with regard to each state in the universe of discourse in order to reach the ''truth'' of <math>J,\!</math> that is, the region of the universe where <math>J\!</math> is true. This interpretation is visibly clear in the Figures above and appeals to the imagination in a satisfying way but it has the added benefit of giving fresh meaning to the original name of the shift operator <math>\mathrm{E}.\!</math> Namely, <math>\mathrm{E}J\!</math> can be read as a proposition that ''enlarges'' on the meaning of <math>J,\!</math> in the sense of explaining its practical bearings and clarifying what it means in terms of actions and effects &mdash; the available options for differential action and the consequential effects that result from each choice.<br />
<br />
Read this way, the enlargement <math>\mathrm{E}J\!</math> has strong ties to the normal use of <math>J,\!</math> no matter whether it is understood as a proposition or a function, namely, to act as a figurative device for indicating the models of <math>J,\!</math> in effect, pointing to the interpretive elements in its fiber of truth <math>J^{-1}(1).\!</math> It is this kind of &ldquo;use&rdquo; that is often contrasted with the &ldquo;mention&rdquo; of a proposition, and thereby hangs a tale.<br />
<br />
=====Digression : Reflection on Use and Mention=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Reflection is turning a topic over in various aspects and in various lights so that nothing significant about it shall be overlooked &mdash; almost as one might turn a stone over to see what its hidden side is like or what is covered by it.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 57]<br />
|}<br />
<br />
The contrast drawn in logic between the ''use'' and the ''mention'' of a proposition corresponds to the difference that we observe in functional terms between using <math>{}^{\backprime\backprime} J \, {}^{\prime\prime}\!</math> to indicate the region <math>J^{-1}(1)\!</math> and using <math>{}^{\backprime\backprime} J \, {}^{\prime\prime}\!</math> to indicate the function <math>J.\!</math> You may think that one of these uses ought to be proscribed, and logicians are quick to prescribe against their confusion. But there seems to be no likelihood in practice that their interactions can be avoided. If the name <math>{}^{\backprime\backprime} J \, {}^{\prime\prime}\!</math> is used as a sign of the function <math>J,\!</math> and if the function <math>J\!</math> has its use in signifying something else, as would constantly be the case when some future theory of signs has given a functional meaning to every sign whatsoever, then is not <math>J,\!</math> by transitivity a sign of the thing itself? There are, of course, two answers to this question. Not every act of signifying or referring need be transitive. Not every warrant or guarantee or certificate is automatically transferable, indeed, not many. Not every feature of a feature is a feature of the featuree. Otherwise, if a buffalo is white, and white is a color, then a buffalo would ''be'' a color.<br />
<br />
The logical or pragmatic distinction between use and mention is cogent and necessary, and so is the analogous functional distinction between determining a value and determining what determines that value, but so are the normal techniques that we use to make these distinctions apply flexibly in practice. The way that the hue and cry about use and mention is raised in logical discussions, you might be led to think that this single dimension of choices embraces the only kinds of use worth mentioning and the only kinds of mention worth using. It will constitute the expeditionary and taxonomic tasks of that future theory of signs to explore and to classify the many other constellations and dimensions of use and mention that are yet to be opened up by the generative potential of full-fledged sign relations.<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
The well-known capacity that thoughts have &mdash; as doctors have discovered &mdash; for dissolving and dispersing those hard lumps of deep, ingrowing, morbidly entangled conflict that arise out of gloomy regions of the self probably rests on nothing other than their social and worldly nature, which links the individual being with other people and things; but unfortunately what gives them their power of healing seems to be the same as what diminishes the quality of personal experience in them.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 130]<br />
|}<br />
<br />
=====Difference Map of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
&ldquo;It doesn't matter what one does,&rdquo; the Man Without Qualities said to himself, shrugging his shoulders. &ldquo;In a tangle of forces like this it doesn't make a scrap of difference.&rdquo; He turned away like a man who has learned renunciation, almost indeed like a sick man who shrinks from any intensity of contact. And then, striding through his adjacent dressing-room, he passed a punching-ball that hung there; he gave it a blow far swifter and harder than is usual in moods of resignation or states of weakness.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 8]<br />
|}<br />
<br />
With the tacit extension map <math>\boldsymbol\varepsilon J\!</math> and the enlargement map <math>\mathrm{E}J\!</math> well in place, the difference map <math>\mathrm{D}J\!</math> can be computed along the lines displayed in Table&nbsp;41, ending up with an expansion of <math>\mathrm{D}J\!</math> over the cells of <math>[u, v].\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 41.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & \mathrm{E}J<br />
& + & \boldsymbol\varepsilon J<br />
\\[6pt]<br />
& = & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}<br />
& + & J_{(u, v)}<br />
\\[6pt]<br />
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \cdot v<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = &<br />
u \cdot v \cdot \qquad 0<br />
\\[6pt]<br />
& + &<br />
u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v<br />
& + &<br />
u \cdot \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v<br />
\\[6pt]<br />
& + &<br />
u \cdot v \cdot \texttt{~} \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
&&& + &<br />
\texttt{(} u \texttt{)} \cdot v \cdot \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
& + &<br />
u \cdot v \cdot \texttt{~} \mathrm{d}u \;\cdot\; \mathrm{d}v \texttt{~}<br />
&&&&& + &<br />
\texttt{(} u \texttt{)} \cdot \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = &<br />
u \cdot v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
u \cdot \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v<br />
& + &<br />
\texttt{(} u \texttt{)} \cdot v \cdot \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)} \cdot \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Alternatively, the difference map <math>\mathrm{D}J\!</math> can be expanded over the cells of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> to arrive at the formulation shown in Table&nbsp;42. The same development would be obtained from the previous Table by collecting terms in an alternate manner, along the rows rather than the columns in the middle portion of the Table.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 42.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & \boldsymbol\varepsilon J<br />
& + & \mathrm{E}J<br />
\\[6pt]<br />
& = & J_{(u, v)}<br />
& + & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}<br />
\\[6pt]<br />
& = & u \cdot v<br />
& + & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
& = & 0<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}J<br />
& = & 0<br />
& + & u \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Even more simply, the same result is reached by matching up the propositional coefficients of <math>\boldsymbol\varepsilon J</math> and <math>\mathrm{E}J\!</math> along the cells of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> and adding the pairs under boolean addition, that is, &ldquo;mod 2&rdquo;, where 1&nbsp;+&nbsp;1&nbsp;=&nbsp;0, as shown in Table&nbsp;43.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 43.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 3)}\!</math><br />
|<br />
<math>\begin{array}{*{5}{l}}<br />
\mathrm{D}J & = & \boldsymbol\varepsilon J & + & \mathrm{E}J<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\boldsymbol\varepsilon J<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ u \,\cdot\, v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~ u \;\cdot\; v \;\cdot\; \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u ~ \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} ~ v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot\, \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & ~~ 0 ~~ \,\cdot\, ~ \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~~ u ~ \,\cdot\, ~~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ ~ v ~~ \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
The difference map <math>\mathrm{D}J\!</math> can also be given a ''dispositional'' interpretation. First, recall that <math>\boldsymbol\varepsilon J\!</math> exhibits the dispositions to change from anywhere in <math>J\!</math> to anywhere at all in the universe of discourse and <math>\mathrm{E}J\!</math> exhibits the dispositions to change from anywhere in the universe to anywhere in <math>J.\!</math> Next, observe that each of these classes of dispositions may be divided in accordance with the case of <math>J\!</math> versus <math>\texttt{(} J \texttt{)}\!</math> that applies to their points of departure and destination, as shown below. Then, since the dispositions corresponding to <math>\boldsymbol\varepsilon J</math> and <math>\mathrm{E}J\!</math> have in common the dispositions to preserve <math>J,\!</math> their symmetric difference <math>\texttt{(} \boldsymbol\varepsilon J, \mathrm{E}J \texttt{)}\!</math> is made up of all the remaining dispositions, which are in fact disposed to cross the boundary of <math>J\!</math> in one direction or the other. In other words, we may conclude that <math>\mathrm{D}J\!</math> expresses the collective disposition to make a definite change with respect to <math>J,\!</math> no matter what value it holds in the current state of affairs.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
\boldsymbol\varepsilon J<br />
& = & \{ \text{Dispositions from}~ J ~\text{to}~ J \}<br />
& + & \{ \text{Dispositions from}~ J ~\text{to}~ \texttt{(} J \texttt{)} \}<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = & \{ \text{Dispositions from}~ J ~\text{to}~ J \}<br />
& + & \{ \text{Dispositions from}~ \texttt{(} J \texttt{)} ~\text{to}~ J \}<br />
\\[6pt]<br />
\mathrm{D}J<br />
& = & \{ \text{Dispositions from}~ J ~\text{to}~ \texttt{(} J \texttt{)} \}<br />
& + & \{ \text{Dispositions from}~ \texttt{(} J \texttt{)} ~\text{to}~ J \}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;44-a through 44-d illustrate the difference proposition <math>\mathrm{D}J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-a -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-a.} ~~ \text{Difference Map of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-b -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-b.} ~~ \text{Difference Map of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-c -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-c.} ~~ \text{Difference Map of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-d -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-d.} ~~ \text{Difference Map of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
=====Differential of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
By deploying discourse throughout a calendar, and by giving a date to each of its elements, one does not obtain a definitive hierarchy of precessions and originalities; this hierarchy is never more than relative to the systems of discourse that it sets out to evaluate.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Archaeology of Knowledge'', [Fou, 143]<br />
|}<br />
<br />
Finally, at long last, the differential proposition <math>\mathrm{d}J\!</math> can be gleaned from the difference proposition <math>\mathrm{D}J\!</math> by ranging over the cells of <math>[u, v]\!</math> and picking out the linear proposition of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> that is &ldquo;closest&rdquo; to the portion of <math>\mathrm{D}J\!</math> that touches on each point. The idea of distance that would give this definition unequivocal sense has been referred to in cautionary quotes, the kind we use to distance ourselves from taking a final position. There are obvious notions of approximation that suggest themselves, but finding one that can be justified as ultimately correct is not as straightforward as it seems.<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
He had drifted into the very heart of the world. From him to the distant beloved was as far as to the next tree.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 144]<br />
|}<br />
<br />
Let us venture a guess as to where these developments might be heading. From the present vantage point it appears that the ultimate answer to the quandary of distances and the question of a fitting measure may be that, rather than having the constitution of an analytic series depend on our familiar notions of approach, proximity, and approximation, it will be found preferable, and perhaps unavoidable, to turn the tables and let the orders of approximation be defined in terms of our favored and operative notions of formal analysis. Only the aftermath of this conversion, if it does converge, could be hoped to prove whether this hortatory form of analysis and the cohort idea of an analytic form &mdash; the limitary concept of a self-corrective process and the coefficient concept of a completable product &mdash; are truly (in practical reality) the more inceptive and persistent of principles and really (for all practical purposes) the more effective and regulative of ideas.<br />
<br />
Awaiting that determination, I proceed with what seems like the obvious course, and compute <math>\mathrm{d}J\!</math> according to the pattern in Table&nbsp;45.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 45.} ~~ \text{Computation of}~ \mathrm{d}J\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}J<br />
& = &<br />
u\!\cdot\!v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
u \, \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \, \mathrm{d}v<br />
& + &<br />
\texttt{(} u \texttt{)} \, v \cdot \mathrm{d}u \, \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \!\cdot\! \mathrm{d}v \texttt{~}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}J<br />
& = &<br />
u\!\cdot\!v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + &<br />
u \, \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + &<br />
\texttt{(} u \texttt{)} \, v \cdot \mathrm{d}u<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;46-a through 46-d illustrate the proposition <math>{\mathrm{d}J},\!</math> rounded out in our usual array of prospects. This proposition of <math>\mathrm{E}U^\bullet\!</math> is what we refer to as the (first order) differential of <math>J,\!</math> and normally regard as ''the'' differential proposition corresponding to <math>J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-a -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-a.} ~~ \text{Differential of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-b -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-b.} ~~ \text{Differential of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-c -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-c.} ~~ \text{Differential of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-d -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-d.} ~~ \text{Differential of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
=====Remainder of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
<p>I bequeath myself to the dirt to grow from the grass I love,<br><br />
If you want me again look for me under your bootsoles.</p><br />
<br />
<p>You will hardly know who I am or what I mean,<br><br />
But I shall be good health to you nevertheless,<br><br />
And filter and fibre your blood.</p><br />
<br />
<p>Failing to fetch me at first keep encouraged,<br><br />
Missing me one place search another,<br><br />
I stop some where waiting for you</p><br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]<br />
|}<br />
<br />
<br><br />
<br />
Let us recapitulate the story so far. We have in effect been carrying out a decomposition of the enlarged proposition <math>\mathrm{E}J\!</math> in a series of stages. First, we considered the equation <math>\mathrm{E}J = \boldsymbol\varepsilon J + \mathrm{D}J,\!</math> which was involved in the definition of <math>\mathrm{D}J\!</math> as the difference <math>\mathrm{E}J - \boldsymbol\varepsilon J.\!</math> Next, we contemplated the equation <math>\mathrm{D}J = \mathrm{d}J + \mathrm{r}J,\!</math> which expresses <math>\mathrm{D}J\!</math> in terms of two components, the differential <math>\mathrm{d}J\!</math> that was just extracted and the residual component <math>\mathrm{r}J = \mathrm{D}J - \mathrm{d}J.~\!</math> This remaining proposition <math>\mathrm{r}J\!</math> can be computed as shown in Table&nbsp;47.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 47.} ~~ \text{Computation of}~ \mathrm{r}J\!</math><br />
|<br />
<math>\begin{array}{*{5}{l}}<br />
\mathrm{r}J & = & \mathrm{D}J & + & \mathrm{d}J<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}J<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{r}J ~<br />
& = & u \!\cdot\! v \cdot ~ \mathrm{d}u \cdot \mathrm{d}v ~ ~ ~ ~ ~<br />
& + & u \texttt{(} v \texttt{)} \cdot \, \mathrm{d}u \cdot \mathrm{d}v \,<br />
& + & \texttt{(} u \texttt{)} v \cdot \, \mathrm{d}u \cdot \mathrm{d}v \,<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \, \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
As it happens, the remainder <math>\mathrm{r}J\!</math> falls under the description of a second order differential <math>\mathrm{r}J = \mathrm{d}^2 J.\!</math> This means that the expansion of <math>\mathrm{E}J\!</math> in the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|<br />
<math>\begin{array}{*{7}{l}}<br />
\mathrm{E}J<br />
& = & \boldsymbol\varepsilon J<br />
& + & \mathrm{D}J<br />
\\[6pt]<br />
& = & \boldsymbol\varepsilon J<br />
& + & \mathrm{d}J<br />
& + & \mathrm{r}J<br />
\\[6pt]<br />
& = & \mathrm{d}^0 J<br />
& + & \mathrm{d}^1 J<br />
& + & \mathrm{d}^2 J<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
which is nothing other than the propositional analogue of a Taylor series, is a decomposition that terminates in a finite number of steps.<br />
<br />
Figures&nbsp;48-a through 48-d illustrate the proposition <math>\mathrm{r}J = \mathrm{d}^2 J,\!</math> which forms the remainder map of <math>J\!</math> and also, in this instance, the second order differential of <math>J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-a -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-a.} ~~ \text{Remainder of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-b -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-b.} ~~ \text{Remainder of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-c -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-c.} ~~ \text{Remainder of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-d -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-d.} ~~ \text{Remainder of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
=====Summary of Conjunction=====<br />
<br />
To establish a convenient reference point for further discussion, Table&nbsp;49 summarizes the operator actions that have been computed for the form of conjunction, as exemplified by the proposition <math>J.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 49.} ~~ \text{Computation Summary for}~ J~\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon J<br />
& = & u \!\cdot\! v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}J<br />
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}J<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{r}J<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Analytic Series : Coordinate Method====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
And if he is told that something ''is'' the way it is, then he thinks: Well, it could probably just as easily be some other way. So the sense of possibility might be defined outright as the capacity to think how everything could &ldquo;just as easily&rdquo; be, and to attach no more importance to what is than to what is not.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 12]<br />
|}<br />
<br />
Table&nbsp;50 exhibits a truth table method for computing the analytic series (or the differential expansion) of a proposition in terms of coordinates.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:70%"<br />
|+ style="height:30px" | <math>\text{Table 50.} ~~ \text{Computation of an Analytic Series in Terms of Coordinates}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:8%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:8%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:8%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}u\!</math><br />
| style="width:8%; border-bottom:1px solid black" | <math>\mathrm{d}v\!</math><br />
| style="width:8%; border-bottom:1px solid black; border-left:1px solid black" | <math>u'\!</math><br />
| style="width:8%; border-bottom:1px solid black" | <math>v'\!</math><br />
| style="width:10%; border-bottom:1px solid black; border-left:4px double black" | <math>\boldsymbol\varepsilon J\!</math><br />
| style="width:10%; border-bottom:1px solid black" | <math>\mathrm{E}J\!</math><br />
| style="width:12%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{D}J\!</math><br />
| style="width:10%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}J\!</math><br />
| style="width:10%; border-bottom:1px solid black" | <math>\mathrm{d}^2\!J\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
The first six columns of the Table, taken as a whole, represent the variables of a construct called the ''contingent universe'' <math>[u, v, \mathrm{d}u, \mathrm{d}v, u', v'],\!</math> or the bundle of ''contingency spaces'' <math>[\mathrm{d}u, \mathrm{d}v, u', v']\!</math> over the universe <math>[u, v].\!</math> Their placement to the left of the double bar indicates that all of them amount to independent variables, but there is a co-dependency among them, as described by the following equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
u' & = & u + \mathrm{d}u & = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\\[8pt]<br />
v' & = & v + \mathrm{d}v & = & \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
These relations correspond to the formal substitutions that are made in defining <math>\mathrm{E}J\!</math> and <math>\mathrm{D}J.\!</math> For now, the whole rigamarole of contingency spaces can be regarded as a technical device for achieving the effect of these substitutions, adapted to a setting where functional compositions and other symbolic manipulations are difficult to contemplate and execute.<br />
<br />
The five columns to the right of the double bar in Table&nbsp;50 contain the values of the dependent variables <math>\{ \boldsymbol\varepsilon J, ~\mathrm{E}J, ~\mathrm{D}J, ~\mathrm{d}J, ~\mathrm{d}^2\!J \}.\!</math> These are normally interpreted as values of functions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B}\!</math> or as values of propositions in the extended universe <math>[u, v, \mathrm{d}u, \mathrm{d}v]\!</math> but the dependencies prevailing in the contingent universe make it possible to regard these same final values as arising via functions on alternative lists of arguments, for example, the set <math>\{ u, v, u', v' \}.\!</math><br />
<br />
The column for <math>\boldsymbol\varepsilon J\!</math> is computed as <math>J(u, v) = uv\!</math> and together with the columns for <math>u\!</math> and <math>v\!</math> illustrates how we &ldquo;share structure&rdquo; in the Table by listing only the first entries of each constant block.<br />
<br />
The column for <math>\mathrm{E}J\!</math> is computed by means of the following chain of identities, where the contingent variables <math>u'\!</math> and <math>v'\!</math> are defined as <math>u' = u + \mathrm{d}u\!</math> and <math>v' = v + \mathrm{d}v.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathrm{E}J(u, v, \mathrm{d}u, \mathrm{d}v)<br />
& = &<br />
J(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& = &<br />
J(u', v')<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
This makes it easy to determine <math>\mathrm{E}J\!</math> by inspection, computing the conjunction <math>J(u', v') = u'v'\!</math> from the columns headed <math>u'\!</math> and <math>v'.\!</math> Since each of these forms expresses the same proposition <math>\mathrm{E}J\!</math> in <math>\mathrm{E}U^\bullet,\!</math> the dependence on <math>\mathrm{d}u\!</math> and <math>\mathrm{d}v\!</math> is still present but merely left implicit in the final variant <math>J(u', v').\!</math><br />
<br />
* '''Note.''' On occasion, it is tempting to use the further notation <math>J'(u, v) = J(u', v'),\!</math> especially to suggest a transformation that acts on whole propositions, for example, taking the proposition <math>J\!</math> into the proposition <math>J' = \mathrm{E}J.\!</math> The prime <math>( {}^{\prime} )\!</math> then signifies an action that is mediated by a field of choices, namely, the values that are picked out for the contingent variables in sweeping through the initial universe. But this heaps an unwieldy lot of construed intentions on a rather slight character and puts too high a premium on the constant correctness of its interpretation. In practice, therefore, it is best to avoid this usage.<br />
<br />
Given the values of <math>\boldsymbol\varepsilon J\!</math> and <math>\mathrm{E}J,\!</math> the columns for the remaining functions can be filled in quickly. The difference map is computed according to the relation <math>\mathrm{D}J = \boldsymbol\varepsilon J + \mathrm{E}J.\!</math> The first order differential <math>\mathrm{d}J\!</math> is found by looking in each block of constant argument pairs <math>u, v\!</math> and choosing the linear function of <math>\mathrm{d}u, \mathrm{d}v\!</math> that best approximates <math>\mathrm{D}J\!</math> in that block. Finally, the remainder is computed as <math>\mathrm{r}J = \mathrm{D}J + \mathrm{d}J,\!</math> in this case yielding the second order differential <math>\mathrm{d}^2\!J.\!</math><br />
<br />
====Analytic Series : Recap====<br />
<br />
Let us now summarize the results of Table&nbsp;50 by writing down for each column and for each block of constant argument pairs <math>u, v\!</math> a reasonably canonical symbolic expression for the function of <math>\mathrm{d}u, \mathrm{d}v\!</math> that appears there. The synopsis formed in this way is presented in Table&nbsp;51. As one has a right to expect, it confirms the results that were obtained previously by operating solely in terms of the formal calculus.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:70%"<br />
|+ style="height:30px" | <math>\text{Table 51.} ~~ \text{Computation of an Analytic Series in Symbolic Terms}\!</math><br />
|- style="height:35px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black" | <math>J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{E}J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{D}J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{d}J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{d}^2\!J\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}u \!\;\cdot\;\! \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}u \!\;\cdot\;\! \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
\mathrm{d}u<br />
\\[4pt]<br />
\mathrm{d}v<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;52 and 53 provide a quick overview of the analysis performed so far, giving the successive decompositions of <math>\mathrm{E}J = J + \mathrm{D}J\!</math> and <math>\mathrm{D}J = \mathrm{d}J + \mathrm{r}J\!</math> in two different styles of diagram.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 52 -- Decomposition of EJ.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 52.} ~~ \text{Decomposition of}~ \mathrm{E}J\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 53 -- Decomposition of DJ.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 53.} ~~ \text{Decomposition of}~ \mathrm{D}J\!</math><br />
|}<br />
<br />
====Terminological Interlude====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Lastly, my attention was especially attracted, not so much to the scene, as to the mirrors that produced it. These mirrors were broken in parts. Yes, they were marked and scratched; they had been &ldquo;starred&rdquo;, in spite of their solidity &hellip;<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 230]<br />
|}<br />
<br />
At this point several issues of terminology have accrued enough substance to intrude on our discussion. The remarks of this Subsection are intended to accomplish two goals. First, we call attention to significant aspects of the previous series of Figures, translating into literal terms what they depict in iconic forms, and we re-stress the most important structural elements they indicate. Next, we prepare the way for taking on more complex examples of transformations, those whose target universes have more than one dimension.<br />
<br />
In talking about the actions of operators it is important to keep in mind the distinctions between the operators per&nbsp;se, their operands, and their results. Furthermore, in working with composite forms of operators <math>\mathrm{W} = (\mathrm{W}_1, \ldots, \mathrm{W}_n),\!</math> transformations <math>\mathrm{F} = (\mathrm{F}_1, \ldots, \mathrm{F}_n),\!</math> and target domains <math>X^\bullet = [x_1, \ldots, x_n],\!</math> we need to preserve a clear distinction between the compound entity of each given type and any one of its separate components. It is curious, given the usefulness of the concepts ''operator'' and ''operand'', that we seem to lack a generic term, formed on the same root, for the corresponding result of an operation. Following the obvious paradigm would lead to words like ''opus'', ''opera'', and ''operant'', but these words are too affected with clang associations to work well at present, though they might be adapted in time. One current usage gets around this problem by using the substantive ''map'' as a systematic epithet to express the result of each operator's action. We will follow this practice as far as possible, for example, using the phrase ''tangent map'' to denote the end product of the tangent functor acting on its operand map.<br />
<br />
* '''Scholium.''' See [JGH, 6-9] for a good account of tangent functors and tangent maps in ordinary analysis and for examples of their use in mechanics. This work as a whole is a model of clarity in applying functorial principles to problems in physical dynamics.<br />
<br />
Whenever we focus on isolated propositions, on single components of composite operators, or on the portions of transformations that have <math>1\!</math>-dimensional ranges, we are free to shift between the native form of a proposition <math>J : U \to \mathbb{B}\!</math> and the thematized form of a mapping <math>J : U^\bullet \to [x]\!</math> without much trouble. In these cases we are able to tolerate a higher degree of ambiguity about the precise nature of an operator's input and output domains than we otherwise might. For example, in the preceding treatment of the example <math>J,\!</math> and for each operator <math>\mathrm{W}\!</math> in the set <math>\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \},\!</math> both the operand <math>J\!</math> and the result <math>\mathrm{W}J\!</math> could be viewed in either one of two ways. On one hand we may treat them as propositions <math>J : U \to \mathbb{B}\!</math> and <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B},\!</math> ignoring the distinction between the range <math>[x] \cong \mathbb{B}\!</math> of <math>\boldsymbol\varepsilon J\!</math> and the range <math>[\mathrm{d}x] \cong \mathbb{D}\!</math> of the other types of <math>\mathrm{W}J.\!</math> This is what we usually do when we content ourselves with simply coloring in regions of venn diagrams. On the other hand we may view these entities as maps <math>J : U^\bullet \to [x] = X^\bullet\!</math> and <math>\boldsymbol\varepsilon J : \mathrm{E}U^\bullet \to [x] \subseteq \mathrm{E}X^\bullet\!</math> or <math>\mathrm{W}J : \mathrm{E}U^\bullet \to [\mathrm{d}x] \subseteq \mathrm{E}X^\bullet,\!</math> in which case the qualitative characters of the output features are not ignored.<br />
<br />
At the beginning of this Section we recast the natural form of a proposition <math>J : U \to \mathbb{B}\!</math> into the thematic role of a transformation <math>J : U^\bullet \to [x],\!</math> where <math>x\!</math> was a variable recruited to express the newly independent <math>\check{J}.\!</math> However, in our computations and representations of operator actions we immediately lapsed back to viewing the results as native elements of the extended universe <math>\mathrm{E}U^\bullet,\!</math> in other words, as propositions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B},\!</math> where <math>\mathrm{W}\!</math> ranged over the set <math>\{ \boldsymbol\varepsilon, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}.\!</math> That is as it should be. We have worked hard to devise a language that gives us these advantages &mdash; the flexibility to exchange terms and types of equal information value and the capacity to reflect as quickly and as wittingly as a controlled reflex on the fibers of our propositions, independently of whether they express amusements, beliefs, or conjectures.<br />
<br />
As we take on target spaces of increasing dimension, however, these types of confusions (and confusions of types) become less and less permissible. For this reason, Tables&nbsp;54 and 55 present a rather detailed summary of the notation and the terminology we are using, as applied to the case <math>J = uv.\!</math> The rationale of these Tables is not so much to train more elephant guns on this poor drosophila of a concrete example but to invest our paradigm with enough solidity to bear the weight of abstraction to come.<br />
<br />
Table&nbsp;54 provides basic notation and descriptive information for the objects and operators used in this Example, giving the generic type (or broadest defined type) for each entity. Here, the sans&nbsp;serif operators <math>\mathsf{W} \in \{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{d}, \mathsf{r} \}\!</math> and their components <math>\mathrm{W} \in \{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}\!</math> both have the same broad type <math>\mathsf{W}, \mathrm{W} : (U^\bullet \to X^\bullet) \to (\mathrm{E}U^\bullet \to \mathrm{E}X^\bullet),\!</math> as appropriate to operators that map transformations <math>J : U^\bullet \to X^\bullet\!</math> to extended transformations <math>\mathsf{W}J, \mathrm{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 54.} ~~ \text{Cast of Characters : Expansive Subtypes of Objects and Operators}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| align="center" | <math>\text{Symbol}\!</math><br />
| align="center" | <math>\text{Notation}\!</math><br />
| align="center" | <math>\text{Description}\!</math><br />
| align="center" | <math>\text{Type}\!</math><br />
|-<br />
| align="center" | <math>U^\bullet\!</math><br />
| <math>= [u, v]\!</math><br />
| <math>\text{Source universe}\!</math><br />
| <math>[\mathbb{B}^2]\!</math><br />
|-<br />
| align="center" | <math>X^\bullet~\!</math><br />
| <math>= [x]\!</math><br />
| <math>\text{Target universe}\!</math><br />
| <math>[\mathbb{B}^1]~\!</math><br />
|-<br />
| align="center" | <math>\mathrm{E}U^\bullet\!</math><br />
| <math>= [u, v, \mathrm{d}u, \mathrm{d}v]\!</math><br />
| <math>\text{Extended source universe}\!</math><br />
| <math>[\mathbb{B}^2 \!\times\! \mathbb{D}^2]</math><br />
|-<br />
| align="center" | <math>\mathrm{E}X^\bullet\!</math><br />
| <math>= [x, \mathrm{d}x]~\!</math><br />
| <math>\text{Extended target universe}\!</math><br />
| <math>[\mathbb{B}^1 \!\times\! \mathbb{D}^1]</math><br />
|-<br />
| align="center" | <math>J\!</math><br />
| <math>J : U \!\to\! \mathbb{B}\!</math><br />
| <math>\text{Proposition}\!</math><br />
| <math>(\mathbb{B}^2 \!\to\! \mathbb{B}) \in [\mathbb{B}^2]\!</math><br />
|-<br />
| align="center" | <math>J\!</math><br />
| <math>J : U^\bullet \!\to\! X^\bullet\!</math><br />
| <math>\text{Transformation or Map}\!</math><br />
| <math>[\mathbb{B}^2] \!\to\! [\mathbb{B}^1]\!</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\boldsymbol\varepsilon<br />
\\<br />
\eta<br />
\\<br />
\mathrm{E}<br />
\\<br />
\mathrm{D}<br />
\\<br />
\mathrm{d}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{W} : U^\bullet \!\to\! \mathrm{E}U^\bullet,<br />
\\<br />
\mathrm{W} : X^\bullet \!\to\! \mathrm{E}X^\bullet,<br />
\\<br />
\mathrm{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\\<br />
\text{for each}~ \mathrm{W} ~\text{in the set:}<br />
\\<br />
\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{ll}<br />
\text{Tacit extension operator} & \boldsymbol\varepsilon<br />
\\<br />
\text{Trope extension operator} & \eta<br />
\\<br />
\text{Enlargement operator} & \mathrm{E}<br />
\\<br />
\text{Difference operator} & \mathrm{D}<br />
\\<br />
\text{Differential operator} & \mathrm{d}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^2] \!\to\! [\mathbb{B}^2 \!\times\! \mathbb{D}^2]},<br />
\\<br />
{[\mathbb{B}^1] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]},<br />
\\\\<br />
([\mathbb{B}^2] \!\to\! [\mathbb{B}^1]) \!\to\!<br />
\\<br />
([\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1])<br />
\end{array}</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\mathsf{e}<br />
\\<br />
\mathsf{E}<br />
\\<br />
\mathsf{D}<br />
\\<br />
\mathsf{T}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{W} : U^\bullet \!\to\! \mathsf{T}U^\bullet = \mathrm{E}U^\bullet,<br />
\\<br />
\mathsf{W} : X^\bullet \!\to\! \mathsf{T}X^\bullet = \mathrm{E}X^\bullet,<br />
\\<br />
\mathsf{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathsf{T}U^\bullet \!\to\! \mathsf{T}X^\bullet)<br />
\\<br />
\text{for each}~ \mathsf{W} ~\text{in the set:}<br />
\\<br />
\{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{T} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{lll}<br />
\text{Radius operator} & \mathsf{e} & = (\boldsymbol\varepsilon, \eta)<br />
\\<br />
\text{Secant operator} & \mathsf{E} & = (\boldsymbol\varepsilon, \mathrm{E})<br />
\\<br />
\text{Chord operator} & \mathsf{D} & = (\boldsymbol\varepsilon, \mathrm{D})<br />
\\<br />
\text{Tangent functor} & \mathsf{T} & = (\boldsymbol\varepsilon, \mathrm{d})<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^2] \!\to\! [\mathbb{B}^2 \!\times\! \mathbb{D}^2]},<br />
\\<br />
{[\mathbb{B}^1] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]},<br />
\\\\<br />
([\mathbb{B}^2] \!\to\! [\mathbb{B}^1]) \!\to\!<br />
\\<br />
([\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1])<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;55 supplies a more detailed outline of terminology for operators and their results. Here, we list the restrictive subtype (or narrowest defined subtype) that applies to each entity and we indicate across the span of the Table the whole spectrum of alternative types that color the interpretation of each symbol. For example, all the component operator maps <math>\mathrm{W}J\!</math> have <math>1\!</math>-dimensional ranges, either <math>\mathbb{B}^1\!</math> or <math>\mathbb{D}^1,\!</math> and so they can be viewed either as propositions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B}\!</math> or as logical transformations <math>\mathrm{W}J : \mathrm{E}U^\bullet \to X^\bullet.\!</math> As a rule, the plan of the Table allows us to name each entry by detaching the underlined adjective at the left of its row and prefixing it to the generic noun at the top of its column. In one case, however, it is customary to depart from this scheme. Because the phrase ''differential proposition'', applied to the result <math>\mathrm{d}J : \mathrm{E}U \to \mathbb{D},\!</math> does not distinguish it from the general run of differential propositions <math>\mathrm{G}: \mathrm{E}U \to \mathbb{B},\!</math> it is usual to single out <math>\mathrm{d}J\!</math> as the ''tangent proposition'' of <math>J.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 55.} ~~ \text{Synopsis of Terminology : Restrictive and Alternative Subtypes}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| &nbsp;<br />
| align="center" | <math>\text{Operator}\!</math><br />
| align="center" | <math>\text{Proposition}\!</math><br />
| align="center" | <math>\text{Map}\!</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tacit}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\boldsymbol\varepsilon : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\boldsymbol\varepsilon : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{B} \\<br />
\boldsymbol\varepsilon J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{B}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x] \\<br />
\boldsymbol\varepsilon J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Trope}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\eta : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\eta : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\eta J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\eta J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Enlargement}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{E} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{E}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{E}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Difference}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{D} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{D}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{D}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Differential}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{d} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{d} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{d}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}\!</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Remainder}}\\\text{operator}\end{matrix}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{r} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{r} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{r}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{r}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Radius}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e} = (\boldsymbol\varepsilon, \eta) \\<br />
\mathsf{e} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{e}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Secant}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}) \\<br />
\mathsf{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{E}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Chord}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}) \\<br />
\mathsf{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{D}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tangent}}\\\text{functor}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T} = (\boldsymbol\varepsilon, \mathrm{d}) \\<br />
\mathsf{T} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{T}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====End of Perfunctory Chatter : Time to Roll the Clip!====<br />
<br />
Two steps remain to finish the analysis of <math>J\!</math> that we began so long ago. First, we need to paste our accumulated heap of flat pictures into the frames of transformations, filling out the shapes of the operator maps <math>\mathsf{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet.~\!</math> This scheme is executed in two styles, using the ''areal views'' in Figures&nbsp;56-a and the ''box views'' in Figures&nbsp;56-b. Finally, in Figures&nbsp;57-1 to 57-4 we put all the pieces together to construct the full operator diagrams for <math>\mathsf{W} : J \to \mathsf{W}J.\!</math> There is a considerable amount of redundancy among the following three series of Figures but that will hopefully provide a fuller picture of the operations under review, enabling these snapshots to serve as successive frames in the animation of logic they are meant to become.<br />
<br />
=====Operator Maps : Areal Views=====<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a1 -- Radius Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a1.} ~~ \text{Radius Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a2 -- Secant Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a2.} ~~ \text{Secant Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a3 -- Chord Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a3.} ~~ \text{Chord Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a4 -- Tangent Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a4.} ~~ \text{Tangent Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
=====Operator Maps : Box Views=====<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b1 -- Radius Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b1.} ~~ \text{Radius Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b2 -- Secant Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b2.} ~~ \text{Secant Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b3 -- Chord Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b3.} ~~ \text{Chord Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b4 -- Tangent Map of J ISW.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b4.} ~~ \text{Tangent Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
=====Operator Diagrams for the Conjunction J = uv=====<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-1 -- Radius Operator Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-1.} ~~ \text{Radius Operator Diagram for the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-2 -- Secant Operator Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-2.} ~~ \text{Secant Operator Diagram for the Conjunction}~ J = uv~\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-3 -- Chord Operator Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-3.} ~~ \text{Chord Operator Diagram for the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-4 -- Tangent Functor Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-4.} ~~ \text{Tangent Functor Diagram for the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
===Taking Aim at Higher Dimensional Targets===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
The past and present wilt . . . . I have filled them and<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;emptied them,<br><br />
And proceed to fill my next fold of the future.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 87]<br />
|}<br />
<br />
In the next Section we consider a transformation <math>F\!</math> of concrete type <math>F : [u, v] \to [x, y]\!</math> and abstract type <math>F : [\mathbb{B}^2] \to [\mathbb{B}^2].\!</math> From the standpoint of propositional calculus we naturally approach the task of understanding such a transformation by parsing it into component maps with <math>1\!</math>-dimensional ranges, as follows:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{ccccccl}<br />
F & = & (F_1, F_2) & = & (f, g) & : & [u, v] \to [x, y],<br />
\\[6pt]<br />
&& F_1 & = & f & : & [u, v] \to [x],<br />
\\[6pt]<br />
&& F_2 & = & g & : & [u, v] \to [y].<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Then we tackle the separate components, now viewed as propositions <math>F_i : U \to \mathbb{B},\!</math> one at a time. At the completion of this analytic phase, we return to the task of synthesizing these partial and transient impressions into an agile form of integrity, a solidly coordinated and deeply integrated comprehension of the ongoing transformation. (Very often, of course, in tangling with refractory cases, we never get as far as the beginning again.)<br />
<br />
Let us now refer to the dimension of the target space or codomain as the ''toll'' (or ''tole'') of a transformation, as distinguished from the dimension of the range or image that is customarily called the ''rank''. When we keep to transformations with a toll of <math>1,\!</math> as <math>J : [u, v] \to [x],\!</math> we tend to get lazy about distinguishing a logical transformation from its component propositions. However, if we deal with transformations of a higher toll, this form of indolence can no longer be tolerated.<br />
<br />
Well, perhaps we can carry it a little further. After all, the operator result <math>\mathrm{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet\!</math> is a map of toll <math>2,\!</math> and cannot be unfolded in one piece as a proposition. But when a map has rank <math>1,\!</math> like <math>\boldsymbol\varepsilon J : \mathrm{E}U \to X \subseteq \mathrm{E}X\!</math> or <math>\mathrm{d}J : \mathrm{E}U \to \mathrm{d}X \subseteq \mathrm{E}X,\!</math> we naturally choose to concentrate on the <math>1\!</math>-dimensional range of the operator result <math>\mathrm{W}J,\!</math> ignoring the final difference in quality between the spaces <math>X\!</math> and <math>\mathrm{d}X,\!</math> and view <math>\mathrm{W}J\!</math> as a proposition about <math>\mathrm{E}U.\!</math><br />
<br />
In this way, an initial ambivalence about the role of the operand <math>J\!</math> conveys a double duty to the result <math>\mathrm{W}J.\!</math> The pivot that is formed by our focus of attention is essential to the linkage that transfers this double moment, as the whole process takes its bearing and wheels around the precise measure of a narrow bead that we can draw on the range of <math>\mathrm{W}J.\!</math> This is the escapement that it takes to get away with what may otherwise seem to be a simple duplicity, and this is the tolerance that is needed to counterbalance a certain arrogance of equivocation, by all of which machinations we make ourselves free to indicate the operator results <math>\mathrm{W}J\!</math> as propositions or as transformations, indifferently.<br />
<br />
But that's it, and no further. Neglect of these distinctions in range and target universes of higher dimensions is bound to cause a hopeless confusion. To guard against these adverse prospects, Tables&nbsp;58 and 59 lay the groundwork for discussing a typical map <math>F : [\mathbb{B}^2] \to [\mathbb{B}^2],\!</math> and begin to pave the way to some extent for discussing any transformation of the form <math>F : [\mathbb{B}^n] \to [\mathbb{B}^k].\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 58.} ~~ \text{Cast of Characters : Expansive Subtypes of Objects and Operators}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| align="center" | <math>\text{Symbol}\!</math><br />
| align="center" | <math>\text{Notation}\!</math><br />
| align="center" | <math>\text{Description}\!</math><br />
| align="center" | <math>\text{Type}\!</math><br />
|-<br />
| align="center" | <math>U^\bullet\!</math><br />
| <math>= [u, v]\!</math><br />
| <math>\text{Source universe}\!</math><br />
| <math>[\mathbb{B}^n]\!</math><br />
|-<br />
| align="center" | <math>X^\bullet~\!</math><br />
| <math>\begin{array}{l}<br />
= [x, y] \\<br />
= [f, g]<br />
\end{array}</math><br />
| <math>\text{Target universe}\!</math><br />
| <math>[\mathbb{B}^k]\!</math><br />
|-<br />
| align="center" | <math>\mathrm{E}U^\bullet\!</math><br />
| <math>= [u, v, \mathrm{d}u, \mathrm{d}v]\!</math><br />
| <math>\text{Extended source universe}\!</math><br />
| <math>[\mathbb{B}^n \!\times\! \mathbb{D}^n]\!</math><br />
|-<br />
| align="center" | <math>\mathrm{E}X^\bullet\!</math><br />
| <math>\begin{array}{l}<br />
= [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
= [f, g, \mathrm{d}f, \mathrm{d}g]<br />
\end{array}</math><br />
| <math>\text{Extended target universe}\!</math><br />
| <math>[\mathbb{B}^k \!\times\! \mathbb{D}^k]\!</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
f \\ g<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{ll}<br />
f : U \!\to\! [x] \cong \mathbb{B} \\<br />
g : U \!\to\! [y] \cong \mathbb{B}<br />
\end{array}</math><br />
| <math>\text{Proposition}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathbb{B}^n \!\to\! \mathbb{B} \\<br />
\in (\mathbb{B}^n, \mathbb{B}^n \!\to\! \mathbb{B}) = [\mathbb{B}^n]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>F\!</math><br />
| <math>F = (f, g) : U^\bullet \!\to\! X^\bullet\!</math><br />
| <math>\text{Transformation of Map}\!</math><br />
| <math>[\mathbb{B}^n] \!\to\! [\mathbb{B}^k]</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\boldsymbol\varepsilon<br />
\\<br />
\eta<br />
\\<br />
\mathrm{E}<br />
\\<br />
\mathrm{D}<br />
\\<br />
\mathrm{d}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{W} : U^\bullet \!\to\! \mathrm{E}U^\bullet,<br />
\\<br />
\mathrm{W} : X^\bullet \!\to\! \mathrm{E}X^\bullet,<br />
\\<br />
\mathrm{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\\<br />
\text{for each}~ \mathrm{W} ~\text{in the set:}<br />
\\<br />
\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{ll}<br />
\text{Tacit extension operator} & \boldsymbol\varepsilon<br />
\\<br />
\text{Trope extension operator} & \eta<br />
\\<br />
\text{Enlargement operator} & \mathrm{E}<br />
\\<br />
\text{Difference operator} & \mathrm{D}<br />
\\<br />
\text{Differential operator} & \mathrm{d}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^n] \!\to\! [\mathbb{B}^n \!\times\! \mathbb{D}^n]},<br />
\\<br />
{[\mathbb{B}^k] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]},<br />
\\\\<br />
([\mathbb{B}^n] \!\to\! [\mathbb{B}^k]) \!\to\!<br />
\\<br />
([\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k])<br />
\end{array}</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\mathsf{e}<br />
\\<br />
\mathsf{E}<br />
\\<br />
\mathsf{D}<br />
\\<br />
\mathsf{T}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{W} : U^\bullet \!\to\! \mathsf{T}U^\bullet = \mathrm{E}U^\bullet,<br />
\\<br />
\mathsf{W} : X^\bullet \!\to\! \mathsf{T}X^\bullet = \mathrm{E}X^\bullet,<br />
\\<br />
\mathsf{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathsf{T}U^\bullet \!\to\! \mathsf{T}X^\bullet)<br />
\\<br />
\text{for each}~ \mathsf{W} ~\text{in the set:}<br />
\\<br />
\{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{T} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{lll}<br />
\text{Radius operator} & \mathsf{e} & = (\boldsymbol\varepsilon, \eta)<br />
\\<br />
\text{Secant operator} & \mathsf{E} & = (\boldsymbol\varepsilon, \mathrm{E})<br />
\\<br />
\text{Chord operator} & \mathsf{D} & = (\boldsymbol\varepsilon, \mathrm{D})<br />
\\<br />
\text{Tangent functor} & \mathsf{T} & = (\boldsymbol\varepsilon, \mathrm{d})<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^n] \!\to\! [\mathbb{B}^n \!\times\! \mathbb{D}^n]},<br />
\\<br />
{[\mathbb{B}^k] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]},<br />
\\\\<br />
([\mathbb{B}^n] \!\to\! [\mathbb{B}^k]) \!\to\!<br />
\\<br />
([\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k])<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 59.} ~~ \text{Synopsis of Terminology : Restrictive and Alternative Subtypes}~\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| &nbsp;<br />
| align="center" | <math>\begin{matrix}\text{Operator}\\\text{or}\\\text{Operand}\end{matrix}</math><br />
| align="center" | <math>\begin{matrix}\text{Proposition}\\\text{or}\\\text{Component}\end{matrix}</math><br />
| align="center" | <math>\begin{matrix}\text{Transformation}\\\text{or}\\\text{Map}\end{matrix}</math><br />
|-<br />
| align="center" | <math>\underline{\text{Operand}}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
F = (F_1, F_2) \\<br />
F = (f, g) : U \!\to\! X<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
F_i : \langle u, v \rangle \!\to\! \mathbb{B} \\<br />
F_i : \mathbb{B}^n \!\to\! \mathbb{B}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
F : [u, v] \!\to\! [x, y] \\<br />
F : [\mathbb{B}^n] \!\to\! [\mathbb{B}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tacit}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\boldsymbol\varepsilon : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\boldsymbol\varepsilon : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{B} \\<br />
\boldsymbol\varepsilon F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{B}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y] \\<br />
\boldsymbol\varepsilon F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Trope}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\eta : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\eta : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\eta F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\eta F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Enlargement}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{E} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{E}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{E}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Difference}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{D} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{D}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{D}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Differential}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{d} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{d} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{d}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}\!</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Remainder}}\\\text{operator}\end{matrix}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{r} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{r} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{r}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{r}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Radius}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e} = (\boldsymbol\varepsilon, \eta) \\<br />
\mathsf{e} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{e}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Secant}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}) \\<br />
\mathsf{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{E}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Chord}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}) \\<br />
\mathsf{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{D}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tangent}}\\\text{functor}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T} = (\boldsymbol\varepsilon, \mathrm{d}) \\<br />
\mathsf{T} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{T}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
===Transformations of Type '''B'''<sup>2</sup> &rarr; '''B'''<sup>2</sup>===<br />
<br />
To take up a slightly more complex example, but one that remains simple enough to pursue through a complete series of developments, consider the transformation from <math>U^\bullet = [u, v]\!</math> to <math>X^\bullet = [x, y]\!</math> that is defined by the following system of equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
x<br />
& = & f(u, v)<br />
& = & \texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[8pt]<br />
y<br />
& = & g(u, v)<br />
& = & \texttt{((} u \texttt{,} v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
The component notation <math>F = (F_1, F_2) = (f, g) : U^\bullet \to X^\bullet\!</math> allows us to give a name and a type to this transformation and permits defining it by the compact description that follows:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
(x, y)<br />
& = & F(u, v)<br />
& = & (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Logical Transformations====<br />
<br />
The information that defines the logical transformation <math>F\!</math> can be represented in the form of a truth table, as shown in Table&nbsp;60. To cut down on subscripts in this example we continue to use plain letter equivalents for all components of spaces and maps.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:60%"<br />
|+ style="height:30px" | <math>\text{Table 60.} ~~ \text{A Propositional Transformation}\!</math><br />
|- style="height:40px; background:ghostwhite; width:100%"<br />
| style="width:25%" | <math>u\!</math><br />
| style="width:25%" | <math>v\!</math><br />
| style="width:25%; border-left:1px solid black" | <math>f\!</math><br />
| style="width:25%" | <math>g\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="border-top:1px solid black" | <math>u\!</math><br />
| style="border-top:1px solid black" | <math>v\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;61 shows how we might paint a picture of the transformation <math>F\!</math> in the manner of Figure&nbsp;30.<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 61 -- Propositional Transformation.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 61.} ~~ \text{A Propositional Transformation}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;62 extracts the gist of Figure&nbsp;61, exhibiting a style of diagram that is adequate for most purposes.<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 62 -- Propositional Transformation (Short Form).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 62.} ~~ \text{A Propositional Transformation (Short Form)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Local Transformations====<br />
<br />
Figure&nbsp;63 gives a more complete picture of the transformation <math>F,\!</math> showing how the points of <math>U^\bullet\!</math> are transformed into points of <math>X^\bullet.\!</math> The bold lines crossing from one universe to the other trace the action that <math>F\!</math> induces on points, in other words, they show how the transformation acts as a mapping from points to points and chart its effects on the elements that are variously called cells, points, positions, or singular propositions.<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 63 -- Transformation of Positions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 63.} ~~ \text{A Transformation of Positions}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;64 shows how the action of <math>F\!</math> on cells or points can be computed in terms of coordinates.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 64.} ~~ \text{A Transformation of Positions}\!</math><br />
|- style="height:40px; background:ghostwhite; width:100%"<br />
| style="width:8%" | <math>u\!</math><br />
| style="width:8%" | <math>v\!</math><br />
| style="width:12%; border-left:1px solid black" | <math>x\!</math><br />
| style="width:12%" | <math>y\!</math><br />
| style="width:10%; border-left:1px solid black" | <math>x~y\!</math><br />
| style="width:10%" | <math>x \texttt{(} y \texttt{)}\!</math><br />
| style="width:10%" | <math>\texttt{(} x \texttt{)} y\!</math><br />
| style="width:10%" | <math>\texttt{(} x \texttt{)(} y \texttt{)}\!</math><br />
| style="width:20%; border-left:1px solid black" | <math>X^\bullet = [x, y]\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\uparrow<br />
\\[4pt]<br />
F =<br />
\\[4pt]<br />
(f, g)<br />
\\[4pt]<br />
\uparrow<br />
\end{matrix}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="border-top:1px solid black" | <math>u\!</math><br />
| style="border-top:1px solid black" | <math>v\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
| style="border-top:1px solid black" | <math>\texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>u~v\!</math><br />
| style="border-top:1px solid black" | <math>\texttt{(} u \texttt{,} v \texttt{)}\!</math><br />
| style="border-top:1px solid black" | <math>\texttt{(} u \texttt{)(} v \texttt{)}\!</math><br />
| style="border-top:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>U^\bullet = [u, v]\!</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;65 extends this scheme from single cells to arbitrary regions, showing how we might compute the action of a logical transformation on arbitrary propositions in the universe of discourse. The effect of a point-transformation on arbitrary propositions, or any other structures erected on points, is referred to as the ''induced action'' of the transformation on the structures in question.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 65-a.} ~~ \text{An Induced Transformation on Propositions}\!</math><br />
|- style="height:50px; background:ghostwhite"<br />
| style="width:20%" | <math>X^\bullet~\!</math><br />
| style="width:20%; border-right:none" | <math>\longleftarrow\!</math><br />
| style="width:20%; border-left:none; border-right:none" | <math>F = (f, g)\!</math><br />
| style="width:20%; border-left:none" | <math>\longleftarrow\!</math><br />
| style="width:20%" | <math>U^\bullet~\!</math><br />
|- style="background:ghostwhite"<br />
| rowspan="2" | <math>f_i (x, y)\!</math><br />
| align="right" | <math>\begin{matrix}u = \\ v =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~0~0\\1~0~1~0\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= u \\ = v\end{matrix}</math><br />
| rowspan="2" style="width:20%" | <math>f_j (u, v)\!</math><br />
|- style="background:ghostwhite"<br />
| align="right" | <math>\begin{matrix}x = \\ y =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~1~0\\1~0~0~1\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= f(u, v) \\ = g(u, v)\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{2}<br />
\\[2pt]<br />
f_{3}<br />
\\[2pt]<br />
f_{4}<br />
\\[2pt]<br />
f_{5}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{7}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)~ ~}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{~ ~(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{,~} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{~~} y \texttt{)}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~0<br />
\\[2pt]<br />
0~0~0~0<br />
\\[2pt]<br />
0~0~0~1<br />
\\[2pt]<br />
0~0~0~1<br />
\\[2pt]<br />
0~1~1~0<br />
\\[2pt]<br />
0~1~1~0<br />
\\[2pt]<br />
0~1~1~1<br />
\\[2pt]<br />
0~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{,~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{,~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{~~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{~~} v \texttt{)}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[2pt]<br />
f_{0}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{7}<br />
\\[2pt]<br />
f_{7}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{8}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{10}<br />
\\[2pt]<br />
f_{11}<br />
\\[2pt]<br />
f_{12}<br />
\\[2pt]<br />
f_{13}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{15}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[2pt]<br />
\texttt{~ ~ ~ ~} y \texttt{~~}<br />
\\[2pt]<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[2pt]<br />
\texttt{~~} x \texttt{~ ~ ~ ~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
1~0~0~0<br />
\\[2pt]<br />
1~0~0~0<br />
\\[2pt]<br />
1~0~0~1<br />
\\[2pt]<br />
1~0~0~1<br />
\\[2pt]<br />
1~1~1~0<br />
\\[2pt]<br />
1~1~1~0<br />
\\[2pt]<br />
1~1~1~1<br />
\\[2pt]<br />
1~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~~} u \texttt{~~} v \texttt{~~}<br />
\\[2pt]<br />
\texttt{~~} u \texttt{~~} v \texttt{~~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{8}<br />
\\[2pt]<br />
f_{8}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{15}<br />
\\[2pt]<br />
f_{15}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 65-b.} ~~ \text{An Induced Transformation on Propositions}\!</math><br />
|- style="height:50px; background:ghostwhite"<br />
| style="width:20%" | <math>X^\bullet~\!</math><br />
| style="width:20%; border-right:none" | <math>\longleftarrow\!</math><br />
| style="width:20%; border-left:none; border-right:none" | <math>F = (f, g)\!</math><br />
| style="width:20%; border-left:none" | <math>\longleftarrow\!</math><br />
| style="width:20%" | <math>U^\bullet~\!</math><br />
|- style="background:ghostwhite"<br />
| rowspan="2" | <math>f_i (x, y)\!</math><br />
| align="right" | <math>\begin{matrix}u = \\ v =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~0~0\\1~0~1~0\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= u \\ = v\end{matrix}</math><br />
| rowspan="2" style="width:20%" | <math>f_j (u, v)\!</math><br />
|- style="background:ghostwhite"<br />
| align="right" | <math>\begin{matrix}x = \\ y =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~1~0\\1~0~0~1\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= f(u, v) \\ = g(u, v)\end{matrix}</math><br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>f_{0}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[2pt]<br />
f_{2}<br />
\\[2pt]<br />
f_{4}<br />
\\[2pt]<br />
f_{8}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~0<br />
\\[2pt]<br />
0~0~0~1<br />
\\[2pt]<br />
0~1~1~0<br />
\\[2pt]<br />
1~0~0~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{,~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{~} u \texttt{~~} v \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{8}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{3}<br />
\\[2pt]<br />
f_{12}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[2pt]<br />
1~1~1~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} u \texttt{)(} v \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[2pt]<br />
f_{14}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[2pt]<br />
f_{9}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[2pt]<br />
1~0~0~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{~~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{~} u \texttt{~~} v \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[2pt]<br />
f_{8}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{5}<br />
\\[2pt]<br />
f_{10}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[2pt]<br />
1~0~0~1<br />
\end{matrix}~\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} u \texttt{,~} v \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[2pt]<br />
f_{9}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[2pt]<br />
f_{11}<br />
\\[2pt]<br />
f_{13}<br />
\\[2pt]<br />
f_{14}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[2pt]<br />
1~0~0~1<br />
\\[2pt]<br />
1~1~1~0<br />
\\[2pt]<br />
1~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} u \texttt{~~} v \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{15}<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>f_{15}\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Difference Operators and Tangent Functors====<br />
<br />
Given the alphabets <math>\mathcal{U} = \{ u, v \}\!</math> and <math>\mathcal{X} = \{ x, y \},\!</math> along with the corresponding universes of discourse <math>U^\bullet, X^\bullet \cong [\mathbb{B}^2],\!</math> how many logical transformations of the general form <math>G = (G_1, G_2) : U^\bullet \to X^\bullet\!</math> are there? Since <math>G_1\!</math> and <math>G_2\!</math> can be any propositions of the type <math>\mathbb{B}^2 \to \mathbb{B},\!</math> there are <math>2^4 = 16\!</math> choices for each of the maps <math>G_1\!</math> and <math>G_2\!</math> and thus there are <math>2^4 \cdot 2^4 = 2^8 = 256\!</math> different mappings altogether of the form <math>G : U^\bullet \to X^\bullet.\!</math> The set of functions of a given type is denoted by placing its type indicator in parentheses, in the present instance writing <math>(U^\bullet \to X^\bullet) = \{ G : U^\bullet \to X^\bullet \},\!</math> and so the cardinality of the ''function space'' <math>(U^\bullet \to X^\bullet)\!</math> is summed up by writing <math>|(U^\bullet \to X^\bullet)| = |(\mathbb{B}^2 \to \mathbb{B}^2)| = 4^4 = 256.\!</math><br />
<br />
Given a transformation <math>G = (G_1, G_2) : U^\bullet \to X^\bullet\!</math> of this type, we proceed to define a pair of further transformations, related to <math>G,\!</math> that operate between the extended universes, <math>\mathrm{E}U^\bullet\!</math> and <math>\mathrm{E}X^\bullet,\!</math> of its source and target domains.<br />
<br />
First, the ''enlargement map'' (or ''secant transformation'') <math>\mathrm{E}G = (\mathrm{E}G_1, \mathrm{E}G_2) : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet\!</math> is defined by the following set of component equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}G_i<br />
& = & G_i (u + \mathrm{d}u, v + \mathrm{d}v)<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Next, the ''difference map'' (or ''chordal transformation'') <math>\mathrm{D}G = (\mathrm{D}G_1, \mathrm{D}G_2) : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet~\!</math> is defined in component-wise fashion as the boolean sum of the initial proposition <math>G_i\!</math> and the enlarged proposition <math>\mathrm{E}G_i,\!</math> for <math>i = 1, 2,\!</math> according to the following set of equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
\mathrm{D}G_i<br />
& = & G_i (u, v)<br />
& + & \mathrm{E}G_i (u, v, \mathrm{d}u, \mathrm{d}v)<br />
\\[8pt]<br />
& = & G_i (u, v)<br />
& + & G_i (u + \mathrm{d}u, v + \mathrm{d}v)<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Maintaining a strict analogy with ordinary difference calculus would perhaps have us write <math>\mathrm{D}G_i = \mathrm{E}G_i - G_i,\!</math> but the sum and difference operations are the same thing in boolean arithmetic. It is more often natural in the logical context to consider an initial proposition <math>q,\!</math> then to compute the enlargement <math>\mathrm{E}q,\!</math> and finally to determine the difference <math>\mathrm{D}q = q + \mathrm{E}q,\!</math> so we let the variant order of terms reflect this sequence of considerations.<br />
<br />
Viewed in this light the difference operator <math>\mathrm{D}\!</math> is imagined to be a function of very wide scope and polymorphic application, one that is able to realize the association between each transformation <math>G\!</math> and its difference map <math>\mathrm{D}G,\!</math> for example, taking the function space <math>(U^\bullet \to X^\bullet)\!</math> into <math>(\mathrm{E}U^\bullet \to \mathrm{E}X^\bullet).\!</math> When we consider the variety of interpretations permitted to propositions over the contexts in which we put them to use, it should be clear that an operator of this scope is not at all a trivial matter to define in general and that it may take some trouble to work out. For the moment we content ourselves with returning to particular cases.<br />
<br />
Acting on the logical transformation <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;),\!</math> the operators <math>\mathrm{E}\!</math> and <math>\mathrm{D}\!</math> yield the enlarged map <math>\mathrm{E}F = (\mathrm{E}f, \mathrm{E}g)\!</math> and the difference map <math>\mathrm{D}F = (\mathrm{D}f, \mathrm{D}g),\!</math> respectively, whose components are given as follows.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}f<br />
& = & \texttt{((} u + \mathrm{d}u \texttt{)(} v + \mathrm{d}v \texttt{))}<br />
\\[8pt]<br />
\mathrm{E}g<br />
& = & \texttt{((} u + \mathrm{d}u \texttt{,~} v + \mathrm{d}v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
\mathrm{D}f<br />
& = & \texttt{((} u \texttt{)(} v \texttt{))}<br />
& + & \texttt{((} u + \mathrm{d}u \texttt{)(} v + \mathrm{d}v \texttt{))}<br />
\\[8pt]<br />
\mathrm{D}g<br />
& = & \texttt{((} u \texttt{,~} v \texttt{))}<br />
& + & \texttt{((} u + \mathrm{d}u \texttt{,~} v + \mathrm{d}v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
But these initial formulas are purely definitional, and help us little in understanding either the purpose of the operators or the meaning of their results. Working symbolically, let us apply the same method to the separate components <math>f\!</math> and <math>g\!</math> that we earlier used on <math>J.\!</math> This work is recorded in Appendix&nbsp;3 and a summary of the results is presented in Tables&nbsp;66-i and 66-ii.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 66-i.} ~~ \text{Computation Summary for}~ f(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f<br />
& = & u \!\cdot\! v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 1<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}f<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \cdot \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{d}f<br />
& = & u \!\cdot\! v \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}f<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 66-ii.} ~~ \text{Computation Summary for}~ g(u, v) = \texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon g<br />
& = & u \!\cdot\! v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}g<br />
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}g<br />
& = & u \!\cdot\! v \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;67 shows how to compute the analytic series for <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math> in terms of coordinates, and Table&nbsp;68 recaps these results in symbolic terms, agreeing with earlier derivations.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 67.} ~~ \text{Computation of an Analytic Series in Terms of Coordinates}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:6%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}u\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>\mathrm{d}v\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>u'\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>v'\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:4px double black" | <math>f\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>g\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{E}f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{E}g}\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{D}f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{D}g}\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{d}f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{d}g}\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{d}^2\!f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{d}^2\!g}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 68.} ~~ \text{Computation of an Analytic Series in Symbolic Terms}\!</math><br />
|- style="height:40px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black" | <math>f\!</math><br />
| <math>g\!</math><br />
| style="border-left:1px solid black" | <math>{\mathrm{D}f}\!</math><br />
| <math>{\mathrm{D}g}\!</math><br />
| style="border-left:1px solid black" | <math>{\mathrm{d}f}\!</math><br />
| <math>{\mathrm{d}g}\!</math><br />
| style="border-left:1px solid black" | <math>{\mathrm{d}^2\!f}\!</math><br />
| <math>{\mathrm{d}^2\!g}\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[4pt]<br />
\texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
\\[4pt]<br />
\texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
\\[4pt]<br />
\texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;69 gives a graphical picture of the difference map <math>\mathrm{D}F = (\mathrm{D}f, \mathrm{D}g)\!</math> for the transformation <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math> This represents the same information about <math>\mathrm{D}f~\!</math> and <math>\mathrm{D}g~\!</math> that was given in the corresponding rows of Tables&nbsp;66-i and 66-ii, for ease of reference repeated below.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[8pt]<br />
\mathrm{D}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 69 -- Difference Map (Short Form).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 69.} ~~ \text{Difference Map of}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;70-a shows a way of visualizing the tangent functor map <math>\mathrm{d}F = (\mathrm{d}f, \mathrm{d}g)\!</math> for the transformation <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math> This amounts to the same information about <math>\mathrm{d}f~\!</math> and <math>\mathrm{d}g~\!</math> that was given in Tables&nbsp;66-i and 66-ii, the corresponding rows of which are repeated below.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{d}f<br />
& = & u \!\cdot\! v \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[8pt]<br />
\mathrm{d}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 70-a -- Tangent Functor Diagram.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 70-a.} ~~ \text{Tangent Functor Diagram for}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math><br />
|}<br />
<br />
<br><br />
<br />
Figure 70-b shows another way to picture the action of the tangent functor on the logical transformation <math>F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 70-b -- Tangent Functor Diagram.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 70-b.} ~~ \text{Tangent Functor Ferris Wheel for}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math><br />
|}<br />
<br />
<br><br />
<br />
* '''Note.''' The original Figure&nbsp;70-b lost some of its labeling in a succession of platform metamorphoses over the years, so we have included an ASCII version below to indicate where the missing labels go.<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| /////\ /////\ | | /XXXX\ /XXXX\ | | /\\\\\ /\\\\\ |<br />
| ///////o//////\ | | /XXXXXXoXXXXXX\ | | /\\\\\\o\\\\\\\ |<br />
| //////// \//////\ | | /XXXXXX/ \XXXXXX\ | | /\\\\\\/ \\\\\\\\ |<br />
| o/////// \//////o | | oXXXXXX/ \XXXXXXo | | o\\\\\\/ \\\\\\\o |<br />
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |<br />
| |/du//| |//dv/| | | |XXXXX| |XXXXX| | | |\du\\| |\\dv\| |<br />
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |<br />
| o//////\ ///////o | | oXXXXXX\ /XXXXXXo | | o\\\\\\\ /\\\\\\o |<br />
| \//////\ //////// | | \XXXXXX\ /XXXXXX/ | | \\\\\\\\ /\\\\\\/ |<br />
| \//////o/////// | | \XXXXXXoXXXXXX/ | | \\\\\\\o\\\\\\/ |<br />
| \///// \///// | | \XXXX/ \XXXX/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ (u)(v) o-----------------------o dv' @ (u)(v) =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| / \ /////\ | | /\\\\\ /XXXX\ | | /\\\\\ /\\\\\ |<br />
| / o//////\ | | /\\\\\\oXXXXXX\ | | /\\\\\\o\\\\\\\ |<br />
| / //\//////\ | | /\\\\\\//\XXXXXX\ | | /\\\\\\/ \\\\\\\\ |<br />
| o ////\//////o | | o\\\\\\////\XXXXXXo | | o\\\\\\/ \\\\\\\o |<br />
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |<br />
| | du |/////|//dv/| | | |\\\\\|/////|XXXXX| | | |\du\\| |\\dv\| |<br />
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |<br />
| o \//////////o | | o\\\\\\\////XXXXXXo | | o\\\\\\\ /\\\\\\o |<br />
| \ \///////// | | \\\\\\\\//XXXXXX/ | | \\\\\\\\ /\\\\\\/ |<br />
| \ o/////// | | \\\\\\\oXXXXXX/ | | \\\\\\\o\\\\\\/ |<br />
| \ / \///// | | \\\\\/ \XXXX/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ (u) v o-----------------------o dv' @ (u) v =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| /////\ / \ | | /XXXX\ /\\\\\ | | /\\\\\ /\\\\\ |<br />
| ///////o \ | | /XXXXXXo\\\\\\\ | | /\\\\\\o\\\\\\\ |<br />
| /////////\ \ | | /XXXXXX//\\\\\\\\ | | /\\\\\\/ \\\\\\\\ |<br />
| o//////////\ o | | oXXXXXX////\\\\\\\o | | o\\\\\\/ \\\\\\\o |<br />
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |<br />
| |/du//|/////| dv | | | |XXXXX|/////|\\\\\| | | |\du\\| |\\dv\| |<br />
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |<br />
| o//////\//// o | | oXXXXXX\////\\\\\\o | | o\\\\\\\ /\\\\\\o |<br />
| \//////\// / | | \XXXXXX\//\\\\\\/ | | \\\\\\\\ /\\\\\\/ |<br />
| \//////o / | | \XXXXXXo\\\\\\/ | | \\\\\\\o\\\\\\/ |<br />
| \///// \ / | | \XXXX/ \\\\\/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ u (v) o-----------------------o dv' @ u (v) =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| / \ / \ | | /\\\\\ /\\\\\ | | /\\\\\ /\\\\\ |<br />
| / o \ | | /\\\\\\o\\\\\\\ | | /\\\\\\o\\\\\\\ |<br />
| / / \ \ | | /\\\\\\/ \\\\\\\\ | | /\\\\\\/ \\\\\\\\ |<br />
| o / \ o | | o\\\\\\/ \\\\\\\o | | o\\\\\\/ \\\\\\\o |<br />
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |<br />
| | du | | dv | | | |\\\\\| |\\\\\| | | |\du\\| |\\dv\| |<br />
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |<br />
| o \ / o | | o\\\\\\\ /\\\\\\o | | o\\\\\\\ /\\\\\\o |<br />
| \ \ / / | | \\\\\\\\ /\\\\\\/ | | \\\\\\\\ /\\\\\\/ |<br />
| \ o / | | \\\\\\\o\\\\\\/ | | \\\\\\\o\\\\\\/ |<br />
| \ / \ / | | \\\\\/ \\\\\/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ u v o-----------------------o dv' @ u v =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| U | |\U\\\\\\\\\\\\\\\\\\\\\| |\U\\\\\\\\\\\\\\\\\\\\\|<br />
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|<br />
| /////\ /////\ | |\\\\\/////\\/////\\\\\\| |\\\\\/ \\/ \\\\\\|<br />
| ///////o//////\ | |\\\\///////o//////\\\\\| |\\\\/ o \\\\\|<br />
| /////////\//////\ | |\\\////////X\//////\\\\| |\\\/ /\\ \\\\|<br />
| o//////////\//////o | |\\o///////XXX\//////o\\| |\\o /\\\\ o\\|<br />
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|<br />
| |//u//|/////|//v//| | |\\|//u//|XXXXX|//v//|\\| |\\| u |\\\\\| v |\\|<br />
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|<br />
| o//////\//////////o | |\\o//////\XXX///////o\\| |\\o \\\\/ o\\|<br />
| \//////\///////// | |\\\\//////\X////////\\\| |\\\\ \\/ /\\\|<br />
| \//////o/////// | |\\\\\//////o///////\\\\| |\\\\\ o /\\\\|<br />
| \///// \///// | |\\\\\\/////\\/////\\\\\| |\\\\\\ /\\ /\\\\\|<br />
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|<br />
| | |\\\\\\\\\\\\\\\\\\\\\\\| |\\\\\\\\\\\\\\\\\\\\\\\|<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= u' o-----------------------o v' =<br />
= | U' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/u'//|XXXXX|\\v'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
Figure 70-b. Tangent Functor Ferris Wheel for F<u, v> = <((u)(v)), ((u, v))><br />
</pre><br />
|}<br />
<br />
==Epilogue, Enchoiry, Exodus==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
It is time to explain myself . . . . let us stand up.<br />
| width="4%" | &nbsp;<br />
|- <br />
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 79]<br />
|}<br />
<br />
==Appendices==<br />
<br />
===Appendix 1. Propositional Forms and Differential Expansions===<br />
<br />
====Table A1. Propositional Forms on Two Variables====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A1.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="background:ghostwhite"<br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}</math><br />
| width="25%" | <math>\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{0}\\f_{1}\\f_{2}\\f_{3}\\f_{4}\\f_{5}\\f_{6}\\f_{7}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0000}\\f_{0001}\\f_{0010}\\f_{0011}\\f_{0100}\\f_{0101}\\f_{0110}\\f_{0111}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~0\\0~0~0~1\\0~0~1~0\\0~0~1~1\\0~1~0~0\\0~1~0~1\\0~1~1~0\\0~1~1~1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)~ ~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~ ~(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{,~} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{~~} y \texttt{)}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{false}<br />
\\<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\<br />
y ~\text{without}~ x<br />
\\<br />
\text{not}~ x<br />
\\<br />
x ~\text{without}~ y<br />
\\<br />
\text{not}~ y<br />
\\<br />
x ~\text{not equal to}~ y<br />
\\<br />
\text{not both}~ x ~\text{and}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\<br />
\lnot x \land \lnot y<br />
\\<br />
\lnot x \land y<br />
\\<br />
\lnot x<br />
\\<br />
x \land \lnot y<br />
\\<br />
\lnot y<br />
\\<br />
x \ne y<br />
\\<br />
\lnot x \lor \lnot y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{8}\\f_{9}\\f_{10}\\f_{11}\\f_{12}\\f_{13}\\f_{14}\\f_{15}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{1000}\\f_{1001}\\f_{1010}\\f_{1011}\\f_{1100}\\f_{1101}\\f_{1110}\\f_{1111}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1~0~0~0\\1~0~0~1\\1~0~1~0\\1~0~1~1\\1~1~0~0\\1~1~0~1\\1~1~1~0\\1~1~1~1<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~ ~ ~ ~} y \texttt{~~}<br />
\\<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{~~} x \texttt{~ ~ ~ ~}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{((~))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{and}~ y<br />
\\<br />
x ~\text{equal to}~ y<br />
\\<br />
y<br />
\\<br />
\text{not}~ x ~\text{without}~ y<br />
\\<br />
x<br />
\\<br />
\text{not}~ y ~\text{without}~ x<br />
\\<br />
x ~\text{or}~ y<br />
\\<br />
\text{true}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x \land y<br />
\\<br />
x = y<br />
\\<br />
y<br />
\\<br />
x \Rightarrow y<br />
\\<br />
x<br />
\\<br />
x \Leftarrow y<br />
\\<br />
x \lor y<br />
\\<br />
1<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A2. Propositional Forms on Two Variables====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A2.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="background:ghostwhite"<br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}</math><br />
| width="25%" | <math>\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>f_{0000}\!</math><br />
| <math>0~0~0~0</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0001}\\f_{0010}\\f_{0100}\\f_{1000}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~1\\0~0~1~0\\0~1~0~0\\1~0~0~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\<br />
y ~\text{without}~ x<br />
\\<br />
x ~\text{without}~ y<br />
\\<br />
x ~\text{and}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot x \land \lnot y<br />
\\<br />
\lnot x \land y<br />
\\<br />
x \land \lnot y<br />
\\<br />
x \land y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{3}\\f_{12}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0011}\\f_{1100}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~1~1\\1~1~0~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ x<br />
\\<br />
x<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot x<br />
\\<br />
x<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{6}\\f_{9}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0110}\\f_{1001}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~0\\1~0~0~1<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{not equal to}~ y<br />
\\<br />
x ~\text{equal to}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x \ne y<br />
\\<br />
x = y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{5}\\f_{10}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0101}\\f_{1010}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~0~1\\1~0~1~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ y<br />
\\<br />
y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot y<br />
\\<br />
y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{7}\\f_{11}\\f_{13}\\f_{14}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0111}\\f_{1011}\\f_{1101}\\f_{1110}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~1\\1~0~1~1\\1~1~0~1\\1~1~1~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not both}~ x ~\text{and}~ y<br />
\\<br />
\text{not}~ x ~\text{without}~ y<br />
\\<br />
\text{not}~ y ~\text{without}~ x<br />
\\<br />
x ~\text{or}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot x \lor \lnot y<br />
\\<br />
x \Rightarrow y<br />
\\<br />
x \Leftarrow y<br />
\\<br />
x \lor y<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>f_{1111}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A3. E''f'' Expanded Over Differential Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A3.} ~~ \mathrm{E}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{T}_{11}f\\\mathrm{E}f|_{\mathrm{d}x ~ \mathrm{d}y}\end{matrix}</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{T}_{10}f\\\mathrm{E}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}\end{matrix}</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{T}_{01}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}\end{matrix}</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{T}_{00}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{3}\\f_{12}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{6}\\f_{9}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{5}\\f_{10}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{7}\\f_{11}\\f_{13}\\f_{14}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
|- style="background:ghostwhite"<br />
| style="border-top:1px solid black" colspan="2" | <math>\text{Fixed Point Total}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>4\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>4\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>4\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>16\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A4. D''f'' Expanded Over Differential Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A4.} ~~ \mathrm{D}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\mathrm{D}f|_{\mathrm{d}x ~ \mathrm{d}y}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}~\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
y<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
y<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
x<br />
\\<br />
x<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
x<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
y<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
y<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <br />
<math>\begin{matrix}<br />
x<br />
\\<br />
x<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A5. E''f'' Expanded Over Ordinary Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A5.} ~~ \mathrm{E}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\mathrm{E}f|_{xy}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{E}f|_{x \texttt{(} y \texttt{)}}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{E}f|_{\texttt{(} x \texttt{)} y}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{E}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{3}\\f_{12}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{6}\\f_{9}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{5}\\f_{10}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{7}\\f_{11}\\f_{13}\\f_{14}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A6. D''f'' Expanded Over Ordinary Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A6.} ~~ \mathrm{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\mathrm{D}f|_{xy}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{x \texttt{(} y \texttt{)}}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} x \texttt{)} y}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\texttt{(} x \texttt{)}\\\texttt{~} x \texttt{~}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Appendix 2. Differential Forms===<br />
<br />
The actions of the difference operator <math>\mathrm{D}\!</math> and the tangent operator <math>\mathrm{d}\!</math> on the 16 bivariate propositions are shown in Tables&nbsp;A7 and A8.<br />
<br />
Table A7 expands the differential forms that result over a ''logical basis'':<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
|<br />
<math>\{~ \texttt{(}\mathrm{d}x\texttt{)(}\mathrm{d}y\texttt{)}, ~\mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}, ~\texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.\!</math><br />
|}<br />
<br />
This set consists of the singular propositions in the first order differential variables, indicating mutually exclusive and exhaustive ''cells'' of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the cell-basis, point-basis, or singular differential basis. In this setting it is frequently convenient to use the following abbreviations:<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
|<br />
<math>\partial x ~=~ \mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}\!</math> &nbsp; &nbsp; and &nbsp; &nbsp; <math>\partial y ~=~ \texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y.\!</math><br />
|}<br />
<br />
Table A8 expands the differential forms that result over an ''algebraic basis'':<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
| <math>\{~ 1, ~\mathrm{d}x, ~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.\!</math><br />
|}<br />
<br />
This set consists of the ''positive propositions'' in the first order differential variables, indicating overlapping positive regions of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the ''positive differential basis''.<br />
<br />
====Table A7. Differential Forms Expanded on a Logical Basis====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A7.} ~~ \text{Differential Forms Expanded on a Logical Basis}\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| &nbsp;<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" | <math>\mathrm{D}f~\!</math><br />
| <math>\mathrm{d}f~\!</math><br />
|-<br />
| <math>f_{0}\!</math><br />
| style="border-right:none" | <math>\texttt{(~)}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x) ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\\<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\\<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\\<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\partial x<br />
\\<br />
\partial x<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
\\<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\partial x & + & \partial y<br />
\\<br />
\partial x & + & \partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\partial y<br />
\\<br />
\partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x) ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\\<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\\<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\\<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| style="border-right:none" | <math>\texttt{((~))}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A8. Differential Forms Expanded on an Algebraic Basis====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A8.} ~~ \text{Differential Forms Expanded on an Algebraic Basis}\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| &nbsp;<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" | <math>\mathrm{D}f~\!</math><br />
| <math>\mathrm{d}f~\!</math><br />
|-<br />
| <math>f_{0}\!</math><br />
| style="border-right:none" | <math>\texttt{(~)}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x<br />
\\<br />
\mathrm{d}x<br />
\end{matrix}\!</math><br />
| <math>\begin{matrix}<br />
\mathrm{d}x<br />
\\<br />
\mathrm{d}x<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\\<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\\<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}y<br />
\\<br />
\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y<br />
\\<br />
\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| style="border-right:none" | <math>\texttt{((~))}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A9. Tangent Proposition as Pointwise Linear Approximation====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A9.} ~~ \text{Tangent Proposition}~ \mathrm{d}f = \text{Pointwise Linear Approximation to the Difference Map}~ \mathrm{D}f\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}f =<br />
\\[2pt]<br />
\partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}^2\!f =<br />
\\[2pt]<br />
\partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\mathrm{d}f|_{x \, y}</math><br />
| <math>\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math><br />
|-<br />
| style="border-right:none" | <math>f_0\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{5}\\f_{10}\end{matrix}\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\end{matrix}\!</math><br />
| <math>\begin{matrix}<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>f_{15}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A10. Taylor Series Expansion Df = d''f'' + d<sup>2</sup>''f''====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" |<br />
<math>\text{Table A10.} ~~ \text{Taylor Series Expansion}~ {\mathrm{D}f = \mathrm{d}f + \mathrm{d}^2\!f}\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{D}f<br />
\\<br />
= & \mathrm{d}f & + & \mathrm{d}^2\!f<br />
\\<br />
= & \partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y & + & \partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\mathrm{d}f|_{x \, y}</math><br />
| <math>\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math><br />
|-<br />
| style="border-right:none" | <math>f_0\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>f_{15}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A11. Partial Differentials and Relative Differentials====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A11.} ~~ \text{Partial Differentials and Relative Differentials}\!</math><br />
|- style="background:ghostwhite; height:50px"<br />
| &nbsp;<br />
| <math>f\!</math><br />
| <math>\frac{\partial f}{\partial x}\!</math><br />
| <math>\frac{\partial f}{\partial y}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}f =<br />
\\[2pt]<br />
\partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\left. \frac{\partial x}{\partial y} \right| f\!</math><br />
| <math>\left. \frac{\partial y}{\partial x} \right| f\!</math><br />
|-<br />
| <math>f_0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A12. Detail of Calculation for the Difference Map====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:4px double black; border-left:4px double black; border-right:4px double black; border-top:4px double black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A12.} ~~ \text{Detail of Calculation for}~ {\mathrm{E}f + f = \mathrm{D}f}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:6%" | &nbsp;<br />
| style="width:14%; border-left:1px solid black" | <math>f\!</math><br />
| style="width:20%; border-left:4px double black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\mathrm{d}x ~ \mathrm{d}y}<br />
\\[4pt]<br />
+ & f|_{\mathrm{d}x ~ \mathrm{d}y}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\mathrm{d}x ~ \mathrm{d}y}<br />
\end{array}</math><br />
| style="width:20%; border-left:1px solid black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}<br />
\\[4pt]<br />
+ & f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}<br />
\end{array}</math><br />
| style="width:20%; border-left:1px solid black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
+ & f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}<br />
\end{array}</math><br />
| style="width:20%; border-left:1px solid black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
+ & f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}<br />
\end{array}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{0}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:4px double black; border-left:4px double black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{1}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{)(} y \texttt{)~}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{2}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{)~} y \texttt{~~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{4}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~~} x \texttt{~(} y \texttt{)~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{8}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~~} x \texttt{~~} y \texttt{~~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{3}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{(} x \texttt{)}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{12}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>x\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{6}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{,~} y \texttt{)~}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{9}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} x \texttt{,~} y \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{5}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{(} y \texttt{)}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{10}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>y\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{7}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{~~} y \texttt{)~}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{11}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{~(} y \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{13}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} x \texttt{)~} y \texttt{)~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{14}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} x \texttt{)(} y \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{15}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:4px double black; border-left:4px double black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Appendix 3. Computational Details===<br />
<br />
====Operator Maps for the Logical Conjunction ''f''<sub>8</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{8}~\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{8}<br />
& = && f_{8}(u, v)<br />
\\[4pt]<br />
& = && uv<br />
\\[4pt]<br />
& = && uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & uv \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & uv \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{8}<br />
& = && uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & uv \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & uv \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & uv \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.2-i} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{E}f_{8}<br />
& = & f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{8}(\mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{8}(\mathrm{d}u, \mathrm{d}v)<br />
\\[4pt]<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[20pt]<br />
\mathrm{E}f_{8}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
\\[4pt]<br />
&&&&& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&&&&&& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.2-ii} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{c}}<br />
\mathrm{E}f_{8}<br />
& = & (u + \mathrm{d}u) \cdot (v + \mathrm{d}v)<br />
\\[6pt]<br />
& = & u \cdot v<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}f_{8}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{8}<br />
& = && \mathrm{E}f_{8}<br />
& + & \boldsymbol\varepsilon f_{8}<br />
\\[4pt]<br />
& = && f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{8}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & uv<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{~}<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}f_{8}<br />
& = & \boldsymbol\varepsilon f_{8}<br />
& + & \mathrm{E}f_{8}<br />
\\[6pt]<br />
& = & f_{8}(u, v)<br />
& + & f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[6pt]<br />
& = & uv<br />
& + & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
& = & 0<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}f_{8}<br />
& = & 0<br />
& + & u \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.3-iii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 3)}\!</math><br />
|<br />
<math>\begin{array}{c*{9}{l}}<br />
\mathrm{D}f_{8}<br />
& = & \boldsymbol\varepsilon f_{8} ~+~ \mathrm{E}f_{8}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{8}<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ u \,\cdot\, v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~ u \;\cdot\; v \;\cdot\; \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}f_{8}<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u ~ \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} ~ v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot\, \mathrm{d}u ~ \mathrm{d}v<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = & ~ ~ 0 ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~ ~ u ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ ~ ~ v ~~ \cdot ~ \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
=====Computation of d''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.4} ~~ \text{Computation of}~ \mathrm{d}f_{8}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{8}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.5} ~~ \text{Computation of}~ \mathrm{r}f_{8}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{8} & = & \mathrm{D}f_{8} ~+~ \mathrm{d}f_{8}<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[20pt]<br />
\mathrm{r}f_{8}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Conjunction=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.6} ~~ \text{Computation Summary for}~ f_{8}(u, v) = uv\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{8}<br />
& = & uv \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}f_{8}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{r}f_{8}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Operator Maps for the Logical Equality ''f''<sub>9</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{9}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{9}<br />
& = && f_{9}(u, v)<br />
\\[4pt]<br />
& = && \texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{9}(1, 1)<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{9}(1, 0)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{9}(0, 1)<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{9}(0, 0)<br />
\\[4pt]<br />
& = && u v & + & 0 & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{9}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.2} ~~ \text{Computation of}~ \mathrm{E}f_{9}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{E}f_{9}<br />
& = && f_{9}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! f_{9}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! f_{9}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{ } \mathrm{d}v \texttt{ })<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! f_{9}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! f_{9}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{ } \mathrm{d}v \texttt{ })<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{ (} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{) }<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{ (} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{) }<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{E}f_{9}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{9}<br />
& = && \mathrm{E}f_{9}<br />
& + & \boldsymbol\varepsilon f_{9}<br />
\\[4pt]<br />
& = && f_{9}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{9}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{9}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[20pt]<br />
\mathrm{D}f_{9}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}f_{9}<br />
& = & 0 \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & 1 \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & 1 \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & 0 \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of d''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.4} ~~ \text{Computation of}~ \mathrm{d}f_{9}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.5} ~~ \text{Computation of}~ \mathrm{r}f_{9}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{9} & = & \mathrm{D}f_{9} ~+~ \mathrm{d}f_{9}<br />
\\[20pt]<br />
\mathrm{D}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[20pt]<br />
\mathrm{r}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Equality=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.6} ~~ \text{Computation Summary for}~ f_{9}(u, v) = \texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{9}<br />
& = & uv \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}f_{9}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f_{9}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{9}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}f_{9}<br />
& = & uv \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Operator Maps for the Logical Implication ''f''<sub>11</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{11}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{11}<br />
& = && f_{11}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(} u \texttt{(} v \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{11}(1, 1)<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{11}(1, 0)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{11}(0, 1)<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{11}(0, 0)<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ }<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ }<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{11}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.2} ~~ \text{Computation of}~ \mathrm{E}f_{11}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{E}f_{11}<br />
& = && f_{11}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = &&<br />
\texttt{(}<br />
\\<br />
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\\<br />
&&& \texttt{(}<br />
\\<br />
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\<br />
&&& \texttt{))}<br />
\\[4pt]<br />
& = &&<br />
u v<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)((} \mathrm{d}v \texttt{)))}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u \texttt{((} \mathrm{d}v \texttt{)))}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[4pt]<br />
& = &&<br />
u v<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{E}f_{11}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{11}<br />
& = && \mathrm{E}f_{11}<br />
& + & \boldsymbol\varepsilon f_{11}<br />
\\[4pt]<br />
& = && f_{11}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{11}(u, v)<br />
\\[4pt]<br />
& = &&<br />
\texttt{(} \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\texttt{(} \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{(} v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & u v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & 0<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & 0<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = && u v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{c*{9}{l}}<br />
\mathrm{D}f_{11}<br />
& = & \boldsymbol\varepsilon f_{11} ~+~ \mathrm{E}f_{11}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{11}<br />
& = & u v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}f_{11}<br />
& = &<br />
u v<br />
\cdot<br />
\texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\cdot<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\cdot<br />
\texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\cdot<br />
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = &<br />
u v<br />
\cdot<br />
\texttt{~(} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{~}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\cdot<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\cdot<br />
\texttt{~} \mathrm{d}u ~ \mathrm{d}v \texttt{~}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\cdot<br />
\texttt{~} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of d''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.4} ~~ \text{Computation of}~ \mathrm{d}f_{11}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{11}<br />
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{11}<br />
& = & u v \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.5} ~~ \text{Computation of}~ \mathrm{r}f_{11}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{11} & = & \mathrm{D}f_{11} ~+~ \mathrm{d}f_{11}<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{11}<br />
& = & u v \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u<br />
\\[20pt]<br />
\mathrm{r}f_{11}<br />
& = & u v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Implication=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.6} ~~ \text{Computation Summary for}~ f_{11}(u, v) = \texttt{(} u \texttt{(} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{11}<br />
& = & u v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}f_{11}<br />
& = & u v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f_{11}<br />
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{11}<br />
& = & u v \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u<br />
\\[6pt]<br />
\mathrm{r}f_{11}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Operator Maps for the Logical Disjunction ''f''<sub>14</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{14}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{14}<br />
& = && f_{14}(u, v)<br />
\\[4pt]<br />
& = && \texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{14}(1, 1)<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{14}(1, 0)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{14}(0, 1)<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{14}(0, 0)<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ }<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ }<br />
& + & 0<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{14}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.2} ~~ \text{Computation of}~ \mathrm{E}f_{14}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{E}f_{14}<br />
& = && f_{14}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = &&<br />
\texttt{((}<br />
\\<br />
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\\<br />
&&& \texttt{)(}<br />
\\<br />
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\<br />
&&& \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! f_{14}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! f_{14}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{ } \mathrm{d}v \texttt{ })<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! f_{14}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! f_{14}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{ } \mathrm{d}v \texttt{ })<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{E}f_{14}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{14}<br />
& = && \mathrm{E}f_{14}<br />
& + & \boldsymbol\varepsilon f_{14}<br />
\\[4pt]<br />
& = && f_{14}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{14}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{))((} v \texttt{,} \mathrm{d}v \texttt{)))}<br />
& + & \texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{14}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
\\[4pt]<br />
&& + & 0<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
\\[4pt]<br />
&& + & uv \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
\\[20pt]<br />
\mathrm{D}f_{14}<br />
& = && uv \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}f_{14}<br />
& = & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & 0 \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of d''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.4} ~~ \text{Computation of}~ \mathrm{d}f_{14}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.5} ~~ \text{Computation of}~ \mathrm{r}f_{14}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{14} & = & \mathrm{D}f_{14} ~+~ \mathrm{d}f_{14}<br />
\\[20pt]<br />
\mathrm{D}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{d}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[20pt]<br />
\mathrm{r}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Disjunction=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.6} ~~ \text{Computation Summary for}~ f_{14}(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{14}<br />
& = & uv \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 1<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}f_{14}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f_{14}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{d}f_{14}<br />
& = & uv \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}f_{14}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
===Appendix 4. Source Materials===<br />
<br />
===Appendix 5. Various Definitions of the Tangent Vector===<br />
<br />
==References==<br />
<br />
===Works Cited===<br />
<br />
{| cellpadding=3<br />
| valign=top | [AuM]<br />
| Auslander, L., and MacKenzie, R.E., ''Introduction to Differentiable Manifolds'', McGraw-Hill, 1963. Reprinted, Dover, New York, NY, 1977.<br />
|-<br />
| valign=top | [BiG]<br />
| Bishop, R.L., and Goldberg, S.I., ''Tensor Analysis on Manifolds'', Macmillan, 1968. Reprinted, Dover, New York, NY, 1980.<br />
|-<br />
| valign=top | [Boo]<br />
| Boole, G., ''An Investigation of The Laws of Thought'', Macmillan, 1854. Reprinted, Dover, New York, NY, 1958.<br />
|-<br />
| valign=top | [BoT]<br />
| Bott, R., and Tu, L.W., ''Differential Forms in Algebraic Topology'', Springer-Verlag, New York, NY, 1982.<br />
|-<br />
| valign=top | [dCa]<br />
| do Carmo, M.P., ''Riemannian Geometry''. Originally published in Portuguese, 1st editiom 1979, 2nd edition 1988. Translated by F. Flaherty, Birkhäuser, Boston, MA, 1992.<br />
|-<br />
| valign=top | [Che46]<br />
| Chevalley, C., ''Theory of Lie Groups'', Princeton University Press, Princeton, NJ, 1946.<br />
|-<br />
| valign=top | [Che56]<br />
| Chevalley, C., ''Fundamental Concepts of Algebra'', Academic Press, 1956.<br />
|-<br />
| valign=top | [Cho86]<br />
| Chomsky, N., ''Knowledge of Language : Its Nature, Origin, and Use'', Praeger, New York, NY, 1986.<br />
|-<br />
| valign=top | [Cho93]<br />
| Chomsky, N., ''Language and Thought'', Moyer Bell, Wakefield, RI, 1993.<br />
|-<br />
| valign=top | [DoM]<br />
| Doolin, B.F., and Martin, C.F., ''Introduction to Differential Geometry for Engineers'', Marcel Dekker, New York, NY, 1990.<br />
|-<br />
| valign=top | [Fuji]<br />
| Fujiwara, H., ''Logic Testing and Design for Testability'', MIT Press, Cambridge, MA, 1985.<br />
|-<br />
| valign=top | [Hic]<br />
| Hicks, N.J., ''Notes on Differential Geometry'', Van Nostrand, Princeton, NJ, 1965.<br />
|-<br />
| valign=top | [Hir]<br />
| Hirsch, M.W., ''Differential Topology'', Springer-Verlag, New York, NY, 1976.<br />
|-<br />
| valign=top | [How]<br />
| Howard, W.A., "The Formulae-as-Types Notion of Construction", Notes circulated from 1969. Reprinted in [SeH, 479-490].<br />
|-<br />
| valign=top | [JGH]<br />
| Jones, A., Gray, A., and Hutton, R., ''Manifolds and Mechanics'', Cambridge University Press, Cambridge, UK, 1987.<br />
|-<br />
| valign=top | [KoA]<br />
| Kosinski, A.A., ''Differential Manifolds'', Academic Press, San Diego, CA, 1993.<br />
|-<br />
| valign=top | [Koh]<br />
| Kohavi, Z., ''Switching and Finite Automata Theory'', 2nd edition, McGraw-Hill, New York, NY, 1978.<br />
|-<br />
| valign=top | [LaS]<br />
| Lambek, J., and Scott, P.J., ''Introduction to Higher Order Categorical Logic'', Cambridge University Press, Cambridge, UK, 1986.<br />
|-<br />
| valign=top | [La83]<br />
| Lang, S., ''Real Analysis'', 2nd edition, Addison-Wesley, Reading, MA, 1983.<br />
|-<br />
| valign=top | [La84]<br />
| Lang, S., ''Algebra'', 2nd edition, Addison-Wesley, Menlo Park, CA, 1984.<br />
|-<br />
| valign=top | [La85]<br />
| Lang, S., ''Differential Manifolds'', Springer-Verlag, New York, NY, 1985.<br />
|-<br />
| valign=top | [La93]<br />
| Lang, S., ''Real and Functional Analysis'', 3rd edition, Springer-Verlag, New York, NY, 1993.<br />
|-<br />
| valign=top | [Lie80]<br />
| Lie, S., "Sophus Lie's 1880 Transformation Group Paper", in ''Lie Groups : History, Frontiers, and Applications, Volume 1'', translated by M. Ackerman, comments by R. Hermann, Math Sci Press, Brookline, MA, 1975. Original paper 1880.<br />
|-<br />
| valign=top | [Lie84]<br />
| Lie, S., "Sophus Lie's 1884 Differential Invariant Paper", in ''Lie Groups : History, Frontiers, and Applications, Volume 3'', translated by M. Ackerman, comments by R. Hermann, Math Sci Press, Brookline, MA, 1976. Original paper 1884.<br />
|-<br />
| valign=top | [LoS]<br />
| Loomis, L.H., and Sternberg, S., ''Advanced Calculus'', Addison-Wesley, Reading, MA, 1968.<br />
|-<br />
| valign=top | [Mel]<br />
| Melzak, Z.A., ''Companion to Concrete Mathematics, Volume 2 : Mathematical Ideas, Modeling, and Applications'', John Wiley amd Sons, New York, NY, 1976.<br />
|-<br />
| valign=top | [Men]<br />
| Menabrea, L.F., "Sketch of the Analytical Engine Invented by Charles Babbage" with Notes by the Translator, Ada Augusta (Byron), Countess of Lovelace'', in [M&M, 225–297]. Originally published 1842.<br />
|-<br />
| valign=top | [M&M]<br />
| Morrison, P., and Morrison, E. (eds.), ''Charles Babbage on the Principles and Development of the Calculator, and Other Seminal Writings by Charles Babbage and Others, With an Introduction by the Editors'', Dover, Mineola, NY, 1961.<br />
|-<br />
| valign=top | [P1]<br />
| Peirce, C.S., ''Collected Papers of Charles Sanders Peirce'', vols. 1–8, C. Hartshorne, P. Weiss, and A.W. Burks (eds.), Harvard University Press, Cambridge, MA, 1931–1960. Cited as CP [volume].[paragraph].<br />
|-<br />
| valign=top | [P2]<br />
| Peirce, C.S., "Qualitative Logic", in ''The New Elements of Mathematics, Volume 4'', C. Eisele (ed.), Mouton, The Hague, 1976. Cited as NE [volume], [page].<br />
|-<br />
| valign=top | [Rob]<br />
| Roberts, D.D., ''The Existential Graphs of Charles S. Peirce'', Mouton, The Hague, 1973.<br />
|-<br />
| valign=top | [SeH]<br />
| Seldin, J.P., and Hindley, J.R. (eds.), ''To H.B. Curry : Essays on Combinatory Logic, Lambda Calculus, and Formalism'', Academic Press, London, UK, 1980.<br />
|-<br />
| valign=top | [SpB]<br />
| Spencer-Brown, G., ''Laws of Form'', George Allen and Unwin, London, UK, 1969.<br />
|-<br />
| valign=top | [Sp65]<br />
| Spivak, M., ''Calculus on Manifolds : A Modern Approach to Classical Theorems of Advanced Calculus'', W.A. Benjamin, New York, NY, 1965.<br />
|-<br />
| valign=top | [Sp79]<br />
| Spivak, M., ''A Comprehensive Introduction to Differential Geometry'', vols. 1–2. 1st edition 1970. 2nd edition, Publish or Perish Inc., Houston, TX, 1979.<br />
|-<br />
| valign=top | [Sty]<br />
| Styazhkin, N.I., ''History of Mathematical Logic from Leibniz to Peano'', 1st published in Russian, Nauka, Moscow, 1964. MIT Press, Cambridge, MA, 1969.<br />
|-<br />
| valign=top | [Wie]<br />
| Wiener, N., ''Cybernetics : or Control and Communication in the Animal and the Machine'', 1st edition 1948. 2nd edition, MIT Press, Cambridge, MA, 1961.<br />
|}<br />
<br />
===Works Consulted===<br />
<br />
{| cellpadding=3<br />
| valign=top | [Ami]<br />
| Amit, D.J., ''Modeling Brain Function : The World of Attractor Neural Networks'', Cambridge University Press, Cambridge, UK, 1989.<br />
|-<br />
| valign=top | [Ed87]<br />
| Edelman, G.M., ''Neural Darwinism : The Theory of Neuronal Group Selection'', Basic Books, New York, NY, 1987.<br />
|-<br />
| valign=top | [Ed88]<br />
| Edelman, G.M., ''Topobiology : An Introduction to Molecular Embryology'', Basic Books, New York, NY, 1988.<br />
|-<br />
| valign=top | [Fla]<br />
| Flanders, H., ''Differential Forms with Applications to the Physical Sciences'', Academic Press, 1963. Reprinted, Dover, Mineola, NY, 1989. <br />
|-<br />
| valign=top | [Has]<br />
| Hassoun, M.H. (ed.), ''Associative Neural Memories : Theory and Implementation'', Oxford University Press, New York, NY, 1993.<br />
|-<br />
| valign=top | [KoB]<br />
| Kosko, B., ''Neural Networks and Fuzzy Systems : A Dynamical Systems Approach to Machine Intelligence'', Prentice-Hall, Englewood Cliffs, NJ, 1992.<br />
|-<br />
| valign=top | [MaB]<br />
| Mac Lane, S., and Birkhoff, G., ''Algebra'', 3rd edition, Chelsea, New York, NY, 1993.<br />
|-<br />
| valign=top | [Mac]<br />
| Mac Lane, S., ''Categories for the Working Mathematician'', Springer-Verlag, New York, NY, 1971.<br />
|-<br />
| valign=top | [McC]<br />
| McCulloch, W.S., ''Embodiments of Mind'', MIT Press, Cambridge, MA, 1965.<br />
|-<br />
| valign=top | [Mc1]<br />
| McCulloch, W.S., "A Heterarchy of Values Determined by the Topology of Nervous Nets", Bulletin of Mathematical Biophysics, vol. 7 (1945), pp. 89–93. Reprinted in [McC].<br />
|-<br />
| valign=top | [MiP]<br />
| Minsky, M.L., and Papert, S.A., ''Perceptrons : An Introduction to Computational Geometry'', MIT Press, Cambridge, MA, 1969. 2nd printing 1972. Expanded edition 1988.<br />
|-<br />
| valign=top | [Rum]<br />
| Rumelhart, D.E., Hinton, G.E., and McClelland, J.L., "A General Framework for Parallel Distributed Processing" = Chapter 2 in Rumelhart, McClelland, and the PDP Research Group, ''Parallel Distributed Processing, Explorations in the Microstructure of Cognition, Volume 1 : Foundations'', MIT Press, Cambridge, MA, 1986.<br />
|}<br />
<br />
===Incidental Works===<br />
<br />
{| cellpadding=3<br />
| valign=top | [Dew]<br />
| Dewey, John, ''How We Think'', D.C. Heath, Lexington, MA, 1910. Reprinted, Prometheus Books, Buffalo, NY, 1991.<br />
|-<br />
| valign=top | [Fou]<br />
| Foucault, Michel, ''The Archaeology of Knowledge and The Discourse on Language'', A.M. Sheridan-Smith and Rupert Swyer (trans.), Pantheon, New York, NY, 1972. Originally published as ''L´Archéologie du Savoir et L´ordre du discours'', Editions Gallimard, 1969 & 1971.<br />
|-<br />
| valign=top | [Hom]<br />
| Homer, ''The Odyssey'', with an English translation by A.T. Murray, Loeb Classical Library, Harvard University Press, Cambridge, MA, 1980. First printed 1919.<br />
|-<br />
| valign=top | [Jam]<br />
| James, William, ''Pragmatism : A New Name for Some Old Ways of Thinking'', Longmans, Green, and Company, New York, NY, 1907.<br />
|-<br />
| valign=top | [Ler]<br />
| Leroux, Gaston, ''The Phantom of the Opera'', foreword by P. Haining, Dorset Press, New York, NY, 1988. Originally published in French, 1911.<br />
|-<br />
| valign=top | [Mus]<br />
| Musil, Robert, ''The Man Without Qualities'', 3 volumes, translated with a foreword by Eithne Wilkins and Ernst Kaiser, Pan Books, London, UK, 1979. English edition first published by Secker and Warburg, 1954. Originally published in German, ''Der Mann ohne Eigenschaften'', 1930 & 1932.<br />
|-<br />
| valign=top | [PlaR]<br />
| Plato, ''The Republic'', with an English translation by Paul Shorey, Loeb Classical Library, Harvard University Press, Cambridge, MA, 1980. First printed 1930 & 1935.<br />
|-<br />
| valign=top | [PlaS]<br />
| Plato, ''The Sophist'', Loeb Classical Library, William Heinemann, London, 1921, 1987.<br />
|-<br />
| valign=top | [Qui]<br />
| Quine, W.V., ''Mathematical Logic'', 1st edition, 1940. Revised edition, 1951. Harvard University Press, Cambridge, MA, 1981.<br />
|-<br />
| valign=top | [SaD]<br />
| de Santillana, Giorgio, and von Dechend, Hertha, ''Hamlet's Mill : An Essay on Myth and the Frame of Time'', David R. Godine, Publisher, Boston, MA, 1977. 1st published 1969.<br />
|-<br />
| valign=top | [Sha]<br />
| Shakespeare, William, '' William Shakespeare : The Complete Works'', Compact Edition, S. Wells and G. Taylor (eds.), Oxford University Press, Oxford, UK, 1988.<br />
|-<br />
| valign=top | [Sh1]<br />
| Shakespeare, William, ''A Midsummer Night's Dream'', Washington Square Press, New York, NY, 1958.<br />
|-<br />
| valign=top | [Sh2]<br />
| Shakespeare, William, ''The Tragedy of Hamlet, Prince of Denmark'', In [Sha], pp. 654&ndash;690.<br />
|-<br />
| valign=top | [Sh3]<br />
| Shakespeare, William, ''Measure for Measure'', Washington Square Press, New York, NY, 1965.<br />
|-<br />
| valign=top | [Web]<br />
| ''Webster's Ninth New Collegiate Dictionary'', Merriam-Webster, Springfield, MA, 1983.<br />
|-<br />
| valign=top | [Whi]<br />
| Whitman, Walt, ''Leaves of Grass'', Vintage Books / The Library of America, New York, NY, 1992. Originally published in numerous editions, 1855&ndash;1892.<br />
|-<br />
| valign=top | [Wil]<br />
| Wilhelm, R., and Baynes, C.F. (trans.), ''The I Ching, or Book of Changes'', foreword by C.G. Jung, preface by H. Wilhelm, 3rd edition, Bollingen Series XIX, Princeton University Press, Princeton, NJ, 1967.<br />
|}<br />
<br />
==Document History==<br />
<br />
<pre><br />
Author: Jon Awbrey<br />
Created: 16 Dec 1993<br />
Relayed: 31 Oct 1994<br />
Revised: 03 Jun 2003<br />
Recoded: 03 Jun 2007<br />
</pre><br />
<br />
[[Category:Adaptive Systems]]<br />
[[Category:Artificial Intelligence]]<br />
[[Category:Boolean Algebra]]<br />
[[Category:Boolean Functions]]<br />
[[Category:Charles Sanders Peirce]]<br />
[[Category:Combinatorics]]<br />
[[Category:Computer Science]]<br />
[[Category:Cybernetics]]<br />
[[Category:Differential Logic]]<br />
[[Category:Discrete Systems]]<br />
[[Category:Dynamical Systems]]<br />
[[Category:Formal Languages]]<br />
[[Category:Formal Sciences]]<br />
[[Category:Formal Systems]]<br />
[[Category:Functional Logic]]<br />
[[Category:Graph Theory]]<br />
[[Category:Group Theory]]<br />
[[Category:Inquiry]]<br />
[[Category:Knowledge Representation]]<br />
[[Category:Linguistics]]<br />
[[Category:Logic]]<br />
[[Category:Logical Graphs]]<br />
[[Category:Mathematics]]<br />
[[Category:Mathematical Systems Theory]]<br />
[[Category:Philosophy]]<br />
[[Category:Science]]<br />
[[Category:Semiotics]]<br />
[[Category:Systems Science]]<br />
[[Category:Visualization]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Differential_Logic_and_Dynamic_Systems&diff=469884Differential Logic and Dynamic Systems2021-01-14T21:19:17Z<p>Jon Awbrey: update</p>
<hr />
<div>'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''<br />
<br />
{| align="center" cellpadding="10"<br />
| [[File:Tangent Functor Ferris Wheel.jpg]]<br />
|}<br />
<br />
{| style="height:36px; width:100%"<br />
| align="left" | ''Stand and unfold yourself.''<br />
| align="right" | Hamlet: Francsico&mdash;1.1.2<br />
|}<br />
<br />
This article develops a differential extension of propositional calculus and applies it to a context of problems arising in dynamic systems.&nbsp; The work pursued here is coordinated with a parallel application that focuses on neural network systems, but the dependencies are arranged to make the present article the main and the more self-contained work, to serve as a conceptual frame and a technical background for the network project.<br />
<br />
==Review and Transition==<br />
<br />
This note continues a previous discussion on the problem of dealing with change and diversity in logic-based intelligent systems. It is useful to begin by summarizing essential material from previous reports.<br />
<br />
Table 1 outlines a notation for propositional calculus based on two types of logical connectives, both of variable <math>k\!</math>-ary scope.<br />
<br />
* A bracketed list of propositional expressions in the form <math>\texttt{(} e_1, e_2, \ldots, e_{k-1}, e_k \texttt{)}\!</math> indicates that exactly one of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> is false.<br />
<br />
* A concatenation of propositional expressions in the form <math>e_1 ~ e_2 ~ \ldots ~ e_{k-1} ~ e_k</math> indicates that all of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> are true, in other words, that their [[logical conjunction]] is true.<br />
<br />
All other propositional connectives can be obtained in a very efficient style of representation through combinations of these two forms. Strictly speaking, the concatenation form is dispensable in light of the bracketed form, but it is convenient to maintain it as an abbreviation of more complicated bracket expressions.<br />
<br />
This treatment of propositional logic is derived from the work of C.S. Peirce [P1, P2], who gave this approach an extensive development in his graphical systems of predicate, relational, and modal logic [Rob]. More recently, these ideas were revived and supplemented in an alternative interpretation by George Spencer-Brown [SpB]. Both of these authors used other forms of enclosure where I use parentheses, but the structural topologies of expression and the functional varieties of interpretation are fundamentally the same.<br />
<br />
While working with expressions solely in propositional calculus, it is easiest to use plain parentheses for logical connectives. In contexts where parentheses are needed for other purposes &ldquo;teletype&rdquo; parentheses <math>\texttt{(} \ldots \texttt{)}\!</math> or barred parentheses <math>(\!| \ldots |\!)</math> may be used for logical operators.<br />
<br />
The briefest expression for logical truth is the empty word, usually denoted by <math>{}^{\backprime\backprime} \boldsymbol\varepsilon {}^{\prime\prime}\!</math> or <math>{}^{\backprime\backprime} \boldsymbol\lambda {}^{\prime\prime}\!</math> in formal languages, where it forms the identity element for concatenation. To make it visible in this text, it may be denoted by the equivalent expression <math>{}^{\backprime\backprime} \texttt{((~))} {}^{\prime\prime},\!</math> or, especially if operating in an algebraic context, by a simple <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}.\!</math> Also when working in an algebraic mode, the plus sign <math>{}^{\backprime\backprime} + {}^{\prime\prime}\!</math> may be used for [[exclusive disjunction]]. For example, we have the following paraphrases of algebraic expressions by bracket expressions:<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
|<br />
<math>\begin{matrix}<br />
x + y ~=~ \texttt{(} x, y \texttt{)}<br />
\\[6pt]<br />
x + y + z ~=~ \texttt{((} x, y \texttt{)}, z \texttt{)} ~=~ \texttt{(} x, \texttt{(} y, z \texttt{))}<br />
\end{matrix}</math><br />
|}<br />
<br />
It is important to note that the last expressions are not equivalent to the triple bracket <math>\texttt{(} x, y, z \texttt{)}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 1.} ~~ \text{Syntax and Semantics of a Calculus for Propositional Logic}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Expression}~\!</math><br />
| <math>\text{Interpretation}\!</math><br />
| <math>\text{Other Notations}\!</math><br />
|-<br />
| &nbsp;<br />
| <math>\text{True}\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{False}\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>x\!</math><br />
| <math>x\!</math><br />
| <math>x\!</math><br />
|-<br />
| <math>\texttt{(} x \texttt{)}\!</math><br />
| <math>\text{Not}~ x\!</math><br />
|<br />
<math>\begin{matrix}<br />
x'<br />
\\<br />
\tilde{x}<br />
\\<br />
\lnot x<br />
\end{matrix}\!</math><br />
|-<br />
| <math>x~y~z\!</math><br />
| <math>x ~\text{and}~ y ~\text{and}~ z\!</math><br />
| <math>x \land y \land z\!</math><br />
|-<br />
| <math>\texttt{((} x \texttt{)(} y \texttt{)(} z \texttt{))}\!</math><br />
| <math>x ~\text{or}~ y ~\text{or}~ z\!</math><br />
| <math>x \lor y \lor z\!</math><br />
|-<br />
| <math>\texttt{(} x ~ \texttt{(} y \texttt{))}\!</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{implies}~ y<br />
\\<br />
\mathrm{If}~ x ~\text{then}~ y<br />
\end{matrix}</math><br />
| <math>x \Rightarrow y\!</math><br />
|-<br />
| <math>\texttt{(} x \texttt{,} y \texttt{)}\!</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{not equal to}~ y<br />
\\<br />
x ~\text{exclusive or}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x \ne y<br />
\\<br />
x + y<br />
\end{matrix}</math><br />
|-<br />
| <math>\texttt{((} x \texttt{,} y \texttt{))}\!</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{is equal to}~ y<br />
\\<br />
x ~\text{if and only if}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x = y<br />
\\<br />
x \Leftrightarrow y<br />
\end{matrix}</math><br />
|-<br />
| <math>\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Just one of}<br />
\\<br />
x, y, z<br />
\\<br />
\text{is false}.<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x'y~z~ & \lor<br />
\\<br />
x~y'z~ & \lor<br />
\\<br />
x~y~z' &<br />
\end{matrix}</math><br />
|-<br />
| <math>\texttt{((} x \texttt{),(} y \texttt{),(} z \texttt{))}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Just one of}<br />
\\<br />
x, y, z<br />
\\<br />
\text{is true}.<br />
\\<br />
&<br />
\\<br />
\text{Partition all}<br />
\\<br />
\text{into}~ x, y, z.<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x~y'z' & \lor<br />
\\<br />
x'y~z' & \lor<br />
\\<br />
x'y'z~ &<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,} y \texttt{),} z \texttt{)}<br />
\\<br />
&<br />
\\<br />
\texttt{(} x \texttt{,(} y \texttt{,} z \texttt{))}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Oddly many of}<br />
\\<br />
x, y, z<br />
\\<br />
\text{are true}.<br />
\end{matrix}\!</math><br />
|<br />
<p><math>x + y + z\!</math></p><br />
<br><br />
<p><math>\begin{matrix}<br />
x~y~z~ & \lor<br />
\\<br />
x~y'z' & \lor<br />
\\<br />
x'y~z' & \lor<br />
\\<br />
x'y'z~ &<br />
\end{matrix}\!</math></p><br />
|-<br />
| <math>\texttt{(} w \texttt{,(} x \texttt{),(} y \texttt{),(} z \texttt{))}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Partition}~ w<br />
\\<br />
\text{into}~ x, y, z.<br />
\\<br />
&<br />
\\<br />
\text{Genus}~ w ~\text{comprises}<br />
\\<br />
\text{species}~ x, y, z.<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
w'x'y'z' & \lor<br />
\\<br />
w~x~y'z' & \lor<br />
\\<br />
w~x'y~z' & \lor<br />
\\<br />
w~x'y'z~ &<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
'''Note.''' The usage that one often sees, of a plus sign "<math>+\!</math>" to represent inclusive disjunction, and the reference to this operation as ''boolean addition'', is a misnomer on at least two counts. Boole used the plus sign to represent exclusive disjunction (at any rate, an operation of aggregation restricted in its logical interpretation to cases where the represented sets are disjoint (Boole, 32)), as any mathematician with a sensitivity to the ring and field properties of algebra would do:<br />
<br />
<blockquote><br />
The expression <math>x + y\!</math> seems indeed uninterpretable, unless it be assumed that the things represented by <math>x\!</math> and the things represented by <math>y\!</math> are entirely separate; that they embrace no individuals in common. (Boole, 66).<br />
</blockquote><br />
<br />
It was only later that Peirce and Jevons treated inclusive disjunction as a fundamental operation, but these authors, with a respect for the algebraic properties that were already associated with the plus sign, used a variety of other symbols for inclusive disjunction (Sty, 177, 189). It seems to have been Schröder who later reassigned the plus sign to inclusive disjunction (Sty, 208). Additional information, discussion, and references can be found in (Boole) and (Sty, 177&ndash;263). Aside from these historical points, which never really count against a current practice that has gained a life of its own, this usage does have a further disadvantage of cutting or confounding the lines of communication between algebra and logic. For this reason, it will be avoided here.<br />
<br />
==A Functional Conception of Propositional Calculus==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Out of the dimness opposite equals advance . . . .<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Always substance and increase,<br><br />
Always a knit of identity . . . . always distinction . . . .<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;always a breed of life.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]<br />
|}<br />
<br />
In the general case, we start with a set of logical features <math>\{a_1, \ldots, a_n\}</math> that represent properties of objects or propositions about the world. In concrete examples the features <math>\{a_i\!\}</math> commonly appear as capital letters from an ''alphabet'' like <math>\{A, B, C, \ldots\}</math> or as meaningful words from a linguistic ''vocabulary'' of codes. This language can be drawn from any sources, whether natural, technical, or artificial in character and interpretation. In the application to dynamic systems we tend to use the letters <math>\{x_1, \ldots, x_n\}</math> as our coordinate propositions, and to interpret them as denoting properties of a system's ''state'', that is, as propositions about its location in configuration space. Because I have to consider non-deterministic systems from the outset, I often use the word ''state'' in a loose sense, to denote the position or configuration component of a contemplated state vector, whether or not it ever gets a deterministic completion.<br />
<br />
The set of logical features <math>\{a_1, \ldots, a_n\}</math> provides a basis for generating an <math>n\!</math>-dimensional ''[[universe of discourse]]'' that I denote as <math>[a_1, \ldots, a_n].</math> It is useful to consider each universe of discourse as a unified categorical object that incorporates both the set of points <math>\langle a_1, \ldots, a_n \rangle</math> and the set of propositions <math>f : \langle a_1, \ldots, a_n \rangle \to \mathbb{B}</math> that are implicit with the ordinary picture of a venn diagram on <math>n\!</math> features. Thus, we may regard the universe of discourse <math>[a_1, \ldots, a_n]</math> as an ordered pair having the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}),</math> and we may abbreviate this last type designation as <math>\mathbb{B}^n\ +\!\to \mathbb{B},</math> or even more succinctly as <math>[\mathbb{B}^n].</math> (Used this way, the angle brackets <math>\langle\ldots\rangle</math> are referred to as ''generator brackets''.)<br />
<br />
Table 2 exhibits the scheme of notation I use to formalize the domain of propositional calculus, corresponding to the logical content of truth tables and venn diagrams. Although it overworks the square brackets a bit, I also use either one of the equivalent notations <math>[n]\!</math> or <math>\mathbf{n}</math> to denote the data type of a finite set on <math>n\!</math> elements.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 2.} ~~ \text{Propositional Calculus : Basic Notation}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Symbol}\!</math><br />
| <math>\text{Notation}\!</math><br />
| <math>\text{Description}\!</math><br />
| <math>\text{Type}\!</math><br />
|-<br />
| <math>\mathfrak{A}\!</math><br />
| <math>\{ {}^{\backprime\backprime} a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} a_n {}^{\prime\prime} \}\!</math><br />
| <math>\text{Alphabet}\!</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>\mathcal{A}\!</math><br />
| <math>\{ a_1, \ldots, a_n \}\!</math><br />
| <math>\text{Basis}\!</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>A_i\!</math><br />
| <math>\{ \texttt{(} a_i \texttt{)}, a_i \}\!</math><br />
| <math>\text{Dimension}~ i\!</math><br />
| <math>\mathbb{B}\!</math><br />
|-<br />
| <math>A\!</math><br />
|<br />
<math>\begin{matrix}<br />
\langle \mathcal{A} \rangle<br />
\\[2pt]<br />
\langle a_1, \ldots, a_n \rangle<br />
\\[2pt]<br />
\{ (a_1, \ldots, a_n) \}<br />
\\[2pt]<br />
A_1 \times \ldots \times A_n<br />
\\[2pt]<br />
\textstyle \prod_{i=1}^n A_i<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Set of cells},<br />
\\[2pt]<br />
\text{coordinate tuples},<br />
\\[2pt]<br />
\text{points, or vectors}<br />
\\[2pt]<br />
\text{in the universe}<br />
\\[2pt]<br />
\text{of discourse}<br />
\end{matrix}</math><br />
| <math>\mathbb{B}^n\!</math><br />
|-<br />
| <math>A^*\!</math><br />
| <math>(\mathrm{hom} : A \to \mathbb{B})\!</math><br />
| <math>\text{Linear functions}\!</math><br />
| <math>(\mathbb{B}^n)^* \cong \mathbb{B}^n\!</math><br />
|-<br />
| <math>A^\uparrow\!</math><br />
| <math>(A \to \mathbb{B})\!</math><br />
| <math>\text{Boolean functions}\!</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}\!</math><br />
|-<br />
| <math>A^\bullet\!</math><br />
|<br />
<math>\begin{matrix}<br />
[\mathcal{A}]<br />
\\[2pt]<br />
(A, A^\uparrow)<br />
\\[2pt]<br />
(A ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
(A, (A \to \mathbb{B}))<br />
\\[2pt]<br />
[a_1, \ldots, a_n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Universe of discourse}<br />
\\[2pt]<br />
\text{based on the features}<br />
\\[2pt]<br />
\{ a_1, \ldots, a_n \}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))<br />
\\[2pt]<br />
(\mathbb{B}^n ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
[\mathbb{B}^n]<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
===Qualitative Logic and Quantitative Analogy===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
''Logical'', however, is used in a third sense, which is at once more vital and more practical; to denote, namely, the systematic care, negative and positive, taken to safeguard reflection so that it may yield the best results under the given conditions.<br />
| width="4%" | &nbsp;<br />
|- <br />
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 56]<br />
|}<br />
<br />
These concepts and notations may now be explained in greater detail. In order to begin as simply as possible, let us distinguish two levels of analysis and set out initially on the easier path. On the first level of analysis we take spaces like <math>\mathbb{B},</math> <math>\mathbb{B}^n,</math> and <math>(\mathbb{B}^n \to \mathbb{B})</math> at face value and treat them as the primary objects of interest. On the second level of analysis we use these spaces as coordinate charts for talking about points and functions in more fundamental spaces.<br />
<br />
A pair of spaces, of types <math>\mathbb{B}^n</math> and <math>(\mathbb{B}^n \to \mathbb{B}),</math> give typical expression to everything that we commonly associate with the ordinary picture of a venn diagram. The dimension, <math>n,\!</math> counts the number of "circles" or simple closed curves that are inscribed in the universe of discourse, corresponding to its relevant logical features or basic propositions. Elements of type <math>\mathbb{B}^n</math> correspond to what are often called propositional ''interpretations'' in logic, that is, the different assignments of truth values to sentence letters. Relative to a given universe of discourse, these interpretations are visualized as its ''cells'', in other words, the smallest enclosed areas or undivided regions of the venn diagram. The functions <math>f : \mathbb{B}^n \to \mathbb{B}</math> correspond to the different ways of shading the venn diagram to indicate arbitrary propositions, regions, or sets. Regions included under a shading indicate the ''models'', and regions excluded represent the ''non-models'' of a proposition. To recognize and formalize the natural cohesion of these two layers of concepts into a single universe of discourse, we introduce the type notations <math>[\mathbb{B}^n] = \mathbb{B}^n\ +\!\to \mathbb{B}</math> to stand for the pair of types <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})).</math> The resulting "stereotype" serves to frame the universe of discourse as a unified categorical object, and makes it subject to prescribed sets of evaluations and transformations (categorical morphisms or ''arrows'') that affect the universe of discourse as an integrated whole.<br />
<br />
Most of the time we can serve the algebraic, geometric, and logical interests of our study without worrying about their occasional conflicts and incidental divergences. The conventions and definitions already set down will continue to cover most of the algebraic and functional aspects of our discussion, but to handle the logical and qualitative aspects we will need to add a few more. In general, abstract sets may be denoted by gothic, greek, or script capital variants of <math>A, B, C,\!</math> and so on, with their elements being denoted by a corresponding set of subscripted letters in plain lower case, for example, <math>\mathcal{A} = \{a_i\}.</math> Most of the time, a set such as <math>\mathcal{A} = \{a_i\}\!</math> will be employed as the ''alphabet'' of a [[formal language]]. These alphabet letters serve to name the logical features (properties or propositions) that generate a particular universe of discourse. When we want to discuss the particular features of a universe of discourse, beyond the abstract designation of a type like <math>(\mathbb{B}^n\ +\!\to \mathbb{B}),</math> then we may use the following notations. If <math>\mathcal{A} = \{a_1, \ldots, a_n\}</math> is an alphabet of logical features, then <math>A = \langle \mathcal{A} \rangle = \langle a_1, \ldots, a_n \rangle</math> is the set of interpretations, <math>A^\uparrow = (A \to \mathbb{B})</math> is the set of propositions, and <math>A^\bullet = [\mathcal{A}] = [a_1, \ldots, a_n]</math> is the combination of these interpretations and propositions into the universe of discourse that is based on the features <math>\{a_1, \ldots, a_n\}.</math><br />
<br />
As always, especially in concrete examples, these rules may be dropped whenever necessary, reverting to a free assortment of feature labels. However, when we need to talk about the logical aspects of a space that is already named as a vector space, it will be necessary to make special provisions. At any rate, these elaborations can be deferred until actually needed.<br />
<br />
===Philosophy of Notation : Formal Terms and Flexible Types===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Where number is irrelevant, regimented mathematical technique has hitherto tended to be lacking. Thus it is that the progress of natural science has depended so largely upon the discernment of measurable quantity of one sort or another.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]<br />
|}<br />
<br />
For much of our discussion propositions and boolean functions are treated as the same formal objects, or as different interpretations of the same formal calculus. This rule of interpretation has exceptions, though. There is a distinctively logical interest in the use of propositional calculus that is not exhausted by its functional interpretation. It is part of our task in this study to deal with these uniquely logical characteristics as they present themselves both in our subject matter and in our formal calculus. Just to provide a hint of what's at stake: In logic, as opposed to the more imaginative realms of mathematics, we consider it a good thing to always know what we are talking about. Where mathematics encourages tolerance for uninterpreted symbols as intermediate terms, logic exerts a keener effort to interpret directly each oblique carrier of meaning, no matter how slight, and to unfold the complicities of every indirection in the flow of information. Translated into functional terms, this means that we want to maintain a continual, immediate, and persistent sense of both the converse relation <math>f^{-1} \subseteq \mathbb{B} \times \mathbb{B}^n,</math> or what is the same thing, <math>f^{-1} : \mathbb{B} \to \mathcal{P}(\mathbb{B}^n),</math> and the ''fibers'' or inverse images <math>f^{-1}(0)\!</math> and <math>f^{-1}(1),\!</math> associated with each boolean function <math>f : \mathbb{B}^n \to \mathbb{B}</math> that we use. In practical terms, the desired implementation of a propositional interpreter should incorporate our intuitive recognition that the induced partition of the functional domain into level sets <math>f^{-1}(b),\!</math> for <math>b \in \mathbb{B},</math> is part and parcel of understanding the denotative uses of each propositional function <math>f.\!</math><br />
<br />
===Special Classes of Propositions===<br />
<br />
It is important to remember that the coordinate propositions <math>\{a_i\},\!</math> besides being projection maps <math>a_i : \mathbb{B}^n \to \mathbb{B},</math> are propositions on an equal footing with all others, even though employed as a basis in a particular moment. This set of <math>n\!</math> propositions may sometimes be referred to as the ''basic propositions'', the ''coordinate propositions'', or the ''simple propositions'' that found a universe of discourse. Either one of the equivalent notations, <math>\{a_i : \mathbb{B}^n \to \mathbb{B}\}</math> or <math>(\mathbb{B}^n \xrightarrow{i} \mathbb{B}),</math> may be used to indicate the adoption of the propositions <math>a_i\!</math> as a basis for describing a universe of discourse.<br />
<br />
Among the <math>2^{2^n}</math> propositions in <math>[a_1, \ldots, a_n]</math> are several families of <math>2^n\!</math> propositions each that take on special forms with respect to the basis <math>\{ a_1, \ldots, a_n \}.</math> Three of these families are especially prominent in the present context, the ''linear'', the ''positive'', and the ''singular'' propositions. Each family is naturally parameterized by the coordinate <math>n\!</math>-tuples in <math>\mathbb{B}^n</math> and falls into <math>n + 1\!</math> ranks, with a binomial coefficient <math>\tbinom{n}{k}</math> giving the number of propositions that have rank or weight <math>k.\!</math><br />
<br />
<ul><br />
<br />
<li><br />
<p>The ''linear propositions'', <math>\{ \ell : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{\ell} \mathbb{B}),\!</math> may be written as sums:</p><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\sum_{i=1}^n e_i ~=~ e_1 + \ldots + e_n<br />
~\text{where}~<br />
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 0 \end{matrix}\right\}<br />
~\text{for}~ i = 1 ~\text{to}~ n.\!</math><br />
|}<br />
</li><br />
<br />
<li><br />
<p>The ''positive propositions'', <math>\{ p : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{p} \mathbb{B}),\!</math> may be written as products:</p><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\prod_{i=1}^n e_i ~=~ e_1 \cdot \ldots \cdot e_n<br />
~\text{where}~<br />
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 1 \end{matrix}\right\}<br />
~\text{for}~ i = 1 ~\text{to}~ n.\!</math><br />
|}<br />
</li><br />
<br />
<li><br />
<p>The ''singular propositions'', <math>\{ \mathbf{x} : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{s} \mathbb{B}),\!</math> may be written as products:</p><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\prod_{i=1}^n e_i ~=~ e_1 \cdot \ldots \cdot e_n<br />
~\text{where}~<br />
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = \texttt{(} a_i \texttt{)} \end{matrix}\right\}<br />
~\text{for}~ i = 1 ~\text{to}~ n.\!</math><br />
|}<br />
</li><br />
<br />
</ul><br />
<br />
In each case the rank <math>k\!</math> ranges from <math>0\!</math> to <math>n\!</math> and counts the number of positive appearances of the coordinate propositions <math>a_1, \ldots, a_n\!</math> in the resulting expression. For example, for <math>{n = 3},\!</math> the linear proposition of rank <math>0\!</math> is <math>0,\!</math> the positive proposition of rank <math>0\!</math> is <math>1,\!</math> and the singular proposition of rank <math>0\!</math> is <math>\texttt{(} a_1 \texttt{)(} a_2 \texttt{)(} a_3\texttt{)}.\!</math><br />
<br />
The basic propositions <math>a_i : \mathbb{B}^n \to \mathbb{B}</math> are both linear and positive. So these two kinds of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions.<br />
<br />
Linear propositions and positive propositions are generated by taking boolean sums and products, respectively, over selected subsets of basic propositions, so both families of propositions are parameterized by the powerset <math>\mathcal{P}(\mathcal{I}),</math> that is, the set of all subsets <math>J\!</math> of the basic index set <math>\mathcal{I} = \{1, \ldots, n\}.\!</math><br />
<br />
Let us define <math>\mathcal{A}_J</math> as the subset of <math>\mathcal{A}</math> that is given by <math>\{a_i : i \in J\}.\!</math> Then we may comprehend the action of the linear and the positive propositions in the following terms:<br />
<br />
* The linear proposition <math>\ell_J : \mathbb{B}^n \to \mathbb{B}</math> evaluates each cell <math>\mathbf{x}</math> of <math>\mathbb{B}^n</math> by looking at the coefficients of <math>\mathbf{x}</math> with respect to the features that <math>\ell_J</math> "likes", namely those in <math>\mathcal{A}_J,</math> and then adds them up in <math>\mathbb{B}.</math> Thus, <math>\ell_J(\mathbf{x})</math> computes the parity of the number of features that <math>\mathbf{x}</math> has in <math>\mathcal{A}_J,</math> yielding one for odd and zero for even. Expressed in this idiom, <math>\ell_J(\mathbf{x}) = 1</math> says that <math>\mathbf{x}</math> seems ''odd'' (or ''oddly true'') to <math>\mathcal{A}_J,</math> whereas <math>\ell_J(\mathbf{x}) = 0</math> says that <math>\mathbf{x}</math> seems ''even'' (or ''evenly true'') to <math>\mathcal{A}_J,</math> so long as we recall that ''zero times'' is evenly often, too.<br />
<br />
* The positive proposition <math>p_J : \mathbb{B}^n \to \mathbb{B}</math> evaluates each cell <math>\mathbf{x}</math> of <math>\mathbb{B}^n</math> by looking at the coefficients of <math>\mathbf{x}</math> with regard to the features that <math>p_J\!</math> "likes", namely those in <math>\mathcal{A}_J,</math> and then takes their product in <math>\mathbb{B}.</math> Thus, <math>p_J(\mathbf{x})</math> assesses the unanimity of the multitude of features that <math>\mathbf{x}</math> has in <math>\mathcal{A}_J,</math> yielding one for all and aught for else. In these consensual or contractual terms, <math>p_J(\mathbf{x}) = 1</math> means that <math>\mathbf{x}</math> is ''AOK'' or congruent with all of the conditions of <math>\mathcal{A}_J,</math> while <math>p_J(\mathbf{x}) = 0</math> means that <math>\mathbf{x}</math> defaults or dissents from some condition of <math>\mathcal{A}_J.</math><br />
<br />
===Basis Relativity and Type Ambiguity===<br />
<br />
Finally, two things are important to keep in mind with regard to the simplicity, linearity, positivity, and singularity of propositions.<br />
<br />
First, all of these properties are relative to a particular basis. For example, a singular proposition with respect to a basis <math>\mathcal{A}</math> will not remain singular if <math>\mathcal{A}</math> is extended by a number of new and independent features. Even if we stick to the original set of pairwise options <math>\{a_i\} \cup \{ \texttt{(} a_i \texttt{)} \}\!</math> to select a new basis, the sets of linear and positive propositions are determined by the choice of simple propositions, and this determination is tantamount to the conventional choice of a cell as origin.<br />
<br />
Second, the singular propositions <math>\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B},</math> picking out as they do a single cell or a coordinate tuple <math>\mathbf{x}</math> of <math>\mathbb{B}^n,</math> become the carriers or the vehicles of a certain type-ambiguity that vacillates between the dual forms <math>\mathbb{B}^n</math> and <math>(\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B})</math> and infects the whole hierarchy of types built on them. In other words, the terms that signify the interpretations <math>\mathbf{x} : \mathbb{B}^n</math> and the singular propositions <math>\mathbf{x} : \mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B}</math> are fully equivalent in information, and this means that every token of the type <math>\mathbb{B}^n</math> can be reinterpreted as an appearance of the subtype <math>\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B}.</math> And vice versa, the two types can be exchanged with each other everywhere that they turn up. In practical terms, this allows the use of singular propositions as a way of denoting points, forming an alternative to coordinate tuples.<br />
<br />
For example, relative to the universe of discourse <math>[a_1, a_2, a_3]\!</math> the singular proposition <math>a_1 a_2 a_3 : \mathbb{B}^3 \xrightarrow{s} \mathbb{B}</math> could be explicitly retyped as <math>a_1 a_2 a_3 : \mathbb{B}^3</math> to indicate the point <math>(1, 1, 1)\!</math> but in most cases the proper interpretation could be gathered from context. Both notations remain dependent on a particular basis, but the code that is generated under the singular option has the advantage in its self-commenting features, in other words, it constantly reminds us of its basis in the process of denoting points. When the time comes to put a multiplicity of different bases into play, and to search for objects and properties that remain invariant under the transformations between them, this infinitesimal potential advantage may well evolve into an overwhelming practical necessity.<br />
<br />
===The Analogy Between Real and Boolean Types===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Measurement consists in correlating our subject matter with the series of real numbers; and such correlations are desirable because, once they are set up, all the well-worked theory of numerical mathematics lies ready at hand as a tool for our further reasoning.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]<br />
|}<br />
<br />
There are two further reasons why it useful to spend time on a careful treatment of types, and they both have to do with our being able to take full computational advantage of certain dimensions of flexibility in the types that apply to terms. First, the domains of differential geometry and logic programming are connected by analogies between real and boolean types of the same pattern. Second, the types involved in these patterns have important isomorphisms connecting them that apply on both the real and the boolean sides of the picture.<br />
<br />
Amazingly enough, these isomorphisms are themselves schematized by the axioms and theorems of propositional logic. This fact is known as the ''propositions as types'' analogy or the Curry&ndash;Howard isomorphism [How]. In another formulation it says that terms are to types as proofs are to propositions. See [LaS, 42&ndash;46] and [SeH] for a good discussion and further references. To anticipate the bearing of these issues on our immediate topic, Table&nbsp;3 sketches a partial overview of the Real to Boolean analogy that may serve to illustrate the paradigm in question.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 3.} ~~ \text{Analogy Between Real and Boolean Types}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Real Domain} ~ \mathbb{R}\!</math><br />
| <math>\longleftrightarrow\!</math><br />
| <math>\text{Boolean Domain} ~ \mathbb{B}\!</math><br />
|-<br />
| <math>\mathbb{R}^n\!</math><br />
| <math>\text{Basic Space}\!</math><br />
| <math>\mathbb{B}^n\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}\!</math><br />
| <math>\text{Function Space}\!</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R}\!</math><br />
| <math>\text{Tangent Vector}\!</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to ((\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R})\!</math><br />
| <math>\text{Vector Field}\!</math><br />
| <math>\mathbb{B}^n \to ((\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B})\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \times (\mathbb{R}^n \to \mathbb{R})) \to \mathbb{R}\!</math><br />
| '''<font size="4">"</font>'''<br />
| <math>(\mathbb{B}^n \times (\mathbb{B}^n \to \mathbb{B})) \to \mathbb{B}\!</math><br />
|-<br />
| <math>((\mathbb{R}^n \to \mathbb{R}) \times \mathbb{R}^n) \to \mathbb{R}\!</math><br />
| '''<font size="4">"</font>'''<br />
| <math>((\mathbb{B}^n \to \mathbb{B}) \times \mathbb{B}^n) \to \mathbb{B}\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^n \to \mathbb{R})~\!</math><br />
| <math>\text{Derivation}\!</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^n \to \mathbb{B})\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}^m\!</math><br />
| <math>\begin{matrix}\text{Basic}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}^m\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^m \to \mathbb{R})\!</math><br />
| <math>\begin{matrix}\text{Function}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^m \to \mathbb{B})\!</math><br />
|}<br />
<br />
<br><br />
<br />
The Table exhibits a sample of likely parallels between the real and boolean domains. The central column gives a selection of terminology that is borrowed from differential geometry and extended in its meaning to the logical side of the Table. These are the varieties of spaces that come up when we turn to analyzing the dynamics of processes that pursue their courses through the states of an arbitrary space <math>X.\!</math> Moreover, when it becomes necessary to approach situations of overwhelming dynamic complexity in a succession of qualitative reaches, then the methods of logic that are afforded by the boolean domains, with their declarative means of synthesis and deductive modes of analysis, supply a natural battery of tools for the task.<br />
<br />
It is usually expedient to take these spaces two at a time, in dual pairs of the form <math>X\!</math> and <math>(X \to \mathbb{K}).</math> In general, one creates pairs of type schemas by replacing any space <math>X\!</math> with its dual <math>(X \to \mathbb{K}),</math> for example, pairing the type <math>X \to Y</math> with the type <math>(X \to \mathbb{K}) \to (Y \to \mathbb{K}),</math> and <math>X \times Y</math> with <math>(X \to \mathbb{K}) \times (Y \to \mathbb{K}).</math> The word ''dual'' is used here in its broader sense to mean all of the functionals, not just the linear ones. Given any function <math>f : X \to \mathbb{K},</math> the ''converse'' or inverse relation corresponding to <math>f\!</math> is denoted <math>f^{-1},\!</math> and the subsets of <math>X\!</math> that are defined by <math>f^{-1}(k),\!</math> taken over <math>k\!</math> in <math>\mathbb{K},</math> are called the ''fibers'' or the ''level sets'' of the function <math>f.\!</math><br />
<br />
===Theory of Control and Control of Theory===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
You will hardly know who I am or what I mean,<br><br />
But I shall be good health to you nevertheless,<br><br />
And filter and fibre your blood.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]<br />
|}<br />
<br />
In the boolean context a function <math>f : X \to \mathbb{B}</math> is tantamount to a ''proposition'' about elements of <math>X,\!</math> and the elements of <math>X\!</math> constitute the ''interpretations'' of that proposition. The fiber <math>f^{-1}(1)\!</math> comprises the set of ''models'' of <math>f,\!</math> or examples of elements in <math>X\!</math> satisfying the proposition <math>f.\!</math> The fiber <math>f^{-1}(0)\!</math> collects the complementary set of ''anti-models'', or the exceptions to the proposition <math>f\!</math> that exist in <math>X.\!</math> Of course, the space of functions <math>(X \to \mathbb{B})\!</math> is isomorphic to the set of all subsets of <math>X,\!</math> called the ''power set'' of <math>X,\!</math> and often denoted <math>\mathcal{P}(X)</math> or <math>2^X.\!</math><br />
<br />
The operation of replacing <math>X\!</math> by <math>(X \to \mathbb{B})\!</math> in a type schema corresponds to a certain shift of attitude towards the space <math>X,\!</math> in which one passes from a focus on the ostensibly individual elements of <math>X\!</math> to a concern with the states of information and uncertainty that one possesses about objects and situations in <math>X.\!</math> The conceptual obstacles in the path of this transition can be smoothed over by using singular functions <math>(\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B})</math> as stepping stones. First of all, it's an easy step from an element <math>\mathbf{x}</math> of type <math>\mathbb{B}^n</math> to the equivalent information of a singular proposition <math>\mathbf{x} : X \xrightarrow{s} \mathbb{B}, </math> and then only a small jump of generalization remains to reach the type of an arbitrary proposition <math>f : X \to \mathbb{B},</math> perhaps understood to indicate a relaxed constraint on the singularity of points or a neighborhood circumscribing the original <math>\mathbf{x}.</math> This is frequently a useful transformation, communicating between the ''objective'' and the ''intentional'' perspectives, in spite perhaps of the open objection that this distinction is transient in the mean time and ultimately superficial.<br />
<br />
It is hoped that this measure of flexibility, allowing us to stretch a point into a proposition, can be useful in the examination of inquiry driven systems, where the differences between empirical, intentional, and theoretical propositions constitute the discrepancies and the distributions that drive experimental activity. I can give this model of inquiry a cybernetic cast by realizing that theory change and theory evolution, as well as the choice and the evaluation of experiments, are actions that are taken by a system or its agent in response to the differences that are detected between observational contents and theoretical coverage.<br />
<br />
All of the above notwithstanding, there are several points that distinguish these two tasks, namely, the ''theory of control'' and the ''control of theory'', features that are often obscured by too much precipitation in the quickness with which we understand their similarities. In the control of uncertainty through inquiry, some of the actuators that we need to be concerned with are axiom changers and theory modifiers, operators with the power to compile and to revise the theories that generate expectations and predictions, effectors that form and edit our grammars for the languages of observational data, and agencies that rework the proposed model to fit the actual sequences of events and the realized relationships of values that are observed in the environment. Moreover, when steps must be taken to carry out an experimental action, there must be something about the particular shape of our uncertainty that guides us in choosing what directions to explore, and this impression is more than likely influenced by previous accumulations of experience. Thus it must be anticipated that much of what goes into scientific progress, or any sustainable effort toward a goal of knowledge, is necessarily predicated on long term observation and modal expectations, not only on the more local or short term prediction and correction.<br />
<br />
===Propositions as Types and Higher Order Types===<br />
<br />
The types collected in Table&nbsp;3 (repeated below) serve to illustrate the themes of ''higher order propositional expressions'' and the ''propositions as types'' (PAT) analogy.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 3.} ~~ \text{Analogy Between Real and Boolean Types}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Real Domain} ~ \mathbb{R}\!</math><br />
| <math>\longleftrightarrow\!</math><br />
| <math>\text{Boolean Domain} ~ \mathbb{B}\!</math><br />
|-<br />
| <math>\mathbb{R}^n\!</math><br />
| <math>\text{Basic Space}\!</math><br />
| <math>\mathbb{B}^n\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}\!</math><br />
| <math>\text{Function Space}\!</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R}\!</math><br />
| <math>\text{Tangent Vector}\!</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to ((\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R})\!</math><br />
| <math>\text{Vector Field}\!</math><br />
| <math>\mathbb{B}^n \to ((\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B})\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \times (\mathbb{R}^n \to \mathbb{R})) \to \mathbb{R}\!</math><br />
| '''<font size="4">"</font>'''<br />
| <math>(\mathbb{B}^n \times (\mathbb{B}^n \to \mathbb{B})) \to \mathbb{B}\!</math><br />
|-<br />
| <math>((\mathbb{R}^n \to \mathbb{R}) \times \mathbb{R}^n) \to \mathbb{R}\!</math><br />
| '''<font size="4">"</font>'''<br />
| <math>((\mathbb{B}^n \to \mathbb{B}) \times \mathbb{B}^n) \to \mathbb{B}\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^n \to \mathbb{R})~\!</math><br />
| <math>\text{Derivation}\!</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^n \to \mathbb{B})\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}^m\!</math><br />
| <math>\begin{matrix}\text{Basic}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}^m\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^m \to \mathbb{R})\!</math><br />
| <math>\begin{matrix}\text{Function}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^m \to \mathbb{B})\!</math><br />
|}<br />
<br />
<br><br />
<br />
First, observe that the type of a ''tangent vector at a point'', also known as a ''directional derivative'' at that point, has the form <math>(\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K},</math> where <math>\mathbb{K}</math> is the chosen ground field, in the present case either <math>\mathbb{R}</math> or <math>\mathbb{B}.</math> At a point in a space of type <math>\mathbb{K}^n,</math> a directional derivative operator <math>\vartheta\!</math> takes a function on that space, an <math>f\!</math> of type <math>(\mathbb{K}^n \to \mathbb{K}),</math> and maps it to a ground field value of type <math>\mathbb{K}.</math> This value is known as the ''derivative'' of <math>f\!</math> in the direction <math>\vartheta\!</math> [Che46, 76&ndash;77]. In the boolean case <math>\vartheta : (\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math> has the form of a proposition about propositions, in other words, a proposition of the next higher type.<br />
<br />
Next, by way of illustrating the propositions as types idea, consider a proposition of the form <math>X \Rightarrow (Y \Rightarrow Z).</math> One knows from propositional calculus that this is logically equivalent to a proposition of the form <math>(X \land Y) \Rightarrow Z.</math> But this equivalence should remind us of the functional isomorphism that exists between a construction of the type <math>X \to (Y \to Z)</math> and a construction of the type <math>(X \times Y) \to Z.</math> The propositions as types analogy permits us to take a functional type like this and, under the right conditions, replace the functional arrows &ldquo;<math>\to~\!</math>&rdquo; and products &ldquo;<math>\times\!</math>&rdquo; with the respective logical arrows &ldquo;<math>\Rightarrow\!</math>&rdquo; and products &ldquo;<math>\land\!</math>&rdquo;. Accordingly, viewing the result as a proposition, we can employ axioms and theorems of propositional calculus to suggest appropriate isomorphisms among the categorical and functional constructions.<br />
<br />
Finally, examine the middle four rows of Table&nbsp;3. These display a series of isomorphic types that stretch from the categories that are labeled ''Vector Field'' to those that are labeled ''Derivation''. A ''vector field'', also known as an ''infinitesimal transformation'', associates a tangent vector at a point with each point of a space. In symbols, a vector field is a function of the form <math>\textstyle \xi : X \to \bigcup_{x \in X} \xi_x\!</math> that assigns to each point <math>x\!</math> of the space <math>X\!</math> a tangent vector to <math>X\!</math> at that point, namely, the tangent vector <math>\xi_x\!</math> [Che46, 82&ndash;83]. If <math>X\!</math> is of the type <math>\mathbb{K}^n,</math> then <math>\xi\!</math> is of the type <math>\mathbb{K}^n \to ((\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K}).</math> This has the pattern <math>X \to (Y \to Z),</math> with <math>X = \mathbb{K}^n,</math> <math>Y = (\mathbb{K}^n \to \mathbb{K}),</math> and <math>Z = \mathbb{K}.</math><br />
<br />
Applying the propositions as types analogy, one can follow this pattern through a series of metamorphoses from the type of a vector field to the type of a derivation, as traced out in Table&nbsp;4. Observe how the function <math>f : X \to \mathbb{K},</math> associated with the place of <math>Y\!</math> in the pattern, moves through its paces from the second to the first position. In this way, the vector field <math>\xi,\!</math> initially viewed as attaching each tangent vector <math>\xi_x\!</math> to the site <math>x\!</math> where it acts in <math>X,\!</math> now comes to be seen as acting on each scalar potential <math>f : X \to \mathbb{K}</math> like a generalized species of differentiation, producing another function <math>\xi f : X \to \mathbb{K}</math> of the same type.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 4.} ~~ \text{An Equivalence Based on the Propositions as Types Analogy}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Pattern}\!</math><br />
| <math>\text{Construct}\!</math><br />
| <math>\text{Instance}\!</math><br />
|-<br />
| <math>X \to (Y \to Z)\!</math><br />
| <math>\text{Vector Field}\!</math><br />
| <math>\mathbb{K}^n \to ((\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K})\!</math><br />
|-<br />
| <math>(X \times Y) \to Z\!</math><br />
| <math>\Uparrow\!</math><br />
| <math>(\mathbb{K}^n \times (\mathbb{K}^n \to \mathbb{K})) \to \mathbb{K}\!</math><br />
|-<br />
| <math>(Y \times X) \to Z\!</math><br />
| <math>\Downarrow\!</math><br />
| <math>((\mathbb{K}^n \to \mathbb{K}) \times \mathbb{K}^n) \to \mathbb{K}\!</math><br />
|-<br />
| <math>Y \to (X \to Z)\!</math><br />
| <math>\text{Derivation}\!</math><br />
| <math>(\mathbb{K}^n \to \mathbb{K}) \to (\mathbb{K}^n \to \mathbb{K})\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Reality at the Threshold of Logic===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
But no science can rest entirely on measurement, and many scientific investigations are quite out of reach of that device. To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]<br />
|}<br />
<br />
Table 5 accumulates an array of notation that I hope will not be too distracting. Some of it is rarely needed, but has been filled in for the sake of completeness. Its purpose is simple, to give literal expression to the visual intuitions that come with venn diagrams, and to help build a bridge between our qualitative and quantitative outlooks on dynamic systems.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 5.} ~~ \text{A Bridge Over Troubled Waters}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| align="center" | <math>\text{Linear Space}\!</math><br />
| align="center" | <math>\text{Liminal Space}\!</math><br />
| align="center" | <math>\text{Logical Space}\!</math><br />
|-<br />
| <math>\begin{matrix}\mathcal{X} & = & \{ x_1, \ldots, x_n \}\end{matrix}</math><br />
| <math>\begin{matrix}\underline{\mathcal{X}} & = & \{ \underline{x}_1, \ldots, \underline{x}_n \}\end{matrix}</math><br />
| <math>\begin{matrix}\mathcal{A} & = & \{ a_1, \ldots, a_n \}\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X_i & = & \langle x_i \rangle<br />
\\<br />
& \cong & \mathbb{K}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}_i & = & \{ \texttt{(} \underline{x}_i \texttt{)}, \underline{x}_i \}<br />
\\<br />
& \cong & \mathbb{B}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A_i & = & \{ \texttt{(} a_i \texttt{)}, a_i \}<br />
\\<br />
& \cong & \mathbb{B}<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X<br />
\\<br />
= & \langle \mathcal{X} \rangle<br />
\\<br />
= & \langle x_1, \ldots, x_n \rangle<br />
\\<br />
= & X_1 \times \ldots \times X_n<br />
\\<br />
= & \prod_{i=1}^n X_i<br />
\\<br />
\cong & \mathbb{K}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}<br />
\\<br />
= & \langle \underline{\mathcal{X}} \rangle<br />
\\<br />
= & \langle \underline{x}_1, \ldots, \underline{x}_n \rangle<br />
\\<br />
= & \underline{X}_1 \times \ldots \times \underline{X}_n<br />
\\<br />
= & \prod_{i=1}^n \underline{X}_i<br />
\\<br />
\cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A<br />
\\<br />
= & \langle \mathcal{A} \rangle<br />
\\<br />
= & \langle a_1, \ldots, a_n \rangle<br />
\\<br />
= & A_1 \times \ldots \times A_n<br />
\\<br />
= & \prod_{i=1}^n A_i<br />
\\<br />
\cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X^* & = & (\ell : X \to \mathbb{K})<br />
\\<br />
& \cong & \mathbb{K}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}^* & = & (\ell : \underline{X} \to \mathbb{B})<br />
\\<br />
& \cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A^* & = & (\ell : A \to \mathbb{B})<br />
\\<br />
& \cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X^\uparrow & = & (X \to \mathbb{K})<br />
\\<br />
& \cong & (\mathbb{K}^n \to \mathbb{K})<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}^\uparrow & = & (\underline{X} \to \mathbb{B})<br />
\\<br />
& \cong & (\mathbb{B}^n \to \mathbb{B})<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A^\uparrow & = & (A \to \mathbb{B})<br />
\\<br />
& \cong & (\mathbb{B}^n \to \mathbb{B})<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X^\bullet<br />
\\<br />
= & [\mathcal{X}]<br />
\\<br />
= & [x_1, \ldots, x_n]<br />
\\<br />
= & (X, X^\uparrow)<br />
\\<br />
= & (X ~+\!\to \mathbb{K})<br />
\\<br />
= & (X, (X \to \mathbb{K}))<br />
\\<br />
\cong & (\mathbb{K}^n, (\mathbb{K}^n \to \mathbb{K}))<br />
\\<br />
= & (\mathbb{K}^n ~+\!\to \mathbb{K})<br />
\\<br />
= & [\mathbb{K}^n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}^\bullet<br />
\\<br />
= & [\underline{\mathcal{X}}]<br />
\\<br />
= & [\underline{x}_1, \ldots, \underline{x}_n]<br />
\\<br />
= & (\underline{X}, \underline{X}^\uparrow)<br />
\\<br />
= & (\underline{X} ~+\!\to \mathbb{B})<br />
\\<br />
= & (\underline{X}, (\underline{X} \to \mathbb{B}))<br />
\\<br />
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))<br />
\\<br />
= & (\mathbb{B}^n ~+\!\to \mathbb{B})<br />
\\<br />
= & [\mathbb{B}^n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A^\bullet<br />
\\<br />
= & [\mathcal{A}]<br />
\\<br />
= & [a_1, \ldots, a_n]<br />
\\<br />
= & (A, A^\uparrow)<br />
\\<br />
= & (A ~+\!\to \mathbb{B})<br />
\\<br />
= & (A, (A \to \mathbb{B}))<br />
\\<br />
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))<br />
\\<br />
= & (\mathbb{B}^n ~+\!\to \mathbb{B})<br />
\\<br />
= & [\mathbb{B}^n]<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
The left side of the Table collects mostly standard notation for an <math>n\!</math>-dimensional vector space over a field <math>\mathbb{K}.</math> The right side of the table repeats the first elements of a notation that I sketched above, to be used in further developments of propositional calculus. (I plan to use this notation in the logical analysis of neural network systems.) The middle column of the table is designed as a transitional step from the case of an arbitrary field <math>\mathbb{K},</math> with a special interest in the continuous line <math>\mathbb{R},</math> to the qualitative and discrete situations that are instanced and typified by <math>\mathbb{B}.</math><br />
<br />
I now proceed to explain these concepts in more detail. The most important ideas developed in Table&nbsp;5 are these:<br />
<br />
* The idea of a universe of discourse, which includes both a space of ''points'' and a space of ''maps'' on those points.<br />
<br />
* The idea of passing from a more complex universe to a simpler universe by a process of ''thresholding'' each dimension of variation down to a single bit of information.<br />
<br />
For the sake of concreteness, let us suppose that we start with a continuous <math>n\!</math>-dimensional vector space like <math>X = \langle x_1, \ldots, x_n \rangle \cong \mathbb{R}^n.</math> The coordinate system <math>\mathcal{X} = \{x_i\}</math> is a set of maps <math>x_i : \mathbb{R}^n \to \mathbb{R},</math> also known as the ''coordinate projections''. Given a "dataset" of points <math>\mathbf{x}</math> in <math>\mathbb{R}^n,</math> there are numerous ways of sensibly reducing the data down to one bit for each dimension. One strategy that is general enough for our present purposes is as follows. For each <math>i\!</math> we choose an <math>n\!</math>-ary relation <math>L_i\!</math> on <math>\mathbb{R}^n,</math> that is, a subset of <math>\mathbb{R}^n,</math> and then we define the <math>i^\mathrm{th}\!</math> threshold map, or ''limen'' <math>\underline{x}_i</math> as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\underline{x}_i : \mathbb{R}^n \to \mathbb{B}\ \text{such that:}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
\underline{x}_i(\mathbf{x}) = 1 & \text{if} & \mathbf{x} \in L_i,<br />
\\[4pt]<br />
\underline{x}_i(\mathbf{x}) = 0 & \text{if} & \mathbf{x} \not\in L_i.<br />
\end{matrix}</math><br />
|}<br />
<br />
In other notations that are sometimes used, the operator <math>\chi (\ldots)</math> or the corner brackets <math>\lceil\ldots\rceil</math> can be used to denote a ''characteristic function'', that is, a mapping from statements to their truth values in <math>\mathbb{B}.</math> Finally, it is not uncommon to use the name of the relation itself as a predicate that maps <math>n\!</math>-tuples into truth values. Thus we have the following notational variants of the above definition:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
\underline{x}_i (\mathbf{x}) & = & \chi (\mathbf{x} \in L_i) & = & \lceil \mathbf{x} \in L_i \rceil & = & L_i (\mathbf{x}).<br />
\end{matrix}</math><br />
|}<br />
<br />
Notice that, as defined here, there need be no actual relation between the <math>n\!</math>-dimensional subsets <math>\{L_i\}\!</math> and the coordinate axes corresponding to <math>\{x_i\},\!</math> aside from the circumstance that the two sets have the same cardinality. In concrete cases, though, one usually has some reason for associating these "volumes" with these "lines", for instance, <math>L_i\!</math> is bounded by some hyperplane that intersects the <math>i^\text{th}\!</math> axis at a unique threshold value <math>r_i \in \mathbb{R}.\!</math> Often, the hyperplane is chosen normal to the axis. In recognition of this motive, let us make the following convention. When the set <math>L_i\!</math> has points on the <math>i^\text{th}\!</math> axis, that is, points of the form <math>(0, \ldots, 0, r_i, 0, \ldots, 0)</math> where only the <math>x_i\!</math> coordinate is possibly non-zero, we may pick any one of these coordinate values as a parametric index of the relation. In this case we say that the indexing is ''real'', otherwise the indexing is ''imaginary''. For a knowledge based system <math>X,\!</math> this should serve once again to mark the distinction between ''acquaintance'' and ''opinion''.<br />
<br />
States of knowledge about the location of a system or about the distribution of a population of systems in a state space <math>X = \mathbb{R}^n</math> can now be expressed by taking the set <math>\underline{\mathcal{X}} = \{\underline{x}_i\}</math> as a basis of logical features. In picturesque terms, one may think of the underscore and the subscript as combining to form a subtextual spelling for the <math>i^\text{th}\!</math> threshold map. This can help to remind us that the ''threshold operator'' <math>(\underline{~})_i</math> acts on <math>\mathbf{x}</math> by setting up a kind of a &ldquo;hurdle&rdquo; for it. In this interpretation the coordinate proposition <math>\underline{x}_i</math> asserts that the representative point <math>\mathbf{x}</math> resides ''above'' the <math>i^\mathrm{th}\!</math> threshold.<br />
<br />
Primitive assertions of the form <math>\underline{x}_i (\mathbf{x})</math> may then be negated and joined by means of propositional connectives in the usual ways to provide information about the state <math>\mathbf{x}</math> of a contemplated system or a statistical ensemble of systems. Parentheses <math>\texttt{(} \ldots \texttt{)}\!</math> may be used to indicate logical negation. Eventually one discovers the usefulness of the <math>k\!</math>-ary ''just one false'' operators of the form <math>\texttt{(} a_1 \texttt{,} \ldots \texttt{,} a_k \texttt{)},\!</math> as treated in earlier reports. This much tackle generates a space of points (cells, interpretations), <math>\underline{X} \cong \mathbb{B}^n,</math> and a space of functions (regions, propositions), <math>\underline{X}^\uparrow \cong (\mathbb{B}^n \to \mathbb{B}).</math> Together these form a new universe of discourse <math>\underline{X}^\bullet</math> of the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})),</math> which we may abbreviate as <math>\mathbb{B}^n\ +\!\to \mathbb{B}</math> or most succinctly as <math>[\mathbb{B}^n].</math><br />
<br />
The square brackets have been chosen to recall the rectangular frame of a venn diagram. In thinking about a universe of discourse it is a good idea to keep this picture in mind, graphically illustrating the links among the elementary cells <math>\underline{\mathbf{x}},</math> the defining features <math>\underline{x}_i,</math> and the potential shadings <math>f : \underline{X} \to \mathbb{B}</math> all at the same time, not to mention the arbitrariness of the way we choose to inscribe our distinctions in the medium of a continuous space.<br />
<br />
Finally, let <math>X^*\!</math> denote the space of linear functions, <math>(\ell : X \to \mathbb{K}),</math> which has in the finite case the same dimensionality as <math>X,\!</math> and let the same notation be extended across the Table.<br />
<br />
We have just gone through a lot of work, apparently doing nothing more substantial than spinning a complex spell of notational devices through a labyrinth of baffled spaces and baffling maps. The reason for doing this was to bind together and to constitute the intuitive concept of a universe of discourse into a coherent categorical object, the kind of thing, once grasped, that can be turned over in the mind and considered in all its manifold changes and facets. The effort invested in these preliminary measures is intended to pay off later, when we need to consider the state transformations and the time evolution of neural network systems.<br />
<br />
===Tables of Propositional Forms===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope. It provides explicit techniques for manipulating the most basic ingredients of discourse.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7&ndash;8]<br />
|}<br />
<br />
To prepare for the next phase of discussion, Tables&nbsp;6 and 7 collect and summarize all of the propositional forms on one and two variables. These propositional forms are represented over bases of boolean variables as complete sets of boolean-valued functions. Adjacent to their names and specifications are listed what are roughly the simplest expressions in the ''[[cactus language]]'', the particular syntax for propositional calculus that I use in formal and computational contexts. For the sake of orientation, the English paraphrases and the more common notations are listed in the last two columns. As simple and circumscribed as these low-dimensional universes may appear to be, a careful exploration of their differential extensions will involve us in complexities sufficient to demand our attention for some time to come.<br />
<br />
Propositional forms on one variable correspond to boolean functions <math>f : \mathbb{B}^1 \to \mathbb{B}.</math> In Table&nbsp;6 these functions are listed in a variant form of [[truth table]], one in which the axes of the usual arrangement are rotated through a right angle. Each function <math>f_i\!</math> is indexed by the string of values that it takes on the points of the universe <math>X^\bullet = [x] \cong \mathbb{B}^1.</math> The binary index generated in this way is converted to its decimal equivalent and these are used as conventional names for the <math>f_i,\!</math> as shown in the first column of the Table. In their own right the <math>2^1\!</math> points of the universe <math>X^\bullet</math> are coordinated as a space of type <math>\mathbb{B}^1,</math> this in light of the universe <math>X^\bullet</math> being a functional domain where the coordinate projection <math>x\!</math> takes on its values in <math>\mathbb{B}.</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 6.} ~~ \text{Propositional Forms on One Variable}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_0\!</math><br />
| <math>f_{00}\!</math><br />
| <math>0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>f_1\!</math><br />
| <math>f_{01}\!</math><br />
| <math>0~1\!</math><br />
| <math>\texttt{(} x \texttt{)}\!</math><br />
| <math>\text{not}~ x\!</math><br />
| <math>\lnot x\!</math><br />
|-<br />
| <math>f_2\!</math><br />
| <math>f_{10}\!</math><br />
| <math>1~0\!</math><br />
| <math>x\!</math><br />
| <math>x\!</math><br />
| <math>x\!</math><br />
|-<br />
| <math>f_3\!</math><br />
| <math>f_{11}\!</math><br />
| <math>1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
Propositional forms on two variables correspond to boolean functions <math>f : \mathbb{B}^2 \to \mathbb{B}.</math> In Table&nbsp;7 each function <math>f_i\!</math> is indexed by the values that it takes on the points of the universe <math>X^\bullet = [x, y] \cong \mathbb{B}^2.</math> Converting the binary index thus generated to a decimal equivalent, we obtain the functional nicknames that are listed in the first column. The <math>2^2\!</math> points of the universe <math>X^\bullet</math> are coordinated as a space of type <math>\mathbb{B}^2,</math> as indicated under the heading of the Table, where the coordinate projections <math>x\!</math> and <math>y\!</math> run through the various combinations of their values in <math>\mathbb{B}.</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 7-a.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}\!</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}\!</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}\!</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}\!</math><br />
| style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}\!</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}\!</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[4pt]<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\\[4pt]<br />
f_{3}<br />
\\[4pt]<br />
f_{4}<br />
\\[4pt]<br />
f_{5}<br />
\\[4pt]<br />
f_{6}<br />
\\[4pt]<br />
f_{7}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0000}<br />
\\[4pt]<br />
f_{0001}<br />
\\[4pt]<br />
f_{0010}<br />
\\[4pt]<br />
f_{0011}<br />
\\[4pt]<br />
f_{0100}<br />
\\[4pt]<br />
f_{0101}<br />
\\[4pt]<br />
f_{0110}<br />
\\[4pt]<br />
f_{0111}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~0<br />
\\[4pt]<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~0~1~1<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
0~1~0~1<br />
\\[4pt]<br />
0~1~1~0<br />
\\[4pt]<br />
0~1~1~1<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{)} ~ y ~<br />
\\[4pt]<br />
\texttt{(} x \texttt{)}<br />
\\[4pt]<br />
~ x ~ \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{,} ~ y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x ~ y \texttt{)}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{false}<br />
\\[4pt]<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\[4pt]<br />
y ~\text{without}~ x<br />
\\[4pt]<br />
\text{not}~ x<br />
\\[4pt]<br />
x ~\text{without}~ y<br />
\\[4pt]<br />
\text{not}~ y<br />
\\[4pt]<br />
x ~\text{not equal to}~ y<br />
\\[4pt]<br />
\text{not both}~ x ~\text{and}~ y<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
\lnot x \land \lnot y<br />
\\[4pt]<br />
\lnot x \land y<br />
\\[4pt]<br />
\lnot x<br />
\\[4pt]<br />
x \land \lnot y<br />
\\[4pt]<br />
\lnot y<br />
\\[4pt]<br />
x \ne y<br />
\\[4pt]<br />
\lnot x \lor \lnot y<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{8}<br />
\\[4pt]<br />
f_{9}<br />
\\[4pt]<br />
f_{10}<br />
\\[4pt]<br />
f_{11}<br />
\\[4pt]<br />
f_{12}<br />
\\[4pt]<br />
f_{13}<br />
\\[4pt]<br />
f_{14}<br />
\\[4pt]<br />
f_{15}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1000}<br />
\\[4pt]<br />
f_{1001}<br />
\\[4pt]<br />
f_{1010}<br />
\\[4pt]<br />
f_{1011}<br />
\\[4pt]<br />
f_{1100}<br />
\\[4pt]<br />
f_{1101}<br />
\\[4pt]<br />
f_{1110}<br />
\\[4pt]<br />
f_{1111}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
1~0~0~0<br />
\\[4pt]<br />
1~0~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\\[4pt]<br />
1~1~1~1<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x ~ y<br />
\\[4pt]<br />
\texttt{((} x \texttt{,} ~ y \texttt{))}<br />
\\[4pt]<br />
y<br />
\\[4pt]<br />
\texttt{(} x ~ \texttt{(} y \texttt{))}<br />
\\[4pt]<br />
x<br />
\\[4pt]<br />
\texttt{((} x \texttt{)} ~ y \texttt{)}<br />
\\[4pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
\texttt{((~))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x ~\text{and}~ y<br />
\\[4pt]<br />
x ~\text{equal to}~ y<br />
\\[4pt]<br />
y<br />
\\[4pt]<br />
\text{not}~ x ~\text{without}~ y<br />
\\[4pt]<br />
x<br />
\\[4pt]<br />
\text{not}~ y ~\text{without}~ x<br />
\\[4pt]<br />
x ~\text{or}~ y<br />
\\[4pt]<br />
\text{true}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x \land y<br />
\\[4pt]<br />
x = y<br />
\\[4pt]<br />
y<br />
\\[4pt]<br />
x \Rightarrow y<br />
\\[4pt]<br />
x<br />
\\[4pt]<br />
x \Leftarrow y<br />
\\[4pt]<br />
x \lor y<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 7-b.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}</math><br />
| style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>f_{0000}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\\[4pt]<br />
f_{4}<br />
\\[4pt]<br />
f_{8}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0001}<br />
\\[4pt]<br />
f_{0010}<br />
\\[4pt]<br />
f_{0100}<br />
\\[4pt]<br />
f_{1000}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
1~0~0~0<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{)} ~ y ~<br />
\\[4pt]<br />
~ x ~ \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
~ x ~~ y ~<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\[4pt]<br />
y ~\text{without}~ x<br />
\\[4pt]<br />
x ~\text{without}~ y<br />
\\[4pt]<br />
x ~\text{and}~ y<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot x \land \lnot y<br />
\\[4pt]<br />
\lnot x \land y<br />
\\[4pt]<br />
x \land \lnot y<br />
\\[4pt]<br />
x \land y<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{3}<br />
\\[4pt]<br />
f_{12}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0011}<br />
\\[4pt]<br />
f_{1100}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\[4pt]<br />
x<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{not}~ x<br />
\\[4pt]<br />
x<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot x<br />
\\[4pt]<br />
x<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[4pt]<br />
f_{9}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0110}<br />
\\[4pt]<br />
f_{1001}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[4pt]<br />
1~0~0~1<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{,} y \texttt{)}<br />
\\[4pt]<br />
\texttt{((} x \texttt{,} y \texttt{))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x ~\text{not equal to}~ y<br />
\\[4pt]<br />
x ~\text{equal to}~ y<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x \ne y<br />
\\[4pt]<br />
x = y<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{5}<br />
\\[4pt]<br />
f_{10}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0101}<br />
\\[4pt]<br />
f_{1010}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\[4pt]<br />
y<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{not}~ y<br />
\\[4pt]<br />
y<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot y<br />
\\[4pt]<br />
y<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[4pt]<br />
f_{11}<br />
\\[4pt]<br />
f_{13}<br />
\\[4pt]<br />
f_{14}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0111}<br />
\\[4pt]<br />
f_{1011}<br />
\\[4pt]<br />
f_{1101}<br />
\\[4pt]<br />
f_{1110}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} ~ x ~~ y ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{(} ~ x ~ \texttt{(} y \texttt{))}<br />
\\[4pt]<br />
\texttt{((} x \texttt{)} ~ y ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{not both}~ x ~\text{and}~ y<br />
\\[4pt]<br />
\text{not}~ x ~\text{without}~ y<br />
\\[4pt]<br />
\text{not}~ y ~\text{without}~ x<br />
\\[4pt]<br />
x ~\text{or}~ y<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot x \lor \lnot y<br />
\\[4pt]<br />
x \Rightarrow y<br />
\\[4pt]<br />
x \Leftarrow y<br />
\\[4pt]<br />
x \lor y<br />
\end{matrix}\!</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>f_{1111}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
==A Differential Extension of Propositional Calculus==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Fire over water:<br><br />
The image of the condition before transition.<br><br />
Thus the superior man is careful<br><br />
In the differentiation of things,<br><br />
So that each finds its place.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; ''I Ching'', Hexagram 64, [Wil, 249]<br />
|}<br />
<br />
This much preparation is enough to begin introducing my subject, if I excuse myself from giving full arguments for my definitional choices until some later stage. I am trying to develop a ''differential theory of qualitative equations'' that parallels the application of differential geometry to dynamic systems. The idea of a tangent vector is key to this work and a major goal is to find the right logical analogues of tangent spaces, bundles, and functors. The strategy is taken of looking for the simplest versions of these constructions that can be discovered within the realm of propositional calculus, so long as they serve to fill out the general theme.<br />
<br />
===Differential Propositions : Qualitative Analogues of Differential Equations===<br />
<br />
In order to define the differential extension of a universe of discourse <math>[\mathcal{A}],</math> the initial alphabet <math>\mathcal{A}</math> must be extended to include a collection of symbols for ''differential features'', or basic ''changes'' that are capable of occurring in <math>[\mathcal{A}].</math> Intuitively, these symbols may be construed as denoting primitive features of change, qualitative attributes of motion, or propositions about how things or points in the universe may be changing or moving with respect to the features that are noted in the initial alphabet.<br />
<br />
Therefore, let us define the corresponding ''differential alphabet'' or ''tangent alphabet'' as <math>\mathrm{d}\mathcal{A}\!</math> <math>=\!</math> <math>\{\mathrm{d}a_1, \ldots, \mathrm{d}a_n\},</math> in principle just an arbitrary alphabet of symbols, disjoint from the initial alphabet <math>\mathcal{A}\!</math> <math>=\!</math> <math>\{ a_1, \ldots, a_n \},\!</math> that is intended to be interpreted in the way just indicated. It only remains to be understood that the precise interpretation of the symbols in <math>\mathrm{d}\mathcal{A}\!</math> is often conceived to be changeable from point to point of the underlying space <math>A.\!</math> Indeed, for all we know, the state space <math>A\!</math> might well be the state space of a language interpreter, one that is concerned, among other things, with the idiomatic meanings of the dialect generated by <math>\mathcal{A}</math> and <math>\mathrm{d}\mathcal{A}.\!</math><br />
<br />
The ''tangent space'' to <math>A\!</math> at one of its points <math>x,\!</math> sometimes written <math>\mathrm{T}_x(A),</math> takes the form <math>\mathrm{d}A</math> <math>=\!</math> <math>\langle \mathrm{d}\mathcal{A} \rangle</math> <math>=\!</math> <math>\langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle.\!</math> Strictly speaking, the name ''cotangent space'' is probably more correct for this construction, but the fact that we take up spaces and their duals in pairs to form our universes of discourse allows our language to be pliable here.<br />
<br />
Proceeding as we did with the base space <math>A,\!</math> the tangent space <math>\mathrm{d}A</math> at a point of <math>A\!</math> can be analyzed as a product of distinct and independent factors:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\mathrm{d}A ~=~ \prod_{i=1}^n \mathrm{d}A_i ~=~ \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n.\!</math><br />
|}<br />
<br />
Here, <math>\mathrm{d}A_i\!</math> is a set of two differential propositions, <math>\mathrm{d}A_i = \{ (\mathrm{d}a_i), \mathrm{d}a_i \},\!</math> where <math>\texttt{(} \mathrm{d}a_i \texttt{)}\!</math> is a proposition with the logical value of <math>\text{not} ~ \mathrm{d}a_i.\!</math> Each component <math>\mathrm{d}A_i\!</math> has the type <math>\mathbb{B},\!</math> operating under the ordered correspondence <math>\{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \} \cong \{ 0, 1 \}.\!</math> However, clarity is often served by acknowledging this differential usage with a superficially distinct type <math>\mathbb{D},\!</math> whose intension may be indicated as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\mathbb{D} = \{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \} = \{ \text{same}, \text{different} \} = \{ \text{stay}, \text{change} \} = \{ \text{stop}, \text{step} \}.\!</math><br />
|}<br />
<br />
Viewed within a coordinate representation, spaces of type <math>\mathbb{B}^n\!</math> and <math>\mathbb{D}^n\!</math> may appear to be identical sets of binary vectors, but taking a view at this level of abstraction would be like ignoring the qualitative units and the diverse dimensions that distinguish position and momentum, or the different roles of quantity and impulse.<br />
<br />
===An Interlude on the Path===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
There would have been no beginnings: instead, speech would proceed from me, while I stood in its path &ndash; a slender gap &ndash; the point of its possible disappearance.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
A sense of the relation between <math>\mathbb{B}</math> and <math>\mathbb{D}</math> may be obtained by considering the ''path classifier'' (or the ''equivalence class of curves'') approach to tangent vectors. Consider a universe <math>[\mathcal{X}].\!</math> Given the boolean value system, a path in the space <math>X = \langle \mathcal{X} \rangle</math> is a map <math>q : \mathbb{B} \to X.</math> In this context the set of paths <math>(\mathbb{B} \to X)</math> is isomorphic to the cartesian square <math>X^2 = X \times X,</math> or the set of ordered pairs chosen from <math>X.\!</math><br />
<br />
We may analyze <math>X^2 = \{ (u, v) : u, v \in X \}</math> into two parts, specifically, the ordered pairs <math>(u, v)\!</math> that lie on and off the diagonal:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}X^2 & = & \{ (u, v) : u = v \} & \cup & \{ (u, v) : u \ne v \}.\end{matrix}</math><br />
|}<br />
<br />
This partition may also be expressed in the following symbolic form:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}X^2 & \cong & \operatorname{diag} (X) & + & 2 \binom{X}{2}.\end{matrix}</math><br />
|}<br />
<br />
The separate terms of this formula are defined as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}\operatorname{diag} (X) & = & \{ (x, x) : x \in X \}.\end{matrix}\!</math><br />
|}<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}\binom{X}{k} & = & X ~\text{choose}~ k & = & \{ k\text{-sets from}~ X \}.\end{matrix}\!</math><br />
|}<br />
<br />
Thus we have:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}\binom{X}{2} & = & \{ \{ u, v \} : u, v \in X \}.\end{matrix}</math><br />
|}<br />
<br />
We may now use the features in <math>\mathrm{d}\mathcal{X} = \{ \mathrm{d}x_i \} = \{ \mathrm{d}x_1, \ldots, \mathrm{d}x_n \}</math> to classify the paths of <math>(\mathbb{B} \to X)</math> by way of the pairs in <math>X^2.\!</math> If <math>X \cong \mathbb{B}^n,</math> then a path <math>q\!</math> in <math>X\!</math> has the following form:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
q : (\mathbb{B} \to \mathbb{B}^n) & \cong & \mathbb{B}^n \times \mathbb{B}^n & \cong & \mathbb{B}^{2n} & \cong & (\mathbb{B}^2)^n.<br />
\end{matrix}</math><br />
|}<br />
<br />
Intuitively, we want to map this <math>(\mathbb{B}^2)^n</math> onto <math>\mathbb{D}^n</math> by mapping each component <math>\mathbb{B}^2</math> onto a copy of <math>\mathbb{D}.</math> But in the presenting context <math>{}^{\backprime\backprime} \mathbb{D} {}^{\prime\prime}</math> is just a name associated with, or an incidental quality attributed to, coefficient values in <math>\mathbb{B}</math> when they are attached to features in <math>\mathrm{d}\mathcal{X}.</math><br />
<br />
Taking these intentions into account, define <math>\mathrm{d}x_i : X^2 \to \mathbb{B}</math> in the following manner:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcrcl}<br />
\mathrm{d}x_i(u, v)<br />
& = & \texttt{(} ~ x_i(u) & \texttt{,} & x_i(v) ~ \texttt{)}<br />
\\<br />
& = & x_i(u) & + & x_i(v)<br />
\\<br />
& = & x_i(v) & - & x_i(u).<br />
\end{array}</math><br />
|}<br />
<br />
In the above transcription, the operator bracket of the form <math>\texttt{(} \ldots \texttt{,} \ldots \texttt{)}\!</math> is a ''cactus lobe'', in general signifying that just one of the arguments listed is false. In the case of two arguments this is the same thing as saying that the arguments are not equal. The plus sign signifies boolean addition, in the sense of addition in <math>\mathrm{GF}(2),</math> and thus means the same thing in this context as the minus sign, in the sense of adding the additive inverse.<br />
<br />
The above definition of <math>\mathrm{d}x_i : X^2 \to \mathbb{B}</math> is equivalent to defining <math>\mathrm{d}x_i : (\mathbb{B} \to X) \to \mathbb{B}\!</math> in the following way:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcrcl}<br />
\mathrm{d}x_i (q)<br />
& = & \texttt{(} ~ x_i(q_0) & \texttt{,} & x_i(q_1) ~ \texttt{)}<br />
\\<br />
& = & x_i(q_0) & + & x_i(q_1)<br />
\\<br />
& = & x_i(q_1) & - & x_i(q_0).<br />
\end{array}</math><br />
|}<br />
<br />
In this definition <math>q_b = q(b),\!</math> for each <math>b\!</math> in <math>\mathbb{B}.</math> Thus, the proposition <math>\mathrm{d}x_i</math> is true of the path <math>q = (u, v)\!</math> exactly if the terms of <math>q,\!</math> the endpoints <math>u\!</math> and <math>v,\!</math> lie on different sides of the question <math>x_i.\!</math><br />
<br />
The language of features in <math>\langle \mathrm{d}\mathcal{X} \rangle,</math> indeed the whole calculus of propositions in <math>[\mathrm{d}\mathcal{X}],</math> may now be used to classify paths and sets of paths. In other words, the paths can be taken as models of the propositions <math>g : \mathrm{d}X \to \mathbb{B}.</math> For example, the paths corresponding to <math>\mathrm{diag}(X)</math> fall under the description <math>\texttt{(} \mathrm{d}x_1 \texttt{)} \cdots \texttt{(} \mathrm{d}x_n \texttt{)},\!</math> which says that nothing changes against the backdrop of the coordinate frame <math>\{ x_1, \ldots, x_n \}.\!</math><br />
<br />
Finally, a few words of explanation may be in order. If this concept of a path appears to be described in a roundabout fashion, it is because I am trying to avoid using any assumption of vector space properties for the space <math>X\!</math> that contains its range. In many ways the treatment is still unsatisfactory, but improvements will have to wait for the introduction of substitution operators acting on singular propositions.<br />
<br />
===The Extended Universe of Discourse===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
At the moment of speaking, I would like to have perceived a nameless voice, long preceding me, leaving me merely to enmesh myself in it, taking up its cadence, and to lodge myself, when no one was looking, in its interstices as if it had paused an instant, in suspense, to beckon to me.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
Next we define the ''extended alphabet'' or ''bundled alphabet'' <math>\mathrm{E}\mathcal{A}</math> as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lclcl}<br />
\mathrm{E}\mathcal{A}<br />
& = & \mathcal{A} \cup \mathrm{d}\mathcal{A}<br />
& = & \{ a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}.<br />
\end{array}</math><br />
|}<br />
<br />
This supplies enough material to construct the ''differential extension'' <math>\mathrm{E}A,</math> or the ''tangent bundle'' over the initial space <math>A,\!</math> in the following fashion:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcl}<br />
\mathrm{E}A<br />
& = & \langle \mathrm{E}\mathcal{A} \rangle<br />
\\[4pt]<br />
& = & \langle \mathcal{A} \cup \mathrm{d}\mathcal{A} \rangle<br />
\\[4pt]<br />
& = & \langle a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle,<br />
\end{array}</math><br />
|}<br />
<br />
and also:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcl}<br />
\mathrm{E}A<br />
& = & A \times \mathrm{d}A<br />
\\[4pt]<br />
& = & A_1 \times \ldots \times A_n \times \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n.<br />
\end{array}</math><br />
|}<br />
<br />
This gives <math>\mathrm{E}A</math> the type <math>\mathbb{B}^n \times \mathbb{D}^n.</math><br />
<br />
Finally, the tangent universe <math>\mathrm{E}A^\bullet = [\mathrm{E}\mathcal{A}]\!</math> is constituted from the totality of points and maps, or interpretations and propositions, that are based on the extended set of features <math>\mathrm{E}\mathcal{A},</math> and this fact is summed up in the following notation:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lclcl}<br />
\mathrm{E}A^\bullet<br />
& = & [\mathrm{E}\mathcal{A}]<br />
& = & [a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n].<br />
\end{array}</math><br />
|}<br />
<br />
This gives the tangent universe <math>\mathrm{E}A^\bullet\!</math> the type:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcl}<br />
(\mathbb{B}^n \times \mathbb{D}^n\ +\!\to \mathbb{B})<br />
& = & (\mathbb{B}^n \times \mathbb{D}^n, (\mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B})).<br />
\end{array}</math><br />
|}<br />
<br />
A proposition in the tangent universe <math>[\mathrm{E}\mathcal{A}]</math> is called a ''differential proposition'' and forms the analogue of a system of differential equations, constraints, or relations in ordinary calculus.<br />
<br />
With these constructions, the differential extension <math>\mathrm{E}A</math> and the space of differential propositions <math>(\mathrm{E}A \to \mathbb{B}),\!</math> we have arrived, in main outline, at one of the major subgoals of this study. Table&nbsp;8 summarizes the concepts that have been introduced for working with differentially extended universes of discourse.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 8.} ~~ \text{Differential Extension : Basic Notation}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Symbol}\!</math><br />
| <math>\text{Notation}\!</math><br />
| <math>\text{Description}\!</math><br />
| <math>\text{Type}\!</math><br />
|-<br />
| <math>\mathrm{d}\mathfrak{A}\!</math><br />
| <math>\{ {}^{\backprime\backprime} \mathrm{d}a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} \mathrm{d}a_n {}^{\prime\prime} \}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Alphabet of}<br />
\\[2pt]<br />
\text{differential symbols}<br />
\end{matrix}</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>\mathrm{d}\mathcal{A}\!</math><br />
| <math>\{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Basis of}<br />
\\[2pt]<br />
\text{differential features}<br />
\end{matrix}</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>\mathrm{d}A_i\!</math><br />
| <math>\{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \}\!</math><br />
| <math>\text{Differential dimension}~ i\!</math><br />
| <math>\mathbb{D}\!</math><br />
|-<br />
| <math>\mathrm{d}A\!</math><br />
|<br />
<math>\begin{matrix}<br />
\langle \mathrm{d}\mathcal{A} \rangle<br />
\\[2pt]<br />
\langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle<br />
\\[2pt]<br />
\{ (\mathrm{d}a_1, \ldots, \mathrm{d}a_n) \}<br />
\\[2pt]<br />
\mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n<br />
\\[2pt]<br />
\textstyle \prod_i \mathrm{d}A_i<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Tangent space at a point:}<br />
\\[2pt]<br />
\text{Set of changes, motions,}<br />
\\[2pt]<br />
\text{steps, tangent vectors}<br />
\\[2pt]<br />
\text{at a point}<br />
\end{matrix}</math><br />
| <math>\mathbb{D}^n\!</math><br />
|-<br />
| <math>\mathrm{d}A^*\!</math><br />
| <math>(\mathrm{hom} : \mathrm{d}A \to \mathbb{B})\!</math><br />
| <math>\text{Linear functions on}~ \mathrm{d}A\!</math><br />
| <math>(\mathbb{D}^n)^* \cong \mathbb{D}^n\!</math><br />
|-<br />
| <math>\mathrm{d}A^\uparrow\!</math><br />
| <math>(\mathrm{d}A \to \mathbb{B})\!</math><br />
| <math>\text{Boolean functions on}~ \mathrm{d}A\!</math><br />
| <math>\mathbb{D}^n \to \mathbb{B}\!</math><br />
|-<br />
| <math>\mathrm{d}A^\bullet\!</math><br />
|<br />
<math>\begin{matrix}<br />
[\mathrm{d}\mathcal{A}]<br />
\\[2pt]<br />
(\mathrm{d}A, \mathrm{d}A^\uparrow)<br />
\\[2pt]<br />
(\mathrm{d}A ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
(\mathrm{d}A, (\mathrm{d}A \to \mathbb{B}))<br />
\\[2pt]<br />
[\mathrm{d}a_1, \ldots, \mathrm{d}a_n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Tangent universe at a point of}~ A^\bullet,<br />
\\[2pt]<br />
\text{based on the tangent features}<br />
\\[2pt]<br />
\{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
(\mathbb{D}^n, (\mathbb{D}^n \to \mathbb{B}))<br />
\\[2pt]<br />
(\mathbb{D}^n ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
[\mathbb{D}^n]<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
The adjectives ''differential'' or ''tangent'' are systematically attached to every construct based on the differential alphabet <math>\mathrm{d}\mathfrak{A},</math> taken by itself. Strictly speaking, we probably ought to call <math>\mathrm{d}\mathcal{A}</math> the set of ''cotangent'' features derived from <math>\mathcal{A},</math> but the only time this distinction really seems to matter is when we need to distinguish the tangent vectors as maps of type <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math> from cotangent vectors as elements of type <math>\mathbb{D}^n.</math> In like fashion, having defined <math>\mathrm{E}\mathcal{A} = \mathcal{A} \cup \mathrm{d}\mathcal{A},</math> we can systematically attach the adjective ''extended'' or the substantive ''bundle'' to all of the constructs associated with this full complement of <math>{2n}\!</math> features.<br />
<br />
It eventually becomes necessary to extend the initial alphabet even further, to allow for the discussion of higher order differential expressions. Table&nbsp;9 provides a suggestion of how these further extensions can be carried out.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 9.} ~~ \text{Higher Order Differential Features}\!</math><br />
|<br />
<p><math>\begin{array}{lllll}<br />
\mathrm{d}^0 \mathcal{A} & = & \{ a_1, \ldots, a_n \} & = & \mathcal{A}<br />
\\<br />
\mathrm{d}^1 \mathcal{A} & = & \{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \} & = & \mathrm{d}\mathcal{A}<br />
\end{array}</math></p><br />
<br />
<p><math>\begin{array}{lll}<br />
\mathrm{d}^k \mathcal{A} & = & \{ \mathrm{d}^k a_1, \ldots, \mathrm{d}^k a_n \}<br />
\\<br />
\mathrm{d}^* \mathcal{A} & = & \{ \mathrm{d}^0 \mathcal{A}, \ldots, \mathrm{d}^k \mathcal{A}, \ldots \}<br />
\end{array}</math></p><br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}^0 \mathcal{A} & = & \mathrm{d}^0 \mathcal{A}<br />
\\<br />
\mathrm{E}^1 \mathcal{A} & = & \mathrm{d}^0 \mathcal{A} ~\cup~ \mathrm{d}^1 \mathcal{A}<br />
\\<br />
\mathrm{E}^k \mathcal{A} & = & \mathrm{d}^0 \mathcal{A} ~\cup~ \ldots ~\cup~ \mathrm{d}^k \mathcal{A}<br />
\\<br />
\mathrm{E}^\infty \mathcal{A} & = & \bigcup~ \mathrm{d}^* \mathcal{A}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
===Intentional Propositions===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Do you guess I have some intricate purpose?<br><br />
Well I have . . . . for the April rain has, and the mica on<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the side of a rock has.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 45]<br />
|}<br />
<br />
In order to analyze the behavior of a system at successive moments in time, while staying within the limitations of propositional logic, it is necessary to create independent alphabets of logical features for each moment of time that we contemplate using in our discussion. These moments have reference to typical instances and relative intervals, not actual or absolute times. For example, to discuss ''velocities'' (first order rates of change) we need to consider points of time in pairs. There are a number of natural ways of doing this. Given an initial alphabet, we could use its symbols as a lexical basis to generate successive alphabets of compound symbols, say, with temporal markers appended as suffixes.<br />
<br />
As a standard way of dealing with these situations, the following scheme of notation suggests a way of extending any alphabet of logical features through as many temporal moments as a particular order of analysis may demand. The lexical operators <math>\mathrm{p}^k</math> and <math>\mathrm{Q}^k</math> are convenient in many contexts where the accumulation of prime symbols and union symbols would otherwise be cumbersome.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 10.} ~~ \text{A Realm of Intentional Features}\!</math><br />
|<br />
<p><math>\begin{array}{lllll}<br />
\mathrm{p}^0 \mathcal{A} & = & \{ a_1, \ldots, a_n \} & = & \mathcal{A}<br />
\\<br />
\mathrm{p}^1 \mathcal{A} & = & \{ a_1^\prime, \ldots, a_n^\prime \} & = & \mathcal{A}^\prime<br />
\\<br />
\mathrm{p}^2 \mathcal{A} & = & \{ a_1^{\prime\prime}, \ldots, a_n^{\prime\prime} \} & = & \mathcal{A}^{\prime\prime}<br />
\\<br />
\cdots & & \cdots &<br />
\end{array}</math></p><br />
<br />
<p><math>\begin{array}{lll}<br />
\mathrm{p}^k \mathcal{A} & = & \{ \mathrm{p}^k a_1, \ldots, \mathrm{p}^k a_n \}<br />
\end{array}</math></p><br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{Q}^0 \mathcal{A} & = & \mathcal{A}<br />
\\<br />
\mathrm{Q}^1 \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}'<br />
\\<br />
\mathrm{Q}^2 \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}' \cup \mathcal{A}''<br />
\\<br />
\cdots & & \cdots<br />
\\<br />
\mathrm{Q}^k \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}' \cup \ldots \cup \mathrm{p}^k \mathcal{A}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
The resulting augmentations of our logical basis determine a series of discursive universes that may be called the ''intentional extension'' of propositional calculus. This extension follows a pattern analogous to the differential extension, which was developed in terms of the operators <math>\mathrm{d}^k</math> and <math>\mathrm{E}^k,</math> and there is a natural relation between these two extensions that bears further examination. In contexts displaying this pattern, where a sequence of domains stretches from an anchoring domain <math>X\!</math> through an indefinite number of higher reaches, a particular collection of domains based on <math>X\!</math> will be referred to as a ''realm'' of <math>X,\!</math> and when the succession exhibits a temporal aspect, as a ''reign'' of <math>X.\!</math><br />
<br />
For the purposes of this discussion, an ''intentional proposition'' is defined as a proposition in the universe of discourse <math>\mathrm{Q}X^\bullet = [\mathrm{Q}\mathcal{X}],</math> in other words, a map <math>q : \mathrm{Q}X \to \mathbb{B}.</math> The sense of this definition may be seen if we consider the following facts. First, the equivalence <math>\mathrm{Q}X = X \times X'</math> motivates the following chain of isomorphisms between spaces:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lllcl}<br />
(\mathrm{Q}X \to \mathbb{B})<br />
& \cong & (X & \times & ~X' \to \mathbb{B})<br />
\\[4pt]<br />
& \cong & (X & \to & (X' \to \mathbb{B}))<br />
\\[4pt]<br />
& \cong & (X' & \to & (X~ \to \mathbb{B})).<br />
\end{array}</math><br />
|}<br />
<br />
Viewed in this light, an intentional proposition <math>q\!</math> may be rephrased as a map <math>q : X \times X' \to \mathbb{B},</math> which judges the juxtaposition of states in <math>X\!</math> from one moment to the next. Alternatively, <math>q\!</math> may be parsed in two stages in two different ways, as <math>q : X \to (X' \to \mathbb{B})</math> and as <math>q : X' \to (X \to \mathbb{B}),</math> which associate to each point of <math>X\!</math> or <math>X'\!</math> a proposition about states in <math>X'\!</math> or <math>X,\!</math> respectively. In this way, an intentional proposition embodies a type of value system, in effect, a proposal that places a value on a collection of ends-in-view, or a project that evaluates a set of goals as regarded from each point of view in the state space of a system.<br />
<br />
In sum, the intentional proposition <math>q\!</math> indicates a method for the systematic selection of local goals. As a general form of description, a map of the type <math>q : \mathrm{Q}^i X \to \mathbb{B}\!</math> may be referred to as an "<math>i^\text{th}</math> order intentional proposition". Naturally, when we speak of intentional propositions without qualification, we usually mean first order intentions.<br />
<br />
Many different realms of discourse have the same structure as the extensions that have been indicated here. From a strictly logical point of view, each new layer of terms is composed of independent logical variables that are no different in kind from those that go before, and each further course of logical atoms is treated like so many individual, but otherwise indifferent bricks by the prototype computer program that I use as a propositional interpreter. Thus, the names that I use to single out the differential and the intentional extensions, and the lexical paradigms that I follow to construct them, are meant to suggest the interpretations that I have in mind, but they can only hint at the extra meanings that human communicators may pack into their terms and inflections.<br />
<br />
As applied here, the word ''intentional'' is drawn from common use and may have little bearing on its technical use in other, more properly philosophical, contexts. I am merely using the complex of intentional concepts &mdash; aims, ends, goals, objectives, purposes, and so on &mdash; metaphorically to flesh out and vividly to represent any situation where one needs to contemplate a system in multiple aspects of state and destination, that is, its being in certain states and at the same time acting as if headed through certain states. If confusion arises, more neutral words like ''conative'', ''contingent'', ''discretionary'', ''experimental'', ''kinetic'', ''progressive'', ''tentative'', or ''trial'' would probably serve as well.<br />
<br />
===Life on Easy Street===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Failing to fetch me at first keep encouraged,<br><br />
Missing me one place search another,<br><br />
I stop some where waiting for you<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]<br />
|}<br />
<br />
The finite character of the extended universe <math>[\mathrm{E}\mathcal{A}]</math> makes the problem of solving differential propositions relatively straightforward, at least, in principle. The solution set of the differential proposition <math>q : \mathrm{E}A \to \mathbb{B}</math> is the set of models <math>q^{-1}(1)\!</math> in <math>\mathrm{E}A.</math> Finding all the models of <math>q,\!</math> the extended interpretations in <math>\mathrm{E}A</math> that satisfy <math>q,\!</math> can be carried out by a finite search. Being in possession of complete algorithms for propositional calculus modeling, theorem checking, or theorem proving makes the analytic task fairly simple in principle, though the question of efficiency in the face of arbitrary complexity may always remain another matter entirely. While the fact that propositional satisfiability is NP-complete may be discouraging for the prospects of a single efficient algorithm that covers the whole space <math>[\mathrm{E}\mathcal{A}]</math> with equal facility, there appears to be much room for improvement in classifying special forms and in developing algorithms that are tailored to their practical processing.<br />
<br />
In view of these constraints and contingencies, our focus shifts to the tasks of approximation and interpretation that support intuition, especially in dealing with the natural kinds of differential propositions that arise in applications, and in the effort to understand, in succinct and adaptive forms, their dynamic implications. In the absence of direct insights, these tasks are partially carried out by forging analogies with the familiar situations and customary routines of ordinary calculus. But the indirect approach, going by way of specious analogy and intuitive habit, forces us to remain on guard against the circumstance that occurs when the word ''forging'' takes on its shadier nuance, indicting the constant risk of a counterfeit in the proportion.<br />
<br />
==Back to the Beginning : Exemplary Universes==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
I would have preferred to be enveloped in words, borne way beyond all possible beginnings.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
To anchor our understanding of differential logic, let us look at how the various concepts apply in the simplest possible concrete cases, where the initial dimension is only 1 or 2. In spite of the obvious simplicity of these cases, it is possible to observe how central difficulties of the subject begin to arise already at this stage.<br />
<br />
===A One-Dimensional Universe===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
There was never any more inception than there is now,<br><br />
Nor any more youth or age than there is now;<br><br />
And will never be any more perfection than there is now,<br><br />
Nor any more heaven or hell than there is now.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]<br />
|}<br />
<br />
Let <math>\mathcal{X} = \{ x_1 \} = \{ A \}</math> be an alphabet that represents one boolean variable or a single logical feature. In this example the capital letter <math>{}^{\backprime\backprime} A {}^{\prime\prime}\!</math> is used usual informally, to name a feature and not a space, in departure from our formerly stated formal conventions. At any rate, the basis element <math>A = x_1\!</math> may be interpreted as a simple proposition or a coordinate projection <math>A = x_1 : \mathbb{B} \xrightarrow{i} \mathbb{B}.</math> The space <math>X = \langle A \rangle = \{ \texttt{(} A \texttt{)}, A \}</math> of points (cells, vectors, interpretations) has cardinality <math>2^n = 2^1 = 2\!</math> and is isomorphic to <math>\mathbb{B} = \{ 0, 1 \}.</math> Moreover, <math>X\!</math> may be identified with the set of singular propositions <math>\{ x : \mathbb{B} \xrightarrow{s} \mathbb{B} \}.</math> The space of linear propositions <math>X^* = \{ \mathrm{hom} : \mathbb{B} \xrightarrow{\ell} \mathbb{B} \} = \{ 0, A \}</math> is algebraically dual to <math>X\!</math> and also has cardinality <math>2.\!</math> Here, <math>{}^{\backprime\backprime} 0 {}^{\prime\prime}\!</math> is interpreted as denoting the constant function <math>0 : \mathbb{B} \to \mathbb{B},</math> amounting to the linear proposition of rank <math>0,\!</math> while <math>A\!</math> is the linear proposition of rank <math>1.\!</math> Last but not least we have the positive propositions <math>\{ \mathrm{pos} : \mathbb{B} \xrightarrow{p} \mathbb{B} \} = \{ A, 1 \},\!</math> of rank <math>1\!</math> and <math>0,\!</math> respectively, where <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}\!</math> is understood as denoting the constant function <math>1 : \mathbb{B} \to \mathbb{B}.</math> In sum, there are <math>2^{2^n} = 2^{2^1} = 4</math> propositions altogether in the universe of discourse, comprising the set <math>X^\uparrow = \{ f : X \to \mathbb{B} \} = \{ 0, \texttt{(} A \texttt{)}, A, 1 \} \cong (\mathbb{B} \to \mathbb{B}).</math><br />
<br />
The first order differential extension of <math>\mathcal{X}</math> is <math>\mathrm{E}\mathcal{X} = \{ x_1, \mathrm{d}x_1 \} = \{ A, \mathrm{d}A \}.</math> If the feature <math>A\!</math> is understood as applying to some object or state, then the feature <math>\mathrm{d}A</math> may be interpreted as an attribute of the same object or state that says that it is changing ''significantly'' with respect to the property <math>A,\!</math> or that it has an ''escape velocity'' with respect to the state <math>A.\!</math> In practice, differential features acquire their logical meaning through a class of ''temporal inference rules''.<br />
<br />
For example, relative to a frame of observation that is left implicit for now, one is permitted to make the following sorts of inference: From the fact that <math>A\!</math> and <math>\mathrm{d}A</math> are true at a given moment one may infer that <math>\texttt{(} A \texttt{)}\!</math> will be true in the next moment of observation. Altogether in the present instance, there is the fourfold scheme of inference that is shown below:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:50%"<br />
|<br />
<math>\begin{matrix}<br />
\text{From} & \texttt{(} A \texttt{)}<br />
& \text{and} & \texttt{(} \mathrm{d}A \texttt{)}<br />
& \text{infer} & \texttt{(} A \texttt{)} & \text{next.}<br />
\\[8pt]<br />
\text{From} & \texttt{(} A \texttt{)}<br />
& \text{and} & \mathrm{d}A<br />
& \text{infer} & A & \text{next.}<br />
\\[8pt]<br />
\text{From} & A<br />
& \text{and} & \texttt{(} \mathrm{d}A \texttt{)}<br />
& \text{infer} & A & \text{next.}<br />
\\[8pt]<br />
\text{From} & A<br />
& \text{and} & \mathrm{d}A<br />
& \text{infer} & \texttt{(} A \texttt{)} & \text{next.}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
It might be thought that an independent time variable needs to be brought in at this point, but it is an insight of fundamental importance that the idea of process is logically prior to the notion of time. A time variable is a reference to a ''clock'' &mdash; a canonical, conventional process that is accepted or established as a standard of measurement, but in essence no different than any other process. This raises the question of how different subsystems in a more global process can be brought into comparison, and what it means for one process to serve the function of a local standard for others. But these inquiries only wrap up puzzles in further riddles, and are obviously too involved to be handled at our current level of approximation.<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
The clock indicates the moment . . . . but what does<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;eternity indicate?<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 79]<br />
|}<br />
<br />
Observe that the secular inference rules, used by themselves, involve a loss of information, since nothing in them can tell us whether the momenta <math>\{ \texttt{(} \mathrm{d}A \texttt{)}, \mathrm{d}A \}\!</math> are changed or unchanged in the next instance. In order to know this, one would have to determine <math>\mathrm{d}^2 A,\!</math> and so on, pursuing an infinite regress. Ultimately, in order to rest with a finitely determinate system, it is necessary to make an infinite assumption, for example, that <math>\mathrm{d}^k A = 0\!</math> for all <math>k\!</math> greater than some fixed value <math>M.\!</math> Another way to escape the regress is through the provision of a dynamic law, in typical form making higher order differentials dependent on lower degrees and estates.<br />
<br />
===Example 1. A Square Rigging===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Urge and urge and urge,<br><br />
Always the procreant urge of the world.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]<br />
|}<br />
<br />
By way of example, suppose that we are given the initial condition <math>A = \mathrm{d}A\!</math> and the law <math>\mathrm{d}^2 A = \texttt{(} A \texttt{)}.\!</math> Since the equation <math>A = \mathrm{d}A\!</math> is logically equivalent to the disjunction <math>A ~ \mathrm{d}A ~\text{or}~ \texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)},\!</math> we may infer two possible trajectories, as displayed in Table&nbsp;11. In either case the state <math>A ~ \texttt{(} \mathrm{d}A \texttt{)(} \mathrm{d}^2 A \texttt{)}\!</math> is a stable attractor or a terminal condition for both starting points.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 11.} ~~ \text{A Pair of Commodious Trajectories}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Time}\!</math><br />
| <math>\text{Trajectory 1}\!</math><br />
| <math>\text{Trajectory 2}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
2<br />
\\[4pt]<br />
3<br />
\\[4pt]<br />
4<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A & \mathrm{d}A & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
\texttt{(} A \texttt{)} & \mathrm{d}A & \mathrm{d}^2 A<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
{}^{\shortparallel} & {}^{\shortparallel} & {}^{\shortparallel}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{)} & \texttt{(} \mathrm{d}A \texttt{)} & \mathrm{d}^2 A<br />
\\[4pt]<br />
\texttt{(} A \texttt{)} & \mathrm{d}A & \mathrm{d}^2 A<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
{}^{\shortparallel} & {}^{\shortparallel} & {}^{\shortparallel}<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Because the initial space <math>X = \langle A \rangle\!</math> is one-dimensional, we can easily fit the second order extension <math>\mathrm{E}^2 X = \langle A, \mathrm{d}A, \mathrm{d}^2 A \rangle\!</math> within the compass of a single venn diagram, charting the couple of converging trajectories as shown in Figure&nbsp;12.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 12 -- The Anchor.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 12.} ~~ \text{The Anchor}\!</math><br />
|}<br />
<br />
If we eliminate from view the regions of <math>\mathrm{E}^2 X\!</math> that are ruled out by the dynamic law <math>\mathrm{d}^2 A = \texttt{(} A \texttt{)},\!</math> then what remains is the quotient structure that is shown in Figure&nbsp;13. This picture makes it easy to see that the dynamically allowable portion of the universe is partitioned between the properties <math>A\!</math> and <math>\mathrm{d}^2 A\!.</math> As it happens, this fact might have been expressed &ldquo;right off the bat&rdquo; by an equivalent formulation of the differential law, one that uses the exclusive disjunction to state the law as <math>\texttt{(} A \texttt{,} \mathrm{d}^2 A \texttt{)}\!.</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 13 -- The Tiller.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 13.} ~~ \text{The Tiller}\!</math><br />
|}<br />
<br />
What we have achieved in this example is to give a differential description of a simple dynamic process. In effect, we did this by embedding a directed graph, which can be taken to represent the state transitions of a finite automaton, in a dynamically allotted quotient structure that is created from a boolean lattice or an <math>n\!</math>-cube by nullifying all of the regions that the dynamics outlaws. With growth in the dimensions of our contemplated universes, it becomes essential, both for human comprehension and for computer implementation, that the dynamic structures of interest to us be represented not actually, by acquaintance, but virtually, by description. In our present study, we are using the language of propositional calculus to express the relevant descriptions, and to comprehend the structure that is implicit in the subsets of a <math>n\!</math>-cube without necessarily being forced to actualize all of its points.<br />
<br />
One of the reasons for engaging in this kind of extremely reduced, but explicitly controlled case study is to throw light on the general study of languages, formal and natural, in their full array of syntactic, semantic, and pragmatic aspects. Propositional calculus is one of the last points of departure where we can view these three aspects interacting in a non-trivial way without being immediately and totally overwhelmed by the complexity they generate. Often this complexity causes investigators of formal and natural languages to adopt the strategy of focusing on a single aspect and to abandon all hope of understanding the whole, whether it's the still living natural language or the dynamics of inquiry that lies crystallized in formal logic.<br />
<br />
From the perspective that I find most useful here, a language is a syntactic system that is designed or evolved in part to express a set of descriptions. When the explicit symbols of a language have extensions in its object world that are actually infinite, or when the implicit categories and generative devices of a linguistic theory have extensions in its subject matter that are potentially infinite, then the finite characters of terms, statements, arguments, grammars, logics, and rhetorics force an excess of intension to reside in all these symbols and functions, across the spectrum from the object language to the metalinguistic uses. In the aphorism from W. von Humboldt that Chomsky often cites, for example, in [Cho86, 30] and [Cho93, 49], language requires &ldquo;the infinite use of finite means&rdquo;. This is necessarily true when the extensions are infinite, when the referential symbols and grammatical categories of a language possess infinite sets of models and instances. But it also voices a practical truth when the extensions, though finite at every stage, tend to grow at exponential rates.<br />
<br />
This consequence of dealing with extensions that are &ldquo;practically infinite&rdquo; becomes crucial when one tries to build neural network systems that learn, since the learning competence of any intelligent system is limited to the objects and domains that it is able to represent. If we want to design systems that operate intelligently with the full deck of propositions dealt by intact universes of discourse, then we must supply them with succinct representations and efficient transformations in this domain. Furthermore, in the project of constructing inquiry driven systems, we find ourselves forced to contemplate the level of generality that is embodied in propositions, because the dynamic evolution of these systems is driven by the measurable discrepancies that occur among their expectations, intentions, and observations, and because each of these subsystems or components of knowledge constitutes a propositional modality that can take on the fully generic character of an empirical summary or an axiomatic theory.<br />
<br />
A compression scheme by any other name is a symbolic representation, and this is what the differential extension of propositional calculus, through all of its many universes of discourse, is intended to supply. Why is this particular program of mental calisthenics worth carrying out in general? By providing a uniform logical medium for describing dynamic systems we can make the task of understanding complex systems much easier, both in looking for invariant representations of individual cases and in finding points of comparison among diverse structures that would otherwise appear as isolated systems. All of this goes to facilitate the search for compact knowledge and to adapt what is learned from individual cases to the general realm.<br />
<br />
===Back to the Feature===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
I guess it must be the flag of my disposition, out of hopeful<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;green stuff woven.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 31]<br />
|}<br />
<br />
Let us assume that the sense intended for differential features is well enough established in the intuition, for now, that we may continue with outlining the structure of the differential extension <math>[\mathrm{E}\mathcal{X}] = [A, \mathrm{d}A].\!</math> Over the extended alphabet <math>\mathrm{E}\mathcal{X} = \{ x_1, \mathrm{d}x_1 \} = \{ A, \mathrm{d}A \}\!</math> of cardinality <math>2^n = 2\!</math> we generate the set of points <math>\mathrm{E}X\!</math> of cardinality <math>2^{2n} = 4\!</math> that bears the following chain of equivalent descriptions:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}X & = & \langle A, \mathrm{d}A \rangle<br />
\\[4pt]<br />
& = & \{ \texttt{(} A \texttt{)}, A \} ~\times~ \{ \texttt{(} \mathrm{d}A \texttt{)}, \mathrm{d}A \}<br />
\\[4pt]<br />
& = &<br />
\{<br />
\texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)},~<br />
\texttt{(} A \texttt{)} \mathrm{d}A,~<br />
A \texttt{(} \mathrm{d}A \texttt{)},~<br />
A ~ \mathrm{d}A<br />
\}.<br />
\end{array}</math><br />
|}<br />
<br />
The space <math>\mathrm{E}X\!</math> may be assigned the mnemonic type <math>\mathbb{B} \times \mathbb{D},\!</math> which is really no different than <math>\mathbb{B} \times \mathbb{B} = \mathbb{B}^2.\!</math> An individual element of <math>\mathrm{E}X\!</math> may be regarded as a ''disposition at a point'' or a ''situated direction'', in effect, a singular mode of change occurring at a single point in the universe of discourse. In applications, the modality of this change can be interpreted in various ways, for example, as an expectation, an intention, or an observation with respect to the behavior of a system.<br />
<br />
To complete the construction of the extended universe of discourse <math>\mathrm{E}X^\bullet = [x_1, \mathrm{d}x_1] = [A, \mathrm{d}A]\!</math> one must add the set of differential propositions <math>\mathrm{E}X^\uparrow = \{ g : \mathrm{E}X \to \mathbb{B} \} \cong (\mathbb{B} \times \mathbb{D} \to \mathbb{B})\!</math> to the set of dispositions in <math>\mathrm{E}X.\!</math> There are <math>2^{2^{2n}} = 16\!</math> propositions in <math>\mathrm{E}X^\uparrow,\!</math> as detailed in Table&nbsp;14.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 14.} ~~ \text{Differential Propositions}\!</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>A\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>\mathrm{d}A\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>g_{0}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{1}<br />
\\[4pt]<br />
g_{2}<br />
\\[4pt]<br />
g_{4}<br />
\\[4pt]<br />
g_{8}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
1~0~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
\texttt{(} A \texttt{)} ~ \mathrm{d}A ~<br />
\\[4pt]<br />
~ A ~ \texttt{(} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
~ A ~~ \mathrm{d}A ~<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{neither}~ A ~\text{nor}~ \mathrm{d}A<br />
\\[4pt]<br />
\mathrm{d}A ~\text{and not}~ A<br />
\\[4pt]<br />
A ~\text{and not}~ \mathrm{d}A<br />
\\[4pt]<br />
A ~\text{and}~ \mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot A \land \lnot \mathrm{d}A<br />
\\[4pt]<br />
\lnot A \land \mathrm{d}A<br />
\\[4pt]<br />
A \land \lnot \mathrm{d}A<br />
\\[4pt]<br />
A \land \mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
g_{3}<br />
\\[4pt]<br />
g_{12}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{)}<br />
\\[4pt]<br />
A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ A<br />
\\[4pt]<br />
A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot A<br />
\\[4pt]<br />
A<br />
\end{matrix}\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{6}<br />
\\[4pt]<br />
g_{9}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[4pt]<br />
1~0~0~1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{,} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
\texttt{((} A \texttt{,} \mathrm{d}A \texttt{))}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
A ~\text{not equal to}~ \mathrm{d}A<br />
\\[4pt]<br />
A ~\text{equal to}~ \mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
A \ne \mathrm{d}A<br />
\\[4pt]<br />
A = \mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{5}<br />
\\[4pt]<br />
g_{10}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
\mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ \mathrm{d}A<br />
\\[4pt]<br />
\mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot \mathrm{d}A<br />
\\[4pt]<br />
\mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{7}<br />
\\[4pt]<br />
g_{11}<br />
\\[4pt]<br />
g_{13}<br />
\\[4pt]<br />
g_{14}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} ~ A ~~ \mathrm{d}A ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{(} ~ A ~ \texttt{(} \mathrm{d}A \texttt{))}<br />
\\[4pt]<br />
\texttt{((} A \texttt{)} ~ \mathrm{d}A ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{((} A \texttt{)(} \mathrm{d}A \texttt{))}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not both}~ A ~\text{and}~ \mathrm{d}A<br />
\\[4pt]<br />
\text{not}~ A ~\text{without}~ \mathrm{d}A<br />
\\[4pt]<br />
\text{not}~ \mathrm{d}A ~\text{without}~ A<br />
\\[4pt]<br />
A ~\text{or}~ \mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot A \lor \lnot \mathrm{d}A<br />
\\[4pt]<br />
A \Rightarrow \mathrm{d}A<br />
\\[4pt]<br />
A \Leftarrow \mathrm{d}A<br />
\\[4pt]<br />
A \lor \mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
| <math>f_{3}\!</math><br />
| <math>g_{15}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
Aside from changing the names of variables and shuffling the order of rows, this Table follows the format that was used previously for boolean functions of two variables. The rows are grouped to reflect natural similarity classes among the propositions. In a future discussion, these classes will be given additional explanation and motivation as the orbits of a certain transformation group acting on the set of 16 propositions. Notice that four of the propositions, in their logical expressions, resemble those given in the table for <math>X^\uparrow.\!</math> Thus the first set of propositions <math>\{ f_i \}\!</math> is automatically embedded in the present set <math>\{ g_j \}\!</math> and the corresponding inclusions are indicated at the far left margin of the Table.<br />
<br />
===Tacit Extensions===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
I would really like to have slipped imperceptibly into this lecture, as into all the others I shall be delivering, perhaps over the years ahead.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
Strictly speaking, however, there is a subtle distinction in type between the function <math>f_i : X \to \mathbb{B}</math> and the corresponding function <math>g_j : \mathrm{E}X \to \mathbb{B},</math> even though they share the same logical expression. Naturally, we want to maintain the logical equivalence of expressions that represent the same proposition while appreciating the full diversity of that proposition's functional and typical representatives. Both perspectives, and all the levels of abstraction extending through them, have their reasons, as will develop in time.<br />
<br />
Because this special circumstance points up an important general theme, it is a good idea to discuss it more carefully. Whenever there arises a situation like this, where one alphabet <math>\mathcal{X}</math> is a subset of another alphabet <math>\mathcal{Y},</math> then we say that any proposition <math>f : \langle \mathcal{X} \rangle \to \mathbb{B}</math> has a ''tacit extension'' to a proposition <math>\boldsymbol\varepsilon f : \langle \mathcal{Y} \rangle \to \mathbb{B},\!</math> and that the space <math>(\langle \mathcal{X} \rangle \to \mathbb{B})</math> has an ''automatic embedding'' within the space <math>(\langle \mathcal{Y} \rangle \to \mathbb{B}).</math> The extension is defined in such a way that <math>\boldsymbol\varepsilon f\!</math> puts the same constraint on the variables of <math>\mathcal{X}</math> that are contained in <math>\mathcal{Y}</math> as the proposition <math>f\!</math> initially did, while it puts no constraint on the variables of <math>\mathcal{Y}</math> outside of <math>\mathcal{X},</math> in effect, conjoining the two constraints.<br />
<br />
If the variables in question are indexed as <math>\mathcal{X} = \{ x_1, \ldots, x_n \}</math> and <math>\mathcal{Y} = \{ x_1, \ldots, x_n, \ldots, x_{n+k} \},</math> then the definition of the tacit extension from <math>\mathcal{X}</math> to <math>\mathcal{Y}</math> may be expressed in the form of an equation:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\boldsymbol\varepsilon f(x_1, \ldots, x_n, \ldots, x_{n+k}) ~=~ f(x_1, \ldots, x_n).\!</math><br />
|}<br />
<br />
On formal occasions, such as the present context of definition, the tacit extension from <math>\mathcal{X}</math> to <math>\mathcal{Y}</math> is explicitly symbolized by the operator <math>\boldsymbol\varepsilon : (\langle \mathcal{X} \rangle \to \mathbb{B}) \to (\langle \mathcal{Y} \rangle \to \mathbb{B}),</math> where the appropriate alphabets <math>\mathcal{X}</math> and <math>\mathcal{Y}</math> are understood from context, but normally one may leave the "<math>\boldsymbol\varepsilon\!</math>" silent.<br />
<br />
Let's explore what this means for the present Example. Here, <math>\mathcal{X} = \{ A \}</math> and <math>\mathcal{Y} = \mathrm{E}\mathcal{X} = \{ A, \mathrm{d}A \}.</math> For each of the propositions <math>f_i\!</math> over <math>X\!,</math> specifically, those whose expression <math>e_i\!</math> lies in the collection <math>\{ 0, \texttt{(} A \texttt{)}, A, 1 \},\!</math> the tacit extension <math>\boldsymbol\varepsilon f\!</math> of <math>f\!</math> to <math>\mathrm{E}X</math> can be phrased as a logical conjunction of two factors, <math>f_i = e_i \cdot \tau ~ ,\!</math> where <math>\tau\!</math> is a logical tautology that uses all the variables of <math>\mathcal{Y} - \mathcal{X}.</math> Working in these terms, the tacit extensions <math>\boldsymbol\varepsilon f\!</math> of <math>f\!</math> to <math>\mathrm{E}X</math> may be explicated as shown in Table&nbsp;15.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:60%"<br />
|+ style="height:30px" | <math>\text{Table 15.} ~~ \text{Tacit Extension of}~ [A] ~\text{to}~ [A, \mathrm{d}A]\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0<br />
& = & 0 & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))} & = & & 0<br />
\\[8pt]<br />
\texttt{(} A \texttt{)}<br />
& = & \texttt{(} A \texttt{)} & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))}<br />
& = & \texttt{(} A \texttt{)} \, \mathrm{d}A ~ & + & \texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)}<br />
\\[8pt]<br />
A<br />
& = & ~A~ & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))}<br />
& = & ~A~ ~\mathrm{d}A~ & + & ~A~ \texttt{(} \mathrm{d}A \texttt{)}<br />
\\[8pt]<br />
1<br />
& = & 1 & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))} & = & & 1<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
In its effect on the singular propositions over <math>X,\!</math> this analysis has an interesting interpretation. The tacit extension takes us from thinking about a particular state, like <math>A\!</math> or <math>\texttt{(} A \texttt{)},\!</math> to considering the collection of outcomes, the outgoing changes or the singular dispositions, that spring from that state.<br />
<br />
===Example 2. Drives and Their Vicissitudes===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
I open my scuttle at night and see the far-sprinkled systems,<br><br />
And all I see, multiplied as high as I can cipher, edge but<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the rim of the farther systems.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 81]<br />
|}<br />
<br />
Before we leave the one-feature case let's look at a more substantial example, one that illustrates a general class of curves that can be charted through the extended feature spaces and that provides an opportunity to discuss a number of important themes concerning their structure and dynamics.<br />
<br />
Again, let <math>\mathcal{X} = \{ x_1 \} = \{ A \}.\!</math> In the discussion that follows we will consider a class of trajectories having the property that <math>\mathrm{d}^k A = 0\!</math> for all <math>k\!</math> greater than some fixed <math>m\!</math> and we may indulge in the use of some picturesque terms that describe salient classes of such curves. Given the finite order condition, there is a highest order non-zero difference <math>\mathrm{d}^m A\!</math> exhibited at each point in the course of any determinate trajectory that one may wish to consider. With respect to any point of the corresponding orbit or curve let us call this highest order differential feature <math>\mathrm{d}^m A\!</math> the ''drive'' at that point. Curves of constant drive <math>\mathrm{d}^m A\!</math> are then referred to as ''<math>m^\text{th}\!</math>-gear curves''.<br />
<br />
* '''Scholium.''' The fact that a difference calculus can be developed for boolean functions is well known [Fuji], [Koh, &sect; 8-4] and was probably familiar to Boole, who was an expert in difference equations before he turned to logic. And of course there is the strange but true story of how the Turin machines of the 1840s prefigured the Turing machines of the 1940s [Men, 225-297]. At the very outset of general purpose, mechanized computing we find that the motive power driving the Analytical Engine of Babbage, the kernel of an idea behind all of his wheels, was exactly his notion that difference operations, suitably trained, can serve as universal joints for any conceivable computation [M&M], [Mel, ch. 4].<br />
<br />
Given this language, the Example we take up here can be described as the family of <math>4^\text{th}\!</math>-gear curves through <math>\mathrm{E}^4 X\!</math> <math>=\!</math> <math>\langle A, ~\mathrm{d}A, ~\mathrm{d}^2\!A, ~\mathrm{d}^3\!A, ~\mathrm{d}^4\!A \rangle.</math> These are the trajectories generated subject to the dynamic law <math>\mathrm{d}^4 A = 1,\!</math> where it is understood in such a statement that all higher order differences are equal to <math>0.\!</math> Since <math>\mathrm{d}^4 A\!</math> and all higher <math>\mathrm{d}^k A\!</math> are fixed, the temporal or transitional conditions (initial, mediate, terminal &mdash; transient or stable states) vary only with respect to their projections as points of <math>\mathrm{E}^3 X = \langle A, ~\mathrm{d}A, ~\mathrm{d}^2\!A, ~\mathrm{d}^3\!A \rangle.</math> Thus, there is just enough space in a planar venn diagram to plot all of these orbits and to show how they partition the points of <math>\mathrm{E}^3 X.\!</math> It turns out that there are exactly two possible orbits, of eight points each, as illustrated in Figure&nbsp;16.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 16 -- A Couple of Fourth Gear Orbits.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 16.} ~~ \text{A Couple of Fourth Gear Orbits}\!</math><br />
|}<br />
<br />
With a little thought it is possible to devise an indexing scheme for the general run of dynamic states that allows for comparing universes of discourse that weigh in on different scales of observation. With this end in sight, let us index the states <math>q \in \mathrm{E}^m X\!</math> with the dyadic rationals (or the binary fractions) in the half-open interval <math>[0, 2).\!</math> Formally and canonically, a state <math>q_r\!</math> is indexed by a fraction <math>r = \tfrac{s}{t}\!</math> whose denominator is the power of two <math>t = 2^m\!</math> and whose numerator is a binary numeral formed from the coefficients of state in a manner to be described next. The ''differential coefficients'' of the state <math>q\!</math> are just the values <math>\mathrm{d}^k\!A(q)</math> for <math>k = 0 ~\text{to}~ m,\!</math> where <math>\mathrm{d}^0\!A</math> is defined as being identical to <math>A.\!</math> To form the binary index <math>d_0.d_1 \ldots d_m\!</math> of the state <math>q\!</math> the coefficient <math>\mathrm{d}^k\!A(q)</math> is read off as the binary digit <math>d_k\!</math> associated with the place value <math>2^{-k}.\!</math> Expressed by way of algebraic formulas, the rational index <math>r\!</math> of the state <math>q\!</math> can be given by the following equivalent formulations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
r(q)<br />
& = &<br />
\displaystyle\sum_k d_k \cdot 2^{-k}<br />
& = &<br />
\displaystyle\sum_k \text{d}^k A(q) \cdot 2^{-k}<br />
\\[8pt]<br />
=<br />
\\[8pt]<br />
\displaystyle\frac{s(q)}{t}<br />
& = &<br />
\displaystyle\frac{\sum_k d_k \cdot 2^{(m-k)}}{2^m}<br />
& = &<br />
\displaystyle\frac{\sum_k \text{d}^k A(q) \cdot 2^{(m-k)}}{2^m}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Applied to the example of <math>4^\text{th}\!</math>-gear curves, this scheme results in the data of Tables&nbsp;17-a and 17-b, which exhibit one period for each orbit. The states in each orbit are listed as ordered pairs <math>(p_i, q_j),\!</math> where <math>p_i\!</math> may be read as a temporal parameter that indicates the present time of the state and where <math>j\!</math> is the decimal equivalent of the binary numeral <math>s.\!</math> Informally and more casually, the Tables exhibit the states <math>q_s\!</math> as subscripted with the numerators of their rational indices, taking for granted the constant denominators of <math>2^m\! = 2^4 = 16.\!</math> In this set-up the temporal successions of states can be reckoned as given by a kind of ''parallel round-up rule''. That is, if <math>(d_k, d_{k+1})\!</math> is any pair of adjacent digits in the state index <math>r,\!</math> then the value of <math>d_k\!</math> in the next state is <math>{d_k}' = d_k + d_{k+1}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:55%"<br />
|+ style="height:30px" | <math>\text{Table 17-a.} ~~ \text{A Couple of Orbits in Fourth Gear : Orbit 1}\!</math><br />
|- style="background:ghostwhite"<br />
| <math>\text{Time}\!</math><br />
| <math>\text{State}\!</math><br />
| <math>A\!</math><br />
| <math>\mathrm{d}A\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| <math>p_i\!</math><br />
| <math>q_j\!</math><br />
| <math>\mathrm{d}^0\!A</math><br />
| <math>\mathrm{d}^1\!A</math><br />
| <math>\mathrm{d}^2\!A</math><br />
| <math>\mathrm{d}^3\!A</math><br />
| <math>\mathrm{d}^4\!A</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
p_0<br />
\\[4pt]<br />
p_1<br />
\\[4pt]<br />
p_2<br />
\\[4pt]<br />
p_3<br />
\\[4pt]<br />
p_4<br />
\\[4pt]<br />
p_5<br />
\\[4pt]<br />
p_6<br />
\\[4pt]<br />
p_7<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
q_{01}<br />
\\[4pt]<br />
q_{03}<br />
\\[4pt]<br />
q_{05}<br />
\\[4pt]<br />
q_{15}<br />
\\[4pt]<br />
q_{17}<br />
\\[4pt]<br />
q_{19}<br />
\\[4pt]<br />
q_{21}<br />
\\[4pt]<br />
q_{31}<br />
\end{matrix}\!</math><br />
| colspan="5" |<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="text-align:center; width:100%"<br />
|<br />
<math>\begin{matrix}<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
1.<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:55%"<br />
|+ style="height:30px" | <math>\text{Table 17-b.} ~~ \text{A Couple of Orbits in Fourth Gear : Orbit 2}\!</math><br />
|- style="background:ghostwhite"<br />
| <math>\text{Time}\!</math><br />
| <math>\text{State}\!</math><br />
| <math>A\!</math><br />
| <math>\mathrm{d}A\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| <math>p_i\!</math><br />
| <math>q_j\!</math><br />
| <math>\mathrm{d}^0\!A</math><br />
| <math>\mathrm{d}^1\!A</math><br />
| <math>\mathrm{d}^2\!A</math><br />
| <math>\mathrm{d}^3\!A</math><br />
| <math>\mathrm{d}^4\!A</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
p_0<br />
\\[4pt]<br />
p_1<br />
\\[4pt]<br />
p_2<br />
\\[4pt]<br />
p_3<br />
\\[4pt]<br />
p_4<br />
\\[4pt]<br />
p_5<br />
\\[4pt]<br />
p_6<br />
\\[4pt]<br />
p_7<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
q_{25}<br />
\\[4pt]<br />
q_{11}<br />
\\[4pt]<br />
q_{29}<br />
\\[4pt]<br />
q_{07}<br />
\\[4pt]<br />
q_{09}<br />
\\[4pt]<br />
q_{27}<br />
\\[4pt]<br />
q_{13}<br />
\\[4pt]<br />
q_{23}<br />
\end{matrix}\!</math><br />
| colspan="5" |<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="text-align:center; width:100%"<br />
|<br />
<math>\begin{matrix}<br />
1.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
==Transformations of Discourse==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
It is understandable that an engineer should be completely absorbed in his speciality, instead of pouring himself out into the freedom and vastness of the world of thought, even though his machines are being sent off to the ends of the earth; for he no more needs to be capable of applying to his own personal soul what is daring and new in the soul of his subject than a machine is in fact capable of applying to itself the differential calculus on which it is based. The same thing cannot, however, be said about mathematics; for here we have the new method of thought, pure intellect, the very well-spring of the times, the ''fons et origo'' of an unfathomable transformation.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 39]<br />
|}<br />
<br />
In this section we take up the general study of logical transformations, or maps that relate one universe of discourse to another. In many ways, and especially as applied to the subject of intelligent dynamic systems, my argument develops the antithesis of the statement just quoted. Along the way, if incidental to my ends, I hope this essay can pose a fittingly irenic epitaph to the frankly ironic epigraph inscribed at its head.<br />
<br />
My goal in this section is to answer a single question: What is a propositional tangent functor? In other words, my aim is to develop a clear conception of what manner of thing would pass in the logical realm for a genuine analogue of the tangent functor, an object conceived to generalize as far as possible in the abstract terms of category theory the ordinary notions of functional differentiation and the all too familiar operations of taking derivatives.<br />
<br />
As a first step I discuss the kinds of transformations that we already know as ''extensions'' and ''projections'', and I use these special cases to illustrate several different styles of logical and visual representation that will figure heavily in the sequel.<br />
<br />
===Foreshadowing Transformations : Extensions and Projections of Discourse===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
And, despite the care which she took to look behind her at every moment, she failed to see a shadow which followed her like her own shadow, which stopped when she stopped, which started again when she did, and which made no more noise than a well-conducted shadow should.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 126]<br />
|}<br />
<br />
Many times in our discussion we have occasion to place one universe of discourse in the context of a larger universe of discourse. An embedding of the general type <math>[\mathcal{X}] \to [\mathcal{Y}]\!</math> is implied any time that we make use of one alphabet <math>[\mathcal{X}]\!</math> that happens to be included in another alphabet <math>[\mathcal{Y}].\!</math> When we are discussing differential issues we usually have in mind that the extended alphabet <math>[\mathcal{Y}]\!</math> has a special construction or a specific lexical relation with respect to the initial alphabet <math>[\mathcal{X}],\!</math> one that is marked by characteristic types of accents, indices, or inflected forms.<br />
<br />
====Extension from 1 to 2 Dimensions====<br />
<br />
Figure 18-a lays out the ''angular form'' of venn diagram for universes of 1 and 2 dimensions, indicating the embedding map of type <math>\mathbb{B}^1 \to \mathbb{B}^2\!</math> and detailing the coordinates that are associated with individual cells. Because all points, cells, or logical interpretations are represented as connected geometric areas, we can say that these pictures provide us with an ''areal view'' of each universe of discourse.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-a -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-a.} ~~ \text{Extension from 1 to 2 Dimensions : Areal}\!</math><br />
|}<br />
<br />
Figure 18-b shows the differential extension from <math>X^\bullet = [x]\!</math> to <math>\mathrm{E}X^\bullet = [x, \mathrm{d}x]\!</math> in a ''bundle of boxes'' form of venn diagram. As awkward as it may seem at first, this type of picture is often the most natural and the most easily available representation when we want to conceptualize the localized information or momentary knowledge of an intelligent dynamic system. It gives a ready picture of a ''proposition at a point'', in the present instance, of a proposition about changing states which is itself associated with a particular dynamic state of a system. It is easy to see how this application might be extended to conceive of more general types of instantaneous knowledge that are possessed by a system.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-b -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-b.} ~~ \text{Extension from 1 to 2 Dimensions : Bundle}\!</math><br />
|}<br />
<br />
Figure 18-c shows the same extension in a ''compact'' style of venn diagram, where the differential features at each position are represented by arrows extending from that position that cross or do not cross, as the case may be, the corresponding feature boundaries.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-c -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-c.} ~~ \text{Extension from 1 to 2 Dimensions : Compact}\!</math><br />
|}<br />
<br />
Figure 18-d compresses the picture of the differential extension even further, yielding a directed graph or ''digraph'' form of representation. (Notice that my definition of a digraph allows for loops or ''slings'' at individual points, in addition to arcs or ''arrows'' between the points.)<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-d -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-d.} ~~ \text{Extension from 1 to 2 Dimensions : Digraph}\!</math><br />
|}<br />
<br />
====Extension from 2 to 4 Dimensions====<br />
<br />
Figure 19-a lays out the ''areal view'' or the ''angular form'' of venn diagram for universes of 2 and 4 dimensions, indicating the embedding map of type <math>\mathbb{B}^2 \to \mathbb{B}^4.\!</math> In many ways these pictures are the best kind there is, giving full canvass to an ideal vista. Their style allows the clearest, the fairest, and the plainest view that we can form of a universe of discourse, affording equal representation to all dispositions and maintaining a balance with respect to ordinary and differential features. If only we could extend this view! Unluckily, an obvious difficulty beclouds this prospect, and that is how precipitately we run into the limits of our plane and visual intuitions. Even within the scope of the spare few dimensions that we have scanned up to this point subtle discrepancies have crept in already. The circumstances that bind us and the frameworks that block us, the flat distortion of the planar projection and the inevitable ineffability that precludes us from wrapping its rhomb figure into rings around a torus, all of these factors disguise the underlying but true connectivity of the universe of discourse.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-a -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-a.} ~~ \text{Extension from 2 to 4 Dimensions : Areal}\!</math><br />
|}<br />
<br />
Figure 19-b shows the differential extension from <math>U^\bullet = [u, v]\!</math> to <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v]\!</math> in the ''bundle of boxes'' form of venn diagram.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-b -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-b.} ~~ \text{Extension from 2 to 4 Dimensions : Bundle}\!</math><br />
|}<br />
<br />
As dimensions increase, this factorization of the extended universe along the lines that are marked out by the bundle picture begins to look more and more like a practical necessity. But whenever we use a propositional model to address a real situation in the context of nature we need to remain aware that this articulation into factors, affecting our description, may be wholly artificial in nature and cleave to nothing, no joint in nature, nor any juncture in time to be in or out of joint.<br />
<br />
Figure 19-c illustrates the extension from 2 to 4 dimensions in the ''compact'' style of venn diagram. Here, just the changes with respect to the center cell are shown.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-c -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-c.} ~~ \text{Extension from 2 to 4 Dimensions : Compact}\!</math><br />
|}<br />
<br />
Figure 19-d gives the ''digraph'' form of representation for the differential extension <math>U^\bullet \to \mathrm{E}U^\bullet,\!</math> where the 4 nodes marked with a circle <math>{}^{\bigcirc}\!</math> are the cells <math>uv,\, u \texttt{(} v \texttt{)},\, \texttt{(} u \texttt{)} v,\, \texttt{(} u \texttt{)(} v \texttt{)},\!</math> respectively, and where a 2-headed arc counts as 2 arcs of the differential digraph.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-d -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-d.} ~~ \text{Extension from 2 to 4 Dimensions : Digraph}\!</math><br />
|}<br />
<br />
===Thematization of Functions : And a Declaration of Independence for Variables===<br />
<br />
{| width="100%"<br />
| align="left" |<br />
''And as imagination bodies forth''<br><br />
''The forms of things unknown, the poet's pen''<br><br />
''Turns them to shapes, and gives to airy nothing''<br><br />
''A local habitation and a name.''<br />
| align="right" valign="bottom" | A Midsummer Night's Dream, 5.1.18<br />
|}<br />
<br />
In the representation of propositions as functions it is possible to notice different degrees of explicitness in the way their functional character is symbolized. To indicate what I mean by this, the next series of Figures illustrates a set of graphic conventions that will be put to frequent use in the remainder of this discussion, both to mark the relevant distinctions and to help us convert between related expressions at different levels of explicitness in their functionality.<br />
<br />
====Thematization : Venn Diagrams====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
The known universe has one complete lover and that is the greatest poet. He consumes an eternal passion and is indifferent which chance happens and which possible contingency of fortune or misfortune and persuades daily and hourly his delicious pay.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 11&ndash;12]<br />
|}<br />
<br />
Figure 20-i traces the first couple of steps in this order of ''thematic'' progression, that will gradually run the gamut through a complete series of degrees of functional explicitness in the expression of logical propositions. The first venn diagram represents a situation where the function is indicated by a shaded figure and a logical expression. At this stage one may be thinking of the proposition only as expressed by a formula in a particular language and its content only as a subset of the universe of discourse, as when considering the proposition <math>u\!\cdot\!v</math> in the universe <math>[u, v].\!</math> The second venn diagram depicts a situation in which two significant steps have been taken. First, one has taken the trouble to give the proposition <math>u\!\cdot\!v</math> a distinctive functional name <math>{}^{\backprime\backprime} J {}^{\prime\prime}.\!</math> Second, one has come to think explicitly about the target domain that contains the functional values of <math>J,\!</math> as when writing <math>J : \langle u, v \rangle \to \mathbb{B}.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 20-i -- Thematization of Conjunction (Stage 1).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 20-i.} ~~ \text{Thematization of Conjunction (Stage 1)}\!</math><br />
|}<br />
<br />
In Figure 20-ii the proposition <math>J\!</math> is viewed explicitly as a transformation from one universe of discourse to another.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 20-ii -- Thematization of Conjunction (Stage 2).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 20-ii.} ~~ \text{Thematization of Conjunction (Stage 2)}\!</math><br />
|}<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
o-------------------------------o o-------------------------------o<br />
| | | |<br />
| o-----o o-----o | | o-----o o-----o |<br />
| / \ / \ | | / \ / \ |<br />
| / o \ | | / o \ |<br />
| / /`\ \ | | / /`\ \ |<br />
| o o```o o | | o o```o o |<br />
| | u |```| v | | | | u |```| v | |<br />
| o o```o o | | o o```o o |<br />
| \ \`/ / | | \ \`/ / |<br />
| \ o / | | \ o / |<br />
| \ / \ / | | \ / \ / |<br />
| o-----o o-----o | | o-----o o-----o |<br />
| | | |<br />
o-------------------------------o o-------------------------------o<br />
\ / \ /<br />
\ / \ /<br />
\ / \ J /<br />
\ / \ /<br />
\ / \ /<br />
o----------\---------/----------o o----------\---------/----------o<br />
| \ / | | \ / |<br />
| \ / | | \ / |<br />
| o-----@-----o | | o-----@-----o |<br />
| /`````````````\ | | /`````````````\ |<br />
| /```````````````\ | | /```````````````\ |<br />
| /`````````````````\ | | /`````````````````\ |<br />
| o```````````````````o | | o```````````````````o |<br />
| |```````````````````| | | |```````````````````| |<br />
| |```````` J ````````| | | |```````` x ````````| |<br />
| |```````````````````| | | |```````````````````| |<br />
| o```````````````````o | | o```````````````````o |<br />
| \`````````````````/ | | \`````````````````/ |<br />
| \```````````````/ | | \```````````````/ |<br />
| \`````````````/ | | \`````````````/ |<br />
| o-----------o | | o-----------o |<br />
| | | |<br />
| | | |<br />
o-------------------------------o o-------------------------------o<br />
J = u v x = J<u, v><br />
<br />
Figure 20-ii. Thematization of Conjunction (Stage 2)<br />
</pre><br />
|}<br />
<br />
In the first venn diagram the name that is assigned to a composite proposition, function, or region in the source universe is delegated to a simple feature in the target universe. This can result in a single character or term exceeding the responsibilities it can carry off well. Allowing the name of a function <math>J : \langle u, v \rangle \to \mathbb{B}\!</math> to serve as the name of its dependent variable <math>J : \mathbb{B}\!</math> does not mean that one has to confuse a function with any of its values, but it does put one at risk for a number of obvious problems, and we should not be surprised, on numerous and limiting occasions, when quibbling arises from the attempts of a too original syntax to serve these two masters.<br />
<br />
The second venn diagram circumvents these difficulties by introducing a new variable name for each basic feature of the target universe, as when writing <math>J : \langle u, v \rangle \to \langle x \rangle,\!</math> and thereby assigns a concrete type <math>\langle x \rangle</math> to the abstract codomain <math>\mathbb{B}.\!</math> To make this induction of variables more formal one can append subscripts, as in <math>x_J,\!</math> to indicate the origin or derivation of the new characters. Or we may use a lexical modifier to convert function names into variable names, for example, associating the function name <math>J\!</math> with the variable name <math>\check{J}.\!</math> Thus we may think of <math>x = x_J = \check{J}\!</math> as the ''cache variable'' corresponding to the function <math>J\!</math> or the symbol <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> considered as a contingent variable.<br />
<br />
In Figure 20-iii we arrive at a stage where the functional equations <math>J = u\!\cdot\!v</math> and <math>x = u\!\cdot\!v</math> are regarded as propositions in their own right, reigning in and ruling over the 3-feature universes of discourse <math>[u, v, J]~\!</math> and <math>[u, v, x],\!</math> respectively. Subject to the cautions already noted, the function name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> can be reinterpreted as the name of a feature <math>\check{J}</math> and the equation <math>J = u\!\cdot\!v</math> can be read as the logical equivalence <math>\texttt{((} J, u ~ v \texttt{))}.\!</math> To give it a generic name let us call this newly expressed, collateral proposition the ''thematization'' or the ''thematic extension'' of the original proposition <math>J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 20-iii -- Thematization of Conjunction (Stage 3).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 20-iii.} ~~ \text{Thematization of Conjunction (Stage 3)}\!</math><br />
|}<br />
<br />
The first venn diagram represents the thematization of the conjunction <math>J\!</math> with shading in the appropriate regions of the universe <math>[u, v, J].\!</math> Also, it illustrates a quick way of constructing a thematic extension. First, draw a line, in practice or the imagination, that bisects every cell of the original universe, placing half of each cell under the aegis of the thematized proposition and the other half under its antithesis. Next, on the scene where the theme applies leave the shade wherever it lies, and off the stage, where it plays otherwise, stagger the pattern in a harlequin guise.<br />
<br />
In the final venn diagram of this sequence the thematic progression comes full circle and completes one round of its development. The ambiguities that were occasioned by the changing role of the name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> are resolved by introducing a new variable name <math>{}^{\backprime\backprime} x {}^{\prime\prime}</math> to take the place of <math>\check{J},\!</math> and the region that represents this fresh featured <math>x\!</math> is circumscribed in a more conventional symmetry of form and placement. Just as we once gave the name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> to the proposition <math>u\!\cdot\!v,</math> we now give the name <math>{}^{\backprime\backprime} \iota {}^{\prime\prime}</math> to its thematization <math>\texttt{((} x, u ~ v \texttt{))}.\!</math> Already, again, at this culminating stage of reflection, we begin to think of the newly named proposition as a distinctive individual, a particular function <math>\iota : \langle u, v, x \rangle \to \mathbb{B}.\!</math><br />
<br />
From now on, the terms ''thematic extension'' and ''thematization'' will be used to describe both the process and degree of explication that progresses through this series of pictures, both the operation of increasingly explicit symbolization and the dimension of variation that is swept out by it. To speak of this change in general, that takes us in our current example from <math>J\!</math> to <math>\iota,\!</math> we introduce a class of operators symbolized by the Greek letter <math>\theta,\!</math> writing <math>\iota = \theta J\!</math> in the present instance. The operator <math>\theta,\!</math> in the present situation bearing the type <math>\theta : [u, v] \to [u, v, x],\!</math> provides us with a convenient way of recapitulating and summarizing the complete cycle of thematic developments.<br />
<br />
Figure 21 shows how the thematic extension operator <math>\theta\!</math> acts on two further examples, the disjunction <math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math> and the equality <math>\texttt{((} u, v \texttt{))}.\!</math> Referring to the disjunction as <math>f(u, v)\!</math> and the equality as <math>f(u, v),\!</math> we may express the thematic extensions as <math>\varphi = \theta f\!</math> and <math>\gamma = \theta g.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 21 -- Thematization of Disjunction and Equality.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 21.} ~~ \text{Thematization of Disjunction and Equality}\!</math><br />
|}<br />
<br />
====Thematization : Truth Tables====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
That which distorts honest shapes or which creates unearthly beings or places or contingencies is a nuisance and a revolt.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 19]<br />
|}<br />
<br />
Tables 22 through 25 outline a method for computing the thematic extensions of propositions in terms of their coordinate values.<br />
<br />
A preliminary step, as illustrated in Table&nbsp;22, is to write out the truth table representations of the propositional forms whose thematic extensions one wants to compute, in the present instance, the functions <math>f(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math> and <math>g(u, v) = \texttt{((} u, v \texttt{))}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:50%"<br />
|+ style="height:30px" | <math>\text{Table 22.} ~~ \text{Disjunction}~ f ~\text{and Equality}~ g\!</math><br />
|- style="height:35px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black" | <math>f\!</math><br />
| <math>g\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Next, each propositional form is individually represented in the fashion shown in Tables&nbsp;23-i and 23-ii, using <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} g {}^{\prime\prime}\!</math> as function names and creating new variables <math>x\!</math> and <math>y\!</math> to hold the associated functional values. This pair of Tables outlines the first stage in the transition from the <math>2\!</math>-dimensional universes of <math>f\!</math> and <math>g\!</math> to the <math>3\!</math>-dimensional universes of <math>\theta f\!</math> and <math>\theta g.\!</math> The top halves of the Tables replicate the truth table patterns for <math>f\!</math> and <math>g\!</math> in the form <math>f : [u, v] \to [x]\!</math> and <math>g : [u, v] \to [y].\!</math> The bottom halves of the tables print the negatives of these pictures, as it were, and paste the truth tables for <math>\texttt{(} f \texttt{)}\!</math> and <math>\texttt{(} g \texttt{)}\!</math> under the copies for <math>f\!</math> and <math>g.\!</math> At this stage, the columns for <math>\theta f\!</math> and <math>\theta g\!</math> are appended almost as afterthoughts, amounting to indicator functions for the sets of ordered triples that make up the functions <math>f\!</math> and <math>g.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 23-i and 23-ii.} ~~ \text{Thematics of Disjunction and Equality (1)}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 23-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black" | <math>f\!</math><br />
| <math>x\!</math><br />
| <math>\varphi\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |<br />
<math>\begin{matrix}\to\\\to\\\to\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" | &nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 23-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black" | <math>g\!</math><br />
| <math>y\!</math><br />
| <math>\gamma\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |<br />
<math>\begin{matrix}\to\\\to\\\to\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" | &nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
All the data are now in place to give the truth tables for <math>\theta f\!</math> and <math>\theta g.\!</math> All that remains to be done is to permute the rows and change the roles of <math>x\!</math> and <math>y\!</math> from dependent to independent variables. In Tables&nbsp;24-i and 24-ii the rows are arranged in such a way as to put the 3-tuples <math>(u, v, x)\!</math> and <math>(u, v, y)\!</math> in binary numerical order, suitable for viewing as the arguments of the maps <math>\theta f = \varphi : [u, v, x] \to \mathbb{B}\!</math> and <math>\theta g = \gamma : [u, v, y] \to \mathbb{B}.\!</math> Moreover, the structure of the tables is altered slightly, allowing the now vestigial functions <math>\theta f\!</math> and <math>\theta g\!</math> to be passed over without further attention and shifting the heavy vertical bars a notch to the right. In effect, this clinches the fact that the thematic variables <math>x := \check{f}\!</math> and <math>y := \check{g}\!</math> are now treated as independent variables.<br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 24-i and 24-ii.} ~~ \text{Thematics of Disjunction and Equality (2)}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 24-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>f\!</math><br />
| <math>x\!</math><br />
| style="border-left:1px solid black" | <math>\varphi\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <font size="+2">&nbsp;<br></font><math>\begin{matrix}\to\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 24-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>g\!</math><br />
| <math>y\!</math><br />
| style="border-left:1px solid black" | <math>\gamma\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | &nbsp;<br><math>\begin{matrix}\to\\\to\end{matrix}\!</math><br>&nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
An optional reshuffling of the rows brings additional features of the thematic extensions to light. Leaving the columns in place for the sake of comparison, Tables&nbsp;25-i and 25-ii sort the rows in a different order, in effect treating <math>x\!</math> and <math>y\!</math> as the primary variables in their respective 3-tuples. Regarding the thematic extensions in the form <math>\varphi : [x, u, v] \to \mathbb{B}\!</math> and <math>\gamma : [y, u, v] \to \mathbb{B}\!</math> makes it easier to see in this tabular setting a property that was graphically obvious in the venn diagrams above. Specifically, when the thematic variable <math>\check{F}\!</math> is true then <math>\theta F\!</math> exhibits the pattern of the original <math>F,\!</math> and when <math>\check{F}\!</math> is false then <math>\theta F\!</math> exhibits the pattern of its negation <math>\texttt{(} F \texttt{)}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 25-i and 25-ii.} ~~ \text{Thematics of Disjunction and Equality (3)}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 25-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>f\!</math><br />
| <math>x\!</math><br />
| style="border-left:1px solid black" | <math>\varphi\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>{\to}\!</math><br><font size="+2">&nbsp;<br>&nbsp;<br>&nbsp;<br></font><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <font size="+2">&nbsp;<br></font><math>\begin{matrix}\to\\\to\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 25-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>g\!</math><br />
| <math>y\!</math><br />
| style="border-left:1px solid black" | <math>\gamma\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | &nbsp;<br><math>\begin{matrix}\to\\\to\end{matrix}\!</math><br>&nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
Finally, Tables&nbsp;26-i and 26-ii compare the tacit extensions <math>\boldsymbol\varepsilon : [u, v] \to [u, v, x]\!</math> and <math>\boldsymbol\varepsilon : [u, v] \to [u, v, y]\!</math> with the thematic extensions of the same types, as applied to the propositions <math>f\!</math> and <math>g,\!</math> respectively.<br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 26-i and 26-ii.} ~~ \text{Tacit Extension and Thematization}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 26-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>x\!</math><br />
| style="border-left:1px solid black" | <math>\boldsymbol\varepsilon f\!</math><br />
| <math>\theta f\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 26-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>y\!</math><br />
| style="border-left:1px solid black" | <math>\boldsymbol\varepsilon g\!</math><br />
| <math>\theta g\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
Table 27 summarizes the thematic extensions of all propositions on two variables. Column&nbsp;4 lists the equations of form <math>\texttt{((} \check{f_i}, f_i (u, v) \texttt{))}\!</math> and Column&nbsp;5 simplifies these equations into the form of algebraic expressions. As always, <math>{}^{\backprime\backprime} + {}^{\prime\prime}\!</math> refers to exclusive disjunction and each <math>{}^{\backprime\backprime} \check{f} {}^{\prime\prime}\!</math> appearing in the last two Columns refers to the corresponding variable name <math>{}^{\backprime\backprime} \check{f_i} {}^{\prime\prime}.~\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 27.} ~~ \text{Thematization of Bivariate Propositions}\!</math><br />
|- style="height:30px; background:ghostwhite"<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>{f}\!</math><br />
| <math>\theta f\!</math><br />
| <math>\theta f\!</math><br />
|- style="background:ghostwhite"<br />
| align="right" | <math>u\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| align="right" | <math>v\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| align="left" | <math>\texttt{((} \check{f} \texttt{,~(~)~))}\!</math><br />
| align="left" | <math>\check{f} + 1\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\\[4pt]<br />
f_{4}<br />
\\[4pt]<br />
f_{8}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
1~0~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} u \texttt{)~} v \texttt{~}<br />
\\[4pt]<br />
\texttt{~} u \texttt{~(} v \texttt{)}<br />
\\[4pt]<br />
\texttt{~} u \texttt{~~} v \texttt{~}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(u)(v)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~(u)~v~~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~u~(v)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~u~~v~~))}<br />
\end{array}</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + u + v + uv<br />
\\[4pt]<br />
\check{f} + v + uv + 1<br />
\\[4pt]<br />
\check{f} + u + uv + 1<br />
\\[4pt]<br />
\check{f} + uv + 1<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{3}<br />
\\[4pt]<br />
f_{12}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{)}<br />
\\[4pt]<br />
\texttt{~} u \texttt{~}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(u)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~u~~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + u<br />
\\[4pt]<br />
\check{f} + u + 1<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[4pt]<br />
f_{9}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[4pt]<br />
1~0~0~1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{,} v \texttt{)}<br />
\\[4pt]<br />
\texttt{((} u \texttt{,} v \texttt{))}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~~(} u \texttt{,} v \texttt{)~~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~((} u \texttt{,} v \texttt{))~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + u + v + 1<br />
\\[4pt]<br />
\check{f} + u + v<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{5}<br />
\\[4pt]<br />
f_{10}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} v \texttt{)}<br />
\\[4pt]<br />
\texttt{~} v \texttt{~}<br />
\end{matrix}</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(} v \texttt{)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~} v \texttt{~~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + v<br />
\\[4pt]<br />
\check{f} + v + 1<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[4pt]<br />
f_{11}<br />
\\[4pt]<br />
f_{13}<br />
\\[4pt]<br />
f_{14}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(~} u \texttt{~~} v \texttt{~)}<br />
\\[4pt]<br />
\texttt{(~} u \texttt{~(} v \texttt{))}<br />
\\[4pt]<br />
\texttt{((} u \texttt{)~} v \texttt{~)}<br />
\\[4pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(~} u \texttt{~~} v \texttt{~)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~(~} u \texttt{~(} v \texttt{))~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~((} u \texttt{)~} v \texttt{~)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~((} u \texttt{)(} v \texttt{))~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + uv<br />
\\[4pt]<br />
\check{f} + u + uv<br />
\\[4pt]<br />
\check{f} + v + uv<br />
\\[4pt]<br />
\check{f} + u + v + uv + 1<br />
\end{array}\!</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| align="left" | <math>\texttt{((} \check{f} \texttt{,~((~))~))}\!</math><br />
| align="left" | <math>\check{f}\!</math><br />
|}<br />
<br />
<br><br />
<br />
In order to show what all of the thematic extensions from two dimensions to three dimensions look like in terms of coordinates, Tables&nbsp;28 and 29 present ordinary truth tables for the functions <math>f_i : \mathbb{B}^2 \to \mathbb{B}\!</math> and for the corresponding thematizations <math>\theta f_i = \varphi_i : \mathbb{B}^3 \to \mathbb{B}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 28.} ~~ \text{Propositions on Two Variables}\!</math><br />
|- style="height:35px; background:ghostwhite"<br />
| style="width:5%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:5%; border-bottom:1px solid black; border-left:1px solid black" | <math>f_{0}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{1}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{2}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{3}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{4}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{5}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{6}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{7}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{8}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{9}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{10}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{11}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{12}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{13}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{14}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{15}\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 29.} ~~ \text{Thematic Extensions of Bivariate Propositions}\!</math><br />
|- style="height:35px; background:ghostwhite"<br />
| style="width:5%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\check{f}\!</math><br />
| style="width:5%; border-bottom:1px solid black; border-left:1px solid black" | <math>\varphi_{0}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{1}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{2}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{3}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{4}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{5}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{6}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{7}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{8}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{9}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{10}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{11}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{12}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{13}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{14}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{15}\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
|-<br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Propositional Transformations===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
If only the word &lsquo;artificial&rsquo; were associated with the idea of ''art'', or expert skill gained through voluntary apprenticeship (instead of suggesting the factitious and unreal), we might say that ''logical'' refers to artificial thought.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 56&ndash;57]<br />
|}<br />
<br />
In this section we develop a comprehensive set of concepts for dealing with transformations between universes of discourse. In this most general setting the source and target universes of a transformation are allowed to be different, but may be the same. When we apply these concepts to dynamic systems we focus on the important special case of transformations that map a universe into itself, regarding them as the state transitions of a discrete dynamical process and placing them among the myriad ways that a universe of discourse might change, and by that change turn into itself.<br />
<br />
====Alias and Alibi Transformations====<br />
<br />
There are customarily two modes of understanding a transformation, at least, when we try to interpret its relevance to something in reality. A transformation always refers to a changing prospect, to say it in a unified but equivocal way, but this can be taken to mean either a subjective change in the interpreting observer's point of view or an objective change in the systematic subject of discussion. In practice these variant uses of the transformation concept are distinguished in the following terms:<br />
<br />
# A ''perspectival'' or ''alias'' transformation refers to a shift in perspective or a change in language that takes place in the observer's frame of reference.<br />
# A ''transitional'' or ''alibi'' transformation refers to a change of position or an alteration of state that occurs in the object system as it falls under study.<br />
<br />
(For a recent discussion of the alias vs. alibi issue, as it relates to linear transformations in vector spaces and to other issues of an algebraic nature, see [MaB, 256, 582-4].)<br />
<br />
Naturally, when we are concerned with the dynamical properties of a system, the transitional aspect of transformation is the factor that comes to the fore, and this involves us in contemplating all of the ways of changing a universe into itself while remaining under the rule of established dynamical laws. In the prospective application to dynamic systems, and to neural networks viewed in this light, our interest lies chiefly with the transformations of a state space into itself that constitute the state transitions of a discrete dynamic process. Nevertheless, many important properties of these transformations, and some constructions that we need to see most clearly, are independent of the transitional interpretation and are likely to be confounded with irrelevant features if presented first and only in that association.<br />
<br />
In addition, and in partial contrast, intelligent systems are exactly that species of dynamic agents that have the capacity to have a point of view, and we cannot do justice to their peculiar properties without examining their ability to form and transform their own frames of reference in exposure to the elements of their own experience. In this setting, the perspectival aspect of transformation is the facet that shines most brightly, perhaps too often leaving us fascinated with mere glimmerings of its actual potential. It needs to be emphasized that nothing of the ordinary sort needs be moved in carrying out a transformation under the alias interpretation, that it may only involve a change in the forms of address, an amendment of the terms which are customed to approach and fashioned to describe the very same things in the very same world. But again, working within a discipline of realistic computation, we know how formidably complex and resource-consuming such transformations of perspective can be to implement in practice, much less to endow in the self-governed form of a nascently intelligent dynamical system.<br />
<br />
====Transformations of General Type====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
''Es ist passiert'', &ldquo;it just sort of happened&rdquo;, people said there when other people in other places thought heaven knows what had occurred. It was a peculiar phrase, not known in this sense to the Germans and with no equivalent in other languages, the very breath of it transforming facts and the bludgeonings of fate into something light as eiderdown, as thought itself.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 34]<br />
|}<br />
<br />
Consider the situation illustrated in Figure&nbsp;30, where the alphabets <math>\mathcal{U} = \{ u, v \}\!</math> and <math>\mathcal{X} = \{ x, y, z \}\!</math> are used to label basic features in two different logical universes, <math>U^\bullet = [u, v]\!</math> and <math>X^\bullet = [x, y, z].\!</math><br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
o-------------------------------------------------------o<br />
| U |<br />
| |<br />
| o-----------o o-----------o |<br />
| / \ / \ |<br />
| / o \ |<br />
| / / \ \ |<br />
| / / \ \ |<br />
| o o o o |<br />
| | | | | |<br />
| | u | | v | |<br />
| | | | | |<br />
| o o o o |<br />
| \ \ / / |<br />
| \ \ / / |<br />
| \ o / |<br />
| \ / \ / |<br />
| o-----------o o-----------o |<br />
| |<br />
| |<br />
o---------------------------o---------------------------o<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
o-------------------------o o-------------------------o o-------------------------o<br />
| U | | U | | U |<br />
| o---o o---o | | o---o o---o | | o---o o---o |<br />
| / \ / \ | | / \ / \ | | / \ / \ |<br />
| / o \ | | / o \ | | / o \ |<br />
| / / \ \ | | / / \ \ | | / / \ \ |<br />
| o o o o | | o o o o | | o o o o |<br />
| | u | | v | | | | u | | v | | | | u | | v | |<br />
| o o o o | | o o o o | | o o o o |<br />
| \ \ / / | | \ \ / / | | \ \ / / |<br />
| \ o / | | \ o / | | \ o / |<br />
| \ / \ / | | \ / \ / | | \ / \ / |<br />
| o---o o---o | | o---o o---o | | o---o o---o |<br />
| | | | | |<br />
o-------------------------o o-------------------------o o-------------------------o<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ g | \ f / | h /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ o----------|-----------\-----/-----------|----------o /<br />
\ | X | \ / | | /<br />
\ | | \ / | | /<br />
\ | | o-----o-----o | | /<br />
\| | / \ | |/<br />
\ | / \ | /<br />
|\ | / \ | /|<br />
| \ | / \ | / |<br />
| \ | / \ | / |<br />
| \ | o x o | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \| | | |/ |<br />
| o--o--------o o--------o--o |<br />
| / \ \ / / \ |<br />
| / \ \ / / \ |<br />
| / \ o / \ |<br />
| / \ / \ / \ |<br />
| / \ / \ / \ |<br />
| o o--o-----o--o o |<br />
| | | | | |<br />
| | | | | |<br />
| | | | | |<br />
| | y | | z | |<br />
| | | | | |<br />
| | | | | |<br />
| o o o o |<br />
| \ \ / / |<br />
| \ \ / / |<br />
| \ o / |<br />
| \ / \ / |<br />
| \ / \ / |<br />
| o-----------o o-----------o |<br />
| |<br />
| |<br />
o---------------------------------------------------o<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ p , q /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
o<br />
<br />
Figure 30. Generic Frame of a Logical Transformation<br />
</pre><br />
|}<br />
<br />
Enter the picture, as we usually do, in the middle of things, with features like <math>x, y , z\!</math> that present themselves to be simple enough in their own right and that form a satisfactory, if temporary foundation to provide a basis for discussion. In this universe and on these terms we find expression for various propositions and questions of principal interest to ourselves, as indicated by the maps <math>p, q : X \to \mathbb{B}.\!</math> Then we discover that the simple features <math>\{ x, y, z \}\!</math> are really more complex than we thought at first, and it becomes useful to regard them as functions <math>\{ f, g, h \}\!</math> of other features <math>\{ u, v \}\!</math> that we place in a preface to our original discourse, or suppose as topics of a preliminary universe of discourse <math>U^\bullet = [u, v].\!</math> It may happen that these late-blooming but pre-ambling features are found to lie closer, in a sense that may be our job to determine, to the central nature of the situation of interest, in which case they earn our regard as being more fundamental, but these functions and features are only required to supply a critical stance on the universe of discourse or an alternate perspective on the nature of things in order to be preserved as useful.<br />
<br />
A particular transformation <math>F : [u, v] \to [x, y, z]\!</math> may be expressed by a system of equations, as shown below. Here, <math>F\!</math> is defined by its component maps <math>F = (F_1, F_2, F_3) = (f, g, h),\!</math> where each component map in <math>\{ f, g, h \}\!</math> is a proposition of type <math>\mathbb{B}^n \to \mathbb{B}^1.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:50%"<br />
|<br />
<math>\begin{matrix}<br />
x & = & f(u, v)<br />
\\[10pt]<br />
y & = & g(u, v)<br />
\\[10pt]<br />
z & = & h(u, v)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Regarded as a logical statement, this system of equations expresses a relation between a collection of freely chosen propositions <math>\{ f, g, h \}\!</math> in one universe of discourse and the special collection of simple propositions <math>\{ x, y, z \}\!</math> on which is founded another universe of discourse. Growing familiarity with a particular transformation of discourse, and the desire to achieve a ready understanding of its implications, requires that we be able to convert this information about generals and simples into information about all the main subtypes of propositions, including the linear and singular propositions.<br />
<br />
===Analytic Expansions : Operators and Functors===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Consider what effects that might ''conceivably'' have practical bearings you ''conceive'' the objects of your ''conception'' to have. Then, your ''conception'' of those effects is the whole of your ''conception'' of the object.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; C.S. Peirce, &ldquo;The Maxim of Pragmatism&rdquo;, CP 5.438<br />
|}<br />
<br />
Given the barest idea of a logical transformation, as suggested by the sketch in Figure&nbsp;30, and having conceptualized the universe of discourse, with all of its points and propositions, as a beginning object of discussion, we are ready to enter the next phase of our investigation.<br />
<br />
====Operators on Propositions and Transformations====<br />
<br />
The next step is naturally inclined toward objects of the next higher order, namely, with operators that take in argument lists of logical transformations and that give back specified types of logical transformations as their results. For our present aims, we do not need to consider the most general class of such operators, nor any one of them for its own sake. Rather, we are interested in the special sorts of operators that arise in the study and analysis of logical transformations. Figuratively speaking, these operators serve as instruments for the live tomography (and hopefully not the vivisection) of the forms of change under view. Beyond that, they open up ways to implement the changes of view that we need to grasp all the variations on a transformational theme, or to appreciate enough of its significant features to &ldquo;get the drift&rdquo; of the change occurring, to form a passing acquaintance or a synthetic comprehension of its general character and disposition.<br />
<br />
The simplest type of operator is one that takes a single transformation as an argument and returns a single transformation as a result, and most of the operators explicitly considered in our discussion will be of this kind. Figure&nbsp;31 illustrates the typical situation.<br />
<br />
{| align="center" border="0" cellpadding="20"<br />
|<br />
<pre><br />
o---------------------------------------o<br />
| |<br />
| |<br />
| U% F X% |<br />
| o------------------>o |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| !W! | | !W! |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| v v |<br />
| o------------------>o |<br />
| !W!U% !W!F !W!X% |<br />
| |<br />
| |<br />
o---------------------------------------o<br />
Figure 31. Operator Diagram (1)<br />
</pre><br />
|}<br />
<br />
In this Figure <math>{}^{\backprime\backprime} \mathsf{W} {}^{\prime\prime}\!</math> stands for a generic operator <math>\mathsf{W},\!</math> in this case one that takes a logical transformation <math>F\!</math> of type <math>(U^\bullet \to X^\bullet)\!</math> into a logical transformation <math>\mathsf{W}F\!</math> of type <math>(\mathsf{W}U^\bullet \to \mathsf{W}X^\bullet).\!</math> Thus, the operator <math>\mathsf{W}\!</math> must be viewed as making assignments for both families of objects we have previously considered, that is, for universes of discourse like <math>{U^\bullet}\!</math> and <math>{X^\bullet}\!</math> and for logical transformations like <math>F.\!</math><br />
<br />
'''Note.''' Strictly speaking, an operator like <math>\mathsf{W}\!</math> works between two whole categories of universes and transformations, which we call the ''source'' and the ''target'' categories of <math>\mathsf{W}.\!</math> Given this setting, <math>\mathsf{W}\!</math> specifies for each universe <math>U^\bullet\!</math> in its source category a definite universe <math>\mathsf{W}U^\bullet\!</math> in its target category, and to each transformation <math>F\!</math> in its source category it assigns a unique transformation <math>\mathsf{W}F\!</math> in its target category. Naturally, this only works if <math>\mathsf{W}\!</math> takes the source <math>U^\bullet</math> and the target <math>X^\bullet</math> of the map <math>F\!</math> over to the source <math>\mathsf{W}U^\bullet\!</math> and the target <math>\mathsf{W}X^\bullet\!</math> of the map <math>\mathsf{W}F.\!</math> With luck or care enough, we can avoid ever having to put anything like that in words again, letting diagrams do the work. In the situations of present concern we are usually focused on a single transformation <math>F,\!</math> and thus we can take it for granted that the assignment of universes under <math>\mathsf{W}\!</math> is defined appropriately at the source and target ends of <math>F.\!</math> It is not always the case, though, that we need to use the particular names (like <math>{}^{\backprime\backprime} \mathsf{W}U^\bullet {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \mathsf{W}X^\bullet {}^{\prime\prime}\!</math>) that <math>\mathsf{W}\!</math> assigns by default to its operative image universes. In most contexts we will usually have a prior acquaintance with these universes under other names and it is necessary only that we can tell from the information associated with an operator <math>\mathsf{W}\!</math> what universes they are.<br />
<br />
In Figure&nbsp;31 the maps <math>F\!</math> and <math>\mathsf{W}F\!</math> are displayed horizontally, the way one normally orients functional arrows in a written text, and <math>\mathsf{W}\!</math> rolls the map <math>F\!</math> downward into the images that are associated with <math>\mathsf{W}F.\!</math> In Figure&nbsp;32 the same information is redrawn so that the maps <math>F\!</math> and <math>\mathsf{W}F\!</math> flow down the page, and <math>\mathsf{W}\!</math> unfurls the map <math>F\!</math> rightward into domains that are the eminent purview of <math>\mathsf{W}F.\!</math><br />
<br />
{| align="center" border="0" cellpadding="20"<br />
|<br />
<pre><br />
o---------------------------------------o<br />
| |<br />
| |<br />
| U% !W! !W!U% |<br />
| o------------------>o |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| F | | !W!F |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| v v |<br />
| o------------------>o |<br />
| X% !W! !W!X% |<br />
| |<br />
| |<br />
o---------------------------------------o<br />
Figure 32. Operator Diagram (2)<br />
</pre><br />
|}<br />
<br />
The latter arrangement, as exhibited in Figure&nbsp;32, is more congruent with the thinking about operators that we shall do in the rest of this discussion, since all logical transformations from here on out will be pictured vertically, after the fashion of Figure&nbsp;30.<br />
<br />
====Differential Analysis of Propositions and Transformations====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" | The resultant metaphysical problem now is this: ''Does the man go round the squirrel or not?''<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 43]<br />
|}<br />
<br />
The approach to the differential analysis of logical propositions and transformations of discourse to be pursued here is carried out in terms of particular operators <math>\mathsf{W}\!</math> that act on propositions <math>F\!</math> or on transformations <math>F\!</math> to yield the corresponding operator maps <math>\mathsf{W}F.\!</math> The operator results then become the subject of a series of further stages of analysis, which take them apart into their propositional components, rendering them as a set of purely logical constituents. After this is done, all the parts are then re-integrated to reconstruct the original object in the light of a more complete understanding, at least in ways that enable one to appreciate certain aspects of it with fresh insight.<br />
<br />
* '''Remark on Strategy.''' At this point we run into a set of conceptual difficulties that force us to make a strategic choice in how we proceed. Part of the problem can be remedied by extending our discussion of tacit extensions to the transformational context. But the troubles that remain are much more obstinate and lead us to try two different types of solution. The approach that we develop first makes use of a variant type of extension operator, the ''trope extension'', to be defined below. This method is more conservative and requires less preparation, but has features which make it seem unsatisfactory in the long run. A more radical approach, but one with a better hope of long term success, makes use of the notion of ''contingency spaces''. These are an even more generous type of extended universe than the kind we currently use, but are defined subject to certain internal constraints. The extra work needed to set up this method forces us to put it off to a later stage. However, as a compromise, and to prepare the ground for the next pass, we call attention to the various conceptual difficulties as they arise along the way and try to give an honest estimate of how well our first approach deals with them.<br />
<br />
We now describe in general terms the particular operators that are instrumental to this form of analysis. The main series of operators all have the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathsf{W}<br />
& : &<br />
( U^\bullet \to X^\bullet )<br />
& \to &<br />
( \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet )<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
If we assume that the source universe <math>U^\bullet</math> and the target universe <math>X^\bullet</math> have finite dimensions <math>n\!</math> and <math>k,\!</math> respectively, then each operator <math>\mathsf{W}\!</math> is encompassed by the same abstract type:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathsf{W}<br />
& : &<br />
( [\mathbb{B}^n] \to [\mathbb{B}^k] )<br />
& \to &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k \times \mathbb{D}^k] )<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Since the range features of the operator result <math>\mathsf{W}F : [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k \times \mathbb{D}^k]</math> can be sorted by their ordinary versus differential qualities and the component maps can be examined independently, the complete operator <math>\mathsf{W}\!</math> can be separated accordingly into two components, in the form <math>\mathsf{W} = (\boldsymbol\varepsilon, \mathrm{W}).\!</math> Given a fixed context of source and target universes, <math>\boldsymbol\varepsilon\!</math> is always the same type of operator, a multiple component version of the tacit extension operators that were described earlier. In this context <math>\boldsymbol\varepsilon\!</math> has the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{array}{lccccc}<br />
\text{Concrete type}<br />
& \boldsymbol\varepsilon<br />
& : &<br />
( U^\bullet \to X^\bullet )<br />
& \to &<br />
( \mathrm{E}U^\bullet \to X^\bullet )<br />
\\[10pt]<br />
\text{Abstract type}<br />
& \boldsymbol\varepsilon<br />
& : &<br />
( [\mathbb{B}^n] \to [\mathbb{B}^k] )<br />
& \to &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k] )<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
On the other hand, the operator <math>\mathrm{W}\!</math> is specific to each <math>\mathsf{W}.\!</math> In this context <math>\mathrm{W}\!</math> always has the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{array}{lccccc}<br />
\text{Concrete type}<br />
& W<br />
& : &<br />
( U^\bullet \to X^\bullet )<br />
& \to &<br />
( \mathrm{E}U^\bullet \to \mathrm{d}X^\bullet )<br />
\\[10pt]<br />
\text{Abstract type}<br />
& W<br />
& : &<br />
( [\mathbb{B}^n] \to [\mathbb{B}^k] )<br />
& \to &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{D}^k] )<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
In the types just assigned to <math>\boldsymbol\varepsilon\!</math> and <math>\mathrm{W}\!</math> and by implication to their results <math>\boldsymbol\varepsilon F\!</math> and <math>\mathrm{W}F,\!</math> we have listed the most restrictive ranges defined for them rather than the more expansive target spaces that subsume these ranges. When there is need to recognize both, we may use type indications like the following:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\boldsymbol\varepsilon F<br />
& : &<br />
( \mathrm{E}U^\bullet \to X^\bullet \subseteq \mathrm{E}X^\bullet )<br />
& \cong &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k] \subseteq [\mathbb{B}^k \times \mathbb{D}^k] )<br />
\\[10pt]<br />
WF<br />
& : &<br />
( \mathrm{E}U^\bullet \to \mathrm{d}X^\bullet \subseteq \mathrm{E}X^\bullet )<br />
& \cong &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{D}^k] \subseteq [\mathbb{B}^k \times \mathbb{D}^k] )<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Hopefully, though, a general appreciation of these subsumptions will prevent us from having to make such declarations more often than absolutely necessary.<br />
<br />
In giving names to these operators we try to preserve as much of the traditional nomenclature and as many of the classical associations as possible. The chief difficulty in doing this is occasioned by the distinction between the &ldquo;sans&nbsp;serif&rdquo; operators <math>\mathsf{W}\!</math> and their &ldquo;serified&rdquo; components <math>\mathrm{W},\!</math> which forces us to find two distinct but parallel sets of terminology. Here is a plan to that purpose. First, the component operators <math>\mathrm{W}\!</math> are named by analogy with the corresponding operators in the classical difference calculus. Next, the complete operators <math>\mathsf{W} = (\boldsymbol\varepsilon, \mathrm{W})</math> are assigned titles according to their roles in a geometric or trigonometric allegory, if only to ensure that the tangent functor, that belongs to this family and whose exposition we are still working toward, comes out fit with its customary name. Finally, the operator results <math>\mathsf{W}F\!</math> and <math>\mathrm{W}F\!</math> can be fixed in our frame of reference by tethering the operative adjective for <math>\mathsf{W}\!</math> or <math>\mathrm{W}\!</math> to the anchoring epithet &ldquo;map&rdquo;, in conformity with an already standard practice.<br />
<br />
=====The Secant Operator : '''E'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Mr. Peirce, after pointing out that our beliefs are really rules for action, said that, to develop a thought's meaning, we need only determine what conduct it is fitted to produce: that conduct is for us its sole significance.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 46]<br />
|}<br />
<br />
Figures&nbsp;33-i and 33-ii depict two stages in the form of analysis that will be applied to transformations throughout the remainder of this study. From now on our interest is staked on an operator denoted <math>{}^{\backprime\backprime} \mathsf{E} {}^{\prime\prime},\!</math> which receives the principal investment of analytic attention, and on the constituent parts of <math>\mathsf{E},\!</math> which derive their shares of significance as developed by the analysis. In the sequel, we refer to <math>\mathsf{E}\!</math> as the ''secant operator'', taking it for granted that a context has been chosen that defines its type. The secant operator has the component description <math>\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}),\!</math> and its active ingredient <math>\mathrm{E}\!</math> is known as the ''enlargement operator''. (Here, we name <math>\mathrm{E}\!</math> after the literal ancestor of the shift operator in the calculus of finite differences, defined so that <math>\mathrm{E}f(x) = f(x+1)\!</math> for any suitable function <math>f,\!</math> though of course the logical analogue that we take up here must have a rather different definition.)<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
U% $E$ $E$U% $E$U% $E$U%<br />
o------------------>o============o============o<br />
| | | |<br />
| | | |<br />
| | | |<br />
| | | |<br />
F | | $E$F = | $d$^0.F + | $r$^0.F<br />
| | | |<br />
| | | |<br />
| | | |<br />
v v v v<br />
o------------------>o============o============o<br />
X% $E$ $E$X% $E$X% $E$X%<br />
<br />
Figure 33-i. Analytic Diagram (1)<br />
</pre><br />
|}<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
U% $E$ $E$U% $E$U% $E$U% $E$U%<br />
o------------------>o============o============o============o<br />
| | | | |<br />
| | | | |<br />
| | | | |<br />
| | | | |<br />
F | | $E$F = | $d$^0.F + | $d$^1.F + | $r$^1.F<br />
| | | | |<br />
| | | | |<br />
| | | | |<br />
v v v v v<br />
o------------------>o============o============o============o<br />
X% $E$ $E$X% $E$X% $E$X% $E$X%<br />
<br />
Figure 33-ii. Analytic Diagram (2)<br />
</pre><br />
|}<br />
<br />
In its action on universes <math>\mathsf{E}\!</math> yields the same result as <math>\mathrm{E},\!</math> a fact that can be expressed in equational form by writing <math>\mathsf{E}U^\bullet = \mathrm{E}U^\bullet\!</math> for any universe <math>U^\bullet.\!</math> Notice that the extended universes across the top and bottom of the diagram are indicated to be strictly identical, rather than requiring a corresponding decomposition for them. In a certain sense, the functional parts of <math>\mathsf{E}F\!</math> are partitioned into separate contexts that have to be re-integrated again, but the best image to use is that of making transparent copies of each universe and then overlapping their functional contents once more at the conclusion of the analysis, as suggested by the graphic conventions that are used at the top of Figure&nbsp;30.<br />
<br />
Acting on a transformation <math>F\!</math> from universe <math>U^\bullet\!</math> to universe <math>X^\bullet,\!</math> the operator <math>\mathsf{E}\!</math> determines a transformation <math>\mathsf{E}F\!</math> from <math>\mathsf{E}U^\bullet\!</math> to <math>\mathsf{E}X^\bullet.\!</math> The map <math>\mathsf{E}F\!</math> forms the main body of evidence to be investigated in performing a differential analysis of <math>F.\!</math> Because we shall frequently be focusing on small pieces of this map for considerable lengths of time, and consequently lose sight of the &ldquo;big picture&rdquo;, it is critically important to emphasize that the map <math>\mathsf{E}F\!</math> is a transformation that determines a relation from one extended universe into another. This means that we should not be satisfied with our understanding of a transformation <math>F\!</math> until we can lay out the full &ldquo;parts diagram&rdquo; of <math>\mathsf{E}F\!</math> along the lines of the generic frame in Figure&nbsp;30.<br />
<br />
Working within the confines of propositional calculus, it is possible to give an elementary definition of <math>\mathsf{E}F\!</math> by means of a system of propositional equations, as we now describe.<br />
<br />
Given a transformation<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>F = (F_1, \ldots, F_k) : \mathbb{B}^n \to \mathbb{B}^k\!</math><br />
|}<br />
<br />
of concrete type<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>F : [u_1, \ldots, u_n] \to [x_1, \ldots, x_k],\!</math><br />
|}<br />
<br />
the transformation<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\mathsf{E}F = (F_1, \ldots, F_k, \mathrm{E}F_1, \ldots, \mathrm{E}F_k) : \mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B}^k \times \mathbb{D}^k\!</math><br />
|}<br />
<br />
of concrete type<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\mathsf{E}F : [u_1, \dots, u_n, \mathrm{d}u_1, \dots, \mathrm{d}u_n] \to [x_1, \ldots, x_k, \mathrm{d}x_1, \ldots, \mathrm{d}x_k]\!</math><br />
|}<br />
<br />
is defined by means of the following system of logical equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\\[16pt]<br />
\mathrm{d}x_1<br />
& = & \mathrm{E}F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1 + \mathrm{d}u_1, \ldots, u_n + \mathrm{d}u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
\mathrm{d}x_k<br />
& = & \mathrm{E}F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1 + \mathrm{d}u_1, \ldots, u_n + \mathrm{d}u_n)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
It is important to note that this system of equations can be read as a conjunction of equational propositions, in effect, as a single proposition in the universe of discourse generated by all the named variables. Specifically, this is the universe of discourse over <math>2(n+k)\!</math> variables denoted by:<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
\mathrm{E}[\mathcal{U} \cup \mathcal{X}]<br />
& = &<br />
[u_1, \ldots, u_n, ~ x_1, \ldots, x_k, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n, ~ \mathrm{d}x_1, \ldots, \mathrm{d}x_k].<br />
\end{matrix}</math><br />
|}<br />
<br />
In this light, it should be clear that the system of equations defining <math>\mathsf{E}F\!</math> embodies, in a higher rank and differentially extended version, an analogy with the process of thematization that we treated earlier for propositions of type <math>F : \mathbb{B}^n \to \mathbb{B}.\!</math><br />
<br />
The entire collection of constraints that is represented in the above system of equations may be abbreviated by writing <math>\mathsf{E}F = (\boldsymbol\varepsilon F, \mathrm{E}F),\!</math> for any map <math>F.\!</math> This is tantamount to regarding <math>\mathsf{E}\!</math> as a complex operator, <math>\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}),\!</math> with a form of application that distributes each component of the operator to work on each component of the operand, as follows:<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
\mathsf{E}F<br />
& = &<br />
(\boldsymbol\varepsilon, \mathrm{E})F<br />
& = &<br />
(\boldsymbol\varepsilon F, \mathrm{E}F)<br />
& = &<br />
(\boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k, ~ \mathrm{E}F_1, \ldots, \mathrm{E}F_k).<br />
\end{matrix}</math><br />
|}<br />
<br />
Quite a lot of &ldquo;thematic infrastructure&rdquo; or interpretive information is being swept under the rug in the use of such abbreviations. When confusion arises about the meaning of such constructions, one always has recourse to the defining system of equations, in its totality a purely propositional expression. This means that the parenthesized argument lists, that were used in this context to build an image of multi-component transformations, should not be expected to determine a well-defined product in themselves but only to serve as reminders of the prior thematic decisions (choices of variable names, etc.) that have to be made in order to determine one. Accordingly, the argument list notation can be regarded as a kind of ''thematic frame'', an interpretive storage device that preserves the proper associations of concrete logical features between the extended universes at the source and target of <math>\mathsf{E}F.\!</math><br />
<br />
The generic notations <math>\mathsf{d}^0\!F, \mathsf{d}^1\!F, \ldots, \mathsf{d}^m\!F\!</math> in Figure&nbsp;33 refer to the increasing orders of differentials that are extracted in the course of analyzing <math>F.\!</math> When the analysis is halted at a partial stage of development, notations like <math>\mathsf{r}^0\!F, \mathsf{r}^1\!F, \ldots, \mathsf{r}^m\!F\!</math> may be used to summarize the contributions to <math>\mathsf{E}F\!</math> that remain to be analyzed. The Figure illustrates a convention that makes <math>\mathsf{r}^m\!F,\!</math> in effect, the sum of all differentials of order strictly greater than <math>m.\!</math><br />
<br />
We next discuss the operators that figure into this form of analysis, describing their effects on transformations. In simplified or specialized contexts these operators tend to take on a variety of different names and notations, some of whose number we introduce along the way.<br />
<br />
=====The Radius Operator : '''e'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
And the tangible fact at the root of all our thought-distinctions, however subtle, is that there is no one of them so fine as to consist in anything but a possible difference of practice.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 46]<br />
|}<br />
<br />
The operator identified as <math>\mathrm{d}^0\!</math> in the analytic diagram (Figure&nbsp;33) has the sole purpose of creating a proxy for <math>F\!</math> in the appropriately extended context. Construed in terms of its broadest components, <math>\mathrm{d}^0\!</math> is equivalent to the doubly tacit extension operator <math>(\boldsymbol\varepsilon, \boldsymbol\varepsilon),\!</math> in recognition of which let us redub it as <math>{}^{\backprime\backprime} \mathsf{e} {}^{\prime\prime}.\!</math> Pursuing a geometric analogy, we may refer to <math>\mathsf{e} =(\boldsymbol\varepsilon, \boldsymbol\varepsilon) = \mathrm{d}^0\!</math> as the ''radius operator''. The operation intended by all of these forms is defined by the following equation:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathsf{e}F<br />
& = &<br />
(\boldsymbol\varepsilon, \boldsymbol\varepsilon)F<br />
\\[4pt]<br />
& = &<br />
(\boldsymbol\varepsilon F, ~ \boldsymbol\varepsilon F)<br />
\\[4pt]<br />
& = &<br />
(\boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k, ~ \boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k).<br />
\end{array}</math><br />
|}<br />
<br />
which is tantamount to the system of equations below.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\\[16pt]<br />
\mathrm{d}x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
\mathrm{d}x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
=====The Phantom of the Operators : '''&eta;'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
I was wondering what the reason could be, when I myself raised my head and everything within me seemed drawn towards the Unseen, ''which was playing the most perfect music''!<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 81]<br />
|}<br />
<br />
We now describe an operator whose persistent but elusive action behind the scenes, whose slightly twisted and ambivalent character, and whose fugitive disposition, caught somewhere in flight between the arrantly negative and the positive but errant intent, has cost us some painstaking trouble to detect. In the end we shall place it among the other extensions and projections, as a shade among shadows, of muted tones and motley hue, that adumbrates its own thematic frame and paradoxically lights the way toward a whole new spectrum of values.<br />
<br />
Given a transformation <math>F : [u_1, \ldots, u_n] \to [x_1, \dots, x_k],\!</math> we often have call to consider a family of related transformations, all having the form:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>F^\dagger : [u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n] \to [\mathrm{d}x_1, \dots, \mathrm{d}x_k].\!</math><br />
|}<br />
<br />
The operator <math>\eta</math> is introduced to deal with the simplest one of these maps:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\eta F : [u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n] \to [\mathrm{d}x_1, \ldots \mathrm{d}x_k],\!</math><br />
|}<br />
<br />
which is defined by the following equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
\mathrm{d}x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
In effect, the operator <math>\eta\!</math> is nothing but the stand-alone version of a procedure that is otherwise invoked subordinate to the work of the radius operator <math>\mathsf{e}.\!</math> Operating independently, <math>\eta\!</math> achieves precisely the same results that the second <math>\boldsymbol\varepsilon\!</math> in <math>(\boldsymbol\varepsilon, \boldsymbol\varepsilon)\!</math> accomplishes by working within the context of its ordered pair thematic frame. From this point on, because the use of <math>\boldsymbol\varepsilon\!</math> and <math>\eta\!</math> in this setting combines the aims of both the tacit and the thematic extensions, and because <math>\eta\!</math> reflects in regard to <math>\boldsymbol\varepsilon\!</math> little more than the application of a differential twist, a mere turn of phrase, we refer to <math>\eta\!</math> as the ''trope extension'' operator.<br />
<br />
=====The Chord Operator : '''D'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
What difference would it practically make to any one if this notion rather than that notion were true? If no practical difference whatever can be traced, then the alternatives mean practically the same thing, and all dispute is idle.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 45]<br />
|}<br />
<br />
Next we discuss an operator that is always immanent in this form of analysis, and remains implicitly present in the entire proceeding. It may appear once as a record: a relic or revenant that reprises the reminders of an earlier stage of development. Or it may appear always as a resource: a reserve or redoubt that caches in advance an echo of what remains to be played out, cleared up, and requited in full at a future stage. And all of this remains true whether or not we recall the key at any time, and whether or not the subtending theme is recited explicitly at any stage of play.<br />
<br />
This is the operator that is referred to as <math>\mathsf{r}^0\!</math> in the initial stage of analysis (Figure&nbsp;33-i) and that is expanded as <math>\mathsf{d}^1 + \mathsf{r}^1\!</math> in the subsequent step (Figure&nbsp;33-ii). In congruence, but not quite harmony with our allusions of analogy that are not quite geometry, we call this the ''chord operator'' and denote it <math>\mathsf{D}.\!</math> In the more casual terms that are here introduced, <math>\mathsf{D}</math> is defined as the remainder of <math>\mathsf{E}\!</math> and <math>\mathsf{e}\!</math> and it assigns a due measure to each undertone of accord or discord that is struck between the note of enterprise <math>\mathsf{E}\!</math> and the bar of exigency <math>\mathsf{e}.\!</math><br />
<br />
The tension between these counterposed notions, in balance transient but regular in stridence, may be refracted along familiar lines, though never by any such fraction resolved. In this style we write <math>\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}),\!</math> calling <math>\mathrm{D}\!</math> the ''difference operator'' and noting that it plays a role in this realm of mutable and diverse discourse that is analogous to the part taken by the discrete difference operator in the ordinary difference calculus. Finally, we should note that the chord <math>\mathsf{D}\!</math> is not one that need be lost at any stage of development. At the <math>m^\text{th}\!</math> stage of play it can always be reconstituted in the following form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathsf{D}<br />
& = & \mathsf{E} - \mathsf{e}<br />
\\[6pt]<br />
& = & \mathsf{r}^0<br />
\\[6pt]<br />
& = & \mathsf{d}^1 + \mathsf{r}^1<br />
\\[6pt]<br />
& = & \mathsf{d}^1 + \ldots + \mathsf{d}^m + \mathsf{r}^m<br />
\\[6pt]<br />
& = & \displaystyle \sum_{i=1}^m \mathsf{d}^i + \mathsf{r}^m<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====The Tangent Operator : '''T'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
They take part in scenes of whose significance they have no inkling. They are merely tangent to curves of history the beginnings and ends and forms of which pass wholly beyond their ken. So we are tangent to the wider life of things.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 300]<br />
|}<br />
<br />
The operator tagged as <math>\mathsf{d}^1\!</math> in the analytic diagram (Figure&nbsp;33) is called the ''tangent operator'' and is usually denoted in this text as <math>\mathsf{d}\!</math> or <math>\mathsf{T}.\!</math> Because it has the properties required to qualify as a functor, namely, preserving the identity element of the composition operation and the articulated form of every composition of transformations, it also earns the title of a ''tangent functor''. According to the custom adopted here, we dissect it as <math>\mathsf{T} = \mathsf{d} = (\boldsymbol\varepsilon, \mathrm{d}),\!</math> where <math>\mathrm{d}\!</math> is the operator that yields the first order differential <math>\mathrm{d}F\!</math> when applied to a transformation <math>F,\!</math> and whose name is legion.<br />
<br />
Figure&nbsp;34 illustrates a stage of analysis where we ignore everything but the tangent functor <math>\mathsf{T}\!</math> and attend to it chiefly as it bears on the first order differential <math>\mathrm{d}F\!</math> in the analytic expansion of <math>F.\!</math> In this situation we often refer to the extended universes <math>\mathrm{E}U^\bullet\!</math> and <math>\mathrm{E}X^\bullet\!</math> under the equivalent designations <math>\mathsf{T}U^\bullet\!</math> and <math>\mathsf{T}X^\bullet,\!</math> respectively. The purpose of the tangent functor <math>\mathsf{T}\!</math> is to extract the tangent map <math>\mathsf{T}F\!</math> at each point of <math>U^\bullet,\!</math> and the tangent map <math>\mathsf{T}F = (\boldsymbol\varepsilon, \mathrm{d})F\!</math> tells us not only what the transformation <math>F\!</math> is doing at each point of the universe <math>U^\bullet\!</math> but also what <math>F\!</math> is doing to states in the neighborhood of that point, approximately, linearly, and relatively speaking.<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
U% $T$ $T$U% $T$U%<br />
o------------------>o============o<br />
| | |<br />
| | |<br />
| | |<br />
| | |<br />
F | | $T$F = | <!e!, d> F<br />
| | |<br />
| | |<br />
| | |<br />
v v v<br />
o------------------>o============o<br />
X% $T$ $T$X% $T$X%<br />
<br />
Figure 34. Tangent Functor Diagram<br />
</pre><br />
|}<br />
<br />
* '''NB.''' There is one aspect of the preceding construction that remains especially problematic. Why did we define the operators <math>\mathrm{W}\!</math> in <math>\{ \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}\!</math> so that the ranges of their resulting maps all fall within the realms of differential quality, even fabricating a variant of the tacit extension operator to have that character? Clearly, not all of the operator maps <math>\mathrm{W}F\!</math> have equally good reasons for placing their values in differential stocks. The reason for it appears to be that, without doing this, we cannot justify the comparison and combination of their functional values in the various analytic steps. By default, only those values in the same functional component can be brought into algebraic modes of interaction. Up till now the only mechanism provided for their broader association has been a purely logical one, their common placement in a target universe of discourse, but the task of converting this logical circumstance into algebraic forms of application has not yet been taken up.<br />
<br />
===Transformations of Type '''B'''<sup>2</sup> &rarr; '''B'''<sup>1</sup>===<br />
<br />
To study the effects of these analytic operators in the simplest possible setting, let us revert to a still more primitive case. Consider the singular proposition <math>J(u, v)= u\!\cdot\!v,\!</math> regarded either as the functional product of the maps <math>u\!</math> and <math>v\!</math> or as the logical conjunction of the features <math>u\!</math> and <math>v,\!</math> a map whose fiber of truth <math>J^{-1}(1)\!</math> picks out the single cell of that logical description in the universe of discourse <math>U^\bullet.\!</math> Thus <math>J,\!</math> or <math>u\!\cdot\!v,\!</math> may be treated as another name for the point whose coordinates are <math>(1, 1)\!</math> in <math>U^\bullet.\!</math><br />
<br />
====Analytic Expansion of Conjunction====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
<p>In her sufferings she read a great deal and discovered that she had lost something, the possession of which she had previously not been much aware of: a&nbsp;soul.</p><br />
<br />
<p>What is that? It is easily defined negatively: it is simply what curls up and hides when there is any mention of algebraic series.</p><br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 118]<br />
|}<br />
<br />
Figure&nbsp;35 pictures the form of conjunction <math>J : \mathbb{B}^2 \to \mathbb{B}\!</math> as a transformation from the <math>2\!</math>-dimensional universe <math>[u, v]\!</math> to the <math>1\!</math>-dimensional universe <math>[x].\!</math> This is a subtle but significant change of viewpoint on the proposition, attaching an arbitrary but concrete quality to its functional value. Using the language introduced earlier, we can express this change by saying that the proposition <math>J : \langle u, v \rangle \to \mathbb{B}\!</math> is being recast into the thematized role of a transformation <math>J : [u, v] \to [x],\!</math> where the new variable <math>x\!</math> takes the part of a thematic variable <math>\check{J}.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 35 -- A Conjunction Viewed as a Transformation.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 35.} ~~ \text{Conjunction as Transformation}\!</math><br />
|}<br />
<br />
=====Tacit Extension of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
I teach straying from me, yet who can stray from me?<br><br />
I follow you whoever you are from the present hour;<br><br />
My words itch at your ears till you understand them.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 83]<br />
|}<br />
<br />
Earlier we defined the tacit extension operators <math>\boldsymbol\varepsilon : X^\bullet \to Y^\bullet\!</math> as maps embedding each proposition of a given universe <math>X^\bullet~\!</math> in a more generously given universe <math>Y^\bullet \supset X^\bullet.\!</math> Of immediate interest are the tacit extensions <math>\boldsymbol\varepsilon : U^\bullet \to \mathrm{E}U^\bullet,\!</math> that locate each proposition of <math>U^\bullet\!</math> in the enlarged context of <math>\mathrm{E}U^\bullet.\!</math> In its application to the propositional conjunction <math>J = u\!\cdot\!v</math> in <math>[u, v],\!</math> the tacit extension operator <math>\boldsymbol\varepsilon\!</math> yields the proposition <math>\boldsymbol\varepsilon J\!</math> in <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v].\!</math> The extended proposition <math>\boldsymbol\varepsilon J\!</math> may be computed according to the scheme in Table&nbsp;36, in effect doing nothing more that conjoining a tautology of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> to <math>J\!</math> in <math>U^\bullet.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 36.} ~~ \text{Computation of}~ \boldsymbol\varepsilon J\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\boldsymbol\varepsilon J & = & J {}_{^\langle} u, v {}_{^\rangle}<br />
\\[4pt]<br />
& = & u \cdot v<br />
\\[4pt]<br />
& = & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{ } \mathrm{d}v \texttt{ }<br />
& + & u \cdot v \cdot \texttt{ } \mathrm{d}u \texttt{ } \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \cdot v \cdot \texttt{ } \mathrm{d}u \texttt{ } \cdot \texttt{ } \mathrm{d}v \texttt{ }<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{array}{*{4}{l}}<br />
\boldsymbol\varepsilon J<br />
& = && u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & u \cdot v \cdot \texttt{~} \mathrm{d}u \texttt{~} \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & u \cdot v \cdot \texttt{~} \mathrm{d}u \texttt{~} \cdot \texttt{~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
The lower portion of the Table contains the dispositional features of <math>\boldsymbol\varepsilon J\!</math> arranged in such a way that the variety of ordinary features spreads across the rows and the variety of differential features runs through the columns. This organization serves to facilitate pattern matching in the remainder of our computations. Again, the tacit extension is usually so trivial a concern that we do not always bother to make an explicit note of it, taking it for granted that any function <math>F\!</math> being employed in a differential context is equivalent to <math>\boldsymbol\varepsilon F\!</math> for a suitable <math>\boldsymbol\varepsilon.\!</math><br />
<br />
Figures&nbsp;37-a through 37-d present several pictures of the proposition <math>J\!</math> and its tacit extension <math>\boldsymbol\varepsilon J.\!</math> Notice in these Figures how <math>\boldsymbol\varepsilon J\!</math> in <math>\mathrm{E}U^\bullet\!</math> visibly extends <math>J\!</math> in <math>U^\bullet\!</math> by annexing to the indicated cells of <math>J\!</math> all the arcs that exit from or flow out of them. In effect, this extension attaches to these cells all the dispositions that spring from them, in other words, it attributes to these cells all the conceivable changes that are their issue.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-a -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-a.} ~~ \text{Tacit Extension of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-b -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-b.} ~~ \text{Tacit Extension of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-c -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-c.} ~~ \text{Tacit Extension of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-d -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-d.} ~~ \text{Tacit Extension of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
The computational scheme shown in Table&nbsp;36 treated <math>J\!</math> as a proposition in <math>U^\bullet\!</math> and formed <math>\boldsymbol\varepsilon J\!</math> as a proposition in <math>\mathrm{E}U^\bullet.\!</math> When <math>J\!</math> is regarded as a mapping <math>J : U^\bullet \to X^\bullet\!</math> then <math>\boldsymbol\varepsilon J\!</math> must be obtained as a mapping <math>\boldsymbol\varepsilon J : \mathrm{E}U^\bullet \to X^\bullet.\!</math> By default, the tacit extension of the map <math>J : [u, v] \to [x]\!</math> is naturally taken to be a particular map,<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\boldsymbol\varepsilon J : [u, v, \mathrm{d}u, \mathrm{d}v] \to [x] \subseteq [x, \mathrm{d}x],\!</math><br />
|}<br />
<br />
namely, the one that looks like <math>J\!</math> when painted in the frame of the extended source universe and that takes the same thematic variable in the extended target universe as the one that <math>J\!</math> already takes.<br />
<br />
But the choice of a particular thematic variable, for example <math>x\!</math> for <math>\check{J},\!</math> is a shade more arbitrary than the choice of original variable names <math>\{ u, v \},\!</math> so the map we are calling the ''trope extension'',<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\eta J : [u, v, \mathrm{d}u, \mathrm{d}v] \to [\mathrm{d}x] \subseteq [x, \mathrm{d}x],\!</math><br />
|}<br />
<br />
since it looks just the same as <math>\boldsymbol\varepsilon J\!</math> in the way its fibers paint the source domain, belongs just as fully to the family of tacit extensions, generically considered.<br />
<br />
These considerations have the practical consequence that all of our computations and illustrations of <math>\boldsymbol\varepsilon J\!</math> perform the double duty of capturing <math>\eta J\!</math> as well. In other words, we are saved the work of carrying out calculations and drawing figures for the trope extension <math>\eta J,\!</math> because it would be identical to the work already done for <math>\boldsymbol\varepsilon J.\!</math> Since the computations given for <math>\boldsymbol\varepsilon J\!</math> are expressed solely in terms of the variables <math>\{ u, v, \mathrm{d}u, \mathrm{d}v \},\!</math> they work equally well for finding <math>\eta J.\!</math> Further, since each of the above Figures shows only how the level sets of <math>\boldsymbol\varepsilon J\!</math> partition the extended source universe <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v],\!</math> all of them serve equally well as portraits of <math>\eta J.\!</math><br />
<br />
=====Enlargement Map of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
No one could have established the existence of any details that might not just as well have existed in earlier times too; but all the relations between things had shifted slightly. Ideas that had once been of lean account grew fat.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 62]<br />
|}<br />
<br />
The enlargement map <math>\mathrm{E}J\!</math> is computed from the proposition <math>J\!</math> by making a particular class of formal substitutions for its variables, in this case <math>u + \mathrm{d}u\!</math> for <math>u\!</math> and <math>v + \mathrm{d}v\!</math> for <math>v,\!</math> and afterwards expanding the result in whatever way is found convenient.<br />
<br />
Table&nbsp;38 shows a typical scheme of computation, following a systematic method of exploiting boolean expansions over selected variables and ultimately developing <math>\mathrm{E}J\!</math> over the cells of <math>[u, v].\!</math> The critical step of this procedure uses the facts that <math>\texttt{(} 0, x \texttt{)} = 0 + x = x\!</math> and <math>\texttt{(} 1, x \texttt{)} = 1 + x = \texttt{(} x \texttt{)}\!</math> for any boolean variable <math>x.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 38.} ~~ \text{Computation of}~ \mathrm{E}J ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{E}J & = & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}<br />
\\[4pt]<br />
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
& = & \texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot J_{(1 + \mathrm{d}u, 1 + \mathrm{d}v)}<br />
& + & \texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot J_{(1 + \mathrm{d}u, \mathrm{d}v)}<br />
& + & \texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot J_{(\mathrm{d}u, 1 + \mathrm{d}v)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot J_{(\mathrm{d}u, \mathrm{d}v)}<br />
\\[4pt]<br />
& = &<br />
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot J_{(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})}<br />
& + &<br />
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot J_{(\texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v)}<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot J_{(\mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)})}<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot J_{(\mathrm{d}u, \mathrm{d}v)}<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{E}J<br />
& = &<br />
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&& + &<br />
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{ } \mathrm{d}v \texttt{ }<br />
\\[4pt]<br />
&&&&& + &<br />
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot \texttt{ } \mathrm{d}u \texttt{ } \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&&&&&& + &<br />
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \texttt{ } \mathrm{d}u \texttt{ }~\texttt{ } \mathrm{d}v \texttt{ }<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;39 exhibits another method that happens to work quickly in this particular case, using distributive laws to multiply things out in an algebraic manner, arranging the notations of feature and fluxion according to a scale of simple character and degree. Proceeding this way leads through an intermediate step which, in chiming the changes of ordinary calculus, should take on a familiar ring. Consequential properties of exclusive disjunction then carry us on to the concluding line.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 39.} ~~ \text{Computation of}~ \mathrm{E}J ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{c}}<br />
\mathrm{E}J<br />
& = & (u + \mathrm{d}u) \cdot (v + \mathrm{d}v)<br />
\\[6pt]<br />
& = & u \cdot v<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = &<br />
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{ } \mathrm{d}v \texttt{ }<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot \texttt{ } \mathrm{d}u \texttt{ } \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \texttt{ } \mathrm{d}u \texttt{ }~\texttt{ } \mathrm{d}v \texttt{ }<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;40-a through 40-d present several views of the enlarged proposition <math>\mathrm{E}J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-a -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-a.} ~~ \text{Enlargement of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-b -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-b.} ~~ \text{Enlargement of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-c -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-c.} ~~ \text{Enlargement of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-d -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-d.} ~~ \text{Enlargement of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
An intuitive reading of the proposition <math>\mathrm{E}J\!</math> becomes available at this point. Recall that propositions in the extended universe <math>\mathrm{E}U^\bullet\!</math> express the ''dispositions'' of a system and the constraints that are placed on them. In other words, a differential proposition in <math>\mathrm{E}U^\bullet\!</math> can be read as referring to various changes that a system might undergo in and from its various states. In particular, we can understand <math>\mathrm{E}J\!</math> as a statement that tells us what changes need to be made with regard to each state in the universe of discourse in order to reach the ''truth'' of <math>J,\!</math> that is, the region of the universe where <math>J\!</math> is true. This interpretation is visibly clear in the Figures above and appeals to the imagination in a satisfying way but it has the added benefit of giving fresh meaning to the original name of the shift operator <math>\mathrm{E}.\!</math> Namely, <math>\mathrm{E}J\!</math> can be read as a proposition that ''enlarges'' on the meaning of <math>J,\!</math> in the sense of explaining its practical bearings and clarifying what it means in terms of actions and effects &mdash; the available options for differential action and the consequential effects that result from each choice.<br />
<br />
Read this way, the enlargement <math>\mathrm{E}J\!</math> has strong ties to the normal use of <math>J,\!</math> no matter whether it is understood as a proposition or a function, namely, to act as a figurative device for indicating the models of <math>J,\!</math> in effect, pointing to the interpretive elements in its fiber of truth <math>J^{-1}(1).\!</math> It is this kind of &ldquo;use&rdquo; that is often contrasted with the &ldquo;mention&rdquo; of a proposition, and thereby hangs a tale.<br />
<br />
=====Digression : Reflection on Use and Mention=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Reflection is turning a topic over in various aspects and in various lights so that nothing significant about it shall be overlooked &mdash; almost as one might turn a stone over to see what its hidden side is like or what is covered by it.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 57]<br />
|}<br />
<br />
The contrast drawn in logic between the ''use'' and the ''mention'' of a proposition corresponds to the difference that we observe in functional terms between using <math>{}^{\backprime\backprime} J \, {}^{\prime\prime}\!</math> to indicate the region <math>J^{-1}(1)\!</math> and using <math>{}^{\backprime\backprime} J \, {}^{\prime\prime}\!</math> to indicate the function <math>J.\!</math> You may think that one of these uses ought to be proscribed, and logicians are quick to prescribe against their confusion. But there seems to be no likelihood in practice that their interactions can be avoided. If the name <math>{}^{\backprime\backprime} J \, {}^{\prime\prime}\!</math> is used as a sign of the function <math>J,\!</math> and if the function <math>J\!</math> has its use in signifying something else, as would constantly be the case when some future theory of signs has given a functional meaning to every sign whatsoever, then is not <math>J,\!</math> by transitivity a sign of the thing itself? There are, of course, two answers to this question. Not every act of signifying or referring need be transitive. Not every warrant or guarantee or certificate is automatically transferable, indeed, not many. Not every feature of a feature is a feature of the featuree. Otherwise, if a buffalo is white, and white is a color, then a buffalo would ''be'' a color.<br />
<br />
The logical or pragmatic distinction between use and mention is cogent and necessary, and so is the analogous functional distinction between determining a value and determining what determines that value, but so are the normal techniques that we use to make these distinctions apply flexibly in practice. The way that the hue and cry about use and mention is raised in logical discussions, you might be led to think that this single dimension of choices embraces the only kinds of use worth mentioning and the only kinds of mention worth using. It will constitute the expeditionary and taxonomic tasks of that future theory of signs to explore and to classify the many other constellations and dimensions of use and mention that are yet to be opened up by the generative potential of full-fledged sign relations.<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
The well-known capacity that thoughts have &mdash; as doctors have discovered &mdash; for dissolving and dispersing those hard lumps of deep, ingrowing, morbidly entangled conflict that arise out of gloomy regions of the self probably rests on nothing other than their social and worldly nature, which links the individual being with other people and things; but unfortunately what gives them their power of healing seems to be the same as what diminishes the quality of personal experience in them.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 130]<br />
|}<br />
<br />
=====Difference Map of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
&ldquo;It doesn't matter what one does,&rdquo; the Man Without Qualities said to himself, shrugging his shoulders. &ldquo;In a tangle of forces like this it doesn't make a scrap of difference.&rdquo; He turned away like a man who has learned renunciation, almost indeed like a sick man who shrinks from any intensity of contact. And then, striding through his adjacent dressing-room, he passed a punching-ball that hung there; he gave it a blow far swifter and harder than is usual in moods of resignation or states of weakness.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 8]<br />
|}<br />
<br />
With the tacit extension map <math>\boldsymbol\varepsilon J\!</math> and the enlargement map <math>\mathrm{E}J\!</math> well in place, the difference map <math>\mathrm{D}J\!</math> can be computed along the lines displayed in Table&nbsp;41, ending up with an expansion of <math>\mathrm{D}J\!</math> over the cells of <math>[u, v].\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 41.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & \mathrm{E}J<br />
& + & \boldsymbol\varepsilon J<br />
\\[6pt]<br />
& = & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}<br />
& + & J_{(u, v)}<br />
\\[6pt]<br />
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \cdot v<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = &<br />
u \cdot v \cdot \qquad 0<br />
\\[6pt]<br />
& + &<br />
u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v<br />
& + &<br />
u \cdot \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v<br />
\\[6pt]<br />
& + &<br />
u \cdot v \cdot \texttt{~} \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
&&& + &<br />
\texttt{(} u \texttt{)} \cdot v \cdot \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
& + &<br />
u \cdot v \cdot \texttt{~} \mathrm{d}u \;\cdot\; \mathrm{d}v \texttt{~}<br />
&&&&& + &<br />
\texttt{(} u \texttt{)} \cdot \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = &<br />
u \cdot v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
u \cdot \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v<br />
& + &<br />
\texttt{(} u \texttt{)} \cdot v \cdot \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)} \cdot \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Alternatively, the difference map <math>\mathrm{D}J\!</math> can be expanded over the cells of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> to arrive at the formulation shown in Table&nbsp;42. The same development would be obtained from the previous Table by collecting terms in an alternate manner, along the rows rather than the columns in the middle portion of the Table.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 42.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & \boldsymbol\varepsilon J<br />
& + & \mathrm{E}J<br />
\\[6pt]<br />
& = & J_{(u, v)}<br />
& + & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}<br />
\\[6pt]<br />
& = & u \cdot v<br />
& + & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
& = & 0<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}J<br />
& = & 0<br />
& + & u \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Even more simply, the same result is reached by matching up the propositional coefficients of <math>\boldsymbol\varepsilon J</math> and <math>\mathrm{E}J\!</math> along the cells of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> and adding the pairs under boolean addition, that is, &ldquo;mod 2&rdquo;, where 1&nbsp;+&nbsp;1&nbsp;=&nbsp;0, as shown in Table&nbsp;43.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 43.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 3)}\!</math><br />
|<br />
<math>\begin{array}{*{5}{l}}<br />
\mathrm{D}J & = & \boldsymbol\varepsilon J & + & \mathrm{E}J<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\boldsymbol\varepsilon J<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ u \,\cdot\, v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~ u \;\cdot\; v \;\cdot\; \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u ~ \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} ~ v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot\, \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & ~~ 0 ~~ \,\cdot\, ~ \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~~ u ~ \,\cdot\, ~~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ ~ v ~~ \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
The difference map <math>\mathrm{D}J\!</math> can also be given a ''dispositional'' interpretation. First, recall that <math>\boldsymbol\varepsilon J\!</math> exhibits the dispositions to change from anywhere in <math>J\!</math> to anywhere at all in the universe of discourse and <math>\mathrm{E}J\!</math> exhibits the dispositions to change from anywhere in the universe to anywhere in <math>J.\!</math> Next, observe that each of these classes of dispositions may be divided in accordance with the case of <math>J\!</math> versus <math>\texttt{(} J \texttt{)}\!</math> that applies to their points of departure and destination, as shown below. Then, since the dispositions corresponding to <math>\boldsymbol\varepsilon J</math> and <math>\mathrm{E}J\!</math> have in common the dispositions to preserve <math>J,\!</math> their symmetric difference <math>\texttt{(} \boldsymbol\varepsilon J, \mathrm{E}J \texttt{)}\!</math> is made up of all the remaining dispositions, which are in fact disposed to cross the boundary of <math>J\!</math> in one direction or the other. In other words, we may conclude that <math>\mathrm{D}J\!</math> expresses the collective disposition to make a definite change with respect to <math>J,\!</math> no matter what value it holds in the current state of affairs.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
\boldsymbol\varepsilon J<br />
& = & \{ \text{Dispositions from}~ J ~\text{to}~ J \}<br />
& + & \{ \text{Dispositions from}~ J ~\text{to}~ \texttt{(} J \texttt{)} \}<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = & \{ \text{Dispositions from}~ J ~\text{to}~ J \}<br />
& + & \{ \text{Dispositions from}~ \texttt{(} J \texttt{)} ~\text{to}~ J \}<br />
\\[6pt]<br />
\mathrm{D}J<br />
& = & \{ \text{Dispositions from}~ J ~\text{to}~ \texttt{(} J \texttt{)} \}<br />
& + & \{ \text{Dispositions from}~ \texttt{(} J \texttt{)} ~\text{to}~ J \}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;44-a through 44-d illustrate the difference proposition <math>\mathrm{D}J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-a -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-a.} ~~ \text{Difference Map of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-b -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-b.} ~~ \text{Difference Map of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-c -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-c.} ~~ \text{Difference Map of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-d -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-d.} ~~ \text{Difference Map of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
=====Differential of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
By deploying discourse throughout a calendar, and by giving a date to each of its elements, one does not obtain a definitive hierarchy of precessions and originalities; this hierarchy is never more than relative to the systems of discourse that it sets out to evaluate.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Archaeology of Knowledge'', [Fou, 143]<br />
|}<br />
<br />
Finally, at long last, the differential proposition <math>\mathrm{d}J\!</math> can be gleaned from the difference proposition <math>\mathrm{D}J\!</math> by ranging over the cells of <math>[u, v]\!</math> and picking out the linear proposition of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> that is &ldquo;closest&rdquo; to the portion of <math>\mathrm{D}J\!</math> that touches on each point. The idea of distance that would give this definition unequivocal sense has been referred to in cautionary quotes, the kind we use to distance ourselves from taking a final position. There are obvious notions of approximation that suggest themselves, but finding one that can be justified as ultimately correct is not as straightforward as it seems.<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
He had drifted into the very heart of the world. From him to the distant beloved was as far as to the next tree.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 144]<br />
|}<br />
<br />
Let us venture a guess as to where these developments might be heading. From the present vantage point it appears that the ultimate answer to the quandary of distances and the question of a fitting measure may be that, rather than having the constitution of an analytic series depend on our familiar notions of approach, proximity, and approximation, it will be found preferable, and perhaps unavoidable, to turn the tables and let the orders of approximation be defined in terms of our favored and operative notions of formal analysis. Only the aftermath of this conversion, if it does converge, could be hoped to prove whether this hortatory form of analysis and the cohort idea of an analytic form &mdash; the limitary concept of a self-corrective process and the coefficient concept of a completable product &mdash; are truly (in practical reality) the more inceptive and persistent of principles and really (for all practical purposes) the more effective and regulative of ideas.<br />
<br />
Awaiting that determination, I proceed with what seems like the obvious course, and compute <math>\mathrm{d}J\!</math> according to the pattern in Table&nbsp;45.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 45.} ~~ \text{Computation of}~ \mathrm{d}J\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}J<br />
& = &<br />
u\!\cdot\!v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
u \, \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \, \mathrm{d}v<br />
& + &<br />
\texttt{(} u \texttt{)} \, v \cdot \mathrm{d}u \, \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \!\cdot\! \mathrm{d}v \texttt{~}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}J<br />
& = &<br />
u\!\cdot\!v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + &<br />
u \, \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + &<br />
\texttt{(} u \texttt{)} \, v \cdot \mathrm{d}u<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;46-a through 46-d illustrate the proposition <math>{\mathrm{d}J},\!</math> rounded out in our usual array of prospects. This proposition of <math>\mathrm{E}U^\bullet\!</math> is what we refer to as the (first order) differential of <math>J,\!</math> and normally regard as ''the'' differential proposition corresponding to <math>J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-a -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-a.} ~~ \text{Differential of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-b -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-b.} ~~ \text{Differential of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-c -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-c.} ~~ \text{Differential of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-d -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-d.} ~~ \text{Differential of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
=====Remainder of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
<p>I bequeath myself to the dirt to grow from the grass I love,<br><br />
If you want me again look for me under your bootsoles.</p><br />
<br />
<p>You will hardly know who I am or what I mean,<br><br />
But I shall be good health to you nevertheless,<br><br />
And filter and fibre your blood.</p><br />
<br />
<p>Failing to fetch me at first keep encouraged,<br><br />
Missing me one place search another,<br><br />
I stop some where waiting for you</p><br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]<br />
|}<br />
<br />
<br><br />
<br />
Let us recapitulate the story so far. We have in effect been carrying out a decomposition of the enlarged proposition <math>\mathrm{E}J\!</math> in a series of stages. First, we considered the equation <math>\mathrm{E}J = \boldsymbol\varepsilon J + \mathrm{D}J,\!</math> which was involved in the definition of <math>\mathrm{D}J\!</math> as the difference <math>\mathrm{E}J - \boldsymbol\varepsilon J.\!</math> Next, we contemplated the equation <math>\mathrm{D}J = \mathrm{d}J + \mathrm{r}J,\!</math> which expresses <math>\mathrm{D}J\!</math> in terms of two components, the differential <math>\mathrm{d}J\!</math> that was just extracted and the residual component <math>\mathrm{r}J = \mathrm{D}J - \mathrm{d}J.~\!</math> This remaining proposition <math>\mathrm{r}J\!</math> can be computed as shown in Table&nbsp;47.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 47.} ~~ \text{Computation of}~ \mathrm{r}J\!</math><br />
|<br />
<math>\begin{array}{*{5}{l}}<br />
\mathrm{r}J & = & \mathrm{D}J & + & \mathrm{d}J<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}J<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{r}J ~<br />
& = & u \!\cdot\! v \cdot ~ \mathrm{d}u \cdot \mathrm{d}v ~ ~ ~ ~ ~<br />
& + & u \texttt{(} v \texttt{)} \cdot \, \mathrm{d}u \cdot \mathrm{d}v \,<br />
& + & \texttt{(} u \texttt{)} v \cdot \, \mathrm{d}u \cdot \mathrm{d}v \,<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \, \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
As it happens, the remainder <math>\mathrm{r}J\!</math> falls under the description of a second order differential <math>\mathrm{r}J = \mathrm{d}^2 J.\!</math> This means that the expansion of <math>\mathrm{E}J\!</math> in the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|<br />
<math>\begin{array}{*{7}{l}}<br />
\mathrm{E}J<br />
& = & \boldsymbol\varepsilon J<br />
& + & \mathrm{D}J<br />
\\[6pt]<br />
& = & \boldsymbol\varepsilon J<br />
& + & \mathrm{d}J<br />
& + & \mathrm{r}J<br />
\\[6pt]<br />
& = & \mathrm{d}^0 J<br />
& + & \mathrm{d}^1 J<br />
& + & \mathrm{d}^2 J<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
which is nothing other than the propositional analogue of a Taylor series, is a decomposition that terminates in a finite number of steps.<br />
<br />
Figures&nbsp;48-a through 48-d illustrate the proposition <math>\mathrm{r}J = \mathrm{d}^2 J,\!</math> which forms the remainder map of <math>J\!</math> and also, in this instance, the second order differential of <math>J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-a -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-a.} ~~ \text{Remainder of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-b -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-b.} ~~ \text{Remainder of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-c -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-c.} ~~ \text{Remainder of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-d -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-d.} ~~ \text{Remainder of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
=====Summary of Conjunction=====<br />
<br />
To establish a convenient reference point for further discussion, Table&nbsp;49 summarizes the operator actions that have been computed for the form of conjunction, as exemplified by the proposition <math>J.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 49.} ~~ \text{Computation Summary for}~ J~\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon J<br />
& = & u \!\cdot\! v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}J<br />
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}J<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{r}J<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Analytic Series : Coordinate Method====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
And if he is told that something ''is'' the way it is, then he thinks: Well, it could probably just as easily be some other way. So the sense of possibility might be defined outright as the capacity to think how everything could &ldquo;just as easily&rdquo; be, and to attach no more importance to what is than to what is not.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 12]<br />
|}<br />
<br />
Table&nbsp;50 exhibits a truth table method for computing the analytic series (or the differential expansion) of a proposition in terms of coordinates.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:70%"<br />
|+ style="height:30px" | <math>\text{Table 50.} ~~ \text{Computation of an Analytic Series in Terms of Coordinates}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:8%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:8%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:8%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}u\!</math><br />
| style="width:8%; border-bottom:1px solid black" | <math>\mathrm{d}v\!</math><br />
| style="width:8%; border-bottom:1px solid black; border-left:1px solid black" | <math>u'\!</math><br />
| style="width:8%; border-bottom:1px solid black" | <math>v'\!</math><br />
| style="width:10%; border-bottom:1px solid black; border-left:4px double black" | <math>\boldsymbol\varepsilon J\!</math><br />
| style="width:10%; border-bottom:1px solid black" | <math>\mathrm{E}J\!</math><br />
| style="width:12%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{D}J\!</math><br />
| style="width:10%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}J\!</math><br />
| style="width:10%; border-bottom:1px solid black" | <math>\mathrm{d}^2\!J\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
The first six columns of the Table, taken as a whole, represent the variables of a construct called the ''contingent universe'' <math>[u, v, \mathrm{d}u, \mathrm{d}v, u', v'],\!</math> or the bundle of ''contingency spaces'' <math>[\mathrm{d}u, \mathrm{d}v, u', v']\!</math> over the universe <math>[u, v].\!</math> Their placement to the left of the double bar indicates that all of them amount to independent variables, but there is a co-dependency among them, as described by the following equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
u' & = & u + \mathrm{d}u & = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\\[8pt]<br />
v' & = & v + \mathrm{d}v & = & \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
These relations correspond to the formal substitutions that are made in defining <math>\mathrm{E}J\!</math> and <math>\mathrm{D}J.\!</math> For now, the whole rigamarole of contingency spaces can be regarded as a technical device for achieving the effect of these substitutions, adapted to a setting where functional compositions and other symbolic manipulations are difficult to contemplate and execute.<br />
<br />
The five columns to the right of the double bar in Table&nbsp;50 contain the values of the dependent variables <math>\{ \boldsymbol\varepsilon J, ~\mathrm{E}J, ~\mathrm{D}J, ~\mathrm{d}J, ~\mathrm{d}^2\!J \}.\!</math> These are normally interpreted as values of functions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B}\!</math> or as values of propositions in the extended universe <math>[u, v, \mathrm{d}u, \mathrm{d}v]\!</math> but the dependencies prevailing in the contingent universe make it possible to regard these same final values as arising via functions on alternative lists of arguments, for example, the set <math>\{ u, v, u', v' \}.\!</math><br />
<br />
The column for <math>\boldsymbol\varepsilon J\!</math> is computed as <math>J(u, v) = uv\!</math> and together with the columns for <math>u\!</math> and <math>v\!</math> illustrates how we &ldquo;share structure&rdquo; in the Table by listing only the first entries of each constant block.<br />
<br />
The column for <math>\mathrm{E}J\!</math> is computed by means of the following chain of identities, where the contingent variables <math>u'\!</math> and <math>v'\!</math> are defined as <math>u' = u + \mathrm{d}u\!</math> and <math>v' = v + \mathrm{d}v.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathrm{E}J(u, v, \mathrm{d}u, \mathrm{d}v)<br />
& = &<br />
J(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& = &<br />
J(u', v')<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
This makes it easy to determine <math>\mathrm{E}J\!</math> by inspection, computing the conjunction <math>J(u', v') = u'v'\!</math> from the columns headed <math>u'\!</math> and <math>v'.\!</math> Since each of these forms expresses the same proposition <math>\mathrm{E}J\!</math> in <math>\mathrm{E}U^\bullet,\!</math> the dependence on <math>\mathrm{d}u\!</math> and <math>\mathrm{d}v\!</math> is still present but merely left implicit in the final variant <math>J(u', v').\!</math><br />
<br />
* '''Note.''' On occasion, it is tempting to use the further notation <math>J'(u, v) = J(u', v'),\!</math> especially to suggest a transformation that acts on whole propositions, for example, taking the proposition <math>J\!</math> into the proposition <math>J' = \mathrm{E}J.\!</math> The prime <math>( {}^{\prime} )\!</math> then signifies an action that is mediated by a field of choices, namely, the values that are picked out for the contingent variables in sweeping through the initial universe. But this heaps an unwieldy lot of construed intentions on a rather slight character and puts too high a premium on the constant correctness of its interpretation. In practice, therefore, it is best to avoid this usage.<br />
<br />
Given the values of <math>\boldsymbol\varepsilon J\!</math> and <math>\mathrm{E}J,\!</math> the columns for the remaining functions can be filled in quickly. The difference map is computed according to the relation <math>\mathrm{D}J = \boldsymbol\varepsilon J + \mathrm{E}J.\!</math> The first order differential <math>\mathrm{d}J\!</math> is found by looking in each block of constant argument pairs <math>u, v\!</math> and choosing the linear function of <math>\mathrm{d}u, \mathrm{d}v\!</math> that best approximates <math>\mathrm{D}J\!</math> in that block. Finally, the remainder is computed as <math>\mathrm{r}J = \mathrm{D}J + \mathrm{d}J,\!</math> in this case yielding the second order differential <math>\mathrm{d}^2\!J.\!</math><br />
<br />
====Analytic Series : Recap====<br />
<br />
Let us now summarize the results of Table&nbsp;50 by writing down for each column and for each block of constant argument pairs <math>u, v\!</math> a reasonably canonical symbolic expression for the function of <math>\mathrm{d}u, \mathrm{d}v\!</math> that appears there. The synopsis formed in this way is presented in Table&nbsp;51. As one has a right to expect, it confirms the results that were obtained previously by operating solely in terms of the formal calculus.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:70%"<br />
|+ style="height:30px" | <math>\text{Table 51.} ~~ \text{Computation of an Analytic Series in Symbolic Terms}\!</math><br />
|- style="height:35px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black" | <math>J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{E}J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{D}J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{d}J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{d}^2\!J\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}u \!\;\cdot\;\! \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}u \!\;\cdot\;\! \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
\mathrm{d}u<br />
\\[4pt]<br />
\mathrm{d}v<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;52 and 53 provide a quick overview of the analysis performed so far, giving the successive decompositions of <math>\mathrm{E}J = J + \mathrm{D}J\!</math> and <math>\mathrm{D}J = \mathrm{d}J + \mathrm{r}J\!</math> in two different styles of diagram.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 52 -- Decomposition of EJ.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 52.} ~~ \text{Decomposition of}~ \mathrm{E}J\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 53 -- Decomposition of DJ.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 53.} ~~ \text{Decomposition of}~ \mathrm{D}J\!</math><br />
|}<br />
<br />
====Terminological Interlude====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Lastly, my attention was especially attracted, not so much to the scene, as to the mirrors that produced it. These mirrors were broken in parts. Yes, they were marked and scratched; they had been &ldquo;starred&rdquo;, in spite of their solidity &hellip;<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 230]<br />
|}<br />
<br />
At this point several issues of terminology have accrued enough substance to intrude on our discussion. The remarks of this Subsection are intended to accomplish two goals. First, we call attention to significant aspects of the previous series of Figures, translating into literal terms what they depict in iconic forms, and we re-stress the most important structural elements they indicate. Next, we prepare the way for taking on more complex examples of transformations, those whose target universes have more than one dimension.<br />
<br />
In talking about the actions of operators it is important to keep in mind the distinctions between the operators per&nbsp;se, their operands, and their results. Furthermore, in working with composite forms of operators <math>\mathrm{W} = (\mathrm{W}_1, \ldots, \mathrm{W}_n),\!</math> transformations <math>\mathrm{F} = (\mathrm{F}_1, \ldots, \mathrm{F}_n),\!</math> and target domains <math>X^\bullet = [x_1, \ldots, x_n],\!</math> we need to preserve a clear distinction between the compound entity of each given type and any one of its separate components. It is curious, given the usefulness of the concepts ''operator'' and ''operand'', that we seem to lack a generic term, formed on the same root, for the corresponding result of an operation. Following the obvious paradigm would lead to words like ''opus'', ''opera'', and ''operant'', but these words are too affected with clang associations to work well at present, though they might be adapted in time. One current usage gets around this problem by using the substantive ''map'' as a systematic epithet to express the result of each operator's action. We will follow this practice as far as possible, for example, using the phrase ''tangent map'' to denote the end product of the tangent functor acting on its operand map.<br />
<br />
* '''Scholium.''' See [JGH, 6-9] for a good account of tangent functors and tangent maps in ordinary analysis and for examples of their use in mechanics. This work as a whole is a model of clarity in applying functorial principles to problems in physical dynamics.<br />
<br />
Whenever we focus on isolated propositions, on single components of composite operators, or on the portions of transformations that have <math>1\!</math>-dimensional ranges, we are free to shift between the native form of a proposition <math>J : U \to \mathbb{B}\!</math> and the thematized form of a mapping <math>J : U^\bullet \to [x]\!</math> without much trouble. In these cases we are able to tolerate a higher degree of ambiguity about the precise nature of an operator's input and output domains than we otherwise might. For example, in the preceding treatment of the example <math>J,\!</math> and for each operator <math>\mathrm{W}\!</math> in the set <math>\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \},\!</math> both the operand <math>J\!</math> and the result <math>\mathrm{W}J\!</math> could be viewed in either one of two ways. On one hand we may treat them as propositions <math>J : U \to \mathbb{B}\!</math> and <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B},\!</math> ignoring the distinction between the range <math>[x] \cong \mathbb{B}\!</math> of <math>\boldsymbol\varepsilon J\!</math> and the range <math>[\mathrm{d}x] \cong \mathbb{D}\!</math> of the other types of <math>\mathrm{W}J.\!</math> This is what we usually do when we content ourselves with simply coloring in regions of venn diagrams. On the other hand we may view these entities as maps <math>J : U^\bullet \to [x] = X^\bullet\!</math> and <math>\boldsymbol\varepsilon J : \mathrm{E}U^\bullet \to [x] \subseteq \mathrm{E}X^\bullet\!</math> or <math>\mathrm{W}J : \mathrm{E}U^\bullet \to [\mathrm{d}x] \subseteq \mathrm{E}X^\bullet,\!</math> in which case the qualitative characters of the output features are not ignored.<br />
<br />
At the beginning of this Section we recast the natural form of a proposition <math>J : U \to \mathbb{B}\!</math> into the thematic role of a transformation <math>J : U^\bullet \to [x],\!</math> where <math>x\!</math> was a variable recruited to express the newly independent <math>\check{J}.\!</math> However, in our computations and representations of operator actions we immediately lapsed back to viewing the results as native elements of the extended universe <math>\mathrm{E}U^\bullet,\!</math> in other words, as propositions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B},\!</math> where <math>\mathrm{W}\!</math> ranged over the set <math>\{ \boldsymbol\varepsilon, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}.\!</math> That is as it should be. We have worked hard to devise a language that gives us these advantages &mdash; the flexibility to exchange terms and types of equal information value and the capacity to reflect as quickly and as wittingly as a controlled reflex on the fibers of our propositions, independently of whether they express amusements, beliefs, or conjectures.<br />
<br />
As we take on target spaces of increasing dimension, however, these types of confusions (and confusions of types) become less and less permissible. For this reason, Tables&nbsp;54 and 55 present a rather detailed summary of the notation and the terminology we are using, as applied to the case <math>J = uv.\!</math> The rationale of these Tables is not so much to train more elephant guns on this poor drosophila of a concrete example but to invest our paradigm with enough solidity to bear the weight of abstraction to come.<br />
<br />
Table&nbsp;54 provides basic notation and descriptive information for the objects and operators used in this Example, giving the generic type (or broadest defined type) for each entity. Here, the sans&nbsp;serif operators <math>\mathsf{W} \in \{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{d}, \mathsf{r} \}\!</math> and their components <math>\mathrm{W} \in \{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}\!</math> both have the same broad type <math>\mathsf{W}, \mathrm{W} : (U^\bullet \to X^\bullet) \to (\mathrm{E}U^\bullet \to \mathrm{E}X^\bullet),\!</math> as appropriate to operators that map transformations <math>J : U^\bullet \to X^\bullet\!</math> to extended transformations <math>\mathsf{W}J, \mathrm{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 54.} ~~ \text{Cast of Characters : Expansive Subtypes of Objects and Operators}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| align="center" | <math>\text{Symbol}\!</math><br />
| align="center" | <math>\text{Notation}\!</math><br />
| align="center" | <math>\text{Description}\!</math><br />
| align="center" | <math>\text{Type}\!</math><br />
|-<br />
| align="center" | <math>U^\bullet\!</math><br />
| <math>= [u, v]\!</math><br />
| <math>\text{Source universe}\!</math><br />
| <math>[\mathbb{B}^2]\!</math><br />
|-<br />
| align="center" | <math>X^\bullet~\!</math><br />
| <math>= [x]\!</math><br />
| <math>\text{Target universe}\!</math><br />
| <math>[\mathbb{B}^1]~\!</math><br />
|-<br />
| align="center" | <math>\mathrm{E}U^\bullet\!</math><br />
| <math>= [u, v, \mathrm{d}u, \mathrm{d}v]\!</math><br />
| <math>\text{Extended source universe}\!</math><br />
| <math>[\mathbb{B}^2 \!\times\! \mathbb{D}^2]</math><br />
|-<br />
| align="center" | <math>\mathrm{E}X^\bullet\!</math><br />
| <math>= [x, \mathrm{d}x]~\!</math><br />
| <math>\text{Extended target universe}\!</math><br />
| <math>[\mathbb{B}^1 \!\times\! \mathbb{D}^1]</math><br />
|-<br />
| align="center" | <math>J\!</math><br />
| <math>J : U \!\to\! \mathbb{B}\!</math><br />
| <math>\text{Proposition}\!</math><br />
| <math>(\mathbb{B}^2 \!\to\! \mathbb{B}) \in [\mathbb{B}^2]\!</math><br />
|-<br />
| align="center" | <math>J\!</math><br />
| <math>J : U^\bullet \!\to\! X^\bullet\!</math><br />
| <math>\text{Transformation or Map}\!</math><br />
| <math>[\mathbb{B}^2] \!\to\! [\mathbb{B}^1]\!</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\boldsymbol\varepsilon<br />
\\<br />
\eta<br />
\\<br />
\mathrm{E}<br />
\\<br />
\mathrm{D}<br />
\\<br />
\mathrm{d}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{W} : U^\bullet \!\to\! \mathrm{E}U^\bullet,<br />
\\<br />
\mathrm{W} : X^\bullet \!\to\! \mathrm{E}X^\bullet,<br />
\\<br />
\mathrm{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\\<br />
\text{for each}~ \mathrm{W} ~\text{in the set:}<br />
\\<br />
\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{ll}<br />
\text{Tacit extension operator} & \boldsymbol\varepsilon<br />
\\<br />
\text{Trope extension operator} & \eta<br />
\\<br />
\text{Enlargement operator} & \mathrm{E}<br />
\\<br />
\text{Difference operator} & \mathrm{D}<br />
\\<br />
\text{Differential operator} & \mathrm{d}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^2] \!\to\! [\mathbb{B}^2 \!\times\! \mathbb{D}^2]},<br />
\\<br />
{[\mathbb{B}^1] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]},<br />
\\\\<br />
([\mathbb{B}^2] \!\to\! [\mathbb{B}^1]) \!\to\!<br />
\\<br />
([\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1])<br />
\end{array}</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\mathsf{e}<br />
\\<br />
\mathsf{E}<br />
\\<br />
\mathsf{D}<br />
\\<br />
\mathsf{T}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{W} : U^\bullet \!\to\! \mathsf{T}U^\bullet = \mathrm{E}U^\bullet,<br />
\\<br />
\mathsf{W} : X^\bullet \!\to\! \mathsf{T}X^\bullet = \mathrm{E}X^\bullet,<br />
\\<br />
\mathsf{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathsf{T}U^\bullet \!\to\! \mathsf{T}X^\bullet)<br />
\\<br />
\text{for each}~ \mathsf{W} ~\text{in the set:}<br />
\\<br />
\{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{T} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{lll}<br />
\text{Radius operator} & \mathsf{e} & = (\boldsymbol\varepsilon, \eta)<br />
\\<br />
\text{Secant operator} & \mathsf{E} & = (\boldsymbol\varepsilon, \mathrm{E})<br />
\\<br />
\text{Chord operator} & \mathsf{D} & = (\boldsymbol\varepsilon, \mathrm{D})<br />
\\<br />
\text{Tangent functor} & \mathsf{T} & = (\boldsymbol\varepsilon, \mathrm{d})<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^2] \!\to\! [\mathbb{B}^2 \!\times\! \mathbb{D}^2]},<br />
\\<br />
{[\mathbb{B}^1] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]},<br />
\\\\<br />
([\mathbb{B}^2] \!\to\! [\mathbb{B}^1]) \!\to\!<br />
\\<br />
([\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1])<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;55 supplies a more detailed outline of terminology for operators and their results. Here, we list the restrictive subtype (or narrowest defined subtype) that applies to each entity and we indicate across the span of the Table the whole spectrum of alternative types that color the interpretation of each symbol. For example, all the component operator maps <math>\mathrm{W}J\!</math> have <math>1\!</math>-dimensional ranges, either <math>\mathbb{B}^1\!</math> or <math>\mathbb{D}^1,\!</math> and so they can be viewed either as propositions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B}\!</math> or as logical transformations <math>\mathrm{W}J : \mathrm{E}U^\bullet \to X^\bullet.\!</math> As a rule, the plan of the Table allows us to name each entry by detaching the underlined adjective at the left of its row and prefixing it to the generic noun at the top of its column. In one case, however, it is customary to depart from this scheme. Because the phrase ''differential proposition'', applied to the result <math>\mathrm{d}J : \mathrm{E}U \to \mathbb{D},\!</math> does not distinguish it from the general run of differential propositions <math>\mathrm{G}: \mathrm{E}U \to \mathbb{B},\!</math> it is usual to single out <math>\mathrm{d}J\!</math> as the ''tangent proposition'' of <math>J.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 55.} ~~ \text{Synopsis of Terminology : Restrictive and Alternative Subtypes}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| &nbsp;<br />
| align="center" | <math>\text{Operator}\!</math><br />
| align="center" | <math>\text{Proposition}\!</math><br />
| align="center" | <math>\text{Map}\!</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tacit}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\boldsymbol\varepsilon : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\boldsymbol\varepsilon : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{B} \\<br />
\boldsymbol\varepsilon J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{B}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x] \\<br />
\boldsymbol\varepsilon J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Trope}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\eta : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\eta : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\eta J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\eta J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Enlargement}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{E} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{E}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{E}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Difference}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{D} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{D}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{D}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Differential}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{d} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{d} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{d}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}\!</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Remainder}}\\\text{operator}\end{matrix}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{r} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{r} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{r}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{r}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Radius}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e} = (\boldsymbol\varepsilon, \eta) \\<br />
\mathsf{e} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{e}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Secant}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}) \\<br />
\mathsf{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{E}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Chord}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}) \\<br />
\mathsf{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{D}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tangent}}\\\text{functor}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T} = (\boldsymbol\varepsilon, \mathrm{d}) \\<br />
\mathsf{T} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{T}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====End of Perfunctory Chatter : Time to Roll the Clip!====<br />
<br />
Two steps remain to finish the analysis of <math>J\!</math> that we began so long ago. First, we need to paste our accumulated heap of flat pictures into the frames of transformations, filling out the shapes of the operator maps <math>\mathsf{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet.~\!</math> This scheme is executed in two styles, using the ''areal views'' in Figures&nbsp;56-a and the ''box views'' in Figures&nbsp;56-b. Finally, in Figures&nbsp;57-1 to 57-4 we put all the pieces together to construct the full operator diagrams for <math>\mathsf{W} : J \to \mathsf{W}J.\!</math> There is a considerable amount of redundancy among the following three series of Figures but that will hopefully provide a fuller picture of the operations under review, enabling these snapshots to serve as successive frames in the animation of logic they are meant to become.<br />
<br />
=====Operator Maps : Areal Views=====<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a1 -- Radius Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a1.} ~~ \text{Radius Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a2 -- Secant Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a2.} ~~ \text{Secant Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a3 -- Chord Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a3.} ~~ \text{Chord Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a4 -- Tangent Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a4.} ~~ \text{Tangent Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
=====Operator Maps : Box Views=====<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b1 -- Radius Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b1.} ~~ \text{Radius Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b2 -- Secant Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b2.} ~~ \text{Secant Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b3 -- Chord Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b3.} ~~ \text{Chord Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b4 -- Tangent Map of J ISW.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b4.} ~~ \text{Tangent Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
=====Operator Diagrams for the Conjunction J = uv=====<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-1 -- Radius Operator Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-1.} ~~ \text{Radius Operator Diagram for the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-2 -- Secant Operator Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-2.} ~~ \text{Secant Operator Diagram for the Conjunction}~ J = uv~\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-3 -- Chord Operator Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-3.} ~~ \text{Chord Operator Diagram for the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-4 -- Tangent Functor Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-4.} ~~ \text{Tangent Functor Diagram for the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
===Taking Aim at Higher Dimensional Targets===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
The past and present wilt . . . . I have filled them and<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;emptied them,<br><br />
And proceed to fill my next fold of the future.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 87]<br />
|}<br />
<br />
In the next Section we consider a transformation <math>F\!</math> of concrete type <math>F : [u, v] \to [x, y]\!</math> and abstract type <math>F : [\mathbb{B}^2] \to [\mathbb{B}^2].\!</math> From the standpoint of propositional calculus we naturally approach the task of understanding such a transformation by parsing it into component maps with <math>1\!</math>-dimensional ranges, as follows:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{ccccccl}<br />
F & = & (F_1, F_2) & = & (f, g) & : & [u, v] \to [x, y],<br />
\\[6pt]<br />
&& F_1 & = & f & : & [u, v] \to [x],<br />
\\[6pt]<br />
&& F_2 & = & g & : & [u, v] \to [y].<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Then we tackle the separate components, now viewed as propositions <math>F_i : U \to \mathbb{B},\!</math> one at a time. At the completion of this analytic phase, we return to the task of synthesizing these partial and transient impressions into an agile form of integrity, a solidly coordinated and deeply integrated comprehension of the ongoing transformation. (Very often, of course, in tangling with refractory cases, we never get as far as the beginning again.)<br />
<br />
Let us now refer to the dimension of the target space or codomain as the ''toll'' (or ''tole'') of a transformation, as distinguished from the dimension of the range or image that is customarily called the ''rank''. When we keep to transformations with a toll of <math>1,\!</math> as <math>J : [u, v] \to [x],\!</math> we tend to get lazy about distinguishing a logical transformation from its component propositions. However, if we deal with transformations of a higher toll, this form of indolence can no longer be tolerated.<br />
<br />
Well, perhaps we can carry it a little further. After all, the operator result <math>\mathrm{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet\!</math> is a map of toll <math>2,\!</math> and cannot be unfolded in one piece as a proposition. But when a map has rank <math>1,\!</math> like <math>\boldsymbol\varepsilon J : \mathrm{E}U \to X \subseteq \mathrm{E}X\!</math> or <math>\mathrm{d}J : \mathrm{E}U \to \mathrm{d}X \subseteq \mathrm{E}X,\!</math> we naturally choose to concentrate on the <math>1\!</math>-dimensional range of the operator result <math>\mathrm{W}J,\!</math> ignoring the final difference in quality between the spaces <math>X\!</math> and <math>\mathrm{d}X,\!</math> and view <math>\mathrm{W}J\!</math> as a proposition about <math>\mathrm{E}U.\!</math><br />
<br />
In this way, an initial ambivalence about the role of the operand <math>J\!</math> conveys a double duty to the result <math>\mathrm{W}J.\!</math> The pivot that is formed by our focus of attention is essential to the linkage that transfers this double moment, as the whole process takes its bearing and wheels around the precise measure of a narrow bead that we can draw on the range of <math>\mathrm{W}J.\!</math> This is the escapement that it takes to get away with what may otherwise seem to be a simple duplicity, and this is the tolerance that is needed to counterbalance a certain arrogance of equivocation, by all of which machinations we make ourselves free to indicate the operator results <math>\mathrm{W}J\!</math> as propositions or as transformations, indifferently.<br />
<br />
But that's it, and no further. Neglect of these distinctions in range and target universes of higher dimensions is bound to cause a hopeless confusion. To guard against these adverse prospects, Tables&nbsp;58 and 59 lay the groundwork for discussing a typical map <math>F : [\mathbb{B}^2] \to [\mathbb{B}^2],\!</math> and begin to pave the way to some extent for discussing any transformation of the form <math>F : [\mathbb{B}^n] \to [\mathbb{B}^k].\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 58.} ~~ \text{Cast of Characters : Expansive Subtypes of Objects and Operators}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| align="center" | <math>\text{Symbol}\!</math><br />
| align="center" | <math>\text{Notation}\!</math><br />
| align="center" | <math>\text{Description}\!</math><br />
| align="center" | <math>\text{Type}\!</math><br />
|-<br />
| align="center" | <math>U^\bullet\!</math><br />
| <math>= [u, v]\!</math><br />
| <math>\text{Source universe}\!</math><br />
| <math>[\mathbb{B}^n]\!</math><br />
|-<br />
| align="center" | <math>X^\bullet~\!</math><br />
| <math>\begin{array}{l}<br />
= [x, y] \\<br />
= [f, g]<br />
\end{array}</math><br />
| <math>\text{Target universe}\!</math><br />
| <math>[\mathbb{B}^k]\!</math><br />
|-<br />
| align="center" | <math>\mathrm{E}U^\bullet\!</math><br />
| <math>= [u, v, \mathrm{d}u, \mathrm{d}v]\!</math><br />
| <math>\text{Extended source universe}\!</math><br />
| <math>[\mathbb{B}^n \!\times\! \mathbb{D}^n]\!</math><br />
|-<br />
| align="center" | <math>\mathrm{E}X^\bullet\!</math><br />
| <math>\begin{array}{l}<br />
= [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
= [f, g, \mathrm{d}f, \mathrm{d}g]<br />
\end{array}</math><br />
| <math>\text{Extended target universe}\!</math><br />
| <math>[\mathbb{B}^k \!\times\! \mathbb{D}^k]\!</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
f \\ g<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{ll}<br />
f : U \!\to\! [x] \cong \mathbb{B} \\<br />
g : U \!\to\! [y] \cong \mathbb{B}<br />
\end{array}</math><br />
| <math>\text{Proposition}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathbb{B}^n \!\to\! \mathbb{B} \\<br />
\in (\mathbb{B}^n, \mathbb{B}^n \!\to\! \mathbb{B}) = [\mathbb{B}^n]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>F\!</math><br />
| <math>F = (f, g) : U^\bullet \!\to\! X^\bullet\!</math><br />
| <math>\text{Transformation of Map}\!</math><br />
| <math>[\mathbb{B}^n] \!\to\! [\mathbb{B}^k]</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\boldsymbol\varepsilon<br />
\\<br />
\eta<br />
\\<br />
\mathrm{E}<br />
\\<br />
\mathrm{D}<br />
\\<br />
\mathrm{d}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{W} : U^\bullet \!\to\! \mathrm{E}U^\bullet,<br />
\\<br />
\mathrm{W} : X^\bullet \!\to\! \mathrm{E}X^\bullet,<br />
\\<br />
\mathrm{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\\<br />
\text{for each}~ \mathrm{W} ~\text{in the set:}<br />
\\<br />
\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{ll}<br />
\text{Tacit extension operator} & \boldsymbol\varepsilon<br />
\\<br />
\text{Trope extension operator} & \eta<br />
\\<br />
\text{Enlargement operator} & \mathrm{E}<br />
\\<br />
\text{Difference operator} & \mathrm{D}<br />
\\<br />
\text{Differential operator} & \mathrm{d}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^n] \!\to\! [\mathbb{B}^n \!\times\! \mathbb{D}^n]},<br />
\\<br />
{[\mathbb{B}^k] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]},<br />
\\\\<br />
([\mathbb{B}^n] \!\to\! [\mathbb{B}^k]) \!\to\!<br />
\\<br />
([\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k])<br />
\end{array}</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\mathsf{e}<br />
\\<br />
\mathsf{E}<br />
\\<br />
\mathsf{D}<br />
\\<br />
\mathsf{T}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{W} : U^\bullet \!\to\! \mathsf{T}U^\bullet = \mathrm{E}U^\bullet,<br />
\\<br />
\mathsf{W} : X^\bullet \!\to\! \mathsf{T}X^\bullet = \mathrm{E}X^\bullet,<br />
\\<br />
\mathsf{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathsf{T}U^\bullet \!\to\! \mathsf{T}X^\bullet)<br />
\\<br />
\text{for each}~ \mathsf{W} ~\text{in the set:}<br />
\\<br />
\{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{T} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{lll}<br />
\text{Radius operator} & \mathsf{e} & = (\boldsymbol\varepsilon, \eta)<br />
\\<br />
\text{Secant operator} & \mathsf{E} & = (\boldsymbol\varepsilon, \mathrm{E})<br />
\\<br />
\text{Chord operator} & \mathsf{D} & = (\boldsymbol\varepsilon, \mathrm{D})<br />
\\<br />
\text{Tangent functor} & \mathsf{T} & = (\boldsymbol\varepsilon, \mathrm{d})<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^n] \!\to\! [\mathbb{B}^n \!\times\! \mathbb{D}^n]},<br />
\\<br />
{[\mathbb{B}^k] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]},<br />
\\\\<br />
([\mathbb{B}^n] \!\to\! [\mathbb{B}^k]) \!\to\!<br />
\\<br />
([\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k])<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 59.} ~~ \text{Synopsis of Terminology : Restrictive and Alternative Subtypes}~\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| &nbsp;<br />
| align="center" | <math>\begin{matrix}\text{Operator}\\\text{or}\\\text{Operand}\end{matrix}</math><br />
| align="center" | <math>\begin{matrix}\text{Proposition}\\\text{or}\\\text{Component}\end{matrix}</math><br />
| align="center" | <math>\begin{matrix}\text{Transformation}\\\text{or}\\\text{Map}\end{matrix}</math><br />
|-<br />
| align="center" | <math>\underline{\text{Operand}}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
F = (F_1, F_2) \\<br />
F = (f, g) : U \!\to\! X<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
F_i : \langle u, v \rangle \!\to\! \mathbb{B} \\<br />
F_i : \mathbb{B}^n \!\to\! \mathbb{B}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
F : [u, v] \!\to\! [x, y] \\<br />
F : [\mathbb{B}^n] \!\to\! [\mathbb{B}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tacit}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\boldsymbol\varepsilon : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\boldsymbol\varepsilon : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{B} \\<br />
\boldsymbol\varepsilon F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{B}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y] \\<br />
\boldsymbol\varepsilon F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Trope}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\eta : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\eta : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\eta F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\eta F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Enlargement}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{E} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{E}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{E}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Difference}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{D} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{D}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{D}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Differential}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{d} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{d} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{d}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}\!</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Remainder}}\\\text{operator}\end{matrix}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{r} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{r} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{r}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{r}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Radius}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e} = (\boldsymbol\varepsilon, \eta) \\<br />
\mathsf{e} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{e}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Secant}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}) \\<br />
\mathsf{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{E}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Chord}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}) \\<br />
\mathsf{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{D}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tangent}}\\\text{functor}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T} = (\boldsymbol\varepsilon, \mathrm{d}) \\<br />
\mathsf{T} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{T}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
===Transformations of Type '''B'''<sup>2</sup> &rarr; '''B'''<sup>2</sup>===<br />
<br />
To take up a slightly more complex example, but one that remains simple enough to pursue through a complete series of developments, consider the transformation from <math>U^\bullet = [u, v]\!</math> to <math>X^\bullet = [x, y]\!</math> that is defined by the following system of equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
x<br />
& = & f(u, v)<br />
& = & \texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[8pt]<br />
y<br />
& = & g(u, v)<br />
& = & \texttt{((} u \texttt{,} v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
The component notation <math>F = (F_1, F_2) = (f, g) : U^\bullet \to X^\bullet\!</math> allows us to give a name and a type to this transformation and permits defining it by the compact description that follows:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
(x, y)<br />
& = & F(u, v)<br />
& = & (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Logical Transformations====<br />
<br />
The information that defines the logical transformation <math>F\!</math> can be represented in the form of a truth table, as shown in Table&nbsp;60. To cut down on subscripts in this example we continue to use plain letter equivalents for all components of spaces and maps.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:60%"<br />
|+ style="height:30px" | <math>\text{Table 60.} ~~ \text{A Propositional Transformation}\!</math><br />
|- style="height:40px; background:ghostwhite; width:100%"<br />
| style="width:25%" | <math>u\!</math><br />
| style="width:25%" | <math>v\!</math><br />
| style="width:25%; border-left:1px solid black" | <math>f\!</math><br />
| style="width:25%" | <math>g\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="border-top:1px solid black" | <math>u\!</math><br />
| style="border-top:1px solid black" | <math>v\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;61 shows how we might paint a picture of the transformation <math>F\!</math> in the manner of Figure&nbsp;30.<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 61 -- Propositional Transformation.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 61.} ~~ \text{A Propositional Transformation}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;62 extracts the gist of Figure&nbsp;61, exhibiting a style of diagram that is adequate for most purposes.<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 62 -- Propositional Transformation (Short Form).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 62.} ~~ \text{A Propositional Transformation (Short Form)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Local Transformations====<br />
<br />
Figure&nbsp;63 gives a more complete picture of the transformation <math>F,\!</math> showing how the points of <math>U^\bullet\!</math> are transformed into points of <math>X^\bullet.\!</math> The bold lines crossing from one universe to the other trace the action that <math>F\!</math> induces on points, in other words, they show how the transformation acts as a mapping from points to points and chart its effects on the elements that are variously called cells, points, positions, or singular propositions.<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 63 -- Transformation of Positions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 63.} ~~ \text{A Transformation of Positions}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;64 shows how the action of <math>F\!</math> on cells or points can be computed in terms of coordinates.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 64.} ~~ \text{A Transformation of Positions}\!</math><br />
|- style="height:40px; background:ghostwhite; width:100%"<br />
| style="width:8%" | <math>u\!</math><br />
| style="width:8%" | <math>v\!</math><br />
| style="width:12%; border-left:1px solid black" | <math>x\!</math><br />
| style="width:12%" | <math>y\!</math><br />
| style="width:10%; border-left:1px solid black" | <math>x~y\!</math><br />
| style="width:10%" | <math>x \texttt{(} y \texttt{)}\!</math><br />
| style="width:10%" | <math>\texttt{(} x \texttt{)} y\!</math><br />
| style="width:10%" | <math>\texttt{(} x \texttt{)(} y \texttt{)}\!</math><br />
| style="width:20%; border-left:1px solid black" | <math>X^\bullet = [x, y]\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\uparrow<br />
\\[4pt]<br />
F =<br />
\\[4pt]<br />
(f, g)<br />
\\[4pt]<br />
\uparrow<br />
\end{matrix}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="border-top:1px solid black" | <math>u\!</math><br />
| style="border-top:1px solid black" | <math>v\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
| style="border-top:1px solid black" | <math>\texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>u~v\!</math><br />
| style="border-top:1px solid black" | <math>\texttt{(} u \texttt{,} v \texttt{)}\!</math><br />
| style="border-top:1px solid black" | <math>\texttt{(} u \texttt{)(} v \texttt{)}\!</math><br />
| style="border-top:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>U^\bullet = [u, v]\!</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;65 extends this scheme from single cells to arbitrary regions, showing how we might compute the action of a logical transformation on arbitrary propositions in the universe of discourse. The effect of a point-transformation on arbitrary propositions, or any other structures erected on points, is referred to as the ''induced action'' of the transformation on the structures in question.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 65-a.} ~~ \text{An Induced Transformation on Propositions}\!</math><br />
|- style="height:50px; background:ghostwhite"<br />
| style="width:20%" | <math>X^\bullet~\!</math><br />
| style="width:20%; border-right:none" | <math>\longleftarrow\!</math><br />
| style="width:20%; border-left:none; border-right:none" | <math>F = (f, g)\!</math><br />
| style="width:20%; border-left:none" | <math>\longleftarrow\!</math><br />
| style="width:20%" | <math>U^\bullet~\!</math><br />
|- style="background:ghostwhite"<br />
| rowspan="2" | <math>f_i (x, y)\!</math><br />
| align="right" | <math>\begin{matrix}u = \\ v =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~0~0\\1~0~1~0\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= u \\ = v\end{matrix}</math><br />
| rowspan="2" style="width:20%" | <math>f_j (u, v)\!</math><br />
|- style="background:ghostwhite"<br />
| align="right" | <math>\begin{matrix}x = \\ y =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~1~0\\1~0~0~1\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= f(u, v) \\ = g(u, v)\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{2}<br />
\\[2pt]<br />
f_{3}<br />
\\[2pt]<br />
f_{4}<br />
\\[2pt]<br />
f_{5}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{7}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)~ ~}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{~ ~(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{,~} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{~~} y \texttt{)}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~0<br />
\\[2pt]<br />
0~0~0~0<br />
\\[2pt]<br />
0~0~0~1<br />
\\[2pt]<br />
0~0~0~1<br />
\\[2pt]<br />
0~1~1~0<br />
\\[2pt]<br />
0~1~1~0<br />
\\[2pt]<br />
0~1~1~1<br />
\\[2pt]<br />
0~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{,~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{,~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{~~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{~~} v \texttt{)}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[2pt]<br />
f_{0}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{7}<br />
\\[2pt]<br />
f_{7}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{8}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{10}<br />
\\[2pt]<br />
f_{11}<br />
\\[2pt]<br />
f_{12}<br />
\\[2pt]<br />
f_{13}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{15}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[2pt]<br />
\texttt{~ ~ ~ ~} y \texttt{~~}<br />
\\[2pt]<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[2pt]<br />
\texttt{~~} x \texttt{~ ~ ~ ~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
1~0~0~0<br />
\\[2pt]<br />
1~0~0~0<br />
\\[2pt]<br />
1~0~0~1<br />
\\[2pt]<br />
1~0~0~1<br />
\\[2pt]<br />
1~1~1~0<br />
\\[2pt]<br />
1~1~1~0<br />
\\[2pt]<br />
1~1~1~1<br />
\\[2pt]<br />
1~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~~} u \texttt{~~} v \texttt{~~}<br />
\\[2pt]<br />
\texttt{~~} u \texttt{~~} v \texttt{~~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{8}<br />
\\[2pt]<br />
f_{8}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{15}<br />
\\[2pt]<br />
f_{15}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 65-b.} ~~ \text{An Induced Transformation on Propositions}\!</math><br />
|- style="height:50px; background:ghostwhite"<br />
| style="width:20%" | <math>X^\bullet~\!</math><br />
| style="width:20%; border-right:none" | <math>\longleftarrow\!</math><br />
| style="width:20%; border-left:none; border-right:none" | <math>F = (f, g)\!</math><br />
| style="width:20%; border-left:none" | <math>\longleftarrow\!</math><br />
| style="width:20%" | <math>U^\bullet~\!</math><br />
|- style="background:ghostwhite"<br />
| rowspan="2" | <math>f_i (x, y)\!</math><br />
| align="right" | <math>\begin{matrix}u = \\ v =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~0~0\\1~0~1~0\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= u \\ = v\end{matrix}</math><br />
| rowspan="2" style="width:20%" | <math>f_j (u, v)\!</math><br />
|- style="background:ghostwhite"<br />
| align="right" | <math>\begin{matrix}x = \\ y =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~1~0\\1~0~0~1\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= f(u, v) \\ = g(u, v)\end{matrix}</math><br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>f_{0}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[2pt]<br />
f_{2}<br />
\\[2pt]<br />
f_{4}<br />
\\[2pt]<br />
f_{8}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~0<br />
\\[2pt]<br />
0~0~0~1<br />
\\[2pt]<br />
0~1~1~0<br />
\\[2pt]<br />
1~0~0~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{,~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{~} u \texttt{~~} v \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{8}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{3}<br />
\\[2pt]<br />
f_{12}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[2pt]<br />
1~1~1~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} u \texttt{)(} v \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[2pt]<br />
f_{14}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[2pt]<br />
f_{9}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[2pt]<br />
1~0~0~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{~~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{~} u \texttt{~~} v \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[2pt]<br />
f_{8}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{5}<br />
\\[2pt]<br />
f_{10}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[2pt]<br />
1~0~0~1<br />
\end{matrix}~\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} u \texttt{,~} v \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[2pt]<br />
f_{9}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[2pt]<br />
f_{11}<br />
\\[2pt]<br />
f_{13}<br />
\\[2pt]<br />
f_{14}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[2pt]<br />
1~0~0~1<br />
\\[2pt]<br />
1~1~1~0<br />
\\[2pt]<br />
1~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} u \texttt{~~} v \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{15}<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>f_{15}\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Difference Operators and Tangent Functors====<br />
<br />
Given the alphabets <math>\mathcal{U} = \{ u, v \}\!</math> and <math>\mathcal{X} = \{ x, y \},\!</math> along with the corresponding universes of discourse <math>U^\bullet, X^\bullet \cong [\mathbb{B}^2],\!</math> how many logical transformations of the general form <math>G = (G_1, G_2) : U^\bullet \to X^\bullet\!</math> are there? Since <math>G_1\!</math> and <math>G_2\!</math> can be any propositions of the type <math>\mathbb{B}^2 \to \mathbb{B},\!</math> there are <math>2^4 = 16\!</math> choices for each of the maps <math>G_1\!</math> and <math>G_2\!</math> and thus there are <math>2^4 \cdot 2^4 = 2^8 = 256\!</math> different mappings altogether of the form <math>G : U^\bullet \to X^\bullet.\!</math> The set of functions of a given type is denoted by placing its type indicator in parentheses, in the present instance writing <math>(U^\bullet \to X^\bullet) = \{ G : U^\bullet \to X^\bullet \},\!</math> and so the cardinality of the ''function space'' <math>(U^\bullet \to X^\bullet)\!</math> is summed up by writing <math>|(U^\bullet \to X^\bullet)| = |(\mathbb{B}^2 \to \mathbb{B}^2)| = 4^4 = 256.\!</math><br />
<br />
Given a transformation <math>G = (G_1, G_2) : U^\bullet \to X^\bullet\!</math> of this type, we proceed to define a pair of further transformations, related to <math>G,\!</math> that operate between the extended universes, <math>\mathrm{E}U^\bullet\!</math> and <math>\mathrm{E}X^\bullet,\!</math> of its source and target domains.<br />
<br />
First, the ''enlargement map'' (or ''secant transformation'') <math>\mathrm{E}G = (\mathrm{E}G_1, \mathrm{E}G_2) : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet\!</math> is defined by the following set of component equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}G_i<br />
& = & G_i (u + \mathrm{d}u, v + \mathrm{d}v)<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Next, the ''difference map'' (or ''chordal transformation'') <math>\mathrm{D}G = (\mathrm{D}G_1, \mathrm{D}G_2) : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet~\!</math> is defined in component-wise fashion as the boolean sum of the initial proposition <math>G_i\!</math> and the enlarged proposition <math>\mathrm{E}G_i,\!</math> for <math>i = 1, 2,\!</math> according to the following set of equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
\mathrm{D}G_i<br />
& = & G_i (u, v)<br />
& + & \mathrm{E}G_i (u, v, \mathrm{d}u, \mathrm{d}v)<br />
\\[8pt]<br />
& = & G_i (u, v)<br />
& + & G_i (u + \mathrm{d}u, v + \mathrm{d}v)<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Maintaining a strict analogy with ordinary difference calculus would perhaps have us write <math>\mathrm{D}G_i = \mathrm{E}G_i - G_i,\!</math> but the sum and difference operations are the same thing in boolean arithmetic. It is more often natural in the logical context to consider an initial proposition <math>q,\!</math> then to compute the enlargement <math>\mathrm{E}q,\!</math> and finally to determine the difference <math>\mathrm{D}q = q + \mathrm{E}q,\!</math> so we let the variant order of terms reflect this sequence of considerations.<br />
<br />
Viewed in this light the difference operator <math>\mathrm{D}\!</math> is imagined to be a function of very wide scope and polymorphic application, one that is able to realize the association between each transformation <math>G\!</math> and its difference map <math>\mathrm{D}G,\!</math> for example, taking the function space <math>(U^\bullet \to X^\bullet)\!</math> into <math>(\mathrm{E}U^\bullet \to \mathrm{E}X^\bullet).\!</math> When we consider the variety of interpretations permitted to propositions over the contexts in which we put them to use, it should be clear that an operator of this scope is not at all a trivial matter to define in general and that it may take some trouble to work out. For the moment we content ourselves with returning to particular cases.<br />
<br />
Acting on the logical transformation <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;),\!</math> the operators <math>\mathrm{E}\!</math> and <math>\mathrm{D}\!</math> yield the enlarged map <math>\mathrm{E}F = (\mathrm{E}f, \mathrm{E}g)\!</math> and the difference map <math>\mathrm{D}F = (\mathrm{D}f, \mathrm{D}g),\!</math> respectively, whose components are given as follows.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}f<br />
& = & \texttt{((} u + \mathrm{d}u \texttt{)(} v + \mathrm{d}v \texttt{))}<br />
\\[8pt]<br />
\mathrm{E}g<br />
& = & \texttt{((} u + \mathrm{d}u \texttt{,~} v + \mathrm{d}v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
\mathrm{D}f<br />
& = & \texttt{((} u \texttt{)(} v \texttt{))}<br />
& + & \texttt{((} u + \mathrm{d}u \texttt{)(} v + \mathrm{d}v \texttt{))}<br />
\\[8pt]<br />
\mathrm{D}g<br />
& = & \texttt{((} u \texttt{,~} v \texttt{))}<br />
& + & \texttt{((} u + \mathrm{d}u \texttt{,~} v + \mathrm{d}v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
But these initial formulas are purely definitional, and help us little in understanding either the purpose of the operators or the meaning of their results. Working symbolically, let us apply the same method to the separate components <math>f\!</math> and <math>g\!</math> that we earlier used on <math>J.\!</math> This work is recorded in Appendix&nbsp;3 and a summary of the results is presented in Tables&nbsp;66-i and 66-ii.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 66-i.} ~~ \text{Computation Summary for}~ f(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f<br />
& = & u \!\cdot\! v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 1<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}f<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \cdot \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{d}f<br />
& = & u \!\cdot\! v \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}f<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 66-ii.} ~~ \text{Computation Summary for}~ g(u, v) = \texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon g<br />
& = & u \!\cdot\! v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}g<br />
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}g<br />
& = & u \!\cdot\! v \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;67 shows how to compute the analytic series for <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math> in terms of coordinates, and Table&nbsp;68 recaps these results in symbolic terms, agreeing with earlier derivations.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 67.} ~~ \text{Computation of an Analytic Series in Terms of Coordinates}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:6%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}u\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>\mathrm{d}v\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>u'\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>v'\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:4px double black" | <math>f\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>g\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{E}f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{E}g}\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{D}f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{D}g}\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{d}f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{d}g}\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{d}^2\!f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{d}^2\!g}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 68.} ~~ \text{Computation of an Analytic Series in Symbolic Terms}\!</math><br />
|- style="height:40px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black" | <math>f\!</math><br />
| <math>g\!</math><br />
| style="border-left:1px solid black" | <math>{\mathrm{D}f}\!</math><br />
| <math>{\mathrm{D}g}\!</math><br />
| style="border-left:1px solid black" | <math>{\mathrm{d}f}\!</math><br />
| <math>{\mathrm{d}g}\!</math><br />
| style="border-left:1px solid black" | <math>{\mathrm{d}^2\!f}\!</math><br />
| <math>{\mathrm{d}^2\!g}\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[4pt]<br />
\texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
\\[4pt]<br />
\texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
\\[4pt]<br />
\texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;69 gives a graphical picture of the difference map <math>\mathrm{D}F = (\mathrm{D}f, \mathrm{D}g)\!</math> for the transformation <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math> This represents the same information about <math>\mathrm{D}f~\!</math> and <math>\mathrm{D}g~\!</math> that was given in the corresponding rows of Tables&nbsp;66-i and 66-ii, for ease of reference repeated below.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[8pt]<br />
\mathrm{D}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 69 -- Difference Map (Short Form).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 69.} ~~ \text{Difference Map of}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;70-a shows a way of visualizing the tangent functor map <math>\mathrm{d}F = (\mathrm{d}f, \mathrm{d}g)\!</math> for the transformation <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math> This amounts to the same information about <math>\mathrm{d}f~\!</math> and <math>\mathrm{d}g~\!</math> that was given in Tables&nbsp;66-i and 66-ii, the corresponding rows of which are repeated below.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{d}f<br />
& = & u \!\cdot\! v \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[8pt]<br />
\mathrm{d}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 70-a -- Tangent Functor Diagram.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 70-a.} ~~ \text{Tangent Functor Diagram for}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math><br />
|}<br />
<br />
<br><br />
<br />
Figure 70-b shows another way to picture the action of the tangent functor on the logical transformation <math>F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 70-b -- Tangent Functor Diagram.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 70-b.} ~~ \text{Tangent Functor Ferris Wheel for}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math><br />
|}<br />
<br />
<br><br />
<br />
* '''Note.''' The original Figure&nbsp;70-b lost some of its labeling in a succession of platform metamorphoses over the years, so we have included an ASCII version below to indicate where the missing labels go.<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| /////\ /////\ | | /XXXX\ /XXXX\ | | /\\\\\ /\\\\\ |<br />
| ///////o//////\ | | /XXXXXXoXXXXXX\ | | /\\\\\\o\\\\\\\ |<br />
| //////// \//////\ | | /XXXXXX/ \XXXXXX\ | | /\\\\\\/ \\\\\\\\ |<br />
| o/////// \//////o | | oXXXXXX/ \XXXXXXo | | o\\\\\\/ \\\\\\\o |<br />
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |<br />
| |/du//| |//dv/| | | |XXXXX| |XXXXX| | | |\du\\| |\\dv\| |<br />
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |<br />
| o//////\ ///////o | | oXXXXXX\ /XXXXXXo | | o\\\\\\\ /\\\\\\o |<br />
| \//////\ //////// | | \XXXXXX\ /XXXXXX/ | | \\\\\\\\ /\\\\\\/ |<br />
| \//////o/////// | | \XXXXXXoXXXXXX/ | | \\\\\\\o\\\\\\/ |<br />
| \///// \///// | | \XXXX/ \XXXX/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ (u)(v) o-----------------------o dv' @ (u)(v) =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| / \ /////\ | | /\\\\\ /XXXX\ | | /\\\\\ /\\\\\ |<br />
| / o//////\ | | /\\\\\\oXXXXXX\ | | /\\\\\\o\\\\\\\ |<br />
| / //\//////\ | | /\\\\\\//\XXXXXX\ | | /\\\\\\/ \\\\\\\\ |<br />
| o ////\//////o | | o\\\\\\////\XXXXXXo | | o\\\\\\/ \\\\\\\o |<br />
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |<br />
| | du |/////|//dv/| | | |\\\\\|/////|XXXXX| | | |\du\\| |\\dv\| |<br />
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |<br />
| o \//////////o | | o\\\\\\\////XXXXXXo | | o\\\\\\\ /\\\\\\o |<br />
| \ \///////// | | \\\\\\\\//XXXXXX/ | | \\\\\\\\ /\\\\\\/ |<br />
| \ o/////// | | \\\\\\\oXXXXXX/ | | \\\\\\\o\\\\\\/ |<br />
| \ / \///// | | \\\\\/ \XXXX/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ (u) v o-----------------------o dv' @ (u) v =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| /////\ / \ | | /XXXX\ /\\\\\ | | /\\\\\ /\\\\\ |<br />
| ///////o \ | | /XXXXXXo\\\\\\\ | | /\\\\\\o\\\\\\\ |<br />
| /////////\ \ | | /XXXXXX//\\\\\\\\ | | /\\\\\\/ \\\\\\\\ |<br />
| o//////////\ o | | oXXXXXX////\\\\\\\o | | o\\\\\\/ \\\\\\\o |<br />
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |<br />
| |/du//|/////| dv | | | |XXXXX|/////|\\\\\| | | |\du\\| |\\dv\| |<br />
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |<br />
| o//////\//// o | | oXXXXXX\////\\\\\\o | | o\\\\\\\ /\\\\\\o |<br />
| \//////\// / | | \XXXXXX\//\\\\\\/ | | \\\\\\\\ /\\\\\\/ |<br />
| \//////o / | | \XXXXXXo\\\\\\/ | | \\\\\\\o\\\\\\/ |<br />
| \///// \ / | | \XXXX/ \\\\\/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ u (v) o-----------------------o dv' @ u (v) =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| / \ / \ | | /\\\\\ /\\\\\ | | /\\\\\ /\\\\\ |<br />
| / o \ | | /\\\\\\o\\\\\\\ | | /\\\\\\o\\\\\\\ |<br />
| / / \ \ | | /\\\\\\/ \\\\\\\\ | | /\\\\\\/ \\\\\\\\ |<br />
| o / \ o | | o\\\\\\/ \\\\\\\o | | o\\\\\\/ \\\\\\\o |<br />
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |<br />
| | du | | dv | | | |\\\\\| |\\\\\| | | |\du\\| |\\dv\| |<br />
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |<br />
| o \ / o | | o\\\\\\\ /\\\\\\o | | o\\\\\\\ /\\\\\\o |<br />
| \ \ / / | | \\\\\\\\ /\\\\\\/ | | \\\\\\\\ /\\\\\\/ |<br />
| \ o / | | \\\\\\\o\\\\\\/ | | \\\\\\\o\\\\\\/ |<br />
| \ / \ / | | \\\\\/ \\\\\/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ u v o-----------------------o dv' @ u v =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| U | |\U\\\\\\\\\\\\\\\\\\\\\| |\U\\\\\\\\\\\\\\\\\\\\\|<br />
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|<br />
| /////\ /////\ | |\\\\\/////\\/////\\\\\\| |\\\\\/ \\/ \\\\\\|<br />
| ///////o//////\ | |\\\\///////o//////\\\\\| |\\\\/ o \\\\\|<br />
| /////////\//////\ | |\\\////////X\//////\\\\| |\\\/ /\\ \\\\|<br />
| o//////////\//////o | |\\o///////XXX\//////o\\| |\\o /\\\\ o\\|<br />
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|<br />
| |//u//|/////|//v//| | |\\|//u//|XXXXX|//v//|\\| |\\| u |\\\\\| v |\\|<br />
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|<br />
| o//////\//////////o | |\\o//////\XXX///////o\\| |\\o \\\\/ o\\|<br />
| \//////\///////// | |\\\\//////\X////////\\\| |\\\\ \\/ /\\\|<br />
| \//////o/////// | |\\\\\//////o///////\\\\| |\\\\\ o /\\\\|<br />
| \///// \///// | |\\\\\\/////\\/////\\\\\| |\\\\\\ /\\ /\\\\\|<br />
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|<br />
| | |\\\\\\\\\\\\\\\\\\\\\\\| |\\\\\\\\\\\\\\\\\\\\\\\|<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= u' o-----------------------o v' =<br />
= | U' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/u'//|XXXXX|\\v'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
Figure 70-b. Tangent Functor Ferris Wheel for F<u, v> = <((u)(v)), ((u, v))><br />
</pre><br />
|}<br />
<br />
==Epilogue, Enchoiry, Exodus==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
It is time to explain myself . . . . let us stand up.<br />
| width="4%" | &nbsp;<br />
|- <br />
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 79]<br />
|}<br />
<br />
==Appendices==<br />
<br />
===Appendix 1. Propositional Forms and Differential Expansions===<br />
<br />
====Table A1. Propositional Forms on Two Variables====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A1.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="background:ghostwhite"<br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}</math><br />
| width="25%" | <math>\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{0}\\f_{1}\\f_{2}\\f_{3}\\f_{4}\\f_{5}\\f_{6}\\f_{7}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0000}\\f_{0001}\\f_{0010}\\f_{0011}\\f_{0100}\\f_{0101}\\f_{0110}\\f_{0111}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~0\\0~0~0~1\\0~0~1~0\\0~0~1~1\\0~1~0~0\\0~1~0~1\\0~1~1~0\\0~1~1~1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)~ ~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~ ~(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{,~} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{~~} y \texttt{)}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{false}<br />
\\<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\<br />
y ~\text{without}~ x<br />
\\<br />
\text{not}~ x<br />
\\<br />
x ~\text{without}~ y<br />
\\<br />
\text{not}~ y<br />
\\<br />
x ~\text{not equal to}~ y<br />
\\<br />
\text{not both}~ x ~\text{and}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\<br />
\lnot x \land \lnot y<br />
\\<br />
\lnot x \land y<br />
\\<br />
\lnot x<br />
\\<br />
x \land \lnot y<br />
\\<br />
\lnot y<br />
\\<br />
x \ne y<br />
\\<br />
\lnot x \lor \lnot y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{8}\\f_{9}\\f_{10}\\f_{11}\\f_{12}\\f_{13}\\f_{14}\\f_{15}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{1000}\\f_{1001}\\f_{1010}\\f_{1011}\\f_{1100}\\f_{1101}\\f_{1110}\\f_{1111}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1~0~0~0\\1~0~0~1\\1~0~1~0\\1~0~1~1\\1~1~0~0\\1~1~0~1\\1~1~1~0\\1~1~1~1<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~ ~ ~ ~} y \texttt{~~}<br />
\\<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{~~} x \texttt{~ ~ ~ ~}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{((~))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{and}~ y<br />
\\<br />
x ~\text{equal to}~ y<br />
\\<br />
y<br />
\\<br />
\text{not}~ x ~\text{without}~ y<br />
\\<br />
x<br />
\\<br />
\text{not}~ y ~\text{without}~ x<br />
\\<br />
x ~\text{or}~ y<br />
\\<br />
\text{true}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x \land y<br />
\\<br />
x = y<br />
\\<br />
y<br />
\\<br />
x \Rightarrow y<br />
\\<br />
x<br />
\\<br />
x \Leftarrow y<br />
\\<br />
x \lor y<br />
\\<br />
1<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A2. Propositional Forms on Two Variables====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A2.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="background:ghostwhite"<br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}</math><br />
| width="25%" | <math>\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>f_{0000}\!</math><br />
| <math>0~0~0~0</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0001}\\f_{0010}\\f_{0100}\\f_{1000}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~1\\0~0~1~0\\0~1~0~0\\1~0~0~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\<br />
y ~\text{without}~ x<br />
\\<br />
x ~\text{without}~ y<br />
\\<br />
x ~\text{and}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot x \land \lnot y<br />
\\<br />
\lnot x \land y<br />
\\<br />
x \land \lnot y<br />
\\<br />
x \land y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{3}\\f_{12}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0011}\\f_{1100}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~1~1\\1~1~0~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ x<br />
\\<br />
x<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot x<br />
\\<br />
x<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{6}\\f_{9}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0110}\\f_{1001}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~0\\1~0~0~1<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{not equal to}~ y<br />
\\<br />
x ~\text{equal to}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x \ne y<br />
\\<br />
x = y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{5}\\f_{10}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0101}\\f_{1010}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~0~1\\1~0~1~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ y<br />
\\<br />
y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot y<br />
\\<br />
y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{7}\\f_{11}\\f_{13}\\f_{14}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0111}\\f_{1011}\\f_{1101}\\f_{1110}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~1\\1~0~1~1\\1~1~0~1\\1~1~1~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not both}~ x ~\text{and}~ y<br />
\\<br />
\text{not}~ x ~\text{without}~ y<br />
\\<br />
\text{not}~ y ~\text{without}~ x<br />
\\<br />
x ~\text{or}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot x \lor \lnot y<br />
\\<br />
x \Rightarrow y<br />
\\<br />
x \Leftarrow y<br />
\\<br />
x \lor y<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>f_{1111}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A3. E''f'' Expanded Over Differential Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A3.} ~~ \mathrm{E}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{T}_{11}f\\\mathrm{E}f|_{\mathrm{d}x ~ \mathrm{d}y}\end{matrix}</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{T}_{10}f\\\mathrm{E}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}\end{matrix}</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{T}_{01}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}\end{matrix}</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{T}_{00}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{3}\\f_{12}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{6}\\f_{9}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{5}\\f_{10}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{7}\\f_{11}\\f_{13}\\f_{14}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
|- style="background:ghostwhite"<br />
| style="border-top:1px solid black" colspan="2" | <math>\text{Fixed Point Total}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>4\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>4\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>4\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>16\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A4. D''f'' Expanded Over Differential Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A4.} ~~ \mathrm{D}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\mathrm{D}f|_{\mathrm{d}x ~ \mathrm{d}y}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}~\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
y<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
y<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
x<br />
\\<br />
x<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
x<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
y<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
y<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <br />
<math>\begin{matrix}<br />
x<br />
\\<br />
x<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A5. E''f'' Expanded Over Ordinary Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A5.} ~~ \mathrm{E}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\mathrm{E}f|_{xy}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{E}f|_{x \texttt{(} y \texttt{)}}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{E}f|_{\texttt{(} x \texttt{)} y}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{E}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{3}\\f_{12}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{6}\\f_{9}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{5}\\f_{10}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{7}\\f_{11}\\f_{13}\\f_{14}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A6. D''f'' Expanded Over Ordinary Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A6.} ~~ \mathrm{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\mathrm{D}f|_{xy}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{x \texttt{(} y \texttt{)}}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} x \texttt{)} y}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\texttt{(} x \texttt{)}\\\texttt{~} x \texttt{~}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Appendix 2. Differential Forms===<br />
<br />
The actions of the difference operator <math>\mathrm{D}\!</math> and the tangent operator <math>\mathrm{d}\!</math> on the 16 bivariate propositions are shown in Tables&nbsp;A7 and A8.<br />
<br />
Table A7 expands the differential forms that result over a ''logical basis'':<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
|<br />
<math>\{~ \texttt{(}\mathrm{d}x\texttt{)(}\mathrm{d}y\texttt{)}, ~\mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}, ~\texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.\!</math><br />
|}<br />
<br />
This set consists of the singular propositions in the first order differential variables, indicating mutually exclusive and exhaustive ''cells'' of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the cell-basis, point-basis, or singular differential basis. In this setting it is frequently convenient to use the following abbreviations:<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
|<br />
<math>\partial x ~=~ \mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}\!</math> &nbsp; &nbsp; and &nbsp; &nbsp; <math>\partial y ~=~ \texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y.\!</math><br />
|}<br />
<br />
Table A8 expands the differential forms that result over an ''algebraic basis'':<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
| <math>\{~ 1, ~\mathrm{d}x, ~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.\!</math><br />
|}<br />
<br />
This set consists of the ''positive propositions'' in the first order differential variables, indicating overlapping positive regions of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the ''positive differential basis''.<br />
<br />
====Table A7. Differential Forms Expanded on a Logical Basis====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A7.} ~~ \text{Differential Forms Expanded on a Logical Basis}\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| &nbsp;<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" | <math>\mathrm{D}f~\!</math><br />
| <math>\mathrm{d}f~\!</math><br />
|-<br />
| <math>f_{0}\!</math><br />
| style="border-right:none" | <math>\texttt{(~)}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x) ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\\<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\\<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\\<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\partial x<br />
\\<br />
\partial x<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
\\<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\partial x & + & \partial y<br />
\\<br />
\partial x & + & \partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\partial y<br />
\\<br />
\partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x) ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\\<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\\<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\\<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| style="border-right:none" | <math>\texttt{((~))}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A8. Differential Forms Expanded on an Algebraic Basis====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A8.} ~~ \text{Differential Forms Expanded on an Algebraic Basis}\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| &nbsp;<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" | <math>\mathrm{D}f~\!</math><br />
| <math>\mathrm{d}f~\!</math><br />
|-<br />
| <math>f_{0}\!</math><br />
| style="border-right:none" | <math>\texttt{(~)}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x<br />
\\<br />
\mathrm{d}x<br />
\end{matrix}\!</math><br />
| <math>\begin{matrix}<br />
\mathrm{d}x<br />
\\<br />
\mathrm{d}x<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\\<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\\<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}y<br />
\\<br />
\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y<br />
\\<br />
\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| style="border-right:none" | <math>\texttt{((~))}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A9. Tangent Proposition as Pointwise Linear Approximation====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A9.} ~~ \text{Tangent Proposition}~ \mathrm{d}f = \text{Pointwise Linear Approximation to the Difference Map}~ \mathrm{D}f\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}f =<br />
\\[2pt]<br />
\partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}^2\!f =<br />
\\[2pt]<br />
\partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\mathrm{d}f|_{x \, y}</math><br />
| <math>\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math><br />
|-<br />
| style="border-right:none" | <math>f_0\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{5}\\f_{10}\end{matrix}\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\end{matrix}\!</math><br />
| <math>\begin{matrix}<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>f_{15}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A10. Taylor Series Expansion Df = d''f'' + d<sup>2</sup>''f''====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" |<br />
<math>\text{Table A10.} ~~ \text{Taylor Series Expansion}~ {\mathrm{D}f = \mathrm{d}f + \mathrm{d}^2\!f}\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{D}f<br />
\\<br />
= & \mathrm{d}f & + & \mathrm{d}^2\!f<br />
\\<br />
= & \partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y & + & \partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\mathrm{d}f|_{x \, y}</math><br />
| <math>\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math><br />
|-<br />
| style="border-right:none" | <math>f_0\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>f_{15}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A11. Partial Differentials and Relative Differentials====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A11.} ~~ \text{Partial Differentials and Relative Differentials}\!</math><br />
|- style="background:ghostwhite; height:50px"<br />
| &nbsp;<br />
| <math>f\!</math><br />
| <math>\frac{\partial f}{\partial x}\!</math><br />
| <math>\frac{\partial f}{\partial y}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}f =<br />
\\[2pt]<br />
\partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\left. \frac{\partial x}{\partial y} \right| f\!</math><br />
| <math>\left. \frac{\partial y}{\partial x} \right| f\!</math><br />
|-<br />
| <math>f_0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A12. Detail of Calculation for the Difference Map====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:4px double black; border-left:4px double black; border-right:4px double black; border-top:4px double black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A12.} ~~ \text{Detail of Calculation for}~ {\mathrm{E}f + f = \mathrm{D}f}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:6%" | &nbsp;<br />
| style="width:14%; border-left:1px solid black" | <math>f\!</math><br />
| style="width:20%; border-left:4px double black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\mathrm{d}x ~ \mathrm{d}y}<br />
\\[4pt]<br />
+ & f|_{\mathrm{d}x ~ \mathrm{d}y}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\mathrm{d}x ~ \mathrm{d}y}<br />
\end{array}</math><br />
| style="width:20%; border-left:1px solid black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}<br />
\\[4pt]<br />
+ & f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}<br />
\end{array}</math><br />
| style="width:20%; border-left:1px solid black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
+ & f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}<br />
\end{array}</math><br />
| style="width:20%; border-left:1px solid black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
+ & f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}<br />
\end{array}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{0}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:4px double black; border-left:4px double black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{1}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{)(} y \texttt{)~}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{2}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{)~} y \texttt{~~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{4}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~~} x \texttt{~(} y \texttt{)~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{8}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~~} x \texttt{~~} y \texttt{~~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{3}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{(} x \texttt{)}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{12}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>x\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{6}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{,~} y \texttt{)~}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{9}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} x \texttt{,~} y \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{5}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{(} y \texttt{)}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{10}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>y\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{7}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{~~} y \texttt{)~}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{11}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{~(} y \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{13}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} x \texttt{)~} y \texttt{)~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{14}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} x \texttt{)(} y \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{15}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:4px double black; border-left:4px double black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Appendix 3. Computational Details===<br />
<br />
====Operator Maps for the Logical Conjunction ''f''<sub>8</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{8}~\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{8}<br />
& = && f_{8}(u, v)<br />
\\[4pt]<br />
& = && uv<br />
\\[4pt]<br />
& = && uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & uv \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & uv \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{8}<br />
& = && uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & uv \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & uv \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & uv \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.2-i} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{E}f_{8}<br />
& = & f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{8}(\mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{8}(\mathrm{d}u, \mathrm{d}v)<br />
\\[4pt]<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[20pt]<br />
\mathrm{E}f_{8}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
\\[4pt]<br />
&&&&& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&&&&&& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.2-ii} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{c}}<br />
\mathrm{E}f_{8}<br />
& = & (u + \mathrm{d}u) \cdot (v + \mathrm{d}v)<br />
\\[6pt]<br />
& = & u \cdot v<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}f_{8}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{8}<br />
& = && \mathrm{E}f_{8}<br />
& + & \boldsymbol\varepsilon f_{8}<br />
\\[4pt]<br />
& = && f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{8}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & uv<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{~}<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}f_{8}<br />
& = & \boldsymbol\varepsilon f_{8}<br />
& + & \mathrm{E}f_{8}<br />
\\[6pt]<br />
& = & f_{8}(u, v)<br />
& + & f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[6pt]<br />
& = & uv<br />
& + & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
& = & 0<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}f_{8}<br />
& = & 0<br />
& + & u \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.3-iii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 3)}\!</math><br />
|<br />
<math>\begin{array}{c*{9}{l}}<br />
\mathrm{D}f_{8}<br />
& = & \boldsymbol\varepsilon f_{8} ~+~ \mathrm{E}f_{8}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{8}<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ u \,\cdot\, v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~ u \;\cdot\; v \;\cdot\; \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}f_{8}<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u ~ \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} ~ v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot\, \mathrm{d}u ~ \mathrm{d}v<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = & ~ ~ 0 ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~ ~ u ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ ~ ~ v ~~ \cdot ~ \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
=====Computation of d''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.4} ~~ \text{Computation of}~ \mathrm{d}f_{8}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{8}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.5} ~~ \text{Computation of}~ \mathrm{r}f_{8}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{8} & = & \mathrm{D}f_{8} ~+~ \mathrm{d}f_{8}<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[20pt]<br />
\mathrm{r}f_{8}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Conjunction=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.6} ~~ \text{Computation Summary for}~ f_{8}(u, v) = uv\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{8}<br />
& = & uv \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}f_{8}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{r}f_{8}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Operator Maps for the Logical Equality ''f''<sub>9</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{9}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{9}<br />
& = && f_{9}(u, v)<br />
\\[4pt]<br />
& = && \texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{9}(1, 1)<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{9}(1, 0)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{9}(0, 1)<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{9}(0, 0)<br />
\\[4pt]<br />
& = && u v & + & 0 & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{9}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.2} ~~ \text{Computation of}~ \mathrm{E}f_{9}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{E}f_{9}<br />
& = && f_{9}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! f_{9}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! f_{9}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{ } \mathrm{d}v \texttt{ })<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! f_{9}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! f_{9}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{ } \mathrm{d}v \texttt{ })<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{ (} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{) }<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{ (} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{) }<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{E}f_{9}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{9}<br />
& = && \mathrm{E}f_{9}<br />
& + & \boldsymbol\varepsilon f_{9}<br />
\\[4pt]<br />
& = && f_{9}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{9}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{9}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[20pt]<br />
\mathrm{D}f_{9}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}f_{9}<br />
& = & 0 \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & 1 \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & 1 \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & 0 \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of d''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.4} ~~ \text{Computation of}~ \mathrm{d}f_{9}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.5} ~~ \text{Computation of}~ \mathrm{r}f_{9}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{9} & = & \mathrm{D}f_{9} ~+~ \mathrm{d}f_{9}<br />
\\[20pt]<br />
\mathrm{D}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[20pt]<br />
\mathrm{r}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Equality=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.6} ~~ \text{Computation Summary for}~ f_{9}(u, v) = \texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{9}<br />
& = & uv \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}f_{9}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f_{9}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{9}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}f_{9}<br />
& = & uv \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Operator Maps for the Logical Implication ''f''<sub>11</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{11}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{11}<br />
& = && f_{11}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(} u \texttt{(} v \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{11}(1, 1)<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{11}(1, 0)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{11}(0, 1)<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{11}(0, 0)<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ }<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ }<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{11}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.2} ~~ \text{Computation of}~ \mathrm{E}f_{11}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{E}f_{11}<br />
& = && f_{11}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = &&<br />
\texttt{(}<br />
\\<br />
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\\<br />
&&& \texttt{(}<br />
\\<br />
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\<br />
&&& \texttt{))}<br />
\\[4pt]<br />
& = &&<br />
u v<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)((} \mathrm{d}v \texttt{)))}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u \texttt{((} \mathrm{d}v \texttt{)))}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[4pt]<br />
& = &&<br />
u v<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{E}f_{11}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{11}<br />
& = && \mathrm{E}f_{11}<br />
& + & \boldsymbol\varepsilon f_{11}<br />
\\[4pt]<br />
& = && f_{11}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{11}(u, v)<br />
\\[4pt]<br />
& = &&<br />
\texttt{(} \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\texttt{(} \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{(} v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & u v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & 0<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & 0<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = && u v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{c*{9}{l}}<br />
\mathrm{D}f_{11}<br />
& = & \boldsymbol\varepsilon f_{11} ~+~ \mathrm{E}f_{11}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{11}<br />
& = & u v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}f_{11}<br />
& = &<br />
u v<br />
\cdot<br />
\texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\cdot<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\cdot<br />
\texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\cdot<br />
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = &<br />
u v<br />
\cdot<br />
\texttt{~(} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{~}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\cdot<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\cdot<br />
\texttt{~} \mathrm{d}u ~ \mathrm{d}v \texttt{~}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\cdot<br />
\texttt{~} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of d''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.4} ~~ \text{Computation of}~ \mathrm{d}f_{11}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{11}<br />
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{11}<br />
& = & u v \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.5} ~~ \text{Computation of}~ \mathrm{r}f_{11}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{11} & = & \mathrm{D}f_{11} ~+~ \mathrm{d}f_{11}<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{11}<br />
& = & u v \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u<br />
\\[20pt]<br />
\mathrm{r}f_{11}<br />
& = & u v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Implication=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.6} ~~ \text{Computation Summary for}~ f_{11}(u, v) = \texttt{(} u \texttt{(} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{11}<br />
& = & u v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}f_{11}<br />
& = & u v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f_{11}<br />
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{11}<br />
& = & u v \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u<br />
\\[6pt]<br />
\mathrm{r}f_{11}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Operator Maps for the Logical Disjunction ''f''<sub>14</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{14}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{14}<br />
& = && f_{14}(u, v)<br />
\\[4pt]<br />
& = && \texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{14}(1, 1)<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{14}(1, 0)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{14}(0, 1)<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{14}(0, 0)<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ }<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ }<br />
& + & 0<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{14}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.2} ~~ \text{Computation of}~ \mathrm{E}f_{14}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{E}f_{14}<br />
& = && f_{14}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = &&<br />
\texttt{((}<br />
\\<br />
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\\<br />
&&& \texttt{)(}<br />
\\<br />
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\<br />
&&& \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! f_{14}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! f_{14}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{ } \mathrm{d}v \texttt{ })<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! f_{14}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! f_{14}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{ } \mathrm{d}v \texttt{ })<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{E}f_{14}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{14}<br />
& = && \mathrm{E}f_{14}<br />
& + & \boldsymbol\varepsilon f_{14}<br />
\\[4pt]<br />
& = && f_{14}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{14}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{))((} v \texttt{,} \mathrm{d}v \texttt{)))}<br />
& + & \texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{14}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
\\[4pt]<br />
&& + & 0<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
\\[4pt]<br />
&& + & uv \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
\\[20pt]<br />
\mathrm{D}f_{14}<br />
& = && uv \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}f_{14}<br />
& = & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & 0 \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of d''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.4} ~~ \text{Computation of}~ \mathrm{d}f_{14}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.5} ~~ \text{Computation of}~ \mathrm{r}f_{14}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{14} & = & \mathrm{D}f_{14} ~+~ \mathrm{d}f_{14}<br />
\\[20pt]<br />
\mathrm{D}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{d}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[20pt]<br />
\mathrm{r}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Disjunction=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.6} ~~ \text{Computation Summary for}~ f_{14}(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{14}<br />
& = & uv \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 1<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}f_{14}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f_{14}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{d}f_{14}<br />
& = & uv \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}f_{14}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
===Appendix 4. Source Materials===<br />
<br />
===Appendix 5. Various Definitions of the Tangent Vector===<br />
<br />
==References==<br />
<br />
===Works Cited===<br />
<br />
{| cellpadding=3<br />
| valign=top | [AuM]<br />
| Auslander, L., and MacKenzie, R.E., ''Introduction to Differentiable Manifolds'', McGraw-Hill, 1963. Reprinted, Dover, New York, NY, 1977.<br />
|-<br />
| valign=top | [BiG]<br />
| Bishop, R.L., and Goldberg, S.I., ''Tensor Analysis on Manifolds'', Macmillan, 1968. Reprinted, Dover, New York, NY, 1980.<br />
|-<br />
| valign=top | [Boo]<br />
| Boole, G., ''An Investigation of The Laws of Thought'', Macmillan, 1854. Reprinted, Dover, New York, NY, 1958.<br />
|-<br />
| valign=top | [BoT]<br />
| Bott, R., and Tu, L.W., ''Differential Forms in Algebraic Topology'', Springer-Verlag, New York, NY, 1982.<br />
|-<br />
| valign=top | [dCa]<br />
| do Carmo, M.P., ''Riemannian Geometry''. Originally published in Portuguese, 1st editiom 1979, 2nd edition 1988. Translated by F. Flaherty, Birkhäuser, Boston, MA, 1992.<br />
|-<br />
| valign=top | [Che46]<br />
| Chevalley, C., ''Theory of Lie Groups'', Princeton University Press, Princeton, NJ, 1946.<br />
|-<br />
| valign=top | [Che56]<br />
| Chevalley, C., ''Fundamental Concepts of Algebra'', Academic Press, 1956.<br />
|-<br />
| valign=top | [Cho86]<br />
| Chomsky, N., ''Knowledge of Language : Its Nature, Origin, and Use'', Praeger, New York, NY, 1986.<br />
|-<br />
| valign=top | [Cho93]<br />
| Chomsky, N., ''Language and Thought'', Moyer Bell, Wakefield, RI, 1993.<br />
|-<br />
| valign=top | [DoM]<br />
| Doolin, B.F., and Martin, C.F., ''Introduction to Differential Geometry for Engineers'', Marcel Dekker, New York, NY, 1990.<br />
|-<br />
| valign=top | [Fuji]<br />
| Fujiwara, H., ''Logic Testing and Design for Testability'', MIT Press, Cambridge, MA, 1985.<br />
|-<br />
| valign=top | [Hic]<br />
| Hicks, N.J., ''Notes on Differential Geometry'', Van Nostrand, Princeton, NJ, 1965.<br />
|-<br />
| valign=top | [Hir]<br />
| Hirsch, M.W., ''Differential Topology'', Springer-Verlag, New York, NY, 1976.<br />
|-<br />
| valign=top | [How]<br />
| Howard, W.A., "The Formulae-as-Types Notion of Construction", Notes circulated from 1969. Reprinted in [SeH, 479-490].<br />
|-<br />
| valign=top | [JGH]<br />
| Jones, A., Gray, A., and Hutton, R., ''Manifolds and Mechanics'', Cambridge University Press, Cambridge, UK, 1987.<br />
|-<br />
| valign=top | [KoA]<br />
| Kosinski, A.A., ''Differential Manifolds'', Academic Press, San Diego, CA, 1993.<br />
|-<br />
| valign=top | [Koh]<br />
| Kohavi, Z., ''Switching and Finite Automata Theory'', 2nd edition, McGraw-Hill, New York, NY, 1978.<br />
|-<br />
| valign=top | [LaS]<br />
| Lambek, J., and Scott, P.J., ''Introduction to Higher Order Categorical Logic'', Cambridge University Press, Cambridge, UK, 1986.<br />
|-<br />
| valign=top | [La83]<br />
| Lang, S., ''Real Analysis'', 2nd edition, Addison-Wesley, Reading, MA, 1983.<br />
|-<br />
| valign=top | [La84]<br />
| Lang, S., ''Algebra'', 2nd edition, Addison-Wesley, Menlo Park, CA, 1984.<br />
|-<br />
| valign=top | [La85]<br />
| Lang, S., ''Differential Manifolds'', Springer-Verlag, New York, NY, 1985.<br />
|-<br />
| valign=top | [La93]<br />
| Lang, S., ''Real and Functional Analysis'', 3rd edition, Springer-Verlag, New York, NY, 1993.<br />
|-<br />
| valign=top | [Lie80]<br />
| Lie, S., "Sophus Lie's 1880 Transformation Group Paper", in ''Lie Groups : History, Frontiers, and Applications, Volume 1'', translated by M. Ackerman, comments by R. Hermann, Math Sci Press, Brookline, MA, 1975. Original paper 1880.<br />
|-<br />
| valign=top | [Lie84]<br />
| Lie, S., "Sophus Lie's 1884 Differential Invariant Paper", in ''Lie Groups : History, Frontiers, and Applications, Volume 3'', translated by M. Ackerman, comments by R. Hermann, Math Sci Press, Brookline, MA, 1976. Original paper 1884.<br />
|-<br />
| valign=top | [LoS]<br />
| Loomis, L.H., and Sternberg, S., ''Advanced Calculus'', Addison-Wesley, Reading, MA, 1968.<br />
|-<br />
| valign=top | [Mel]<br />
| Melzak, Z.A., ''Companion to Concrete Mathematics, Volume 2 : Mathematical Ideas, Modeling, and Applications'', John Wiley amd Sons, New York, NY, 1976.<br />
|-<br />
| valign=top | [Men]<br />
| Menabrea, L.F., "Sketch of the Analytical Engine Invented by Charles Babbage" with Notes by the Translator, Ada Augusta (Byron), Countess of Lovelace'', in [M&M, 225–297]. Originally published 1842.<br />
|-<br />
| valign=top | [M&M]<br />
| Morrison, P., and Morrison, E. (eds.), ''Charles Babbage on the Principles and Development of the Calculator, and Other Seminal Writings by Charles Babbage and Others, With an Introduction by the Editors'', Dover, Mineola, NY, 1961.<br />
|-<br />
| valign=top | [P1]<br />
| Peirce, C.S., ''Collected Papers of Charles Sanders Peirce'', vols. 1–8, C. Hartshorne, P. Weiss, and A.W. Burks (eds.), Harvard University Press, Cambridge, MA, 1931–1960. Cited as CP [volume].[paragraph].<br />
|-<br />
| valign=top | [P2]<br />
| Peirce, C.S., "Qualitative Logic", in ''The New Elements of Mathematics, Volume 4'', C. Eisele (ed.), Mouton, The Hague, 1976. Cited as NE [volume], [page].<br />
|-<br />
| valign=top | [Rob]<br />
| Roberts, D.D., ''The Existential Graphs of Charles S. Peirce'', Mouton, The Hague, 1973.<br />
|-<br />
| valign=top | [SeH]<br />
| Seldin, J.P., and Hindley, J.R. (eds.), ''To H.B. Curry : Essays on Combinatory Logic, Lambda Calculus, and Formalism'', Academic Press, London, UK, 1980.<br />
|-<br />
| valign=top | [SpB]<br />
| Spencer-Brown, G., ''Laws of Form'', George Allen and Unwin, London, UK, 1969.<br />
|-<br />
| valign=top | [Sp65]<br />
| Spivak, M., ''Calculus on Manifolds : A Modern Approach to Classical Theorems of Advanced Calculus'', W.A. Benjamin, New York, NY, 1965.<br />
|-<br />
| valign=top | [Sp79]<br />
| Spivak, M., ''A Comprehensive Introduction to Differential Geometry'', vols. 1–2. 1st edition 1970. 2nd edition, Publish or Perish Inc., Houston, TX, 1979.<br />
|-<br />
| valign=top | [Sty]<br />
| Styazhkin, N.I., ''History of Mathematical Logic from Leibniz to Peano'', 1st published in Russian, Nauka, Moscow, 1964. MIT Press, Cambridge, MA, 1969.<br />
|-<br />
| valign=top | [Wie]<br />
| Wiener, N., ''Cybernetics : or Control and Communication in the Animal and the Machine'', 1st edition 1948. 2nd edition, MIT Press, Cambridge, MA, 1961.<br />
|}<br />
<br />
===Works Consulted===<br />
<br />
{| cellpadding=3<br />
| valign=top | [Ami]<br />
| Amit, D.J., ''Modeling Brain Function : The World of Attractor Neural Networks'', Cambridge University Press, Cambridge, UK, 1989.<br />
|-<br />
| valign=top | [Ed87]<br />
| Edelman, G.M., ''Neural Darwinism : The Theory of Neuronal Group Selection'', Basic Books, New York, NY, 1987.<br />
|-<br />
| valign=top | [Ed88]<br />
| Edelman, G.M., ''Topobiology : An Introduction to Molecular Embryology'', Basic Books, New York, NY, 1988.<br />
|-<br />
| valign=top | [Fla]<br />
| Flanders, H., ''Differential Forms with Applications to the Physical Sciences'', Academic Press, 1963. Reprinted, Dover, Mineola, NY, 1989. <br />
|-<br />
| valign=top | [Has]<br />
| Hassoun, M.H. (ed.), ''Associative Neural Memories : Theory and Implementation'', Oxford University Press, New York, NY, 1993.<br />
|-<br />
| valign=top | [KoB]<br />
| Kosko, B., ''Neural Networks and Fuzzy Systems : A Dynamical Systems Approach to Machine Intelligence'', Prentice-Hall, Englewood Cliffs, NJ, 1992.<br />
|-<br />
| valign=top | [MaB]<br />
| Mac Lane, S., and Birkhoff, G., ''Algebra'', 3rd edition, Chelsea, New York, NY, 1993.<br />
|-<br />
| valign=top | [Mac]<br />
| Mac Lane, S., ''Categories for the Working Mathematician'', Springer-Verlag, New York, NY, 1971.<br />
|-<br />
| valign=top | [McC]<br />
| McCulloch, W.S., ''Embodiments of Mind'', MIT Press, Cambridge, MA, 1965.<br />
|-<br />
| valign=top | [Mc1]<br />
| McCulloch, W.S., "A Heterarchy of Values Determined by the Topology of Nervous Nets", Bulletin of Mathematical Biophysics, vol. 7 (1945), pp. 89–93. Reprinted in [McC].<br />
|-<br />
| valign=top | [MiP]<br />
| Minsky, M.L., and Papert, S.A., ''Perceptrons : An Introduction to Computational Geometry'', MIT Press, Cambridge, MA, 1969. 2nd printing 1972. Expanded edition 1988.<br />
|-<br />
| valign=top | [Rum]<br />
| Rumelhart, D.E., Hinton, G.E., and McClelland, J.L., "A General Framework for Parallel Distributed Processing" = Chapter 2 in Rumelhart, McClelland, and the PDP Research Group, ''Parallel Distributed Processing, Explorations in the Microstructure of Cognition, Volume 1 : Foundations'', MIT Press, Cambridge, MA, 1986.<br />
|}<br />
<br />
===Incidental Works===<br />
<br />
{| cellpadding=3<br />
| valign=top | [Dew]<br />
| Dewey, John, ''How We Think'', D.C. Heath, Lexington, MA, 1910. Reprinted, Prometheus Books, Buffalo, NY, 1991.<br />
|-<br />
| valign=top | [Fou]<br />
| Foucault, Michel, ''The Archaeology of Knowledge and The Discourse on Language'', A.M. Sheridan-Smith and Rupert Swyer (trans.), Pantheon, New York, NY, 1972. Originally published as ''L´Archéologie du Savoir et L´ordre du discours'', Editions Gallimard, 1969 & 1971.<br />
|-<br />
| valign=top | [Hom]<br />
| Homer, ''The Odyssey'', with an English translation by A.T. Murray, Loeb Classical Library, Harvard University Press, Cambridge, MA, 1980. First printed 1919.<br />
|-<br />
| valign=top | [Jam]<br />
| James, William, ''Pragmatism : A New Name for Some Old Ways of Thinking'', Longmans, Green, and Company, New York, NY, 1907.<br />
|-<br />
| valign=top | [Ler]<br />
| Leroux, Gaston, ''The Phantom of the Opera'', foreword by P. Haining, Dorset Press, New York, NY, 1988. Originally published in French, 1911.<br />
|-<br />
| valign=top | [Mus]<br />
| Musil, Robert, ''The Man Without Qualities'', 3 volumes, translated with a foreword by Eithne Wilkins and Ernst Kaiser, Pan Books, London, UK, 1979. English edition first published by Secker and Warburg, 1954. Originally published in German, ''Der Mann ohne Eigenschaften'', 1930 & 1932.<br />
|-<br />
| valign=top | [PlaR]<br />
| Plato, ''The Republic'', with an English translation by Paul Shorey, Loeb Classical Library, Harvard University Press, Cambridge, MA, 1980. First printed 1930 & 1935.<br />
|-<br />
| valign=top | [PlaS]<br />
| Plato, ''The Sophist'', Loeb Classical Library, William Heinemann, London, 1921, 1987.<br />
|-<br />
| valign=top | [Qui]<br />
| Quine, W.V., ''Mathematical Logic'', 1st edition, 1940. Revised edition, 1951. Harvard University Press, Cambridge, MA, 1981.<br />
|-<br />
| valign=top | [SaD]<br />
| de Santillana, Giorgio, and von Dechend, Hertha, ''Hamlet's Mill : An Essay on Myth and the Frame of Time'', David R. Godine, Publisher, Boston, MA, 1977. 1st published 1969.<br />
|-<br />
| valign=top | [Sha]<br />
| Shakespeare, William, '' William Shakespeare : The Complete Works'', Compact Edition, S. Wells and G. Taylor (eds.), Oxford University Press, Oxford, UK, 1988.<br />
|-<br />
| valign=top | [Sh1]<br />
| Shakespeare, William, ''A Midsummer Night's Dream'', Washington Square Press, New York, NY, 1958.<br />
|-<br />
| valign=top | [Sh2]<br />
| Shakespeare, William, ''The Tragedy of Hamlet, Prince of Denmark'', In [Sha], pp. 654&ndash;690.<br />
|-<br />
| valign=top | [Sh3]<br />
| Shakespeare, William, ''Measure for Measure'', Washington Square Press, New York, NY, 1965.<br />
|-<br />
| valign=top | [Web]<br />
| ''Webster's Ninth New Collegiate Dictionary'', Merriam-Webster, Springfield, MA, 1983.<br />
|-<br />
| valign=top | [Whi]<br />
| Whitman, Walt, ''Leaves of Grass'', Vintage Books / The Library of America, New York, NY, 1992. Originally published in numerous editions, 1855&ndash;1892.<br />
|-<br />
| valign=top | [Wil]<br />
| Wilhelm, R., and Baynes, C.F. (trans.), ''The I Ching, or Book of Changes'', foreword by C.G. Jung, preface by H. Wilhelm, 3rd edition, Bollingen Series XIX, Princeton University Press, Princeton, NJ, 1967.<br />
|}<br />
<br />
==Document History==<br />
<br />
<pre><br />
Author: Jon Awbrey<br />
Created: 16 Dec 1993<br />
Relayed: 31 Oct 1994<br />
Revised: 03 Jun 2003<br />
Recoded: 03 Jun 2007<br />
</pre><br />
<br />
[[Category:Adaptive Systems]]<br />
[[Category:Artificial Intelligence]]<br />
[[Category:Boolean Algebra]]<br />
[[Category:Boolean Functions]]<br />
[[Category:Charles Sanders Peirce]]<br />
[[Category:Combinatorics]]<br />
[[Category:Computer Science]]<br />
[[Category:Cybernetics]]<br />
[[Category:Differential Logic]]<br />
[[Category:Discrete Systems]]<br />
[[Category:Dynamical Systems]]<br />
[[Category:Formal Languages]]<br />
[[Category:Formal Sciences]]<br />
[[Category:Formal Systems]]<br />
[[Category:Functional Logic]]<br />
[[Category:Graph Theory]]<br />
[[Category:Group Theory]]<br />
[[Category:Inquiry]]<br />
[[Category:Knowledge Representation]]<br />
[[Category:Linguistics]]<br />
[[Category:Logic]]<br />
[[Category:Logical Graphs]]<br />
[[Category:Mathematics]]<br />
[[Category:Mathematical Systems Theory]]<br />
[[Category:Philosophy]]<br />
[[Category:Science]]<br />
[[Category:Semiotics]]<br />
[[Category:Systems Science]]<br />
[[Category:Visualization]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Differential_Logic_and_Dynamic_Systems&diff=469883Differential Logic and Dynamic Systems2021-01-14T15:15:51Z<p>Jon Awbrey: try copying to main space (internet archive having problems with directory space)</p>
<hr />
<div>'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''<br />
<br />
{| align="center" cellpadding="10"<br />
| [[Image:Tangent_Functor_Ferris_Wheel.gif]]<br />
|}<br />
<br />
{| style="height:36px; width:100%"<br />
| align="left" | ''Stand and unfold yourself.''<br />
| align="right" | Hamlet: Francsico&mdash;1.1.2<br />
|}<br />
<br />
This article develops a differential extension of propositional calculus and applies it to a context of problems arising in dynamic systems. The work pursued here is coordinated with a parallel application that focuses on neural network systems, but the dependencies are arranged to make the present article the main and the more self-contained work, to serve as a conceptual frame and a technical background for the network project.<br />
<br />
==Review and Transition==<br />
<br />
This note continues a previous discussion on the problem of dealing with change and diversity in logic-based intelligent systems. It is useful to begin by summarizing essential material from previous reports.<br />
<br />
Table 1 outlines a notation for propositional calculus based on two types of logical connectives, both of variable <math>k\!</math>-ary scope.<br />
<br />
* A bracketed list of propositional expressions in the form <math>\texttt{(} e_1, e_2, \ldots, e_{k-1}, e_k \texttt{)}\!</math> indicates that exactly one of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> is false.<br />
<br />
* A concatenation of propositional expressions in the form <math>e_1 ~ e_2 ~ \ldots ~ e_{k-1} ~ e_k</math> indicates that all of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> are true, in other words, that their [[logical conjunction]] is true.<br />
<br />
All other propositional connectives can be obtained in a very efficient style of representation through combinations of these two forms. Strictly speaking, the concatenation form is dispensable in light of the bracketed form, but it is convenient to maintain it as an abbreviation of more complicated bracket expressions.<br />
<br />
This treatment of propositional logic is derived from the work of C.S. Peirce [P1, P2], who gave this approach an extensive development in his graphical systems of predicate, relational, and modal logic [Rob]. More recently, these ideas were revived and supplemented in an alternative interpretation by George Spencer-Brown [SpB]. Both of these authors used other forms of enclosure where I use parentheses, but the structural topologies of expression and the functional varieties of interpretation are fundamentally the same.<br />
<br />
While working with expressions solely in propositional calculus, it is easiest to use plain parentheses for logical connectives. In contexts where parentheses are needed for other purposes &ldquo;teletype&rdquo; parentheses <math>\texttt{(} \ldots \texttt{)}\!</math> or barred parentheses <math>(\!| \ldots |\!)</math> may be used for logical operators.<br />
<br />
The briefest expression for logical truth is the empty word, usually denoted by <math>{}^{\backprime\backprime} \boldsymbol\varepsilon {}^{\prime\prime}\!</math> or <math>{}^{\backprime\backprime} \boldsymbol\lambda {}^{\prime\prime}\!</math> in formal languages, where it forms the identity element for concatenation. To make it visible in this text, it may be denoted by the equivalent expression <math>{}^{\backprime\backprime} \texttt{((~))} {}^{\prime\prime},\!</math> or, especially if operating in an algebraic context, by a simple <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}.\!</math> Also when working in an algebraic mode, the plus sign <math>{}^{\backprime\backprime} + {}^{\prime\prime}\!</math> may be used for [[exclusive disjunction]]. For example, we have the following paraphrases of algebraic expressions by bracket expressions:<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
|<br />
<math>\begin{matrix}<br />
x + y ~=~ \texttt{(} x, y \texttt{)}<br />
\\[6pt]<br />
x + y + z ~=~ \texttt{((} x, y \texttt{)}, z \texttt{)} ~=~ \texttt{(} x, \texttt{(} y, z \texttt{))}<br />
\end{matrix}</math><br />
|}<br />
<br />
It is important to note that the last expressions are not equivalent to the triple bracket <math>\texttt{(} x, y, z \texttt{)}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 1.} ~~ \text{Syntax and Semantics of a Calculus for Propositional Logic}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Expression}~\!</math><br />
| <math>\text{Interpretation}\!</math><br />
| <math>\text{Other Notations}\!</math><br />
|-<br />
| &nbsp;<br />
| <math>\text{True}\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{False}\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>x\!</math><br />
| <math>x\!</math><br />
| <math>x\!</math><br />
|-<br />
| <math>\texttt{(} x \texttt{)}\!</math><br />
| <math>\text{Not}~ x\!</math><br />
|<br />
<math>\begin{matrix}<br />
x'<br />
\\<br />
\tilde{x}<br />
\\<br />
\lnot x<br />
\end{matrix}\!</math><br />
|-<br />
| <math>x~y~z\!</math><br />
| <math>x ~\text{and}~ y ~\text{and}~ z\!</math><br />
| <math>x \land y \land z\!</math><br />
|-<br />
| <math>\texttt{((} x \texttt{)(} y \texttt{)(} z \texttt{))}\!</math><br />
| <math>x ~\text{or}~ y ~\text{or}~ z\!</math><br />
| <math>x \lor y \lor z\!</math><br />
|-<br />
| <math>\texttt{(} x ~ \texttt{(} y \texttt{))}\!</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{implies}~ y<br />
\\<br />
\mathrm{If}~ x ~\text{then}~ y<br />
\end{matrix}</math><br />
| <math>x \Rightarrow y\!</math><br />
|-<br />
| <math>\texttt{(} x \texttt{,} y \texttt{)}\!</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{not equal to}~ y<br />
\\<br />
x ~\text{exclusive or}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x \ne y<br />
\\<br />
x + y<br />
\end{matrix}</math><br />
|-<br />
| <math>\texttt{((} x \texttt{,} y \texttt{))}\!</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{is equal to}~ y<br />
\\<br />
x ~\text{if and only if}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x = y<br />
\\<br />
x \Leftrightarrow y<br />
\end{matrix}</math><br />
|-<br />
| <math>\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Just one of}<br />
\\<br />
x, y, z<br />
\\<br />
\text{is false}.<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x'y~z~ & \lor<br />
\\<br />
x~y'z~ & \lor<br />
\\<br />
x~y~z' &<br />
\end{matrix}</math><br />
|-<br />
| <math>\texttt{((} x \texttt{),(} y \texttt{),(} z \texttt{))}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Just one of}<br />
\\<br />
x, y, z<br />
\\<br />
\text{is true}.<br />
\\<br />
&<br />
\\<br />
\text{Partition all}<br />
\\<br />
\text{into}~ x, y, z.<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x~y'z' & \lor<br />
\\<br />
x'y~z' & \lor<br />
\\<br />
x'y'z~ &<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,} y \texttt{),} z \texttt{)}<br />
\\<br />
&<br />
\\<br />
\texttt{(} x \texttt{,(} y \texttt{,} z \texttt{))}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Oddly many of}<br />
\\<br />
x, y, z<br />
\\<br />
\text{are true}.<br />
\end{matrix}\!</math><br />
|<br />
<p><math>x + y + z\!</math></p><br />
<br><br />
<p><math>\begin{matrix}<br />
x~y~z~ & \lor<br />
\\<br />
x~y'z' & \lor<br />
\\<br />
x'y~z' & \lor<br />
\\<br />
x'y'z~ &<br />
\end{matrix}\!</math></p><br />
|-<br />
| <math>\texttt{(} w \texttt{,(} x \texttt{),(} y \texttt{),(} z \texttt{))}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Partition}~ w<br />
\\<br />
\text{into}~ x, y, z.<br />
\\<br />
&<br />
\\<br />
\text{Genus}~ w ~\text{comprises}<br />
\\<br />
\text{species}~ x, y, z.<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
w'x'y'z' & \lor<br />
\\<br />
w~x~y'z' & \lor<br />
\\<br />
w~x'y~z' & \lor<br />
\\<br />
w~x'y'z~ &<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
'''Note.''' The usage that one often sees, of a plus sign "<math>+\!</math>" to represent inclusive disjunction, and the reference to this operation as ''boolean addition'', is a misnomer on at least two counts. Boole used the plus sign to represent exclusive disjunction (at any rate, an operation of aggregation restricted in its logical interpretation to cases where the represented sets are disjoint (Boole, 32)), as any mathematician with a sensitivity to the ring and field properties of algebra would do:<br />
<br />
<blockquote><br />
The expression <math>x + y\!</math> seems indeed uninterpretable, unless it be assumed that the things represented by <math>x\!</math> and the things represented by <math>y\!</math> are entirely separate; that they embrace no individuals in common. (Boole, 66).<br />
</blockquote><br />
<br />
It was only later that Peirce and Jevons treated inclusive disjunction as a fundamental operation, but these authors, with a respect for the algebraic properties that were already associated with the plus sign, used a variety of other symbols for inclusive disjunction (Sty, 177, 189). It seems to have been Schröder who later reassigned the plus sign to inclusive disjunction (Sty, 208). Additional information, discussion, and references can be found in (Boole) and (Sty, 177&ndash;263). Aside from these historical points, which never really count against a current practice that has gained a life of its own, this usage does have a further disadvantage of cutting or confounding the lines of communication between algebra and logic. For this reason, it will be avoided here.<br />
<br />
==A Functional Conception of Propositional Calculus==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Out of the dimness opposite equals advance . . . .<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Always substance and increase,<br><br />
Always a knit of identity . . . . always distinction . . . .<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;always a breed of life.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]<br />
|}<br />
<br />
In the general case, we start with a set of logical features <math>\{a_1, \ldots, a_n\}</math> that represent properties of objects or propositions about the world. In concrete examples the features <math>\{a_i\!\}</math> commonly appear as capital letters from an ''alphabet'' like <math>\{A, B, C, \ldots\}</math> or as meaningful words from a linguistic ''vocabulary'' of codes. This language can be drawn from any sources, whether natural, technical, or artificial in character and interpretation. In the application to dynamic systems we tend to use the letters <math>\{x_1, \ldots, x_n\}</math> as our coordinate propositions, and to interpret them as denoting properties of a system's ''state'', that is, as propositions about its location in configuration space. Because I have to consider non-deterministic systems from the outset, I often use the word ''state'' in a loose sense, to denote the position or configuration component of a contemplated state vector, whether or not it ever gets a deterministic completion.<br />
<br />
The set of logical features <math>\{a_1, \ldots, a_n\}</math> provides a basis for generating an <math>n\!</math>-dimensional ''[[universe of discourse]]'' that I denote as <math>[a_1, \ldots, a_n].</math> It is useful to consider each universe of discourse as a unified categorical object that incorporates both the set of points <math>\langle a_1, \ldots, a_n \rangle</math> and the set of propositions <math>f : \langle a_1, \ldots, a_n \rangle \to \mathbb{B}</math> that are implicit with the ordinary picture of a venn diagram on <math>n\!</math> features. Thus, we may regard the universe of discourse <math>[a_1, \ldots, a_n]</math> as an ordered pair having the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}),</math> and we may abbreviate this last type designation as <math>\mathbb{B}^n\ +\!\to \mathbb{B},</math> or even more succinctly as <math>[\mathbb{B}^n].</math> (Used this way, the angle brackets <math>\langle\ldots\rangle</math> are referred to as ''generator brackets''.)<br />
<br />
Table 2 exhibits the scheme of notation I use to formalize the domain of propositional calculus, corresponding to the logical content of truth tables and venn diagrams. Although it overworks the square brackets a bit, I also use either one of the equivalent notations <math>[n]\!</math> or <math>\mathbf{n}</math> to denote the data type of a finite set on <math>n\!</math> elements.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 2.} ~~ \text{Propositional Calculus : Basic Notation}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Symbol}\!</math><br />
| <math>\text{Notation}\!</math><br />
| <math>\text{Description}\!</math><br />
| <math>\text{Type}\!</math><br />
|-<br />
| <math>\mathfrak{A}\!</math><br />
| <math>\{ {}^{\backprime\backprime} a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} a_n {}^{\prime\prime} \}\!</math><br />
| <math>\text{Alphabet}\!</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>\mathcal{A}\!</math><br />
| <math>\{ a_1, \ldots, a_n \}\!</math><br />
| <math>\text{Basis}\!</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>A_i\!</math><br />
| <math>\{ \texttt{(} a_i \texttt{)}, a_i \}\!</math><br />
| <math>\text{Dimension}~ i\!</math><br />
| <math>\mathbb{B}\!</math><br />
|-<br />
| <math>A\!</math><br />
|<br />
<math>\begin{matrix}<br />
\langle \mathcal{A} \rangle<br />
\\[2pt]<br />
\langle a_1, \ldots, a_n \rangle<br />
\\[2pt]<br />
\{ (a_1, \ldots, a_n) \}<br />
\\[2pt]<br />
A_1 \times \ldots \times A_n<br />
\\[2pt]<br />
\textstyle \prod_{i=1}^n A_i<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Set of cells},<br />
\\[2pt]<br />
\text{coordinate tuples},<br />
\\[2pt]<br />
\text{points, or vectors}<br />
\\[2pt]<br />
\text{in the universe}<br />
\\[2pt]<br />
\text{of discourse}<br />
\end{matrix}</math><br />
| <math>\mathbb{B}^n\!</math><br />
|-<br />
| <math>A^*\!</math><br />
| <math>(\mathrm{hom} : A \to \mathbb{B})\!</math><br />
| <math>\text{Linear functions}\!</math><br />
| <math>(\mathbb{B}^n)^* \cong \mathbb{B}^n\!</math><br />
|-<br />
| <math>A^\uparrow\!</math><br />
| <math>(A \to \mathbb{B})\!</math><br />
| <math>\text{Boolean functions}\!</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}\!</math><br />
|-<br />
| <math>A^\bullet\!</math><br />
|<br />
<math>\begin{matrix}<br />
[\mathcal{A}]<br />
\\[2pt]<br />
(A, A^\uparrow)<br />
\\[2pt]<br />
(A ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
(A, (A \to \mathbb{B}))<br />
\\[2pt]<br />
[a_1, \ldots, a_n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Universe of discourse}<br />
\\[2pt]<br />
\text{based on the features}<br />
\\[2pt]<br />
\{ a_1, \ldots, a_n \}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))<br />
\\[2pt]<br />
(\mathbb{B}^n ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
[\mathbb{B}^n]<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
===Qualitative Logic and Quantitative Analogy===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
''Logical'', however, is used in a third sense, which is at once more vital and more practical; to denote, namely, the systematic care, negative and positive, taken to safeguard reflection so that it may yield the best results under the given conditions.<br />
| width="4%" | &nbsp;<br />
|- <br />
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 56]<br />
|}<br />
<br />
These concepts and notations may now be explained in greater detail. In order to begin as simply as possible, let us distinguish two levels of analysis and set out initially on the easier path. On the first level of analysis we take spaces like <math>\mathbb{B},</math> <math>\mathbb{B}^n,</math> and <math>(\mathbb{B}^n \to \mathbb{B})</math> at face value and treat them as the primary objects of interest. On the second level of analysis we use these spaces as coordinate charts for talking about points and functions in more fundamental spaces.<br />
<br />
A pair of spaces, of types <math>\mathbb{B}^n</math> and <math>(\mathbb{B}^n \to \mathbb{B}),</math> give typical expression to everything that we commonly associate with the ordinary picture of a venn diagram. The dimension, <math>n,\!</math> counts the number of "circles" or simple closed curves that are inscribed in the universe of discourse, corresponding to its relevant logical features or basic propositions. Elements of type <math>\mathbb{B}^n</math> correspond to what are often called propositional ''interpretations'' in logic, that is, the different assignments of truth values to sentence letters. Relative to a given universe of discourse, these interpretations are visualized as its ''cells'', in other words, the smallest enclosed areas or undivided regions of the venn diagram. The functions <math>f : \mathbb{B}^n \to \mathbb{B}</math> correspond to the different ways of shading the venn diagram to indicate arbitrary propositions, regions, or sets. Regions included under a shading indicate the ''models'', and regions excluded represent the ''non-models'' of a proposition. To recognize and formalize the natural cohesion of these two layers of concepts into a single universe of discourse, we introduce the type notations <math>[\mathbb{B}^n] = \mathbb{B}^n\ +\!\to \mathbb{B}</math> to stand for the pair of types <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})).</math> The resulting "stereotype" serves to frame the universe of discourse as a unified categorical object, and makes it subject to prescribed sets of evaluations and transformations (categorical morphisms or ''arrows'') that affect the universe of discourse as an integrated whole.<br />
<br />
Most of the time we can serve the algebraic, geometric, and logical interests of our study without worrying about their occasional conflicts and incidental divergences. The conventions and definitions already set down will continue to cover most of the algebraic and functional aspects of our discussion, but to handle the logical and qualitative aspects we will need to add a few more. In general, abstract sets may be denoted by gothic, greek, or script capital variants of <math>A, B, C,\!</math> and so on, with their elements being denoted by a corresponding set of subscripted letters in plain lower case, for example, <math>\mathcal{A} = \{a_i\}.</math> Most of the time, a set such as <math>\mathcal{A} = \{a_i\}\!</math> will be employed as the ''alphabet'' of a [[formal language]]. These alphabet letters serve to name the logical features (properties or propositions) that generate a particular universe of discourse. When we want to discuss the particular features of a universe of discourse, beyond the abstract designation of a type like <math>(\mathbb{B}^n\ +\!\to \mathbb{B}),</math> then we may use the following notations. If <math>\mathcal{A} = \{a_1, \ldots, a_n\}</math> is an alphabet of logical features, then <math>A = \langle \mathcal{A} \rangle = \langle a_1, \ldots, a_n \rangle</math> is the set of interpretations, <math>A^\uparrow = (A \to \mathbb{B})</math> is the set of propositions, and <math>A^\bullet = [\mathcal{A}] = [a_1, \ldots, a_n]</math> is the combination of these interpretations and propositions into the universe of discourse that is based on the features <math>\{a_1, \ldots, a_n\}.</math><br />
<br />
As always, especially in concrete examples, these rules may be dropped whenever necessary, reverting to a free assortment of feature labels. However, when we need to talk about the logical aspects of a space that is already named as a vector space, it will be necessary to make special provisions. At any rate, these elaborations can be deferred until actually needed.<br />
<br />
===Philosophy of Notation : Formal Terms and Flexible Types===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Where number is irrelevant, regimented mathematical technique has hitherto tended to be lacking. Thus it is that the progress of natural science has depended so largely upon the discernment of measurable quantity of one sort or another.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]<br />
|}<br />
<br />
For much of our discussion propositions and boolean functions are treated as the same formal objects, or as different interpretations of the same formal calculus. This rule of interpretation has exceptions, though. There is a distinctively logical interest in the use of propositional calculus that is not exhausted by its functional interpretation. It is part of our task in this study to deal with these uniquely logical characteristics as they present themselves both in our subject matter and in our formal calculus. Just to provide a hint of what's at stake: In logic, as opposed to the more imaginative realms of mathematics, we consider it a good thing to always know what we are talking about. Where mathematics encourages tolerance for uninterpreted symbols as intermediate terms, logic exerts a keener effort to interpret directly each oblique carrier of meaning, no matter how slight, and to unfold the complicities of every indirection in the flow of information. Translated into functional terms, this means that we want to maintain a continual, immediate, and persistent sense of both the converse relation <math>f^{-1} \subseteq \mathbb{B} \times \mathbb{B}^n,</math> or what is the same thing, <math>f^{-1} : \mathbb{B} \to \mathcal{P}(\mathbb{B}^n),</math> and the ''fibers'' or inverse images <math>f^{-1}(0)\!</math> and <math>f^{-1}(1),\!</math> associated with each boolean function <math>f : \mathbb{B}^n \to \mathbb{B}</math> that we use. In practical terms, the desired implementation of a propositional interpreter should incorporate our intuitive recognition that the induced partition of the functional domain into level sets <math>f^{-1}(b),\!</math> for <math>b \in \mathbb{B},</math> is part and parcel of understanding the denotative uses of each propositional function <math>f.\!</math><br />
<br />
===Special Classes of Propositions===<br />
<br />
It is important to remember that the coordinate propositions <math>\{a_i\},\!</math> besides being projection maps <math>a_i : \mathbb{B}^n \to \mathbb{B},</math> are propositions on an equal footing with all others, even though employed as a basis in a particular moment. This set of <math>n\!</math> propositions may sometimes be referred to as the ''basic propositions'', the ''coordinate propositions'', or the ''simple propositions'' that found a universe of discourse. Either one of the equivalent notations, <math>\{a_i : \mathbb{B}^n \to \mathbb{B}\}</math> or <math>(\mathbb{B}^n \xrightarrow{i} \mathbb{B}),</math> may be used to indicate the adoption of the propositions <math>a_i\!</math> as a basis for describing a universe of discourse.<br />
<br />
Among the <math>2^{2^n}</math> propositions in <math>[a_1, \ldots, a_n]</math> are several families of <math>2^n\!</math> propositions each that take on special forms with respect to the basis <math>\{ a_1, \ldots, a_n \}.</math> Three of these families are especially prominent in the present context, the ''linear'', the ''positive'', and the ''singular'' propositions. Each family is naturally parameterized by the coordinate <math>n\!</math>-tuples in <math>\mathbb{B}^n</math> and falls into <math>n + 1\!</math> ranks, with a binomial coefficient <math>\tbinom{n}{k}</math> giving the number of propositions that have rank or weight <math>k.\!</math><br />
<br />
<ul><br />
<br />
<li><br />
<p>The ''linear propositions'', <math>\{ \ell : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{\ell} \mathbb{B}),\!</math> may be written as sums:</p><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\sum_{i=1}^n e_i ~=~ e_1 + \ldots + e_n<br />
~\text{where}~<br />
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 0 \end{matrix}\right\}<br />
~\text{for}~ i = 1 ~\text{to}~ n.\!</math><br />
|}<br />
</li><br />
<br />
<li><br />
<p>The ''positive propositions'', <math>\{ p : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{p} \mathbb{B}),\!</math> may be written as products:</p><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\prod_{i=1}^n e_i ~=~ e_1 \cdot \ldots \cdot e_n<br />
~\text{where}~<br />
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 1 \end{matrix}\right\}<br />
~\text{for}~ i = 1 ~\text{to}~ n.\!</math><br />
|}<br />
</li><br />
<br />
<li><br />
<p>The ''singular propositions'', <math>\{ \mathbf{x} : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{s} \mathbb{B}),\!</math> may be written as products:</p><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\prod_{i=1}^n e_i ~=~ e_1 \cdot \ldots \cdot e_n<br />
~\text{where}~<br />
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = \texttt{(} a_i \texttt{)} \end{matrix}\right\}<br />
~\text{for}~ i = 1 ~\text{to}~ n.\!</math><br />
|}<br />
</li><br />
<br />
</ul><br />
<br />
In each case the rank <math>k\!</math> ranges from <math>0\!</math> to <math>n\!</math> and counts the number of positive appearances of the coordinate propositions <math>a_1, \ldots, a_n\!</math> in the resulting expression. For example, for <math>{n = 3},\!</math> the linear proposition of rank <math>0\!</math> is <math>0,\!</math> the positive proposition of rank <math>0\!</math> is <math>1,\!</math> and the singular proposition of rank <math>0\!</math> is <math>\texttt{(} a_1 \texttt{)(} a_2 \texttt{)(} a_3\texttt{)}.\!</math><br />
<br />
The basic propositions <math>a_i : \mathbb{B}^n \to \mathbb{B}</math> are both linear and positive. So these two kinds of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions.<br />
<br />
Linear propositions and positive propositions are generated by taking boolean sums and products, respectively, over selected subsets of basic propositions, so both families of propositions are parameterized by the powerset <math>\mathcal{P}(\mathcal{I}),</math> that is, the set of all subsets <math>J\!</math> of the basic index set <math>\mathcal{I} = \{1, \ldots, n\}.\!</math><br />
<br />
Let us define <math>\mathcal{A}_J</math> as the subset of <math>\mathcal{A}</math> that is given by <math>\{a_i : i \in J\}.\!</math> Then we may comprehend the action of the linear and the positive propositions in the following terms:<br />
<br />
* The linear proposition <math>\ell_J : \mathbb{B}^n \to \mathbb{B}</math> evaluates each cell <math>\mathbf{x}</math> of <math>\mathbb{B}^n</math> by looking at the coefficients of <math>\mathbf{x}</math> with respect to the features that <math>\ell_J</math> "likes", namely those in <math>\mathcal{A}_J,</math> and then adds them up in <math>\mathbb{B}.</math> Thus, <math>\ell_J(\mathbf{x})</math> computes the parity of the number of features that <math>\mathbf{x}</math> has in <math>\mathcal{A}_J,</math> yielding one for odd and zero for even. Expressed in this idiom, <math>\ell_J(\mathbf{x}) = 1</math> says that <math>\mathbf{x}</math> seems ''odd'' (or ''oddly true'') to <math>\mathcal{A}_J,</math> whereas <math>\ell_J(\mathbf{x}) = 0</math> says that <math>\mathbf{x}</math> seems ''even'' (or ''evenly true'') to <math>\mathcal{A}_J,</math> so long as we recall that ''zero times'' is evenly often, too.<br />
<br />
* The positive proposition <math>p_J : \mathbb{B}^n \to \mathbb{B}</math> evaluates each cell <math>\mathbf{x}</math> of <math>\mathbb{B}^n</math> by looking at the coefficients of <math>\mathbf{x}</math> with regard to the features that <math>p_J\!</math> "likes", namely those in <math>\mathcal{A}_J,</math> and then takes their product in <math>\mathbb{B}.</math> Thus, <math>p_J(\mathbf{x})</math> assesses the unanimity of the multitude of features that <math>\mathbf{x}</math> has in <math>\mathcal{A}_J,</math> yielding one for all and aught for else. In these consensual or contractual terms, <math>p_J(\mathbf{x}) = 1</math> means that <math>\mathbf{x}</math> is ''AOK'' or congruent with all of the conditions of <math>\mathcal{A}_J,</math> while <math>p_J(\mathbf{x}) = 0</math> means that <math>\mathbf{x}</math> defaults or dissents from some condition of <math>\mathcal{A}_J.</math><br />
<br />
===Basis Relativity and Type Ambiguity===<br />
<br />
Finally, two things are important to keep in mind with regard to the simplicity, linearity, positivity, and singularity of propositions.<br />
<br />
First, all of these properties are relative to a particular basis. For example, a singular proposition with respect to a basis <math>\mathcal{A}</math> will not remain singular if <math>\mathcal{A}</math> is extended by a number of new and independent features. Even if we stick to the original set of pairwise options <math>\{a_i\} \cup \{ \texttt{(} a_i \texttt{)} \}\!</math> to select a new basis, the sets of linear and positive propositions are determined by the choice of simple propositions, and this determination is tantamount to the conventional choice of a cell as origin.<br />
<br />
Second, the singular propositions <math>\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B},</math> picking out as they do a single cell or a coordinate tuple <math>\mathbf{x}</math> of <math>\mathbb{B}^n,</math> become the carriers or the vehicles of a certain type-ambiguity that vacillates between the dual forms <math>\mathbb{B}^n</math> and <math>(\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B})</math> and infects the whole hierarchy of types built on them. In other words, the terms that signify the interpretations <math>\mathbf{x} : \mathbb{B}^n</math> and the singular propositions <math>\mathbf{x} : \mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B}</math> are fully equivalent in information, and this means that every token of the type <math>\mathbb{B}^n</math> can be reinterpreted as an appearance of the subtype <math>\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B}.</math> And vice versa, the two types can be exchanged with each other everywhere that they turn up. In practical terms, this allows the use of singular propositions as a way of denoting points, forming an alternative to coordinate tuples.<br />
<br />
For example, relative to the universe of discourse <math>[a_1, a_2, a_3]\!</math> the singular proposition <math>a_1 a_2 a_3 : \mathbb{B}^3 \xrightarrow{s} \mathbb{B}</math> could be explicitly retyped as <math>a_1 a_2 a_3 : \mathbb{B}^3</math> to indicate the point <math>(1, 1, 1)\!</math> but in most cases the proper interpretation could be gathered from context. Both notations remain dependent on a particular basis, but the code that is generated under the singular option has the advantage in its self-commenting features, in other words, it constantly reminds us of its basis in the process of denoting points. When the time comes to put a multiplicity of different bases into play, and to search for objects and properties that remain invariant under the transformations between them, this infinitesimal potential advantage may well evolve into an overwhelming practical necessity.<br />
<br />
===The Analogy Between Real and Boolean Types===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Measurement consists in correlating our subject matter with the series of real numbers; and such correlations are desirable because, once they are set up, all the well-worked theory of numerical mathematics lies ready at hand as a tool for our further reasoning.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]<br />
|}<br />
<br />
There are two further reasons why it useful to spend time on a careful treatment of types, and they both have to do with our being able to take full computational advantage of certain dimensions of flexibility in the types that apply to terms. First, the domains of differential geometry and logic programming are connected by analogies between real and boolean types of the same pattern. Second, the types involved in these patterns have important isomorphisms connecting them that apply on both the real and the boolean sides of the picture.<br />
<br />
Amazingly enough, these isomorphisms are themselves schematized by the axioms and theorems of propositional logic. This fact is known as the ''propositions as types'' analogy or the Curry&ndash;Howard isomorphism [How]. In another formulation it says that terms are to types as proofs are to propositions. See [LaS, 42&ndash;46] and [SeH] for a good discussion and further references. To anticipate the bearing of these issues on our immediate topic, Table&nbsp;3 sketches a partial overview of the Real to Boolean analogy that may serve to illustrate the paradigm in question.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 3.} ~~ \text{Analogy Between Real and Boolean Types}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Real Domain} ~ \mathbb{R}\!</math><br />
| <math>\longleftrightarrow\!</math><br />
| <math>\text{Boolean Domain} ~ \mathbb{B}\!</math><br />
|-<br />
| <math>\mathbb{R}^n\!</math><br />
| <math>\text{Basic Space}\!</math><br />
| <math>\mathbb{B}^n\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}\!</math><br />
| <math>\text{Function Space}\!</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R}\!</math><br />
| <math>\text{Tangent Vector}\!</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to ((\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R})\!</math><br />
| <math>\text{Vector Field}\!</math><br />
| <math>\mathbb{B}^n \to ((\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B})\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \times (\mathbb{R}^n \to \mathbb{R})) \to \mathbb{R}\!</math><br />
| '''<font size="4">"</font>'''<br />
| <math>(\mathbb{B}^n \times (\mathbb{B}^n \to \mathbb{B})) \to \mathbb{B}\!</math><br />
|-<br />
| <math>((\mathbb{R}^n \to \mathbb{R}) \times \mathbb{R}^n) \to \mathbb{R}\!</math><br />
| '''<font size="4">"</font>'''<br />
| <math>((\mathbb{B}^n \to \mathbb{B}) \times \mathbb{B}^n) \to \mathbb{B}\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^n \to \mathbb{R})~\!</math><br />
| <math>\text{Derivation}\!</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^n \to \mathbb{B})\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}^m\!</math><br />
| <math>\begin{matrix}\text{Basic}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}^m\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^m \to \mathbb{R})\!</math><br />
| <math>\begin{matrix}\text{Function}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^m \to \mathbb{B})\!</math><br />
|}<br />
<br />
<br><br />
<br />
The Table exhibits a sample of likely parallels between the real and boolean domains. The central column gives a selection of terminology that is borrowed from differential geometry and extended in its meaning to the logical side of the Table. These are the varieties of spaces that come up when we turn to analyzing the dynamics of processes that pursue their courses through the states of an arbitrary space <math>X.\!</math> Moreover, when it becomes necessary to approach situations of overwhelming dynamic complexity in a succession of qualitative reaches, then the methods of logic that are afforded by the boolean domains, with their declarative means of synthesis and deductive modes of analysis, supply a natural battery of tools for the task.<br />
<br />
It is usually expedient to take these spaces two at a time, in dual pairs of the form <math>X\!</math> and <math>(X \to \mathbb{K}).</math> In general, one creates pairs of type schemas by replacing any space <math>X\!</math> with its dual <math>(X \to \mathbb{K}),</math> for example, pairing the type <math>X \to Y</math> with the type <math>(X \to \mathbb{K}) \to (Y \to \mathbb{K}),</math> and <math>X \times Y</math> with <math>(X \to \mathbb{K}) \times (Y \to \mathbb{K}).</math> The word ''dual'' is used here in its broader sense to mean all of the functionals, not just the linear ones. Given any function <math>f : X \to \mathbb{K},</math> the ''converse'' or inverse relation corresponding to <math>f\!</math> is denoted <math>f^{-1},\!</math> and the subsets of <math>X\!</math> that are defined by <math>f^{-1}(k),\!</math> taken over <math>k\!</math> in <math>\mathbb{K},</math> are called the ''fibers'' or the ''level sets'' of the function <math>f.\!</math><br />
<br />
===Theory of Control and Control of Theory===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
You will hardly know who I am or what I mean,<br><br />
But I shall be good health to you nevertheless,<br><br />
And filter and fibre your blood.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]<br />
|}<br />
<br />
In the boolean context a function <math>f : X \to \mathbb{B}</math> is tantamount to a ''proposition'' about elements of <math>X,\!</math> and the elements of <math>X\!</math> constitute the ''interpretations'' of that proposition. The fiber <math>f^{-1}(1)\!</math> comprises the set of ''models'' of <math>f,\!</math> or examples of elements in <math>X\!</math> satisfying the proposition <math>f.\!</math> The fiber <math>f^{-1}(0)\!</math> collects the complementary set of ''anti-models'', or the exceptions to the proposition <math>f\!</math> that exist in <math>X.\!</math> Of course, the space of functions <math>(X \to \mathbb{B})\!</math> is isomorphic to the set of all subsets of <math>X,\!</math> called the ''power set'' of <math>X,\!</math> and often denoted <math>\mathcal{P}(X)</math> or <math>2^X.\!</math><br />
<br />
The operation of replacing <math>X\!</math> by <math>(X \to \mathbb{B})\!</math> in a type schema corresponds to a certain shift of attitude towards the space <math>X,\!</math> in which one passes from a focus on the ostensibly individual elements of <math>X\!</math> to a concern with the states of information and uncertainty that one possesses about objects and situations in <math>X.\!</math> The conceptual obstacles in the path of this transition can be smoothed over by using singular functions <math>(\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B})</math> as stepping stones. First of all, it's an easy step from an element <math>\mathbf{x}</math> of type <math>\mathbb{B}^n</math> to the equivalent information of a singular proposition <math>\mathbf{x} : X \xrightarrow{s} \mathbb{B}, </math> and then only a small jump of generalization remains to reach the type of an arbitrary proposition <math>f : X \to \mathbb{B},</math> perhaps understood to indicate a relaxed constraint on the singularity of points or a neighborhood circumscribing the original <math>\mathbf{x}.</math> This is frequently a useful transformation, communicating between the ''objective'' and the ''intentional'' perspectives, in spite perhaps of the open objection that this distinction is transient in the mean time and ultimately superficial.<br />
<br />
It is hoped that this measure of flexibility, allowing us to stretch a point into a proposition, can be useful in the examination of inquiry driven systems, where the differences between empirical, intentional, and theoretical propositions constitute the discrepancies and the distributions that drive experimental activity. I can give this model of inquiry a cybernetic cast by realizing that theory change and theory evolution, as well as the choice and the evaluation of experiments, are actions that are taken by a system or its agent in response to the differences that are detected between observational contents and theoretical coverage.<br />
<br />
All of the above notwithstanding, there are several points that distinguish these two tasks, namely, the ''theory of control'' and the ''control of theory'', features that are often obscured by too much precipitation in the quickness with which we understand their similarities. In the control of uncertainty through inquiry, some of the actuators that we need to be concerned with are axiom changers and theory modifiers, operators with the power to compile and to revise the theories that generate expectations and predictions, effectors that form and edit our grammars for the languages of observational data, and agencies that rework the proposed model to fit the actual sequences of events and the realized relationships of values that are observed in the environment. Moreover, when steps must be taken to carry out an experimental action, there must be something about the particular shape of our uncertainty that guides us in choosing what directions to explore, and this impression is more than likely influenced by previous accumulations of experience. Thus it must be anticipated that much of what goes into scientific progress, or any sustainable effort toward a goal of knowledge, is necessarily predicated on long term observation and modal expectations, not only on the more local or short term prediction and correction.<br />
<br />
===Propositions as Types and Higher Order Types===<br />
<br />
The types collected in Table&nbsp;3 (repeated below) serve to illustrate the themes of ''higher order propositional expressions'' and the ''propositions as types'' (PAT) analogy.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 3.} ~~ \text{Analogy Between Real and Boolean Types}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Real Domain} ~ \mathbb{R}\!</math><br />
| <math>\longleftrightarrow\!</math><br />
| <math>\text{Boolean Domain} ~ \mathbb{B}\!</math><br />
|-<br />
| <math>\mathbb{R}^n\!</math><br />
| <math>\text{Basic Space}\!</math><br />
| <math>\mathbb{B}^n\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}\!</math><br />
| <math>\text{Function Space}\!</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R}\!</math><br />
| <math>\text{Tangent Vector}\!</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to ((\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R})\!</math><br />
| <math>\text{Vector Field}\!</math><br />
| <math>\mathbb{B}^n \to ((\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B})\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \times (\mathbb{R}^n \to \mathbb{R})) \to \mathbb{R}\!</math><br />
| '''<font size="4">"</font>'''<br />
| <math>(\mathbb{B}^n \times (\mathbb{B}^n \to \mathbb{B})) \to \mathbb{B}\!</math><br />
|-<br />
| <math>((\mathbb{R}^n \to \mathbb{R}) \times \mathbb{R}^n) \to \mathbb{R}\!</math><br />
| '''<font size="4">"</font>'''<br />
| <math>((\mathbb{B}^n \to \mathbb{B}) \times \mathbb{B}^n) \to \mathbb{B}\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^n \to \mathbb{R})~\!</math><br />
| <math>\text{Derivation}\!</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^n \to \mathbb{B})\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}^m\!</math><br />
| <math>\begin{matrix}\text{Basic}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}^m\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^m \to \mathbb{R})\!</math><br />
| <math>\begin{matrix}\text{Function}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^m \to \mathbb{B})\!</math><br />
|}<br />
<br />
<br><br />
<br />
First, observe that the type of a ''tangent vector at a point'', also known as a ''directional derivative'' at that point, has the form <math>(\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K},</math> where <math>\mathbb{K}</math> is the chosen ground field, in the present case either <math>\mathbb{R}</math> or <math>\mathbb{B}.</math> At a point in a space of type <math>\mathbb{K}^n,</math> a directional derivative operator <math>\vartheta\!</math> takes a function on that space, an <math>f\!</math> of type <math>(\mathbb{K}^n \to \mathbb{K}),</math> and maps it to a ground field value of type <math>\mathbb{K}.</math> This value is known as the ''derivative'' of <math>f\!</math> in the direction <math>\vartheta\!</math> [Che46, 76&ndash;77]. In the boolean case <math>\vartheta : (\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math> has the form of a proposition about propositions, in other words, a proposition of the next higher type.<br />
<br />
Next, by way of illustrating the propositions as types idea, consider a proposition of the form <math>X \Rightarrow (Y \Rightarrow Z).</math> One knows from propositional calculus that this is logically equivalent to a proposition of the form <math>(X \land Y) \Rightarrow Z.</math> But this equivalence should remind us of the functional isomorphism that exists between a construction of the type <math>X \to (Y \to Z)</math> and a construction of the type <math>(X \times Y) \to Z.</math> The propositions as types analogy permits us to take a functional type like this and, under the right conditions, replace the functional arrows &ldquo;<math>\to~\!</math>&rdquo; and products &ldquo;<math>\times\!</math>&rdquo; with the respective logical arrows &ldquo;<math>\Rightarrow\!</math>&rdquo; and products &ldquo;<math>\land\!</math>&rdquo;. Accordingly, viewing the result as a proposition, we can employ axioms and theorems of propositional calculus to suggest appropriate isomorphisms among the categorical and functional constructions.<br />
<br />
Finally, examine the middle four rows of Table&nbsp;3. These display a series of isomorphic types that stretch from the categories that are labeled ''Vector Field'' to those that are labeled ''Derivation''. A ''vector field'', also known as an ''infinitesimal transformation'', associates a tangent vector at a point with each point of a space. In symbols, a vector field is a function of the form <math>\textstyle \xi : X \to \bigcup_{x \in X} \xi_x\!</math> that assigns to each point <math>x\!</math> of the space <math>X\!</math> a tangent vector to <math>X\!</math> at that point, namely, the tangent vector <math>\xi_x\!</math> [Che46, 82&ndash;83]. If <math>X\!</math> is of the type <math>\mathbb{K}^n,</math> then <math>\xi\!</math> is of the type <math>\mathbb{K}^n \to ((\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K}).</math> This has the pattern <math>X \to (Y \to Z),</math> with <math>X = \mathbb{K}^n,</math> <math>Y = (\mathbb{K}^n \to \mathbb{K}),</math> and <math>Z = \mathbb{K}.</math><br />
<br />
Applying the propositions as types analogy, one can follow this pattern through a series of metamorphoses from the type of a vector field to the type of a derivation, as traced out in Table&nbsp;4. Observe how the function <math>f : X \to \mathbb{K},</math> associated with the place of <math>Y\!</math> in the pattern, moves through its paces from the second to the first position. In this way, the vector field <math>\xi,\!</math> initially viewed as attaching each tangent vector <math>\xi_x\!</math> to the site <math>x\!</math> where it acts in <math>X,\!</math> now comes to be seen as acting on each scalar potential <math>f : X \to \mathbb{K}</math> like a generalized species of differentiation, producing another function <math>\xi f : X \to \mathbb{K}</math> of the same type.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 4.} ~~ \text{An Equivalence Based on the Propositions as Types Analogy}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Pattern}\!</math><br />
| <math>\text{Construct}\!</math><br />
| <math>\text{Instance}\!</math><br />
|-<br />
| <math>X \to (Y \to Z)\!</math><br />
| <math>\text{Vector Field}\!</math><br />
| <math>\mathbb{K}^n \to ((\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K})\!</math><br />
|-<br />
| <math>(X \times Y) \to Z\!</math><br />
| <math>\Uparrow\!</math><br />
| <math>(\mathbb{K}^n \times (\mathbb{K}^n \to \mathbb{K})) \to \mathbb{K}\!</math><br />
|-<br />
| <math>(Y \times X) \to Z\!</math><br />
| <math>\Downarrow\!</math><br />
| <math>((\mathbb{K}^n \to \mathbb{K}) \times \mathbb{K}^n) \to \mathbb{K}\!</math><br />
|-<br />
| <math>Y \to (X \to Z)\!</math><br />
| <math>\text{Derivation}\!</math><br />
| <math>(\mathbb{K}^n \to \mathbb{K}) \to (\mathbb{K}^n \to \mathbb{K})\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Reality at the Threshold of Logic===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
But no science can rest entirely on measurement, and many scientific investigations are quite out of reach of that device. To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]<br />
|}<br />
<br />
Table 5 accumulates an array of notation that I hope will not be too distracting. Some of it is rarely needed, but has been filled in for the sake of completeness. Its purpose is simple, to give literal expression to the visual intuitions that come with venn diagrams, and to help build a bridge between our qualitative and quantitative outlooks on dynamic systems.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 5.} ~~ \text{A Bridge Over Troubled Waters}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| align="center" | <math>\text{Linear Space}\!</math><br />
| align="center" | <math>\text{Liminal Space}\!</math><br />
| align="center" | <math>\text{Logical Space}\!</math><br />
|-<br />
| <math>\begin{matrix}\mathcal{X} & = & \{ x_1, \ldots, x_n \}\end{matrix}</math><br />
| <math>\begin{matrix}\underline{\mathcal{X}} & = & \{ \underline{x}_1, \ldots, \underline{x}_n \}\end{matrix}</math><br />
| <math>\begin{matrix}\mathcal{A} & = & \{ a_1, \ldots, a_n \}\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X_i & = & \langle x_i \rangle<br />
\\<br />
& \cong & \mathbb{K}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}_i & = & \{ \texttt{(} \underline{x}_i \texttt{)}, \underline{x}_i \}<br />
\\<br />
& \cong & \mathbb{B}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A_i & = & \{ \texttt{(} a_i \texttt{)}, a_i \}<br />
\\<br />
& \cong & \mathbb{B}<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X<br />
\\<br />
= & \langle \mathcal{X} \rangle<br />
\\<br />
= & \langle x_1, \ldots, x_n \rangle<br />
\\<br />
= & X_1 \times \ldots \times X_n<br />
\\<br />
= & \prod_{i=1}^n X_i<br />
\\<br />
\cong & \mathbb{K}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}<br />
\\<br />
= & \langle \underline{\mathcal{X}} \rangle<br />
\\<br />
= & \langle \underline{x}_1, \ldots, \underline{x}_n \rangle<br />
\\<br />
= & \underline{X}_1 \times \ldots \times \underline{X}_n<br />
\\<br />
= & \prod_{i=1}^n \underline{X}_i<br />
\\<br />
\cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A<br />
\\<br />
= & \langle \mathcal{A} \rangle<br />
\\<br />
= & \langle a_1, \ldots, a_n \rangle<br />
\\<br />
= & A_1 \times \ldots \times A_n<br />
\\<br />
= & \prod_{i=1}^n A_i<br />
\\<br />
\cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X^* & = & (\ell : X \to \mathbb{K})<br />
\\<br />
& \cong & \mathbb{K}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}^* & = & (\ell : \underline{X} \to \mathbb{B})<br />
\\<br />
& \cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A^* & = & (\ell : A \to \mathbb{B})<br />
\\<br />
& \cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X^\uparrow & = & (X \to \mathbb{K})<br />
\\<br />
& \cong & (\mathbb{K}^n \to \mathbb{K})<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}^\uparrow & = & (\underline{X} \to \mathbb{B})<br />
\\<br />
& \cong & (\mathbb{B}^n \to \mathbb{B})<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A^\uparrow & = & (A \to \mathbb{B})<br />
\\<br />
& \cong & (\mathbb{B}^n \to \mathbb{B})<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X^\bullet<br />
\\<br />
= & [\mathcal{X}]<br />
\\<br />
= & [x_1, \ldots, x_n]<br />
\\<br />
= & (X, X^\uparrow)<br />
\\<br />
= & (X ~+\!\to \mathbb{K})<br />
\\<br />
= & (X, (X \to \mathbb{K}))<br />
\\<br />
\cong & (\mathbb{K}^n, (\mathbb{K}^n \to \mathbb{K}))<br />
\\<br />
= & (\mathbb{K}^n ~+\!\to \mathbb{K})<br />
\\<br />
= & [\mathbb{K}^n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}^\bullet<br />
\\<br />
= & [\underline{\mathcal{X}}]<br />
\\<br />
= & [\underline{x}_1, \ldots, \underline{x}_n]<br />
\\<br />
= & (\underline{X}, \underline{X}^\uparrow)<br />
\\<br />
= & (\underline{X} ~+\!\to \mathbb{B})<br />
\\<br />
= & (\underline{X}, (\underline{X} \to \mathbb{B}))<br />
\\<br />
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))<br />
\\<br />
= & (\mathbb{B}^n ~+\!\to \mathbb{B})<br />
\\<br />
= & [\mathbb{B}^n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A^\bullet<br />
\\<br />
= & [\mathcal{A}]<br />
\\<br />
= & [a_1, \ldots, a_n]<br />
\\<br />
= & (A, A^\uparrow)<br />
\\<br />
= & (A ~+\!\to \mathbb{B})<br />
\\<br />
= & (A, (A \to \mathbb{B}))<br />
\\<br />
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))<br />
\\<br />
= & (\mathbb{B}^n ~+\!\to \mathbb{B})<br />
\\<br />
= & [\mathbb{B}^n]<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
The left side of the Table collects mostly standard notation for an <math>n\!</math>-dimensional vector space over a field <math>\mathbb{K}.</math> The right side of the table repeats the first elements of a notation that I sketched above, to be used in further developments of propositional calculus. (I plan to use this notation in the logical analysis of neural network systems.) The middle column of the table is designed as a transitional step from the case of an arbitrary field <math>\mathbb{K},</math> with a special interest in the continuous line <math>\mathbb{R},</math> to the qualitative and discrete situations that are instanced and typified by <math>\mathbb{B}.</math><br />
<br />
I now proceed to explain these concepts in more detail. The most important ideas developed in Table&nbsp;5 are these:<br />
<br />
* The idea of a universe of discourse, which includes both a space of ''points'' and a space of ''maps'' on those points.<br />
<br />
* The idea of passing from a more complex universe to a simpler universe by a process of ''thresholding'' each dimension of variation down to a single bit of information.<br />
<br />
For the sake of concreteness, let us suppose that we start with a continuous <math>n\!</math>-dimensional vector space like <math>X = \langle x_1, \ldots, x_n \rangle \cong \mathbb{R}^n.</math> The coordinate system <math>\mathcal{X} = \{x_i\}</math> is a set of maps <math>x_i : \mathbb{R}^n \to \mathbb{R},</math> also known as the ''coordinate projections''. Given a "dataset" of points <math>\mathbf{x}</math> in <math>\mathbb{R}^n,</math> there are numerous ways of sensibly reducing the data down to one bit for each dimension. One strategy that is general enough for our present purposes is as follows. For each <math>i\!</math> we choose an <math>n\!</math>-ary relation <math>L_i\!</math> on <math>\mathbb{R}^n,</math> that is, a subset of <math>\mathbb{R}^n,</math> and then we define the <math>i^\mathrm{th}\!</math> threshold map, or ''limen'' <math>\underline{x}_i</math> as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\underline{x}_i : \mathbb{R}^n \to \mathbb{B}\ \text{such that:}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
\underline{x}_i(\mathbf{x}) = 1 & \text{if} & \mathbf{x} \in L_i,<br />
\\[4pt]<br />
\underline{x}_i(\mathbf{x}) = 0 & \text{if} & \mathbf{x} \not\in L_i.<br />
\end{matrix}</math><br />
|}<br />
<br />
In other notations that are sometimes used, the operator <math>\chi (\ldots)</math> or the corner brackets <math>\lceil\ldots\rceil</math> can be used to denote a ''characteristic function'', that is, a mapping from statements to their truth values in <math>\mathbb{B}.</math> Finally, it is not uncommon to use the name of the relation itself as a predicate that maps <math>n\!</math>-tuples into truth values. Thus we have the following notational variants of the above definition:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
\underline{x}_i (\mathbf{x}) & = & \chi (\mathbf{x} \in L_i) & = & \lceil \mathbf{x} \in L_i \rceil & = & L_i (\mathbf{x}).<br />
\end{matrix}</math><br />
|}<br />
<br />
Notice that, as defined here, there need be no actual relation between the <math>n\!</math>-dimensional subsets <math>\{L_i\}\!</math> and the coordinate axes corresponding to <math>\{x_i\},\!</math> aside from the circumstance that the two sets have the same cardinality. In concrete cases, though, one usually has some reason for associating these "volumes" with these "lines", for instance, <math>L_i\!</math> is bounded by some hyperplane that intersects the <math>i^\text{th}\!</math> axis at a unique threshold value <math>r_i \in \mathbb{R}.\!</math> Often, the hyperplane is chosen normal to the axis. In recognition of this motive, let us make the following convention. When the set <math>L_i\!</math> has points on the <math>i^\text{th}\!</math> axis, that is, points of the form <math>(0, \ldots, 0, r_i, 0, \ldots, 0)</math> where only the <math>x_i\!</math> coordinate is possibly non-zero, we may pick any one of these coordinate values as a parametric index of the relation. In this case we say that the indexing is ''real'', otherwise the indexing is ''imaginary''. For a knowledge based system <math>X,\!</math> this should serve once again to mark the distinction between ''acquaintance'' and ''opinion''.<br />
<br />
States of knowledge about the location of a system or about the distribution of a population of systems in a state space <math>X = \mathbb{R}^n</math> can now be expressed by taking the set <math>\underline{\mathcal{X}} = \{\underline{x}_i\}</math> as a basis of logical features. In picturesque terms, one may think of the underscore and the subscript as combining to form a subtextual spelling for the <math>i^\text{th}\!</math> threshold map. This can help to remind us that the ''threshold operator'' <math>(\underline{~})_i</math> acts on <math>\mathbf{x}</math> by setting up a kind of a &ldquo;hurdle&rdquo; for it. In this interpretation the coordinate proposition <math>\underline{x}_i</math> asserts that the representative point <math>\mathbf{x}</math> resides ''above'' the <math>i^\mathrm{th}\!</math> threshold.<br />
<br />
Primitive assertions of the form <math>\underline{x}_i (\mathbf{x})</math> may then be negated and joined by means of propositional connectives in the usual ways to provide information about the state <math>\mathbf{x}</math> of a contemplated system or a statistical ensemble of systems. Parentheses <math>\texttt{(} \ldots \texttt{)}\!</math> may be used to indicate logical negation. Eventually one discovers the usefulness of the <math>k\!</math>-ary ''just one false'' operators of the form <math>\texttt{(} a_1 \texttt{,} \ldots \texttt{,} a_k \texttt{)},\!</math> as treated in earlier reports. This much tackle generates a space of points (cells, interpretations), <math>\underline{X} \cong \mathbb{B}^n,</math> and a space of functions (regions, propositions), <math>\underline{X}^\uparrow \cong (\mathbb{B}^n \to \mathbb{B}).</math> Together these form a new universe of discourse <math>\underline{X}^\bullet</math> of the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})),</math> which we may abbreviate as <math>\mathbb{B}^n\ +\!\to \mathbb{B}</math> or most succinctly as <math>[\mathbb{B}^n].</math><br />
<br />
The square brackets have been chosen to recall the rectangular frame of a venn diagram. In thinking about a universe of discourse it is a good idea to keep this picture in mind, graphically illustrating the links among the elementary cells <math>\underline{\mathbf{x}},</math> the defining features <math>\underline{x}_i,</math> and the potential shadings <math>f : \underline{X} \to \mathbb{B}</math> all at the same time, not to mention the arbitrariness of the way we choose to inscribe our distinctions in the medium of a continuous space.<br />
<br />
Finally, let <math>X^*\!</math> denote the space of linear functions, <math>(\ell : X \to \mathbb{K}),</math> which has in the finite case the same dimensionality as <math>X,\!</math> and let the same notation be extended across the Table.<br />
<br />
We have just gone through a lot of work, apparently doing nothing more substantial than spinning a complex spell of notational devices through a labyrinth of baffled spaces and baffling maps. The reason for doing this was to bind together and to constitute the intuitive concept of a universe of discourse into a coherent categorical object, the kind of thing, once grasped, that can be turned over in the mind and considered in all its manifold changes and facets. The effort invested in these preliminary measures is intended to pay off later, when we need to consider the state transformations and the time evolution of neural network systems.<br />
<br />
===Tables of Propositional Forms===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope. It provides explicit techniques for manipulating the most basic ingredients of discourse.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7&ndash;8]<br />
|}<br />
<br />
To prepare for the next phase of discussion, Tables&nbsp;6 and 7 collect and summarize all of the propositional forms on one and two variables. These propositional forms are represented over bases of boolean variables as complete sets of boolean-valued functions. Adjacent to their names and specifications are listed what are roughly the simplest expressions in the ''[[cactus language]]'', the particular syntax for propositional calculus that I use in formal and computational contexts. For the sake of orientation, the English paraphrases and the more common notations are listed in the last two columns. As simple and circumscribed as these low-dimensional universes may appear to be, a careful exploration of their differential extensions will involve us in complexities sufficient to demand our attention for some time to come.<br />
<br />
Propositional forms on one variable correspond to boolean functions <math>f : \mathbb{B}^1 \to \mathbb{B}.</math> In Table&nbsp;6 these functions are listed in a variant form of [[truth table]], one in which the axes of the usual arrangement are rotated through a right angle. Each function <math>f_i\!</math> is indexed by the string of values that it takes on the points of the universe <math>X^\bullet = [x] \cong \mathbb{B}^1.</math> The binary index generated in this way is converted to its decimal equivalent and these are used as conventional names for the <math>f_i,\!</math> as shown in the first column of the Table. In their own right the <math>2^1\!</math> points of the universe <math>X^\bullet</math> are coordinated as a space of type <math>\mathbb{B}^1,</math> this in light of the universe <math>X^\bullet</math> being a functional domain where the coordinate projection <math>x\!</math> takes on its values in <math>\mathbb{B}.</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 6.} ~~ \text{Propositional Forms on One Variable}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_0\!</math><br />
| <math>f_{00}\!</math><br />
| <math>0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>f_1\!</math><br />
| <math>f_{01}\!</math><br />
| <math>0~1\!</math><br />
| <math>\texttt{(} x \texttt{)}\!</math><br />
| <math>\text{not}~ x\!</math><br />
| <math>\lnot x\!</math><br />
|-<br />
| <math>f_2\!</math><br />
| <math>f_{10}\!</math><br />
| <math>1~0\!</math><br />
| <math>x\!</math><br />
| <math>x\!</math><br />
| <math>x\!</math><br />
|-<br />
| <math>f_3\!</math><br />
| <math>f_{11}\!</math><br />
| <math>1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
Propositional forms on two variables correspond to boolean functions <math>f : \mathbb{B}^2 \to \mathbb{B}.</math> In Table&nbsp;7 each function <math>f_i\!</math> is indexed by the values that it takes on the points of the universe <math>X^\bullet = [x, y] \cong \mathbb{B}^2.</math> Converting the binary index thus generated to a decimal equivalent, we obtain the functional nicknames that are listed in the first column. The <math>2^2\!</math> points of the universe <math>X^\bullet</math> are coordinated as a space of type <math>\mathbb{B}^2,</math> as indicated under the heading of the Table, where the coordinate projections <math>x\!</math> and <math>y\!</math> run through the various combinations of their values in <math>\mathbb{B}.</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 7-a.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}\!</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}\!</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}\!</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}\!</math><br />
| style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}\!</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}\!</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[4pt]<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\\[4pt]<br />
f_{3}<br />
\\[4pt]<br />
f_{4}<br />
\\[4pt]<br />
f_{5}<br />
\\[4pt]<br />
f_{6}<br />
\\[4pt]<br />
f_{7}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0000}<br />
\\[4pt]<br />
f_{0001}<br />
\\[4pt]<br />
f_{0010}<br />
\\[4pt]<br />
f_{0011}<br />
\\[4pt]<br />
f_{0100}<br />
\\[4pt]<br />
f_{0101}<br />
\\[4pt]<br />
f_{0110}<br />
\\[4pt]<br />
f_{0111}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~0<br />
\\[4pt]<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~0~1~1<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
0~1~0~1<br />
\\[4pt]<br />
0~1~1~0<br />
\\[4pt]<br />
0~1~1~1<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{)} ~ y ~<br />
\\[4pt]<br />
\texttt{(} x \texttt{)}<br />
\\[4pt]<br />
~ x ~ \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{,} ~ y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x ~ y \texttt{)}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{false}<br />
\\[4pt]<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\[4pt]<br />
y ~\text{without}~ x<br />
\\[4pt]<br />
\text{not}~ x<br />
\\[4pt]<br />
x ~\text{without}~ y<br />
\\[4pt]<br />
\text{not}~ y<br />
\\[4pt]<br />
x ~\text{not equal to}~ y<br />
\\[4pt]<br />
\text{not both}~ x ~\text{and}~ y<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
\lnot x \land \lnot y<br />
\\[4pt]<br />
\lnot x \land y<br />
\\[4pt]<br />
\lnot x<br />
\\[4pt]<br />
x \land \lnot y<br />
\\[4pt]<br />
\lnot y<br />
\\[4pt]<br />
x \ne y<br />
\\[4pt]<br />
\lnot x \lor \lnot y<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{8}<br />
\\[4pt]<br />
f_{9}<br />
\\[4pt]<br />
f_{10}<br />
\\[4pt]<br />
f_{11}<br />
\\[4pt]<br />
f_{12}<br />
\\[4pt]<br />
f_{13}<br />
\\[4pt]<br />
f_{14}<br />
\\[4pt]<br />
f_{15}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1000}<br />
\\[4pt]<br />
f_{1001}<br />
\\[4pt]<br />
f_{1010}<br />
\\[4pt]<br />
f_{1011}<br />
\\[4pt]<br />
f_{1100}<br />
\\[4pt]<br />
f_{1101}<br />
\\[4pt]<br />
f_{1110}<br />
\\[4pt]<br />
f_{1111}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
1~0~0~0<br />
\\[4pt]<br />
1~0~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\\[4pt]<br />
1~1~1~1<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x ~ y<br />
\\[4pt]<br />
\texttt{((} x \texttt{,} ~ y \texttt{))}<br />
\\[4pt]<br />
y<br />
\\[4pt]<br />
\texttt{(} x ~ \texttt{(} y \texttt{))}<br />
\\[4pt]<br />
x<br />
\\[4pt]<br />
\texttt{((} x \texttt{)} ~ y \texttt{)}<br />
\\[4pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
\texttt{((~))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x ~\text{and}~ y<br />
\\[4pt]<br />
x ~\text{equal to}~ y<br />
\\[4pt]<br />
y<br />
\\[4pt]<br />
\text{not}~ x ~\text{without}~ y<br />
\\[4pt]<br />
x<br />
\\[4pt]<br />
\text{not}~ y ~\text{without}~ x<br />
\\[4pt]<br />
x ~\text{or}~ y<br />
\\[4pt]<br />
\text{true}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x \land y<br />
\\[4pt]<br />
x = y<br />
\\[4pt]<br />
y<br />
\\[4pt]<br />
x \Rightarrow y<br />
\\[4pt]<br />
x<br />
\\[4pt]<br />
x \Leftarrow y<br />
\\[4pt]<br />
x \lor y<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 7-b.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}</math><br />
| style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>f_{0000}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\\[4pt]<br />
f_{4}<br />
\\[4pt]<br />
f_{8}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0001}<br />
\\[4pt]<br />
f_{0010}<br />
\\[4pt]<br />
f_{0100}<br />
\\[4pt]<br />
f_{1000}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
1~0~0~0<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{)} ~ y ~<br />
\\[4pt]<br />
~ x ~ \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
~ x ~~ y ~<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\[4pt]<br />
y ~\text{without}~ x<br />
\\[4pt]<br />
x ~\text{without}~ y<br />
\\[4pt]<br />
x ~\text{and}~ y<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot x \land \lnot y<br />
\\[4pt]<br />
\lnot x \land y<br />
\\[4pt]<br />
x \land \lnot y<br />
\\[4pt]<br />
x \land y<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{3}<br />
\\[4pt]<br />
f_{12}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0011}<br />
\\[4pt]<br />
f_{1100}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\[4pt]<br />
x<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{not}~ x<br />
\\[4pt]<br />
x<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot x<br />
\\[4pt]<br />
x<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[4pt]<br />
f_{9}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0110}<br />
\\[4pt]<br />
f_{1001}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[4pt]<br />
1~0~0~1<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{,} y \texttt{)}<br />
\\[4pt]<br />
\texttt{((} x \texttt{,} y \texttt{))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x ~\text{not equal to}~ y<br />
\\[4pt]<br />
x ~\text{equal to}~ y<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x \ne y<br />
\\[4pt]<br />
x = y<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{5}<br />
\\[4pt]<br />
f_{10}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0101}<br />
\\[4pt]<br />
f_{1010}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\[4pt]<br />
y<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{not}~ y<br />
\\[4pt]<br />
y<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot y<br />
\\[4pt]<br />
y<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[4pt]<br />
f_{11}<br />
\\[4pt]<br />
f_{13}<br />
\\[4pt]<br />
f_{14}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0111}<br />
\\[4pt]<br />
f_{1011}<br />
\\[4pt]<br />
f_{1101}<br />
\\[4pt]<br />
f_{1110}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} ~ x ~~ y ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{(} ~ x ~ \texttt{(} y \texttt{))}<br />
\\[4pt]<br />
\texttt{((} x \texttt{)} ~ y ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{not both}~ x ~\text{and}~ y<br />
\\[4pt]<br />
\text{not}~ x ~\text{without}~ y<br />
\\[4pt]<br />
\text{not}~ y ~\text{without}~ x<br />
\\[4pt]<br />
x ~\text{or}~ y<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot x \lor \lnot y<br />
\\[4pt]<br />
x \Rightarrow y<br />
\\[4pt]<br />
x \Leftarrow y<br />
\\[4pt]<br />
x \lor y<br />
\end{matrix}\!</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>f_{1111}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
==A Differential Extension of Propositional Calculus==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Fire over water:<br><br />
The image of the condition before transition.<br><br />
Thus the superior man is careful<br><br />
In the differentiation of things,<br><br />
So that each finds its place.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; ''I Ching'', Hexagram 64, [Wil, 249]<br />
|}<br />
<br />
This much preparation is enough to begin introducing my subject, if I excuse myself from giving full arguments for my definitional choices until some later stage. I am trying to develop a ''differential theory of qualitative equations'' that parallels the application of differential geometry to dynamic systems. The idea of a tangent vector is key to this work and a major goal is to find the right logical analogues of tangent spaces, bundles, and functors. The strategy is taken of looking for the simplest versions of these constructions that can be discovered within the realm of propositional calculus, so long as they serve to fill out the general theme.<br />
<br />
===Differential Propositions : Qualitative Analogues of Differential Equations===<br />
<br />
In order to define the differential extension of a universe of discourse <math>[\mathcal{A}],</math> the initial alphabet <math>\mathcal{A}</math> must be extended to include a collection of symbols for ''differential features'', or basic ''changes'' that are capable of occurring in <math>[\mathcal{A}].</math> Intuitively, these symbols may be construed as denoting primitive features of change, qualitative attributes of motion, or propositions about how things or points in the universe may be changing or moving with respect to the features that are noted in the initial alphabet.<br />
<br />
Therefore, let us define the corresponding ''differential alphabet'' or ''tangent alphabet'' as <math>\mathrm{d}\mathcal{A}\!</math> <math>=\!</math> <math>\{\mathrm{d}a_1, \ldots, \mathrm{d}a_n\},</math> in principle just an arbitrary alphabet of symbols, disjoint from the initial alphabet <math>\mathcal{A}\!</math> <math>=\!</math> <math>\{ a_1, \ldots, a_n \},\!</math> that is intended to be interpreted in the way just indicated. It only remains to be understood that the precise interpretation of the symbols in <math>\mathrm{d}\mathcal{A}\!</math> is often conceived to be changeable from point to point of the underlying space <math>A.\!</math> Indeed, for all we know, the state space <math>A\!</math> might well be the state space of a language interpreter, one that is concerned, among other things, with the idiomatic meanings of the dialect generated by <math>\mathcal{A}</math> and <math>\mathrm{d}\mathcal{A}.\!</math><br />
<br />
The ''tangent space'' to <math>A\!</math> at one of its points <math>x,\!</math> sometimes written <math>\mathrm{T}_x(A),</math> takes the form <math>\mathrm{d}A</math> <math>=\!</math> <math>\langle \mathrm{d}\mathcal{A} \rangle</math> <math>=\!</math> <math>\langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle.\!</math> Strictly speaking, the name ''cotangent space'' is probably more correct for this construction, but the fact that we take up spaces and their duals in pairs to form our universes of discourse allows our language to be pliable here.<br />
<br />
Proceeding as we did with the base space <math>A,\!</math> the tangent space <math>\mathrm{d}A</math> at a point of <math>A\!</math> can be analyzed as a product of distinct and independent factors:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\mathrm{d}A ~=~ \prod_{i=1}^n \mathrm{d}A_i ~=~ \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n.\!</math><br />
|}<br />
<br />
Here, <math>\mathrm{d}A_i\!</math> is a set of two differential propositions, <math>\mathrm{d}A_i = \{ (\mathrm{d}a_i), \mathrm{d}a_i \},\!</math> where <math>\texttt{(} \mathrm{d}a_i \texttt{)}\!</math> is a proposition with the logical value of <math>\text{not} ~ \mathrm{d}a_i.\!</math> Each component <math>\mathrm{d}A_i\!</math> has the type <math>\mathbb{B},\!</math> operating under the ordered correspondence <math>\{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \} \cong \{ 0, 1 \}.\!</math> However, clarity is often served by acknowledging this differential usage with a superficially distinct type <math>\mathbb{D},\!</math> whose intension may be indicated as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\mathbb{D} = \{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \} = \{ \text{same}, \text{different} \} = \{ \text{stay}, \text{change} \} = \{ \text{stop}, \text{step} \}.\!</math><br />
|}<br />
<br />
Viewed within a coordinate representation, spaces of type <math>\mathbb{B}^n\!</math> and <math>\mathbb{D}^n\!</math> may appear to be identical sets of binary vectors, but taking a view at this level of abstraction would be like ignoring the qualitative units and the diverse dimensions that distinguish position and momentum, or the different roles of quantity and impulse.<br />
<br />
===An Interlude on the Path===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
There would have been no beginnings: instead, speech would proceed from me, while I stood in its path &ndash; a slender gap &ndash; the point of its possible disappearance.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
A sense of the relation between <math>\mathbb{B}</math> and <math>\mathbb{D}</math> may be obtained by considering the ''path classifier'' (or the ''equivalence class of curves'') approach to tangent vectors. Consider a universe <math>[\mathcal{X}].\!</math> Given the boolean value system, a path in the space <math>X = \langle \mathcal{X} \rangle</math> is a map <math>q : \mathbb{B} \to X.</math> In this context the set of paths <math>(\mathbb{B} \to X)</math> is isomorphic to the cartesian square <math>X^2 = X \times X,</math> or the set of ordered pairs chosen from <math>X.\!</math><br />
<br />
We may analyze <math>X^2 = \{ (u, v) : u, v \in X \}</math> into two parts, specifically, the ordered pairs <math>(u, v)\!</math> that lie on and off the diagonal:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}X^2 & = & \{ (u, v) : u = v \} & \cup & \{ (u, v) : u \ne v \}.\end{matrix}</math><br />
|}<br />
<br />
This partition may also be expressed in the following symbolic form:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}X^2 & \cong & \operatorname{diag} (X) & + & 2 \binom{X}{2}.\end{matrix}</math><br />
|}<br />
<br />
The separate terms of this formula are defined as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}\operatorname{diag} (X) & = & \{ (x, x) : x \in X \}.\end{matrix}\!</math><br />
|}<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}\binom{X}{k} & = & X ~\text{choose}~ k & = & \{ k\text{-sets from}~ X \}.\end{matrix}\!</math><br />
|}<br />
<br />
Thus we have:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}\binom{X}{2} & = & \{ \{ u, v \} : u, v \in X \}.\end{matrix}</math><br />
|}<br />
<br />
We may now use the features in <math>\mathrm{d}\mathcal{X} = \{ \mathrm{d}x_i \} = \{ \mathrm{d}x_1, \ldots, \mathrm{d}x_n \}</math> to classify the paths of <math>(\mathbb{B} \to X)</math> by way of the pairs in <math>X^2.\!</math> If <math>X \cong \mathbb{B}^n,</math> then a path <math>q\!</math> in <math>X\!</math> has the following form:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
q : (\mathbb{B} \to \mathbb{B}^n) & \cong & \mathbb{B}^n \times \mathbb{B}^n & \cong & \mathbb{B}^{2n} & \cong & (\mathbb{B}^2)^n.<br />
\end{matrix}</math><br />
|}<br />
<br />
Intuitively, we want to map this <math>(\mathbb{B}^2)^n</math> onto <math>\mathbb{D}^n</math> by mapping each component <math>\mathbb{B}^2</math> onto a copy of <math>\mathbb{D}.</math> But in the presenting context <math>{}^{\backprime\backprime} \mathbb{D} {}^{\prime\prime}</math> is just a name associated with, or an incidental quality attributed to, coefficient values in <math>\mathbb{B}</math> when they are attached to features in <math>\mathrm{d}\mathcal{X}.</math><br />
<br />
Taking these intentions into account, define <math>\mathrm{d}x_i : X^2 \to \mathbb{B}</math> in the following manner:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcrcl}<br />
\mathrm{d}x_i(u, v)<br />
& = & \texttt{(} ~ x_i(u) & \texttt{,} & x_i(v) ~ \texttt{)}<br />
\\<br />
& = & x_i(u) & + & x_i(v)<br />
\\<br />
& = & x_i(v) & - & x_i(u).<br />
\end{array}</math><br />
|}<br />
<br />
In the above transcription, the operator bracket of the form <math>\texttt{(} \ldots \texttt{,} \ldots \texttt{)}\!</math> is a ''cactus lobe'', in general signifying that just one of the arguments listed is false. In the case of two arguments this is the same thing as saying that the arguments are not equal. The plus sign signifies boolean addition, in the sense of addition in <math>\mathrm{GF}(2),</math> and thus means the same thing in this context as the minus sign, in the sense of adding the additive inverse.<br />
<br />
The above definition of <math>\mathrm{d}x_i : X^2 \to \mathbb{B}</math> is equivalent to defining <math>\mathrm{d}x_i : (\mathbb{B} \to X) \to \mathbb{B}\!</math> in the following way:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcrcl}<br />
\mathrm{d}x_i (q)<br />
& = & \texttt{(} ~ x_i(q_0) & \texttt{,} & x_i(q_1) ~ \texttt{)}<br />
\\<br />
& = & x_i(q_0) & + & x_i(q_1)<br />
\\<br />
& = & x_i(q_1) & - & x_i(q_0).<br />
\end{array}</math><br />
|}<br />
<br />
In this definition <math>q_b = q(b),\!</math> for each <math>b\!</math> in <math>\mathbb{B}.</math> Thus, the proposition <math>\mathrm{d}x_i</math> is true of the path <math>q = (u, v)\!</math> exactly if the terms of <math>q,\!</math> the endpoints <math>u\!</math> and <math>v,\!</math> lie on different sides of the question <math>x_i.\!</math><br />
<br />
The language of features in <math>\langle \mathrm{d}\mathcal{X} \rangle,</math> indeed the whole calculus of propositions in <math>[\mathrm{d}\mathcal{X}],</math> may now be used to classify paths and sets of paths. In other words, the paths can be taken as models of the propositions <math>g : \mathrm{d}X \to \mathbb{B}.</math> For example, the paths corresponding to <math>\mathrm{diag}(X)</math> fall under the description <math>\texttt{(} \mathrm{d}x_1 \texttt{)} \cdots \texttt{(} \mathrm{d}x_n \texttt{)},\!</math> which says that nothing changes against the backdrop of the coordinate frame <math>\{ x_1, \ldots, x_n \}.\!</math><br />
<br />
Finally, a few words of explanation may be in order. If this concept of a path appears to be described in a roundabout fashion, it is because I am trying to avoid using any assumption of vector space properties for the space <math>X\!</math> that contains its range. In many ways the treatment is still unsatisfactory, but improvements will have to wait for the introduction of substitution operators acting on singular propositions.<br />
<br />
===The Extended Universe of Discourse===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
At the moment of speaking, I would like to have perceived a nameless voice, long preceding me, leaving me merely to enmesh myself in it, taking up its cadence, and to lodge myself, when no one was looking, in its interstices as if it had paused an instant, in suspense, to beckon to me.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
Next we define the ''extended alphabet'' or ''bundled alphabet'' <math>\mathrm{E}\mathcal{A}</math> as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lclcl}<br />
\mathrm{E}\mathcal{A}<br />
& = & \mathcal{A} \cup \mathrm{d}\mathcal{A}<br />
& = & \{ a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}.<br />
\end{array}</math><br />
|}<br />
<br />
This supplies enough material to construct the ''differential extension'' <math>\mathrm{E}A,</math> or the ''tangent bundle'' over the initial space <math>A,\!</math> in the following fashion:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcl}<br />
\mathrm{E}A<br />
& = & \langle \mathrm{E}\mathcal{A} \rangle<br />
\\[4pt]<br />
& = & \langle \mathcal{A} \cup \mathrm{d}\mathcal{A} \rangle<br />
\\[4pt]<br />
& = & \langle a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle,<br />
\end{array}</math><br />
|}<br />
<br />
and also:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcl}<br />
\mathrm{E}A<br />
& = & A \times \mathrm{d}A<br />
\\[4pt]<br />
& = & A_1 \times \ldots \times A_n \times \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n.<br />
\end{array}</math><br />
|}<br />
<br />
This gives <math>\mathrm{E}A</math> the type <math>\mathbb{B}^n \times \mathbb{D}^n.</math><br />
<br />
Finally, the tangent universe <math>\mathrm{E}A^\bullet = [\mathrm{E}\mathcal{A}]\!</math> is constituted from the totality of points and maps, or interpretations and propositions, that are based on the extended set of features <math>\mathrm{E}\mathcal{A},</math> and this fact is summed up in the following notation:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lclcl}<br />
\mathrm{E}A^\bullet<br />
& = & [\mathrm{E}\mathcal{A}]<br />
& = & [a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n].<br />
\end{array}</math><br />
|}<br />
<br />
This gives the tangent universe <math>\mathrm{E}A^\bullet\!</math> the type:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcl}<br />
(\mathbb{B}^n \times \mathbb{D}^n\ +\!\to \mathbb{B})<br />
& = & (\mathbb{B}^n \times \mathbb{D}^n, (\mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B})).<br />
\end{array}</math><br />
|}<br />
<br />
A proposition in the tangent universe <math>[\mathrm{E}\mathcal{A}]</math> is called a ''differential proposition'' and forms the analogue of a system of differential equations, constraints, or relations in ordinary calculus.<br />
<br />
With these constructions, the differential extension <math>\mathrm{E}A</math> and the space of differential propositions <math>(\mathrm{E}A \to \mathbb{B}),\!</math> we have arrived, in main outline, at one of the major subgoals of this study. Table&nbsp;8 summarizes the concepts that have been introduced for working with differentially extended universes of discourse.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 8.} ~~ \text{Differential Extension : Basic Notation}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Symbol}\!</math><br />
| <math>\text{Notation}\!</math><br />
| <math>\text{Description}\!</math><br />
| <math>\text{Type}\!</math><br />
|-<br />
| <math>\mathrm{d}\mathfrak{A}\!</math><br />
| <math>\{ {}^{\backprime\backprime} \mathrm{d}a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} \mathrm{d}a_n {}^{\prime\prime} \}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Alphabet of}<br />
\\[2pt]<br />
\text{differential symbols}<br />
\end{matrix}</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>\mathrm{d}\mathcal{A}\!</math><br />
| <math>\{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Basis of}<br />
\\[2pt]<br />
\text{differential features}<br />
\end{matrix}</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>\mathrm{d}A_i\!</math><br />
| <math>\{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \}\!</math><br />
| <math>\text{Differential dimension}~ i\!</math><br />
| <math>\mathbb{D}\!</math><br />
|-<br />
| <math>\mathrm{d}A\!</math><br />
|<br />
<math>\begin{matrix}<br />
\langle \mathrm{d}\mathcal{A} \rangle<br />
\\[2pt]<br />
\langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle<br />
\\[2pt]<br />
\{ (\mathrm{d}a_1, \ldots, \mathrm{d}a_n) \}<br />
\\[2pt]<br />
\mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n<br />
\\[2pt]<br />
\textstyle \prod_i \mathrm{d}A_i<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Tangent space at a point:}<br />
\\[2pt]<br />
\text{Set of changes, motions,}<br />
\\[2pt]<br />
\text{steps, tangent vectors}<br />
\\[2pt]<br />
\text{at a point}<br />
\end{matrix}</math><br />
| <math>\mathbb{D}^n\!</math><br />
|-<br />
| <math>\mathrm{d}A^*\!</math><br />
| <math>(\mathrm{hom} : \mathrm{d}A \to \mathbb{B})\!</math><br />
| <math>\text{Linear functions on}~ \mathrm{d}A\!</math><br />
| <math>(\mathbb{D}^n)^* \cong \mathbb{D}^n\!</math><br />
|-<br />
| <math>\mathrm{d}A^\uparrow\!</math><br />
| <math>(\mathrm{d}A \to \mathbb{B})\!</math><br />
| <math>\text{Boolean functions on}~ \mathrm{d}A\!</math><br />
| <math>\mathbb{D}^n \to \mathbb{B}\!</math><br />
|-<br />
| <math>\mathrm{d}A^\bullet\!</math><br />
|<br />
<math>\begin{matrix}<br />
[\mathrm{d}\mathcal{A}]<br />
\\[2pt]<br />
(\mathrm{d}A, \mathrm{d}A^\uparrow)<br />
\\[2pt]<br />
(\mathrm{d}A ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
(\mathrm{d}A, (\mathrm{d}A \to \mathbb{B}))<br />
\\[2pt]<br />
[\mathrm{d}a_1, \ldots, \mathrm{d}a_n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Tangent universe at a point of}~ A^\bullet,<br />
\\[2pt]<br />
\text{based on the tangent features}<br />
\\[2pt]<br />
\{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
(\mathbb{D}^n, (\mathbb{D}^n \to \mathbb{B}))<br />
\\[2pt]<br />
(\mathbb{D}^n ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
[\mathbb{D}^n]<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
The adjectives ''differential'' or ''tangent'' are systematically attached to every construct based on the differential alphabet <math>\mathrm{d}\mathfrak{A},</math> taken by itself. Strictly speaking, we probably ought to call <math>\mathrm{d}\mathcal{A}</math> the set of ''cotangent'' features derived from <math>\mathcal{A},</math> but the only time this distinction really seems to matter is when we need to distinguish the tangent vectors as maps of type <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math> from cotangent vectors as elements of type <math>\mathbb{D}^n.</math> In like fashion, having defined <math>\mathrm{E}\mathcal{A} = \mathcal{A} \cup \mathrm{d}\mathcal{A},</math> we can systematically attach the adjective ''extended'' or the substantive ''bundle'' to all of the constructs associated with this full complement of <math>{2n}\!</math> features.<br />
<br />
It eventually becomes necessary to extend the initial alphabet even further, to allow for the discussion of higher order differential expressions. Table&nbsp;9 provides a suggestion of how these further extensions can be carried out.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 9.} ~~ \text{Higher Order Differential Features}\!</math><br />
|<br />
<p><math>\begin{array}{lllll}<br />
\mathrm{d}^0 \mathcal{A} & = & \{ a_1, \ldots, a_n \} & = & \mathcal{A}<br />
\\<br />
\mathrm{d}^1 \mathcal{A} & = & \{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \} & = & \mathrm{d}\mathcal{A}<br />
\end{array}</math></p><br />
<br />
<p><math>\begin{array}{lll}<br />
\mathrm{d}^k \mathcal{A} & = & \{ \mathrm{d}^k a_1, \ldots, \mathrm{d}^k a_n \}<br />
\\<br />
\mathrm{d}^* \mathcal{A} & = & \{ \mathrm{d}^0 \mathcal{A}, \ldots, \mathrm{d}^k \mathcal{A}, \ldots \}<br />
\end{array}</math></p><br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}^0 \mathcal{A} & = & \mathrm{d}^0 \mathcal{A}<br />
\\<br />
\mathrm{E}^1 \mathcal{A} & = & \mathrm{d}^0 \mathcal{A} ~\cup~ \mathrm{d}^1 \mathcal{A}<br />
\\<br />
\mathrm{E}^k \mathcal{A} & = & \mathrm{d}^0 \mathcal{A} ~\cup~ \ldots ~\cup~ \mathrm{d}^k \mathcal{A}<br />
\\<br />
\mathrm{E}^\infty \mathcal{A} & = & \bigcup~ \mathrm{d}^* \mathcal{A}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
===Intentional Propositions===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Do you guess I have some intricate purpose?<br><br />
Well I have . . . . for the April rain has, and the mica on<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the side of a rock has.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 45]<br />
|}<br />
<br />
In order to analyze the behavior of a system at successive moments in time, while staying within the limitations of propositional logic, it is necessary to create independent alphabets of logical features for each moment of time that we contemplate using in our discussion. These moments have reference to typical instances and relative intervals, not actual or absolute times. For example, to discuss ''velocities'' (first order rates of change) we need to consider points of time in pairs. There are a number of natural ways of doing this. Given an initial alphabet, we could use its symbols as a lexical basis to generate successive alphabets of compound symbols, say, with temporal markers appended as suffixes.<br />
<br />
As a standard way of dealing with these situations, the following scheme of notation suggests a way of extending any alphabet of logical features through as many temporal moments as a particular order of analysis may demand. The lexical operators <math>\mathrm{p}^k</math> and <math>\mathrm{Q}^k</math> are convenient in many contexts where the accumulation of prime symbols and union symbols would otherwise be cumbersome.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 10.} ~~ \text{A Realm of Intentional Features}\!</math><br />
|<br />
<p><math>\begin{array}{lllll}<br />
\mathrm{p}^0 \mathcal{A} & = & \{ a_1, \ldots, a_n \} & = & \mathcal{A}<br />
\\<br />
\mathrm{p}^1 \mathcal{A} & = & \{ a_1^\prime, \ldots, a_n^\prime \} & = & \mathcal{A}^\prime<br />
\\<br />
\mathrm{p}^2 \mathcal{A} & = & \{ a_1^{\prime\prime}, \ldots, a_n^{\prime\prime} \} & = & \mathcal{A}^{\prime\prime}<br />
\\<br />
\cdots & & \cdots &<br />
\end{array}</math></p><br />
<br />
<p><math>\begin{array}{lll}<br />
\mathrm{p}^k \mathcal{A} & = & \{ \mathrm{p}^k a_1, \ldots, \mathrm{p}^k a_n \}<br />
\end{array}</math></p><br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{Q}^0 \mathcal{A} & = & \mathcal{A}<br />
\\<br />
\mathrm{Q}^1 \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}'<br />
\\<br />
\mathrm{Q}^2 \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}' \cup \mathcal{A}''<br />
\\<br />
\cdots & & \cdots<br />
\\<br />
\mathrm{Q}^k \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}' \cup \ldots \cup \mathrm{p}^k \mathcal{A}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
The resulting augmentations of our logical basis determine a series of discursive universes that may be called the ''intentional extension'' of propositional calculus. This extension follows a pattern analogous to the differential extension, which was developed in terms of the operators <math>\mathrm{d}^k</math> and <math>\mathrm{E}^k,</math> and there is a natural relation between these two extensions that bears further examination. In contexts displaying this pattern, where a sequence of domains stretches from an anchoring domain <math>X\!</math> through an indefinite number of higher reaches, a particular collection of domains based on <math>X\!</math> will be referred to as a ''realm'' of <math>X,\!</math> and when the succession exhibits a temporal aspect, as a ''reign'' of <math>X.\!</math><br />
<br />
For the purposes of this discussion, an ''intentional proposition'' is defined as a proposition in the universe of discourse <math>\mathrm{Q}X^\bullet = [\mathrm{Q}\mathcal{X}],</math> in other words, a map <math>q : \mathrm{Q}X \to \mathbb{B}.</math> The sense of this definition may be seen if we consider the following facts. First, the equivalence <math>\mathrm{Q}X = X \times X'</math> motivates the following chain of isomorphisms between spaces:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lllcl}<br />
(\mathrm{Q}X \to \mathbb{B})<br />
& \cong & (X & \times & ~X' \to \mathbb{B})<br />
\\[4pt]<br />
& \cong & (X & \to & (X' \to \mathbb{B}))<br />
\\[4pt]<br />
& \cong & (X' & \to & (X~ \to \mathbb{B})).<br />
\end{array}</math><br />
|}<br />
<br />
Viewed in this light, an intentional proposition <math>q\!</math> may be rephrased as a map <math>q : X \times X' \to \mathbb{B},</math> which judges the juxtaposition of states in <math>X\!</math> from one moment to the next. Alternatively, <math>q\!</math> may be parsed in two stages in two different ways, as <math>q : X \to (X' \to \mathbb{B})</math> and as <math>q : X' \to (X \to \mathbb{B}),</math> which associate to each point of <math>X\!</math> or <math>X'\!</math> a proposition about states in <math>X'\!</math> or <math>X,\!</math> respectively. In this way, an intentional proposition embodies a type of value system, in effect, a proposal that places a value on a collection of ends-in-view, or a project that evaluates a set of goals as regarded from each point of view in the state space of a system.<br />
<br />
In sum, the intentional proposition <math>q\!</math> indicates a method for the systematic selection of local goals. As a general form of description, a map of the type <math>q : \mathrm{Q}^i X \to \mathbb{B}\!</math> may be referred to as an "<math>i^\text{th}</math> order intentional proposition". Naturally, when we speak of intentional propositions without qualification, we usually mean first order intentions.<br />
<br />
Many different realms of discourse have the same structure as the extensions that have been indicated here. From a strictly logical point of view, each new layer of terms is composed of independent logical variables that are no different in kind from those that go before, and each further course of logical atoms is treated like so many individual, but otherwise indifferent bricks by the prototype computer program that I use as a propositional interpreter. Thus, the names that I use to single out the differential and the intentional extensions, and the lexical paradigms that I follow to construct them, are meant to suggest the interpretations that I have in mind, but they can only hint at the extra meanings that human communicators may pack into their terms and inflections.<br />
<br />
As applied here, the word ''intentional'' is drawn from common use and may have little bearing on its technical use in other, more properly philosophical, contexts. I am merely using the complex of intentional concepts &mdash; aims, ends, goals, objectives, purposes, and so on &mdash; metaphorically to flesh out and vividly to represent any situation where one needs to contemplate a system in multiple aspects of state and destination, that is, its being in certain states and at the same time acting as if headed through certain states. If confusion arises, more neutral words like ''conative'', ''contingent'', ''discretionary'', ''experimental'', ''kinetic'', ''progressive'', ''tentative'', or ''trial'' would probably serve as well.<br />
<br />
===Life on Easy Street===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Failing to fetch me at first keep encouraged,<br><br />
Missing me one place search another,<br><br />
I stop some where waiting for you<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]<br />
|}<br />
<br />
The finite character of the extended universe <math>[\mathrm{E}\mathcal{A}]</math> makes the problem of solving differential propositions relatively straightforward, at least, in principle. The solution set of the differential proposition <math>q : \mathrm{E}A \to \mathbb{B}</math> is the set of models <math>q^{-1}(1)\!</math> in <math>\mathrm{E}A.</math> Finding all the models of <math>q,\!</math> the extended interpretations in <math>\mathrm{E}A</math> that satisfy <math>q,\!</math> can be carried out by a finite search. Being in possession of complete algorithms for propositional calculus modeling, theorem checking, or theorem proving makes the analytic task fairly simple in principle, though the question of efficiency in the face of arbitrary complexity may always remain another matter entirely. While the fact that propositional satisfiability is NP-complete may be discouraging for the prospects of a single efficient algorithm that covers the whole space <math>[\mathrm{E}\mathcal{A}]</math> with equal facility, there appears to be much room for improvement in classifying special forms and in developing algorithms that are tailored to their practical processing.<br />
<br />
In view of these constraints and contingencies, our focus shifts to the tasks of approximation and interpretation that support intuition, especially in dealing with the natural kinds of differential propositions that arise in applications, and in the effort to understand, in succinct and adaptive forms, their dynamic implications. In the absence of direct insights, these tasks are partially carried out by forging analogies with the familiar situations and customary routines of ordinary calculus. But the indirect approach, going by way of specious analogy and intuitive habit, forces us to remain on guard against the circumstance that occurs when the word ''forging'' takes on its shadier nuance, indicting the constant risk of a counterfeit in the proportion.<br />
<br />
==Back to the Beginning : Exemplary Universes==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
I would have preferred to be enveloped in words, borne way beyond all possible beginnings.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
To anchor our understanding of differential logic, let us look at how the various concepts apply in the simplest possible concrete cases, where the initial dimension is only 1 or 2. In spite of the obvious simplicity of these cases, it is possible to observe how central difficulties of the subject begin to arise already at this stage.<br />
<br />
===A One-Dimensional Universe===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
There was never any more inception than there is now,<br><br />
Nor any more youth or age than there is now;<br><br />
And will never be any more perfection than there is now,<br><br />
Nor any more heaven or hell than there is now.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]<br />
|}<br />
<br />
Let <math>\mathcal{X} = \{ x_1 \} = \{ A \}</math> be an alphabet that represents one boolean variable or a single logical feature. In this example the capital letter <math>{}^{\backprime\backprime} A {}^{\prime\prime}\!</math> is used usual informally, to name a feature and not a space, in departure from our formerly stated formal conventions. At any rate, the basis element <math>A = x_1\!</math> may be interpreted as a simple proposition or a coordinate projection <math>A = x_1 : \mathbb{B} \xrightarrow{i} \mathbb{B}.</math> The space <math>X = \langle A \rangle = \{ \texttt{(} A \texttt{)}, A \}</math> of points (cells, vectors, interpretations) has cardinality <math>2^n = 2^1 = 2\!</math> and is isomorphic to <math>\mathbb{B} = \{ 0, 1 \}.</math> Moreover, <math>X\!</math> may be identified with the set of singular propositions <math>\{ x : \mathbb{B} \xrightarrow{s} \mathbb{B} \}.</math> The space of linear propositions <math>X^* = \{ \mathrm{hom} : \mathbb{B} \xrightarrow{\ell} \mathbb{B} \} = \{ 0, A \}</math> is algebraically dual to <math>X\!</math> and also has cardinality <math>2.\!</math> Here, <math>{}^{\backprime\backprime} 0 {}^{\prime\prime}\!</math> is interpreted as denoting the constant function <math>0 : \mathbb{B} \to \mathbb{B},</math> amounting to the linear proposition of rank <math>0,\!</math> while <math>A\!</math> is the linear proposition of rank <math>1.\!</math> Last but not least we have the positive propositions <math>\{ \mathrm{pos} : \mathbb{B} \xrightarrow{p} \mathbb{B} \} = \{ A, 1 \},\!</math> of rank <math>1\!</math> and <math>0,\!</math> respectively, where <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}\!</math> is understood as denoting the constant function <math>1 : \mathbb{B} \to \mathbb{B}.</math> In sum, there are <math>2^{2^n} = 2^{2^1} = 4</math> propositions altogether in the universe of discourse, comprising the set <math>X^\uparrow = \{ f : X \to \mathbb{B} \} = \{ 0, \texttt{(} A \texttt{)}, A, 1 \} \cong (\mathbb{B} \to \mathbb{B}).</math><br />
<br />
The first order differential extension of <math>\mathcal{X}</math> is <math>\mathrm{E}\mathcal{X} = \{ x_1, \mathrm{d}x_1 \} = \{ A, \mathrm{d}A \}.</math> If the feature <math>A\!</math> is understood as applying to some object or state, then the feature <math>\mathrm{d}A</math> may be interpreted as an attribute of the same object or state that says that it is changing ''significantly'' with respect to the property <math>A,\!</math> or that it has an ''escape velocity'' with respect to the state <math>A.\!</math> In practice, differential features acquire their logical meaning through a class of ''temporal inference rules''.<br />
<br />
For example, relative to a frame of observation that is left implicit for now, one is permitted to make the following sorts of inference: From the fact that <math>A\!</math> and <math>\mathrm{d}A</math> are true at a given moment one may infer that <math>\texttt{(} A \texttt{)}\!</math> will be true in the next moment of observation. Altogether in the present instance, there is the fourfold scheme of inference that is shown below:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:50%"<br />
|<br />
<math>\begin{matrix}<br />
\text{From} & \texttt{(} A \texttt{)}<br />
& \text{and} & \texttt{(} \mathrm{d}A \texttt{)}<br />
& \text{infer} & \texttt{(} A \texttt{)} & \text{next.}<br />
\\[8pt]<br />
\text{From} & \texttt{(} A \texttt{)}<br />
& \text{and} & \mathrm{d}A<br />
& \text{infer} & A & \text{next.}<br />
\\[8pt]<br />
\text{From} & A<br />
& \text{and} & \texttt{(} \mathrm{d}A \texttt{)}<br />
& \text{infer} & A & \text{next.}<br />
\\[8pt]<br />
\text{From} & A<br />
& \text{and} & \mathrm{d}A<br />
& \text{infer} & \texttt{(} A \texttt{)} & \text{next.}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
It might be thought that an independent time variable needs to be brought in at this point, but it is an insight of fundamental importance that the idea of process is logically prior to the notion of time. A time variable is a reference to a ''clock'' &mdash; a canonical, conventional process that is accepted or established as a standard of measurement, but in essence no different than any other process. This raises the question of how different subsystems in a more global process can be brought into comparison, and what it means for one process to serve the function of a local standard for others. But these inquiries only wrap up puzzles in further riddles, and are obviously too involved to be handled at our current level of approximation.<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
The clock indicates the moment . . . . but what does<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;eternity indicate?<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 79]<br />
|}<br />
<br />
Observe that the secular inference rules, used by themselves, involve a loss of information, since nothing in them can tell us whether the momenta <math>\{ \texttt{(} \mathrm{d}A \texttt{)}, \mathrm{d}A \}\!</math> are changed or unchanged in the next instance. In order to know this, one would have to determine <math>\mathrm{d}^2 A,\!</math> and so on, pursuing an infinite regress. Ultimately, in order to rest with a finitely determinate system, it is necessary to make an infinite assumption, for example, that <math>\mathrm{d}^k A = 0\!</math> for all <math>k\!</math> greater than some fixed value <math>M.\!</math> Another way to escape the regress is through the provision of a dynamic law, in typical form making higher order differentials dependent on lower degrees and estates.<br />
<br />
===Example 1. A Square Rigging===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Urge and urge and urge,<br><br />
Always the procreant urge of the world.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]<br />
|}<br />
<br />
By way of example, suppose that we are given the initial condition <math>A = \mathrm{d}A\!</math> and the law <math>\mathrm{d}^2 A = \texttt{(} A \texttt{)}.\!</math> Since the equation <math>A = \mathrm{d}A\!</math> is logically equivalent to the disjunction <math>A ~ \mathrm{d}A ~\text{or}~ \texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)},\!</math> we may infer two possible trajectories, as displayed in Table&nbsp;11. In either case the state <math>A ~ \texttt{(} \mathrm{d}A \texttt{)(} \mathrm{d}^2 A \texttt{)}\!</math> is a stable attractor or a terminal condition for both starting points.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 11.} ~~ \text{A Pair of Commodious Trajectories}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Time}\!</math><br />
| <math>\text{Trajectory 1}\!</math><br />
| <math>\text{Trajectory 2}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
2<br />
\\[4pt]<br />
3<br />
\\[4pt]<br />
4<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A & \mathrm{d}A & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
\texttt{(} A \texttt{)} & \mathrm{d}A & \mathrm{d}^2 A<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
{}^{\shortparallel} & {}^{\shortparallel} & {}^{\shortparallel}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{)} & \texttt{(} \mathrm{d}A \texttt{)} & \mathrm{d}^2 A<br />
\\[4pt]<br />
\texttt{(} A \texttt{)} & \mathrm{d}A & \mathrm{d}^2 A<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
{}^{\shortparallel} & {}^{\shortparallel} & {}^{\shortparallel}<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Because the initial space <math>X = \langle A \rangle\!</math> is one-dimensional, we can easily fit the second order extension <math>\mathrm{E}^2 X = \langle A, \mathrm{d}A, \mathrm{d}^2 A \rangle\!</math> within the compass of a single venn diagram, charting the couple of converging trajectories as shown in Figure&nbsp;12.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 12 -- The Anchor.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 12.} ~~ \text{The Anchor}\!</math><br />
|}<br />
<br />
If we eliminate from view the regions of <math>\mathrm{E}^2 X\!</math> that are ruled out by the dynamic law <math>\mathrm{d}^2 A = \texttt{(} A \texttt{)},\!</math> then what remains is the quotient structure that is shown in Figure&nbsp;13. This picture makes it easy to see that the dynamically allowable portion of the universe is partitioned between the properties <math>A\!</math> and <math>\mathrm{d}^2 A\!.</math> As it happens, this fact might have been expressed &ldquo;right off the bat&rdquo; by an equivalent formulation of the differential law, one that uses the exclusive disjunction to state the law as <math>\texttt{(} A \texttt{,} \mathrm{d}^2 A \texttt{)}\!.</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 13 -- The Tiller.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 13.} ~~ \text{The Tiller}\!</math><br />
|}<br />
<br />
What we have achieved in this example is to give a differential description of a simple dynamic process. In effect, we did this by embedding a directed graph, which can be taken to represent the state transitions of a finite automaton, in a dynamically allotted quotient structure that is created from a boolean lattice or an <math>n\!</math>-cube by nullifying all of the regions that the dynamics outlaws. With growth in the dimensions of our contemplated universes, it becomes essential, both for human comprehension and for computer implementation, that the dynamic structures of interest to us be represented not actually, by acquaintance, but virtually, by description. In our present study, we are using the language of propositional calculus to express the relevant descriptions, and to comprehend the structure that is implicit in the subsets of a <math>n\!</math>-cube without necessarily being forced to actualize all of its points.<br />
<br />
One of the reasons for engaging in this kind of extremely reduced, but explicitly controlled case study is to throw light on the general study of languages, formal and natural, in their full array of syntactic, semantic, and pragmatic aspects. Propositional calculus is one of the last points of departure where we can view these three aspects interacting in a non-trivial way without being immediately and totally overwhelmed by the complexity they generate. Often this complexity causes investigators of formal and natural languages to adopt the strategy of focusing on a single aspect and to abandon all hope of understanding the whole, whether it's the still living natural language or the dynamics of inquiry that lies crystallized in formal logic.<br />
<br />
From the perspective that I find most useful here, a language is a syntactic system that is designed or evolved in part to express a set of descriptions. When the explicit symbols of a language have extensions in its object world that are actually infinite, or when the implicit categories and generative devices of a linguistic theory have extensions in its subject matter that are potentially infinite, then the finite characters of terms, statements, arguments, grammars, logics, and rhetorics force an excess of intension to reside in all these symbols and functions, across the spectrum from the object language to the metalinguistic uses. In the aphorism from W. von Humboldt that Chomsky often cites, for example, in [Cho86, 30] and [Cho93, 49], language requires &ldquo;the infinite use of finite means&rdquo;. This is necessarily true when the extensions are infinite, when the referential symbols and grammatical categories of a language possess infinite sets of models and instances. But it also voices a practical truth when the extensions, though finite at every stage, tend to grow at exponential rates.<br />
<br />
This consequence of dealing with extensions that are &ldquo;practically infinite&rdquo; becomes crucial when one tries to build neural network systems that learn, since the learning competence of any intelligent system is limited to the objects and domains that it is able to represent. If we want to design systems that operate intelligently with the full deck of propositions dealt by intact universes of discourse, then we must supply them with succinct representations and efficient transformations in this domain. Furthermore, in the project of constructing inquiry driven systems, we find ourselves forced to contemplate the level of generality that is embodied in propositions, because the dynamic evolution of these systems is driven by the measurable discrepancies that occur among their expectations, intentions, and observations, and because each of these subsystems or components of knowledge constitutes a propositional modality that can take on the fully generic character of an empirical summary or an axiomatic theory.<br />
<br />
A compression scheme by any other name is a symbolic representation, and this is what the differential extension of propositional calculus, through all of its many universes of discourse, is intended to supply. Why is this particular program of mental calisthenics worth carrying out in general? By providing a uniform logical medium for describing dynamic systems we can make the task of understanding complex systems much easier, both in looking for invariant representations of individual cases and in finding points of comparison among diverse structures that would otherwise appear as isolated systems. All of this goes to facilitate the search for compact knowledge and to adapt what is learned from individual cases to the general realm.<br />
<br />
===Back to the Feature===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
I guess it must be the flag of my disposition, out of hopeful<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;green stuff woven.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 31]<br />
|}<br />
<br />
Let us assume that the sense intended for differential features is well enough established in the intuition, for now, that we may continue with outlining the structure of the differential extension <math>[\mathrm{E}\mathcal{X}] = [A, \mathrm{d}A].\!</math> Over the extended alphabet <math>\mathrm{E}\mathcal{X} = \{ x_1, \mathrm{d}x_1 \} = \{ A, \mathrm{d}A \}\!</math> of cardinality <math>2^n = 2\!</math> we generate the set of points <math>\mathrm{E}X\!</math> of cardinality <math>2^{2n} = 4\!</math> that bears the following chain of equivalent descriptions:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}X & = & \langle A, \mathrm{d}A \rangle<br />
\\[4pt]<br />
& = & \{ \texttt{(} A \texttt{)}, A \} ~\times~ \{ \texttt{(} \mathrm{d}A \texttt{)}, \mathrm{d}A \}<br />
\\[4pt]<br />
& = &<br />
\{<br />
\texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)},~<br />
\texttt{(} A \texttt{)} \mathrm{d}A,~<br />
A \texttt{(} \mathrm{d}A \texttt{)},~<br />
A ~ \mathrm{d}A<br />
\}.<br />
\end{array}</math><br />
|}<br />
<br />
The space <math>\mathrm{E}X\!</math> may be assigned the mnemonic type <math>\mathbb{B} \times \mathbb{D},\!</math> which is really no different than <math>\mathbb{B} \times \mathbb{B} = \mathbb{B}^2.\!</math> An individual element of <math>\mathrm{E}X\!</math> may be regarded as a ''disposition at a point'' or a ''situated direction'', in effect, a singular mode of change occurring at a single point in the universe of discourse. In applications, the modality of this change can be interpreted in various ways, for example, as an expectation, an intention, or an observation with respect to the behavior of a system.<br />
<br />
To complete the construction of the extended universe of discourse <math>\mathrm{E}X^\bullet = [x_1, \mathrm{d}x_1] = [A, \mathrm{d}A]\!</math> one must add the set of differential propositions <math>\mathrm{E}X^\uparrow = \{ g : \mathrm{E}X \to \mathbb{B} \} \cong (\mathbb{B} \times \mathbb{D} \to \mathbb{B})\!</math> to the set of dispositions in <math>\mathrm{E}X.\!</math> There are <math>2^{2^{2n}} = 16\!</math> propositions in <math>\mathrm{E}X^\uparrow,\!</math> as detailed in Table&nbsp;14.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 14.} ~~ \text{Differential Propositions}\!</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>A\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>\mathrm{d}A\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>g_{0}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{1}<br />
\\[4pt]<br />
g_{2}<br />
\\[4pt]<br />
g_{4}<br />
\\[4pt]<br />
g_{8}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
1~0~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
\texttt{(} A \texttt{)} ~ \mathrm{d}A ~<br />
\\[4pt]<br />
~ A ~ \texttt{(} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
~ A ~~ \mathrm{d}A ~<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{neither}~ A ~\text{nor}~ \mathrm{d}A<br />
\\[4pt]<br />
\mathrm{d}A ~\text{and not}~ A<br />
\\[4pt]<br />
A ~\text{and not}~ \mathrm{d}A<br />
\\[4pt]<br />
A ~\text{and}~ \mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot A \land \lnot \mathrm{d}A<br />
\\[4pt]<br />
\lnot A \land \mathrm{d}A<br />
\\[4pt]<br />
A \land \lnot \mathrm{d}A<br />
\\[4pt]<br />
A \land \mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
g_{3}<br />
\\[4pt]<br />
g_{12}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{)}<br />
\\[4pt]<br />
A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ A<br />
\\[4pt]<br />
A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot A<br />
\\[4pt]<br />
A<br />
\end{matrix}\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{6}<br />
\\[4pt]<br />
g_{9}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[4pt]<br />
1~0~0~1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{,} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
\texttt{((} A \texttt{,} \mathrm{d}A \texttt{))}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
A ~\text{not equal to}~ \mathrm{d}A<br />
\\[4pt]<br />
A ~\text{equal to}~ \mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
A \ne \mathrm{d}A<br />
\\[4pt]<br />
A = \mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{5}<br />
\\[4pt]<br />
g_{10}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
\mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ \mathrm{d}A<br />
\\[4pt]<br />
\mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot \mathrm{d}A<br />
\\[4pt]<br />
\mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{7}<br />
\\[4pt]<br />
g_{11}<br />
\\[4pt]<br />
g_{13}<br />
\\[4pt]<br />
g_{14}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} ~ A ~~ \mathrm{d}A ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{(} ~ A ~ \texttt{(} \mathrm{d}A \texttt{))}<br />
\\[4pt]<br />
\texttt{((} A \texttt{)} ~ \mathrm{d}A ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{((} A \texttt{)(} \mathrm{d}A \texttt{))}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not both}~ A ~\text{and}~ \mathrm{d}A<br />
\\[4pt]<br />
\text{not}~ A ~\text{without}~ \mathrm{d}A<br />
\\[4pt]<br />
\text{not}~ \mathrm{d}A ~\text{without}~ A<br />
\\[4pt]<br />
A ~\text{or}~ \mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot A \lor \lnot \mathrm{d}A<br />
\\[4pt]<br />
A \Rightarrow \mathrm{d}A<br />
\\[4pt]<br />
A \Leftarrow \mathrm{d}A<br />
\\[4pt]<br />
A \lor \mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
| <math>f_{3}\!</math><br />
| <math>g_{15}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
Aside from changing the names of variables and shuffling the order of rows, this Table follows the format that was used previously for boolean functions of two variables. The rows are grouped to reflect natural similarity classes among the propositions. In a future discussion, these classes will be given additional explanation and motivation as the orbits of a certain transformation group acting on the set of 16 propositions. Notice that four of the propositions, in their logical expressions, resemble those given in the table for <math>X^\uparrow.\!</math> Thus the first set of propositions <math>\{ f_i \}\!</math> is automatically embedded in the present set <math>\{ g_j \}\!</math> and the corresponding inclusions are indicated at the far left margin of the Table.<br />
<br />
===Tacit Extensions===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
I would really like to have slipped imperceptibly into this lecture, as into all the others I shall be delivering, perhaps over the years ahead.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
Strictly speaking, however, there is a subtle distinction in type between the function <math>f_i : X \to \mathbb{B}</math> and the corresponding function <math>g_j : \mathrm{E}X \to \mathbb{B},</math> even though they share the same logical expression. Naturally, we want to maintain the logical equivalence of expressions that represent the same proposition while appreciating the full diversity of that proposition's functional and typical representatives. Both perspectives, and all the levels of abstraction extending through them, have their reasons, as will develop in time.<br />
<br />
Because this special circumstance points up an important general theme, it is a good idea to discuss it more carefully. Whenever there arises a situation like this, where one alphabet <math>\mathcal{X}</math> is a subset of another alphabet <math>\mathcal{Y},</math> then we say that any proposition <math>f : \langle \mathcal{X} \rangle \to \mathbb{B}</math> has a ''tacit extension'' to a proposition <math>\boldsymbol\varepsilon f : \langle \mathcal{Y} \rangle \to \mathbb{B},\!</math> and that the space <math>(\langle \mathcal{X} \rangle \to \mathbb{B})</math> has an ''automatic embedding'' within the space <math>(\langle \mathcal{Y} \rangle \to \mathbb{B}).</math> The extension is defined in such a way that <math>\boldsymbol\varepsilon f\!</math> puts the same constraint on the variables of <math>\mathcal{X}</math> that are contained in <math>\mathcal{Y}</math> as the proposition <math>f\!</math> initially did, while it puts no constraint on the variables of <math>\mathcal{Y}</math> outside of <math>\mathcal{X},</math> in effect, conjoining the two constraints.<br />
<br />
If the variables in question are indexed as <math>\mathcal{X} = \{ x_1, \ldots, x_n \}</math> and <math>\mathcal{Y} = \{ x_1, \ldots, x_n, \ldots, x_{n+k} \},</math> then the definition of the tacit extension from <math>\mathcal{X}</math> to <math>\mathcal{Y}</math> may be expressed in the form of an equation:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\boldsymbol\varepsilon f(x_1, \ldots, x_n, \ldots, x_{n+k}) ~=~ f(x_1, \ldots, x_n).\!</math><br />
|}<br />
<br />
On formal occasions, such as the present context of definition, the tacit extension from <math>\mathcal{X}</math> to <math>\mathcal{Y}</math> is explicitly symbolized by the operator <math>\boldsymbol\varepsilon : (\langle \mathcal{X} \rangle \to \mathbb{B}) \to (\langle \mathcal{Y} \rangle \to \mathbb{B}),</math> where the appropriate alphabets <math>\mathcal{X}</math> and <math>\mathcal{Y}</math> are understood from context, but normally one may leave the "<math>\boldsymbol\varepsilon\!</math>" silent.<br />
<br />
Let's explore what this means for the present Example. Here, <math>\mathcal{X} = \{ A \}</math> and <math>\mathcal{Y} = \mathrm{E}\mathcal{X} = \{ A, \mathrm{d}A \}.</math> For each of the propositions <math>f_i\!</math> over <math>X\!,</math> specifically, those whose expression <math>e_i\!</math> lies in the collection <math>\{ 0, \texttt{(} A \texttt{)}, A, 1 \},\!</math> the tacit extension <math>\boldsymbol\varepsilon f\!</math> of <math>f\!</math> to <math>\mathrm{E}X</math> can be phrased as a logical conjunction of two factors, <math>f_i = e_i \cdot \tau ~ ,\!</math> where <math>\tau\!</math> is a logical tautology that uses all the variables of <math>\mathcal{Y} - \mathcal{X}.</math> Working in these terms, the tacit extensions <math>\boldsymbol\varepsilon f\!</math> of <math>f\!</math> to <math>\mathrm{E}X</math> may be explicated as shown in Table&nbsp;15.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:60%"<br />
|+ style="height:30px" | <math>\text{Table 15.} ~~ \text{Tacit Extension of}~ [A] ~\text{to}~ [A, \mathrm{d}A]\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0<br />
& = & 0 & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))} & = & & 0<br />
\\[8pt]<br />
\texttt{(} A \texttt{)}<br />
& = & \texttt{(} A \texttt{)} & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))}<br />
& = & \texttt{(} A \texttt{)} \, \mathrm{d}A ~ & + & \texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)}<br />
\\[8pt]<br />
A<br />
& = & ~A~ & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))}<br />
& = & ~A~ ~\mathrm{d}A~ & + & ~A~ \texttt{(} \mathrm{d}A \texttt{)}<br />
\\[8pt]<br />
1<br />
& = & 1 & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))} & = & & 1<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
In its effect on the singular propositions over <math>X,\!</math> this analysis has an interesting interpretation. The tacit extension takes us from thinking about a particular state, like <math>A\!</math> or <math>\texttt{(} A \texttt{)},\!</math> to considering the collection of outcomes, the outgoing changes or the singular dispositions, that spring from that state.<br />
<br />
===Example 2. Drives and Their Vicissitudes===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
I open my scuttle at night and see the far-sprinkled systems,<br><br />
And all I see, multiplied as high as I can cipher, edge but<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the rim of the farther systems.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 81]<br />
|}<br />
<br />
Before we leave the one-feature case let's look at a more substantial example, one that illustrates a general class of curves that can be charted through the extended feature spaces and that provides an opportunity to discuss a number of important themes concerning their structure and dynamics.<br />
<br />
Again, let <math>\mathcal{X} = \{ x_1 \} = \{ A \}.\!</math> In the discussion that follows we will consider a class of trajectories having the property that <math>\mathrm{d}^k A = 0\!</math> for all <math>k\!</math> greater than some fixed <math>m\!</math> and we may indulge in the use of some picturesque terms that describe salient classes of such curves. Given the finite order condition, there is a highest order non-zero difference <math>\mathrm{d}^m A\!</math> exhibited at each point in the course of any determinate trajectory that one may wish to consider. With respect to any point of the corresponding orbit or curve let us call this highest order differential feature <math>\mathrm{d}^m A\!</math> the ''drive'' at that point. Curves of constant drive <math>\mathrm{d}^m A\!</math> are then referred to as ''<math>m^\text{th}\!</math>-gear curves''.<br />
<br />
* '''Scholium.''' The fact that a difference calculus can be developed for boolean functions is well known [Fuji], [Koh, &sect; 8-4] and was probably familiar to Boole, who was an expert in difference equations before he turned to logic. And of course there is the strange but true story of how the Turin machines of the 1840s prefigured the Turing machines of the 1940s [Men, 225-297]. At the very outset of general purpose, mechanized computing we find that the motive power driving the Analytical Engine of Babbage, the kernel of an idea behind all of his wheels, was exactly his notion that difference operations, suitably trained, can serve as universal joints for any conceivable computation [M&M], [Mel, ch. 4].<br />
<br />
Given this language, the Example we take up here can be described as the family of <math>4^\text{th}\!</math>-gear curves through <math>\mathrm{E}^4 X\!</math> <math>=\!</math> <math>\langle A, ~\mathrm{d}A, ~\mathrm{d}^2\!A, ~\mathrm{d}^3\!A, ~\mathrm{d}^4\!A \rangle.</math> These are the trajectories generated subject to the dynamic law <math>\mathrm{d}^4 A = 1,\!</math> where it is understood in such a statement that all higher order differences are equal to <math>0.\!</math> Since <math>\mathrm{d}^4 A\!</math> and all higher <math>\mathrm{d}^k A\!</math> are fixed, the temporal or transitional conditions (initial, mediate, terminal &mdash; transient or stable states) vary only with respect to their projections as points of <math>\mathrm{E}^3 X = \langle A, ~\mathrm{d}A, ~\mathrm{d}^2\!A, ~\mathrm{d}^3\!A \rangle.</math> Thus, there is just enough space in a planar venn diagram to plot all of these orbits and to show how they partition the points of <math>\mathrm{E}^3 X.\!</math> It turns out that there are exactly two possible orbits, of eight points each, as illustrated in Figure&nbsp;16.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 16 -- A Couple of Fourth Gear Orbits.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 16.} ~~ \text{A Couple of Fourth Gear Orbits}\!</math><br />
|}<br />
<br />
With a little thought it is possible to devise an indexing scheme for the general run of dynamic states that allows for comparing universes of discourse that weigh in on different scales of observation. With this end in sight, let us index the states <math>q \in \mathrm{E}^m X\!</math> with the dyadic rationals (or the binary fractions) in the half-open interval <math>[0, 2).\!</math> Formally and canonically, a state <math>q_r\!</math> is indexed by a fraction <math>r = \tfrac{s}{t}\!</math> whose denominator is the power of two <math>t = 2^m\!</math> and whose numerator is a binary numeral formed from the coefficients of state in a manner to be described next. The ''differential coefficients'' of the state <math>q\!</math> are just the values <math>\mathrm{d}^k\!A(q)</math> for <math>k = 0 ~\text{to}~ m,\!</math> where <math>\mathrm{d}^0\!A</math> is defined as being identical to <math>A.\!</math> To form the binary index <math>d_0.d_1 \ldots d_m\!</math> of the state <math>q\!</math> the coefficient <math>\mathrm{d}^k\!A(q)</math> is read off as the binary digit <math>d_k\!</math> associated with the place value <math>2^{-k}.\!</math> Expressed by way of algebraic formulas, the rational index <math>r\!</math> of the state <math>q\!</math> can be given by the following equivalent formulations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
r(q)<br />
& = &<br />
\displaystyle\sum_k d_k \cdot 2^{-k}<br />
& = &<br />
\displaystyle\sum_k \text{d}^k A(q) \cdot 2^{-k}<br />
\\[8pt]<br />
=<br />
\\[8pt]<br />
\displaystyle\frac{s(q)}{t}<br />
& = &<br />
\displaystyle\frac{\sum_k d_k \cdot 2^{(m-k)}}{2^m}<br />
& = &<br />
\displaystyle\frac{\sum_k \text{d}^k A(q) \cdot 2^{(m-k)}}{2^m}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Applied to the example of <math>4^\text{th}\!</math>-gear curves, this scheme results in the data of Tables&nbsp;17-a and 17-b, which exhibit one period for each orbit. The states in each orbit are listed as ordered pairs <math>(p_i, q_j),\!</math> where <math>p_i\!</math> may be read as a temporal parameter that indicates the present time of the state and where <math>j\!</math> is the decimal equivalent of the binary numeral <math>s.\!</math> Informally and more casually, the Tables exhibit the states <math>q_s\!</math> as subscripted with the numerators of their rational indices, taking for granted the constant denominators of <math>2^m\! = 2^4 = 16.\!</math> In this set-up the temporal successions of states can be reckoned as given by a kind of ''parallel round-up rule''. That is, if <math>(d_k, d_{k+1})\!</math> is any pair of adjacent digits in the state index <math>r,\!</math> then the value of <math>d_k\!</math> in the next state is <math>{d_k}' = d_k + d_{k+1}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:55%"<br />
|+ style="height:30px" | <math>\text{Table 17-a.} ~~ \text{A Couple of Orbits in Fourth Gear : Orbit 1}\!</math><br />
|- style="background:ghostwhite"<br />
| <math>\text{Time}\!</math><br />
| <math>\text{State}\!</math><br />
| <math>A\!</math><br />
| <math>\mathrm{d}A\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| <math>p_i\!</math><br />
| <math>q_j\!</math><br />
| <math>\mathrm{d}^0\!A</math><br />
| <math>\mathrm{d}^1\!A</math><br />
| <math>\mathrm{d}^2\!A</math><br />
| <math>\mathrm{d}^3\!A</math><br />
| <math>\mathrm{d}^4\!A</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
p_0<br />
\\[4pt]<br />
p_1<br />
\\[4pt]<br />
p_2<br />
\\[4pt]<br />
p_3<br />
\\[4pt]<br />
p_4<br />
\\[4pt]<br />
p_5<br />
\\[4pt]<br />
p_6<br />
\\[4pt]<br />
p_7<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
q_{01}<br />
\\[4pt]<br />
q_{03}<br />
\\[4pt]<br />
q_{05}<br />
\\[4pt]<br />
q_{15}<br />
\\[4pt]<br />
q_{17}<br />
\\[4pt]<br />
q_{19}<br />
\\[4pt]<br />
q_{21}<br />
\\[4pt]<br />
q_{31}<br />
\end{matrix}\!</math><br />
| colspan="5" |<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="text-align:center; width:100%"<br />
|<br />
<math>\begin{matrix}<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
1.<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:55%"<br />
|+ style="height:30px" | <math>\text{Table 17-b.} ~~ \text{A Couple of Orbits in Fourth Gear : Orbit 2}\!</math><br />
|- style="background:ghostwhite"<br />
| <math>\text{Time}\!</math><br />
| <math>\text{State}\!</math><br />
| <math>A\!</math><br />
| <math>\mathrm{d}A\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| <math>p_i\!</math><br />
| <math>q_j\!</math><br />
| <math>\mathrm{d}^0\!A</math><br />
| <math>\mathrm{d}^1\!A</math><br />
| <math>\mathrm{d}^2\!A</math><br />
| <math>\mathrm{d}^3\!A</math><br />
| <math>\mathrm{d}^4\!A</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
p_0<br />
\\[4pt]<br />
p_1<br />
\\[4pt]<br />
p_2<br />
\\[4pt]<br />
p_3<br />
\\[4pt]<br />
p_4<br />
\\[4pt]<br />
p_5<br />
\\[4pt]<br />
p_6<br />
\\[4pt]<br />
p_7<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
q_{25}<br />
\\[4pt]<br />
q_{11}<br />
\\[4pt]<br />
q_{29}<br />
\\[4pt]<br />
q_{07}<br />
\\[4pt]<br />
q_{09}<br />
\\[4pt]<br />
q_{27}<br />
\\[4pt]<br />
q_{13}<br />
\\[4pt]<br />
q_{23}<br />
\end{matrix}\!</math><br />
| colspan="5" |<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="text-align:center; width:100%"<br />
|<br />
<math>\begin{matrix}<br />
1.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
==Transformations of Discourse==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
It is understandable that an engineer should be completely absorbed in his speciality, instead of pouring himself out into the freedom and vastness of the world of thought, even though his machines are being sent off to the ends of the earth; for he no more needs to be capable of applying to his own personal soul what is daring and new in the soul of his subject than a machine is in fact capable of applying to itself the differential calculus on which it is based. The same thing cannot, however, be said about mathematics; for here we have the new method of thought, pure intellect, the very well-spring of the times, the ''fons et origo'' of an unfathomable transformation.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 39]<br />
|}<br />
<br />
In this section we take up the general study of logical transformations, or maps that relate one universe of discourse to another. In many ways, and especially as applied to the subject of intelligent dynamic systems, my argument develops the antithesis of the statement just quoted. Along the way, if incidental to my ends, I hope this essay can pose a fittingly irenic epitaph to the frankly ironic epigraph inscribed at its head.<br />
<br />
My goal in this section is to answer a single question: What is a propositional tangent functor? In other words, my aim is to develop a clear conception of what manner of thing would pass in the logical realm for a genuine analogue of the tangent functor, an object conceived to generalize as far as possible in the abstract terms of category theory the ordinary notions of functional differentiation and the all too familiar operations of taking derivatives.<br />
<br />
As a first step I discuss the kinds of transformations that we already know as ''extensions'' and ''projections'', and I use these special cases to illustrate several different styles of logical and visual representation that will figure heavily in the sequel.<br />
<br />
===Foreshadowing Transformations : Extensions and Projections of Discourse===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
And, despite the care which she took to look behind her at every moment, she failed to see a shadow which followed her like her own shadow, which stopped when she stopped, which started again when she did, and which made no more noise than a well-conducted shadow should.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 126]<br />
|}<br />
<br />
Many times in our discussion we have occasion to place one universe of discourse in the context of a larger universe of discourse. An embedding of the general type <math>[\mathcal{X}] \to [\mathcal{Y}]\!</math> is implied any time that we make use of one alphabet <math>[\mathcal{X}]\!</math> that happens to be included in another alphabet <math>[\mathcal{Y}].\!</math> When we are discussing differential issues we usually have in mind that the extended alphabet <math>[\mathcal{Y}]\!</math> has a special construction or a specific lexical relation with respect to the initial alphabet <math>[\mathcal{X}],\!</math> one that is marked by characteristic types of accents, indices, or inflected forms.<br />
<br />
====Extension from 1 to 2 Dimensions====<br />
<br />
Figure 18-a lays out the ''angular form'' of venn diagram for universes of 1 and 2 dimensions, indicating the embedding map of type <math>\mathbb{B}^1 \to \mathbb{B}^2\!</math> and detailing the coordinates that are associated with individual cells. Because all points, cells, or logical interpretations are represented as connected geometric areas, we can say that these pictures provide us with an ''areal view'' of each universe of discourse.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-a -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-a.} ~~ \text{Extension from 1 to 2 Dimensions : Areal}\!</math><br />
|}<br />
<br />
Figure 18-b shows the differential extension from <math>X^\bullet = [x]\!</math> to <math>\mathrm{E}X^\bullet = [x, \mathrm{d}x]\!</math> in a ''bundle of boxes'' form of venn diagram. As awkward as it may seem at first, this type of picture is often the most natural and the most easily available representation when we want to conceptualize the localized information or momentary knowledge of an intelligent dynamic system. It gives a ready picture of a ''proposition at a point'', in the present instance, of a proposition about changing states which is itself associated with a particular dynamic state of a system. It is easy to see how this application might be extended to conceive of more general types of instantaneous knowledge that are possessed by a system.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-b -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-b.} ~~ \text{Extension from 1 to 2 Dimensions : Bundle}\!</math><br />
|}<br />
<br />
Figure 18-c shows the same extension in a ''compact'' style of venn diagram, where the differential features at each position are represented by arrows extending from that position that cross or do not cross, as the case may be, the corresponding feature boundaries.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-c -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-c.} ~~ \text{Extension from 1 to 2 Dimensions : Compact}\!</math><br />
|}<br />
<br />
Figure 18-d compresses the picture of the differential extension even further, yielding a directed graph or ''digraph'' form of representation. (Notice that my definition of a digraph allows for loops or ''slings'' at individual points, in addition to arcs or ''arrows'' between the points.)<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-d -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-d.} ~~ \text{Extension from 1 to 2 Dimensions : Digraph}\!</math><br />
|}<br />
<br />
====Extension from 2 to 4 Dimensions====<br />
<br />
Figure 19-a lays out the ''areal view'' or the ''angular form'' of venn diagram for universes of 2 and 4 dimensions, indicating the embedding map of type <math>\mathbb{B}^2 \to \mathbb{B}^4.\!</math> In many ways these pictures are the best kind there is, giving full canvass to an ideal vista. Their style allows the clearest, the fairest, and the plainest view that we can form of a universe of discourse, affording equal representation to all dispositions and maintaining a balance with respect to ordinary and differential features. If only we could extend this view! Unluckily, an obvious difficulty beclouds this prospect, and that is how precipitately we run into the limits of our plane and visual intuitions. Even within the scope of the spare few dimensions that we have scanned up to this point subtle discrepancies have crept in already. The circumstances that bind us and the frameworks that block us, the flat distortion of the planar projection and the inevitable ineffability that precludes us from wrapping its rhomb figure into rings around a torus, all of these factors disguise the underlying but true connectivity of the universe of discourse.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-a -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-a.} ~~ \text{Extension from 2 to 4 Dimensions : Areal}\!</math><br />
|}<br />
<br />
Figure 19-b shows the differential extension from <math>U^\bullet = [u, v]\!</math> to <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v]\!</math> in the ''bundle of boxes'' form of venn diagram.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-b -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-b.} ~~ \text{Extension from 2 to 4 Dimensions : Bundle}\!</math><br />
|}<br />
<br />
As dimensions increase, this factorization of the extended universe along the lines that are marked out by the bundle picture begins to look more and more like a practical necessity. But whenever we use a propositional model to address a real situation in the context of nature we need to remain aware that this articulation into factors, affecting our description, may be wholly artificial in nature and cleave to nothing, no joint in nature, nor any juncture in time to be in or out of joint.<br />
<br />
Figure 19-c illustrates the extension from 2 to 4 dimensions in the ''compact'' style of venn diagram. Here, just the changes with respect to the center cell are shown.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-c -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-c.} ~~ \text{Extension from 2 to 4 Dimensions : Compact}\!</math><br />
|}<br />
<br />
Figure 19-d gives the ''digraph'' form of representation for the differential extension <math>U^\bullet \to \mathrm{E}U^\bullet,\!</math> where the 4 nodes marked with a circle <math>{}^{\bigcirc}\!</math> are the cells <math>uv,\, u \texttt{(} v \texttt{)},\, \texttt{(} u \texttt{)} v,\, \texttt{(} u \texttt{)(} v \texttt{)},\!</math> respectively, and where a 2-headed arc counts as 2 arcs of the differential digraph.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-d -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-d.} ~~ \text{Extension from 2 to 4 Dimensions : Digraph}\!</math><br />
|}<br />
<br />
===Thematization of Functions : And a Declaration of Independence for Variables===<br />
<br />
{| width="100%"<br />
| align="left" |<br />
''And as imagination bodies forth''<br><br />
''The forms of things unknown, the poet's pen''<br><br />
''Turns them to shapes, and gives to airy nothing''<br><br />
''A local habitation and a name.''<br />
| align="right" valign="bottom" | A Midsummer Night's Dream, 5.1.18<br />
|}<br />
<br />
In the representation of propositions as functions it is possible to notice different degrees of explicitness in the way their functional character is symbolized. To indicate what I mean by this, the next series of Figures illustrates a set of graphic conventions that will be put to frequent use in the remainder of this discussion, both to mark the relevant distinctions and to help us convert between related expressions at different levels of explicitness in their functionality.<br />
<br />
====Thematization : Venn Diagrams====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
The known universe has one complete lover and that is the greatest poet. He consumes an eternal passion and is indifferent which chance happens and which possible contingency of fortune or misfortune and persuades daily and hourly his delicious pay.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 11&ndash;12]<br />
|}<br />
<br />
Figure 20-i traces the first couple of steps in this order of ''thematic'' progression, that will gradually run the gamut through a complete series of degrees of functional explicitness in the expression of logical propositions. The first venn diagram represents a situation where the function is indicated by a shaded figure and a logical expression. At this stage one may be thinking of the proposition only as expressed by a formula in a particular language and its content only as a subset of the universe of discourse, as when considering the proposition <math>u\!\cdot\!v</math> in the universe <math>[u, v].\!</math> The second venn diagram depicts a situation in which two significant steps have been taken. First, one has taken the trouble to give the proposition <math>u\!\cdot\!v</math> a distinctive functional name <math>{}^{\backprime\backprime} J {}^{\prime\prime}.\!</math> Second, one has come to think explicitly about the target domain that contains the functional values of <math>J,\!</math> as when writing <math>J : \langle u, v \rangle \to \mathbb{B}.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 20-i -- Thematization of Conjunction (Stage 1).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 20-i.} ~~ \text{Thematization of Conjunction (Stage 1)}\!</math><br />
|}<br />
<br />
In Figure 20-ii the proposition <math>J\!</math> is viewed explicitly as a transformation from one universe of discourse to another.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 20-ii -- Thematization of Conjunction (Stage 2).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 20-ii.} ~~ \text{Thematization of Conjunction (Stage 2)}\!</math><br />
|}<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
o-------------------------------o o-------------------------------o<br />
| | | |<br />
| o-----o o-----o | | o-----o o-----o |<br />
| / \ / \ | | / \ / \ |<br />
| / o \ | | / o \ |<br />
| / /`\ \ | | / /`\ \ |<br />
| o o```o o | | o o```o o |<br />
| | u |```| v | | | | u |```| v | |<br />
| o o```o o | | o o```o o |<br />
| \ \`/ / | | \ \`/ / |<br />
| \ o / | | \ o / |<br />
| \ / \ / | | \ / \ / |<br />
| o-----o o-----o | | o-----o o-----o |<br />
| | | |<br />
o-------------------------------o o-------------------------------o<br />
\ / \ /<br />
\ / \ /<br />
\ / \ J /<br />
\ / \ /<br />
\ / \ /<br />
o----------\---------/----------o o----------\---------/----------o<br />
| \ / | | \ / |<br />
| \ / | | \ / |<br />
| o-----@-----o | | o-----@-----o |<br />
| /`````````````\ | | /`````````````\ |<br />
| /```````````````\ | | /```````````````\ |<br />
| /`````````````````\ | | /`````````````````\ |<br />
| o```````````````````o | | o```````````````````o |<br />
| |```````````````````| | | |```````````````````| |<br />
| |```````` J ````````| | | |```````` x ````````| |<br />
| |```````````````````| | | |```````````````````| |<br />
| o```````````````````o | | o```````````````````o |<br />
| \`````````````````/ | | \`````````````````/ |<br />
| \```````````````/ | | \```````````````/ |<br />
| \`````````````/ | | \`````````````/ |<br />
| o-----------o | | o-----------o |<br />
| | | |<br />
| | | |<br />
o-------------------------------o o-------------------------------o<br />
J = u v x = J<u, v><br />
<br />
Figure 20-ii. Thematization of Conjunction (Stage 2)<br />
</pre><br />
|}<br />
<br />
In the first venn diagram the name that is assigned to a composite proposition, function, or region in the source universe is delegated to a simple feature in the target universe. This can result in a single character or term exceeding the responsibilities it can carry off well. Allowing the name of a function <math>J : \langle u, v \rangle \to \mathbb{B}\!</math> to serve as the name of its dependent variable <math>J : \mathbb{B}\!</math> does not mean that one has to confuse a function with any of its values, but it does put one at risk for a number of obvious problems, and we should not be surprised, on numerous and limiting occasions, when quibbling arises from the attempts of a too original syntax to serve these two masters.<br />
<br />
The second venn diagram circumvents these difficulties by introducing a new variable name for each basic feature of the target universe, as when writing <math>J : \langle u, v \rangle \to \langle x \rangle,\!</math> and thereby assigns a concrete type <math>\langle x \rangle</math> to the abstract codomain <math>\mathbb{B}.\!</math> To make this induction of variables more formal one can append subscripts, as in <math>x_J,\!</math> to indicate the origin or derivation of the new characters. Or we may use a lexical modifier to convert function names into variable names, for example, associating the function name <math>J\!</math> with the variable name <math>\check{J}.\!</math> Thus we may think of <math>x = x_J = \check{J}\!</math> as the ''cache variable'' corresponding to the function <math>J\!</math> or the symbol <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> considered as a contingent variable.<br />
<br />
In Figure 20-iii we arrive at a stage where the functional equations <math>J = u\!\cdot\!v</math> and <math>x = u\!\cdot\!v</math> are regarded as propositions in their own right, reigning in and ruling over the 3-feature universes of discourse <math>[u, v, J]~\!</math> and <math>[u, v, x],\!</math> respectively. Subject to the cautions already noted, the function name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> can be reinterpreted as the name of a feature <math>\check{J}</math> and the equation <math>J = u\!\cdot\!v</math> can be read as the logical equivalence <math>\texttt{((} J, u ~ v \texttt{))}.\!</math> To give it a generic name let us call this newly expressed, collateral proposition the ''thematization'' or the ''thematic extension'' of the original proposition <math>J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 20-iii -- Thematization of Conjunction (Stage 3).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 20-iii.} ~~ \text{Thematization of Conjunction (Stage 3)}\!</math><br />
|}<br />
<br />
The first venn diagram represents the thematization of the conjunction <math>J\!</math> with shading in the appropriate regions of the universe <math>[u, v, J].\!</math> Also, it illustrates a quick way of constructing a thematic extension. First, draw a line, in practice or the imagination, that bisects every cell of the original universe, placing half of each cell under the aegis of the thematized proposition and the other half under its antithesis. Next, on the scene where the theme applies leave the shade wherever it lies, and off the stage, where it plays otherwise, stagger the pattern in a harlequin guise.<br />
<br />
In the final venn diagram of this sequence the thematic progression comes full circle and completes one round of its development. The ambiguities that were occasioned by the changing role of the name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> are resolved by introducing a new variable name <math>{}^{\backprime\backprime} x {}^{\prime\prime}</math> to take the place of <math>\check{J},\!</math> and the region that represents this fresh featured <math>x\!</math> is circumscribed in a more conventional symmetry of form and placement. Just as we once gave the name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> to the proposition <math>u\!\cdot\!v,</math> we now give the name <math>{}^{\backprime\backprime} \iota {}^{\prime\prime}</math> to its thematization <math>\texttt{((} x, u ~ v \texttt{))}.\!</math> Already, again, at this culminating stage of reflection, we begin to think of the newly named proposition as a distinctive individual, a particular function <math>\iota : \langle u, v, x \rangle \to \mathbb{B}.\!</math><br />
<br />
From now on, the terms ''thematic extension'' and ''thematization'' will be used to describe both the process and degree of explication that progresses through this series of pictures, both the operation of increasingly explicit symbolization and the dimension of variation that is swept out by it. To speak of this change in general, that takes us in our current example from <math>J\!</math> to <math>\iota,\!</math> we introduce a class of operators symbolized by the Greek letter <math>\theta,\!</math> writing <math>\iota = \theta J\!</math> in the present instance. The operator <math>\theta,\!</math> in the present situation bearing the type <math>\theta : [u, v] \to [u, v, x],\!</math> provides us with a convenient way of recapitulating and summarizing the complete cycle of thematic developments.<br />
<br />
Figure 21 shows how the thematic extension operator <math>\theta\!</math> acts on two further examples, the disjunction <math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math> and the equality <math>\texttt{((} u, v \texttt{))}.\!</math> Referring to the disjunction as <math>f(u, v)\!</math> and the equality as <math>f(u, v),\!</math> we may express the thematic extensions as <math>\varphi = \theta f\!</math> and <math>\gamma = \theta g.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 21 -- Thematization of Disjunction and Equality.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 21.} ~~ \text{Thematization of Disjunction and Equality}\!</math><br />
|}<br />
<br />
====Thematization : Truth Tables====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
That which distorts honest shapes or which creates unearthly beings or places or contingencies is a nuisance and a revolt.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 19]<br />
|}<br />
<br />
Tables 22 through 25 outline a method for computing the thematic extensions of propositions in terms of their coordinate values.<br />
<br />
A preliminary step, as illustrated in Table&nbsp;22, is to write out the truth table representations of the propositional forms whose thematic extensions one wants to compute, in the present instance, the functions <math>f(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math> and <math>g(u, v) = \texttt{((} u, v \texttt{))}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:50%"<br />
|+ style="height:30px" | <math>\text{Table 22.} ~~ \text{Disjunction}~ f ~\text{and Equality}~ g\!</math><br />
|- style="height:35px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black" | <math>f\!</math><br />
| <math>g\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Next, each propositional form is individually represented in the fashion shown in Tables&nbsp;23-i and 23-ii, using <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} g {}^{\prime\prime}\!</math> as function names and creating new variables <math>x\!</math> and <math>y\!</math> to hold the associated functional values. This pair of Tables outlines the first stage in the transition from the <math>2\!</math>-dimensional universes of <math>f\!</math> and <math>g\!</math> to the <math>3\!</math>-dimensional universes of <math>\theta f\!</math> and <math>\theta g.\!</math> The top halves of the Tables replicate the truth table patterns for <math>f\!</math> and <math>g\!</math> in the form <math>f : [u, v] \to [x]\!</math> and <math>g : [u, v] \to [y].\!</math> The bottom halves of the tables print the negatives of these pictures, as it were, and paste the truth tables for <math>\texttt{(} f \texttt{)}\!</math> and <math>\texttt{(} g \texttt{)}\!</math> under the copies for <math>f\!</math> and <math>g.\!</math> At this stage, the columns for <math>\theta f\!</math> and <math>\theta g\!</math> are appended almost as afterthoughts, amounting to indicator functions for the sets of ordered triples that make up the functions <math>f\!</math> and <math>g.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 23-i and 23-ii.} ~~ \text{Thematics of Disjunction and Equality (1)}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 23-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black" | <math>f\!</math><br />
| <math>x\!</math><br />
| <math>\varphi\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |<br />
<math>\begin{matrix}\to\\\to\\\to\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" | &nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 23-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black" | <math>g\!</math><br />
| <math>y\!</math><br />
| <math>\gamma\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |<br />
<math>\begin{matrix}\to\\\to\\\to\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" | &nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
All the data are now in place to give the truth tables for <math>\theta f\!</math> and <math>\theta g.\!</math> All that remains to be done is to permute the rows and change the roles of <math>x\!</math> and <math>y\!</math> from dependent to independent variables. In Tables&nbsp;24-i and 24-ii the rows are arranged in such a way as to put the 3-tuples <math>(u, v, x)\!</math> and <math>(u, v, y)\!</math> in binary numerical order, suitable for viewing as the arguments of the maps <math>\theta f = \varphi : [u, v, x] \to \mathbb{B}\!</math> and <math>\theta g = \gamma : [u, v, y] \to \mathbb{B}.\!</math> Moreover, the structure of the tables is altered slightly, allowing the now vestigial functions <math>\theta f\!</math> and <math>\theta g\!</math> to be passed over without further attention and shifting the heavy vertical bars a notch to the right. In effect, this clinches the fact that the thematic variables <math>x := \check{f}\!</math> and <math>y := \check{g}\!</math> are now treated as independent variables.<br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 24-i and 24-ii.} ~~ \text{Thematics of Disjunction and Equality (2)}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 24-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>f\!</math><br />
| <math>x\!</math><br />
| style="border-left:1px solid black" | <math>\varphi\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <font size="+2">&nbsp;<br></font><math>\begin{matrix}\to\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 24-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>g\!</math><br />
| <math>y\!</math><br />
| style="border-left:1px solid black" | <math>\gamma\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | &nbsp;<br><math>\begin{matrix}\to\\\to\end{matrix}\!</math><br>&nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
An optional reshuffling of the rows brings additional features of the thematic extensions to light. Leaving the columns in place for the sake of comparison, Tables&nbsp;25-i and 25-ii sort the rows in a different order, in effect treating <math>x\!</math> and <math>y\!</math> as the primary variables in their respective 3-tuples. Regarding the thematic extensions in the form <math>\varphi : [x, u, v] \to \mathbb{B}\!</math> and <math>\gamma : [y, u, v] \to \mathbb{B}\!</math> makes it easier to see in this tabular setting a property that was graphically obvious in the venn diagrams above. Specifically, when the thematic variable <math>\check{F}\!</math> is true then <math>\theta F\!</math> exhibits the pattern of the original <math>F,\!</math> and when <math>\check{F}\!</math> is false then <math>\theta F\!</math> exhibits the pattern of its negation <math>\texttt{(} F \texttt{)}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 25-i and 25-ii.} ~~ \text{Thematics of Disjunction and Equality (3)}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 25-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>f\!</math><br />
| <math>x\!</math><br />
| style="border-left:1px solid black" | <math>\varphi\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>{\to}\!</math><br><font size="+2">&nbsp;<br>&nbsp;<br>&nbsp;<br></font><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <font size="+2">&nbsp;<br></font><math>\begin{matrix}\to\\\to\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 25-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>g\!</math><br />
| <math>y\!</math><br />
| style="border-left:1px solid black" | <math>\gamma\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | &nbsp;<br><math>\begin{matrix}\to\\\to\end{matrix}\!</math><br>&nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
Finally, Tables&nbsp;26-i and 26-ii compare the tacit extensions <math>\boldsymbol\varepsilon : [u, v] \to [u, v, x]\!</math> and <math>\boldsymbol\varepsilon : [u, v] \to [u, v, y]\!</math> with the thematic extensions of the same types, as applied to the propositions <math>f\!</math> and <math>g,\!</math> respectively.<br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 26-i and 26-ii.} ~~ \text{Tacit Extension and Thematization}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 26-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>x\!</math><br />
| style="border-left:1px solid black" | <math>\boldsymbol\varepsilon f\!</math><br />
| <math>\theta f\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 26-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>y\!</math><br />
| style="border-left:1px solid black" | <math>\boldsymbol\varepsilon g\!</math><br />
| <math>\theta g\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
Table 27 summarizes the thematic extensions of all propositions on two variables. Column&nbsp;4 lists the equations of form <math>\texttt{((} \check{f_i}, f_i (u, v) \texttt{))}\!</math> and Column&nbsp;5 simplifies these equations into the form of algebraic expressions. As always, <math>{}^{\backprime\backprime} + {}^{\prime\prime}\!</math> refers to exclusive disjunction and each <math>{}^{\backprime\backprime} \check{f} {}^{\prime\prime}\!</math> appearing in the last two Columns refers to the corresponding variable name <math>{}^{\backprime\backprime} \check{f_i} {}^{\prime\prime}.~\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 27.} ~~ \text{Thematization of Bivariate Propositions}\!</math><br />
|- style="height:30px; background:ghostwhite"<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>{f}\!</math><br />
| <math>\theta f\!</math><br />
| <math>\theta f\!</math><br />
|- style="background:ghostwhite"<br />
| align="right" | <math>u\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| align="right" | <math>v\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| align="left" | <math>\texttt{((} \check{f} \texttt{,~(~)~))}\!</math><br />
| align="left" | <math>\check{f} + 1\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\\[4pt]<br />
f_{4}<br />
\\[4pt]<br />
f_{8}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
1~0~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} u \texttt{)~} v \texttt{~}<br />
\\[4pt]<br />
\texttt{~} u \texttt{~(} v \texttt{)}<br />
\\[4pt]<br />
\texttt{~} u \texttt{~~} v \texttt{~}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(u)(v)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~(u)~v~~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~u~(v)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~u~~v~~))}<br />
\end{array}</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + u + v + uv<br />
\\[4pt]<br />
\check{f} + v + uv + 1<br />
\\[4pt]<br />
\check{f} + u + uv + 1<br />
\\[4pt]<br />
\check{f} + uv + 1<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{3}<br />
\\[4pt]<br />
f_{12}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{)}<br />
\\[4pt]<br />
\texttt{~} u \texttt{~}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(u)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~u~~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + u<br />
\\[4pt]<br />
\check{f} + u + 1<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[4pt]<br />
f_{9}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[4pt]<br />
1~0~0~1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{,} v \texttt{)}<br />
\\[4pt]<br />
\texttt{((} u \texttt{,} v \texttt{))}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~~(} u \texttt{,} v \texttt{)~~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~((} u \texttt{,} v \texttt{))~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + u + v + 1<br />
\\[4pt]<br />
\check{f} + u + v<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{5}<br />
\\[4pt]<br />
f_{10}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} v \texttt{)}<br />
\\[4pt]<br />
\texttt{~} v \texttt{~}<br />
\end{matrix}</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(} v \texttt{)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~} v \texttt{~~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + v<br />
\\[4pt]<br />
\check{f} + v + 1<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[4pt]<br />
f_{11}<br />
\\[4pt]<br />
f_{13}<br />
\\[4pt]<br />
f_{14}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(~} u \texttt{~~} v \texttt{~)}<br />
\\[4pt]<br />
\texttt{(~} u \texttt{~(} v \texttt{))}<br />
\\[4pt]<br />
\texttt{((} u \texttt{)~} v \texttt{~)}<br />
\\[4pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(~} u \texttt{~~} v \texttt{~)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~(~} u \texttt{~(} v \texttt{))~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~((} u \texttt{)~} v \texttt{~)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~((} u \texttt{)(} v \texttt{))~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + uv<br />
\\[4pt]<br />
\check{f} + u + uv<br />
\\[4pt]<br />
\check{f} + v + uv<br />
\\[4pt]<br />
\check{f} + u + v + uv + 1<br />
\end{array}\!</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| align="left" | <math>\texttt{((} \check{f} \texttt{,~((~))~))}\!</math><br />
| align="left" | <math>\check{f}\!</math><br />
|}<br />
<br />
<br><br />
<br />
In order to show what all of the thematic extensions from two dimensions to three dimensions look like in terms of coordinates, Tables&nbsp;28 and 29 present ordinary truth tables for the functions <math>f_i : \mathbb{B}^2 \to \mathbb{B}\!</math> and for the corresponding thematizations <math>\theta f_i = \varphi_i : \mathbb{B}^3 \to \mathbb{B}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 28.} ~~ \text{Propositions on Two Variables}\!</math><br />
|- style="height:35px; background:ghostwhite"<br />
| style="width:5%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:5%; border-bottom:1px solid black; border-left:1px solid black" | <math>f_{0}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{1}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{2}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{3}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{4}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{5}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{6}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{7}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{8}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{9}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{10}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{11}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{12}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{13}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{14}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{15}\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 29.} ~~ \text{Thematic Extensions of Bivariate Propositions}\!</math><br />
|- style="height:35px; background:ghostwhite"<br />
| style="width:5%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\check{f}\!</math><br />
| style="width:5%; border-bottom:1px solid black; border-left:1px solid black" | <math>\varphi_{0}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{1}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{2}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{3}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{4}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{5}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{6}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{7}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{8}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{9}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{10}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{11}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{12}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{13}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{14}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{15}\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
|-<br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Propositional Transformations===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
If only the word &lsquo;artificial&rsquo; were associated with the idea of ''art'', or expert skill gained through voluntary apprenticeship (instead of suggesting the factitious and unreal), we might say that ''logical'' refers to artificial thought.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 56&ndash;57]<br />
|}<br />
<br />
In this section we develop a comprehensive set of concepts for dealing with transformations between universes of discourse. In this most general setting the source and target universes of a transformation are allowed to be different, but may be the same. When we apply these concepts to dynamic systems we focus on the important special case of transformations that map a universe into itself, regarding them as the state transitions of a discrete dynamical process and placing them among the myriad ways that a universe of discourse might change, and by that change turn into itself.<br />
<br />
====Alias and Alibi Transformations====<br />
<br />
There are customarily two modes of understanding a transformation, at least, when we try to interpret its relevance to something in reality. A transformation always refers to a changing prospect, to say it in a unified but equivocal way, but this can be taken to mean either a subjective change in the interpreting observer's point of view or an objective change in the systematic subject of discussion. In practice these variant uses of the transformation concept are distinguished in the following terms:<br />
<br />
# A ''perspectival'' or ''alias'' transformation refers to a shift in perspective or a change in language that takes place in the observer's frame of reference.<br />
# A ''transitional'' or ''alibi'' transformation refers to a change of position or an alteration of state that occurs in the object system as it falls under study.<br />
<br />
(For a recent discussion of the alias vs. alibi issue, as it relates to linear transformations in vector spaces and to other issues of an algebraic nature, see [MaB, 256, 582-4].)<br />
<br />
Naturally, when we are concerned with the dynamical properties of a system, the transitional aspect of transformation is the factor that comes to the fore, and this involves us in contemplating all of the ways of changing a universe into itself while remaining under the rule of established dynamical laws. In the prospective application to dynamic systems, and to neural networks viewed in this light, our interest lies chiefly with the transformations of a state space into itself that constitute the state transitions of a discrete dynamic process. Nevertheless, many important properties of these transformations, and some constructions that we need to see most clearly, are independent of the transitional interpretation and are likely to be confounded with irrelevant features if presented first and only in that association.<br />
<br />
In addition, and in partial contrast, intelligent systems are exactly that species of dynamic agents that have the capacity to have a point of view, and we cannot do justice to their peculiar properties without examining their ability to form and transform their own frames of reference in exposure to the elements of their own experience. In this setting, the perspectival aspect of transformation is the facet that shines most brightly, perhaps too often leaving us fascinated with mere glimmerings of its actual potential. It needs to be emphasized that nothing of the ordinary sort needs be moved in carrying out a transformation under the alias interpretation, that it may only involve a change in the forms of address, an amendment of the terms which are customed to approach and fashioned to describe the very same things in the very same world. But again, working within a discipline of realistic computation, we know how formidably complex and resource-consuming such transformations of perspective can be to implement in practice, much less to endow in the self-governed form of a nascently intelligent dynamical system.<br />
<br />
====Transformations of General Type====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
''Es ist passiert'', &ldquo;it just sort of happened&rdquo;, people said there when other people in other places thought heaven knows what had occurred. It was a peculiar phrase, not known in this sense to the Germans and with no equivalent in other languages, the very breath of it transforming facts and the bludgeonings of fate into something light as eiderdown, as thought itself.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 34]<br />
|}<br />
<br />
Consider the situation illustrated in Figure&nbsp;30, where the alphabets <math>\mathcal{U} = \{ u, v \}\!</math> and <math>\mathcal{X} = \{ x, y, z \}\!</math> are used to label basic features in two different logical universes, <math>U^\bullet = [u, v]\!</math> and <math>X^\bullet = [x, y, z].\!</math><br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
o-------------------------------------------------------o<br />
| U |<br />
| |<br />
| o-----------o o-----------o |<br />
| / \ / \ |<br />
| / o \ |<br />
| / / \ \ |<br />
| / / \ \ |<br />
| o o o o |<br />
| | | | | |<br />
| | u | | v | |<br />
| | | | | |<br />
| o o o o |<br />
| \ \ / / |<br />
| \ \ / / |<br />
| \ o / |<br />
| \ / \ / |<br />
| o-----------o o-----------o |<br />
| |<br />
| |<br />
o---------------------------o---------------------------o<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
o-------------------------o o-------------------------o o-------------------------o<br />
| U | | U | | U |<br />
| o---o o---o | | o---o o---o | | o---o o---o |<br />
| / \ / \ | | / \ / \ | | / \ / \ |<br />
| / o \ | | / o \ | | / o \ |<br />
| / / \ \ | | / / \ \ | | / / \ \ |<br />
| o o o o | | o o o o | | o o o o |<br />
| | u | | v | | | | u | | v | | | | u | | v | |<br />
| o o o o | | o o o o | | o o o o |<br />
| \ \ / / | | \ \ / / | | \ \ / / |<br />
| \ o / | | \ o / | | \ o / |<br />
| \ / \ / | | \ / \ / | | \ / \ / |<br />
| o---o o---o | | o---o o---o | | o---o o---o |<br />
| | | | | |<br />
o-------------------------o o-------------------------o o-------------------------o<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ g | \ f / | h /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ o----------|-----------\-----/-----------|----------o /<br />
\ | X | \ / | | /<br />
\ | | \ / | | /<br />
\ | | o-----o-----o | | /<br />
\| | / \ | |/<br />
\ | / \ | /<br />
|\ | / \ | /|<br />
| \ | / \ | / |<br />
| \ | / \ | / |<br />
| \ | o x o | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \| | | |/ |<br />
| o--o--------o o--------o--o |<br />
| / \ \ / / \ |<br />
| / \ \ / / \ |<br />
| / \ o / \ |<br />
| / \ / \ / \ |<br />
| / \ / \ / \ |<br />
| o o--o-----o--o o |<br />
| | | | | |<br />
| | | | | |<br />
| | | | | |<br />
| | y | | z | |<br />
| | | | | |<br />
| | | | | |<br />
| o o o o |<br />
| \ \ / / |<br />
| \ \ / / |<br />
| \ o / |<br />
| \ / \ / |<br />
| \ / \ / |<br />
| o-----------o o-----------o |<br />
| |<br />
| |<br />
o---------------------------------------------------o<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ p , q /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
o<br />
<br />
Figure 30. Generic Frame of a Logical Transformation<br />
</pre><br />
|}<br />
<br />
Enter the picture, as we usually do, in the middle of things, with features like <math>x, y , z\!</math> that present themselves to be simple enough in their own right and that form a satisfactory, if temporary foundation to provide a basis for discussion. In this universe and on these terms we find expression for various propositions and questions of principal interest to ourselves, as indicated by the maps <math>p, q : X \to \mathbb{B}.\!</math> Then we discover that the simple features <math>\{ x, y, z \}\!</math> are really more complex than we thought at first, and it becomes useful to regard them as functions <math>\{ f, g, h \}\!</math> of other features <math>\{ u, v \}\!</math> that we place in a preface to our original discourse, or suppose as topics of a preliminary universe of discourse <math>U^\bullet = [u, v].\!</math> It may happen that these late-blooming but pre-ambling features are found to lie closer, in a sense that may be our job to determine, to the central nature of the situation of interest, in which case they earn our regard as being more fundamental, but these functions and features are only required to supply a critical stance on the universe of discourse or an alternate perspective on the nature of things in order to be preserved as useful.<br />
<br />
A particular transformation <math>F : [u, v] \to [x, y, z]\!</math> may be expressed by a system of equations, as shown below. Here, <math>F\!</math> is defined by its component maps <math>F = (F_1, F_2, F_3) = (f, g, h),\!</math> where each component map in <math>\{ f, g, h \}\!</math> is a proposition of type <math>\mathbb{B}^n \to \mathbb{B}^1.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:50%"<br />
|<br />
<math>\begin{matrix}<br />
x & = & f(u, v)<br />
\\[10pt]<br />
y & = & g(u, v)<br />
\\[10pt]<br />
z & = & h(u, v)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Regarded as a logical statement, this system of equations expresses a relation between a collection of freely chosen propositions <math>\{ f, g, h \}\!</math> in one universe of discourse and the special collection of simple propositions <math>\{ x, y, z \}\!</math> on which is founded another universe of discourse. Growing familiarity with a particular transformation of discourse, and the desire to achieve a ready understanding of its implications, requires that we be able to convert this information about generals and simples into information about all the main subtypes of propositions, including the linear and singular propositions.<br />
<br />
===Analytic Expansions : Operators and Functors===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Consider what effects that might ''conceivably'' have practical bearings you ''conceive'' the objects of your ''conception'' to have. Then, your ''conception'' of those effects is the whole of your ''conception'' of the object.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; C.S. Peirce, &ldquo;The Maxim of Pragmatism&rdquo;, CP 5.438<br />
|}<br />
<br />
Given the barest idea of a logical transformation, as suggested by the sketch in Figure&nbsp;30, and having conceptualized the universe of discourse, with all of its points and propositions, as a beginning object of discussion, we are ready to enter the next phase of our investigation.<br />
<br />
====Operators on Propositions and Transformations====<br />
<br />
The next step is naturally inclined toward objects of the next higher order, namely, with operators that take in argument lists of logical transformations and that give back specified types of logical transformations as their results. For our present aims, we do not need to consider the most general class of such operators, nor any one of them for its own sake. Rather, we are interested in the special sorts of operators that arise in the study and analysis of logical transformations. Figuratively speaking, these operators serve as instruments for the live tomography (and hopefully not the vivisection) of the forms of change under view. Beyond that, they open up ways to implement the changes of view that we need to grasp all the variations on a transformational theme, or to appreciate enough of its significant features to &ldquo;get the drift&rdquo; of the change occurring, to form a passing acquaintance or a synthetic comprehension of its general character and disposition.<br />
<br />
The simplest type of operator is one that takes a single transformation as an argument and returns a single transformation as a result, and most of the operators explicitly considered in our discussion will be of this kind. Figure&nbsp;31 illustrates the typical situation.<br />
<br />
{| align="center" border="0" cellpadding="20"<br />
|<br />
<pre><br />
o---------------------------------------o<br />
| |<br />
| |<br />
| U% F X% |<br />
| o------------------>o |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| !W! | | !W! |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| v v |<br />
| o------------------>o |<br />
| !W!U% !W!F !W!X% |<br />
| |<br />
| |<br />
o---------------------------------------o<br />
Figure 31. Operator Diagram (1)<br />
</pre><br />
|}<br />
<br />
In this Figure <math>{}^{\backprime\backprime} \mathsf{W} {}^{\prime\prime}\!</math> stands for a generic operator <math>\mathsf{W},\!</math> in this case one that takes a logical transformation <math>F\!</math> of type <math>(U^\bullet \to X^\bullet)\!</math> into a logical transformation <math>\mathsf{W}F\!</math> of type <math>(\mathsf{W}U^\bullet \to \mathsf{W}X^\bullet).\!</math> Thus, the operator <math>\mathsf{W}\!</math> must be viewed as making assignments for both families of objects we have previously considered, that is, for universes of discourse like <math>{U^\bullet}\!</math> and <math>{X^\bullet}\!</math> and for logical transformations like <math>F.\!</math><br />
<br />
'''Note.''' Strictly speaking, an operator like <math>\mathsf{W}\!</math> works between two whole categories of universes and transformations, which we call the ''source'' and the ''target'' categories of <math>\mathsf{W}.\!</math> Given this setting, <math>\mathsf{W}\!</math> specifies for each universe <math>U^\bullet\!</math> in its source category a definite universe <math>\mathsf{W}U^\bullet\!</math> in its target category, and to each transformation <math>F\!</math> in its source category it assigns a unique transformation <math>\mathsf{W}F\!</math> in its target category. Naturally, this only works if <math>\mathsf{W}\!</math> takes the source <math>U^\bullet</math> and the target <math>X^\bullet</math> of the map <math>F\!</math> over to the source <math>\mathsf{W}U^\bullet\!</math> and the target <math>\mathsf{W}X^\bullet\!</math> of the map <math>\mathsf{W}F.\!</math> With luck or care enough, we can avoid ever having to put anything like that in words again, letting diagrams do the work. In the situations of present concern we are usually focused on a single transformation <math>F,\!</math> and thus we can take it for granted that the assignment of universes under <math>\mathsf{W}\!</math> is defined appropriately at the source and target ends of <math>F.\!</math> It is not always the case, though, that we need to use the particular names (like <math>{}^{\backprime\backprime} \mathsf{W}U^\bullet {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \mathsf{W}X^\bullet {}^{\prime\prime}\!</math>) that <math>\mathsf{W}\!</math> assigns by default to its operative image universes. In most contexts we will usually have a prior acquaintance with these universes under other names and it is necessary only that we can tell from the information associated with an operator <math>\mathsf{W}\!</math> what universes they are.<br />
<br />
In Figure&nbsp;31 the maps <math>F\!</math> and <math>\mathsf{W}F\!</math> are displayed horizontally, the way one normally orients functional arrows in a written text, and <math>\mathsf{W}\!</math> rolls the map <math>F\!</math> downward into the images that are associated with <math>\mathsf{W}F.\!</math> In Figure&nbsp;32 the same information is redrawn so that the maps <math>F\!</math> and <math>\mathsf{W}F\!</math> flow down the page, and <math>\mathsf{W}\!</math> unfurls the map <math>F\!</math> rightward into domains that are the eminent purview of <math>\mathsf{W}F.\!</math><br />
<br />
{| align="center" border="0" cellpadding="20"<br />
|<br />
<pre><br />
o---------------------------------------o<br />
| |<br />
| |<br />
| U% !W! !W!U% |<br />
| o------------------>o |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| F | | !W!F |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| v v |<br />
| o------------------>o |<br />
| X% !W! !W!X% |<br />
| |<br />
| |<br />
o---------------------------------------o<br />
Figure 32. Operator Diagram (2)<br />
</pre><br />
|}<br />
<br />
The latter arrangement, as exhibited in Figure&nbsp;32, is more congruent with the thinking about operators that we shall do in the rest of this discussion, since all logical transformations from here on out will be pictured vertically, after the fashion of Figure&nbsp;30.<br />
<br />
====Differential Analysis of Propositions and Transformations====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" | The resultant metaphysical problem now is this: ''Does the man go round the squirrel or not?''<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 43]<br />
|}<br />
<br />
The approach to the differential analysis of logical propositions and transformations of discourse to be pursued here is carried out in terms of particular operators <math>\mathsf{W}\!</math> that act on propositions <math>F\!</math> or on transformations <math>F\!</math> to yield the corresponding operator maps <math>\mathsf{W}F.\!</math> The operator results then become the subject of a series of further stages of analysis, which take them apart into their propositional components, rendering them as a set of purely logical constituents. After this is done, all the parts are then re-integrated to reconstruct the original object in the light of a more complete understanding, at least in ways that enable one to appreciate certain aspects of it with fresh insight.<br />
<br />
* '''Remark on Strategy.''' At this point we run into a set of conceptual difficulties that force us to make a strategic choice in how we proceed. Part of the problem can be remedied by extending our discussion of tacit extensions to the transformational context. But the troubles that remain are much more obstinate and lead us to try two different types of solution. The approach that we develop first makes use of a variant type of extension operator, the ''trope extension'', to be defined below. This method is more conservative and requires less preparation, but has features which make it seem unsatisfactory in the long run. A more radical approach, but one with a better hope of long term success, makes use of the notion of ''contingency spaces''. These are an even more generous type of extended universe than the kind we currently use, but are defined subject to certain internal constraints. The extra work needed to set up this method forces us to put it off to a later stage. However, as a compromise, and to prepare the ground for the next pass, we call attention to the various conceptual difficulties as they arise along the way and try to give an honest estimate of how well our first approach deals with them.<br />
<br />
We now describe in general terms the particular operators that are instrumental to this form of analysis. The main series of operators all have the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathsf{W}<br />
& : &<br />
( U^\bullet \to X^\bullet )<br />
& \to &<br />
( \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet )<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
If we assume that the source universe <math>U^\bullet</math> and the target universe <math>X^\bullet</math> have finite dimensions <math>n\!</math> and <math>k,\!</math> respectively, then each operator <math>\mathsf{W}\!</math> is encompassed by the same abstract type:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathsf{W}<br />
& : &<br />
( [\mathbb{B}^n] \to [\mathbb{B}^k] )<br />
& \to &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k \times \mathbb{D}^k] )<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Since the range features of the operator result <math>\mathsf{W}F : [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k \times \mathbb{D}^k]</math> can be sorted by their ordinary versus differential qualities and the component maps can be examined independently, the complete operator <math>\mathsf{W}\!</math> can be separated accordingly into two components, in the form <math>\mathsf{W} = (\boldsymbol\varepsilon, \mathrm{W}).\!</math> Given a fixed context of source and target universes, <math>\boldsymbol\varepsilon\!</math> is always the same type of operator, a multiple component version of the tacit extension operators that were described earlier. In this context <math>\boldsymbol\varepsilon\!</math> has the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{array}{lccccc}<br />
\text{Concrete type}<br />
& \boldsymbol\varepsilon<br />
& : &<br />
( U^\bullet \to X^\bullet )<br />
& \to &<br />
( \mathrm{E}U^\bullet \to X^\bullet )<br />
\\[10pt]<br />
\text{Abstract type}<br />
& \boldsymbol\varepsilon<br />
& : &<br />
( [\mathbb{B}^n] \to [\mathbb{B}^k] )<br />
& \to &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k] )<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
On the other hand, the operator <math>\mathrm{W}\!</math> is specific to each <math>\mathsf{W}.\!</math> In this context <math>\mathrm{W}\!</math> always has the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{array}{lccccc}<br />
\text{Concrete type}<br />
& W<br />
& : &<br />
( U^\bullet \to X^\bullet )<br />
& \to &<br />
( \mathrm{E}U^\bullet \to \mathrm{d}X^\bullet )<br />
\\[10pt]<br />
\text{Abstract type}<br />
& W<br />
& : &<br />
( [\mathbb{B}^n] \to [\mathbb{B}^k] )<br />
& \to &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{D}^k] )<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
In the types just assigned to <math>\boldsymbol\varepsilon\!</math> and <math>\mathrm{W}\!</math> and by implication to their results <math>\boldsymbol\varepsilon F\!</math> and <math>\mathrm{W}F,\!</math> we have listed the most restrictive ranges defined for them rather than the more expansive target spaces that subsume these ranges. When there is need to recognize both, we may use type indications like the following:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\boldsymbol\varepsilon F<br />
& : &<br />
( \mathrm{E}U^\bullet \to X^\bullet \subseteq \mathrm{E}X^\bullet )<br />
& \cong &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k] \subseteq [\mathbb{B}^k \times \mathbb{D}^k] )<br />
\\[10pt]<br />
WF<br />
& : &<br />
( \mathrm{E}U^\bullet \to \mathrm{d}X^\bullet \subseteq \mathrm{E}X^\bullet )<br />
& \cong &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{D}^k] \subseteq [\mathbb{B}^k \times \mathbb{D}^k] )<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Hopefully, though, a general appreciation of these subsumptions will prevent us from having to make such declarations more often than absolutely necessary.<br />
<br />
In giving names to these operators we try to preserve as much of the traditional nomenclature and as many of the classical associations as possible. The chief difficulty in doing this is occasioned by the distinction between the &ldquo;sans&nbsp;serif&rdquo; operators <math>\mathsf{W}\!</math> and their &ldquo;serified&rdquo; components <math>\mathrm{W},\!</math> which forces us to find two distinct but parallel sets of terminology. Here is a plan to that purpose. First, the component operators <math>\mathrm{W}\!</math> are named by analogy with the corresponding operators in the classical difference calculus. Next, the complete operators <math>\mathsf{W} = (\boldsymbol\varepsilon, \mathrm{W})</math> are assigned titles according to their roles in a geometric or trigonometric allegory, if only to ensure that the tangent functor, that belongs to this family and whose exposition we are still working toward, comes out fit with its customary name. Finally, the operator results <math>\mathsf{W}F\!</math> and <math>\mathrm{W}F\!</math> can be fixed in our frame of reference by tethering the operative adjective for <math>\mathsf{W}\!</math> or <math>\mathrm{W}\!</math> to the anchoring epithet &ldquo;map&rdquo;, in conformity with an already standard practice.<br />
<br />
=====The Secant Operator : '''E'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Mr. Peirce, after pointing out that our beliefs are really rules for action, said that, to develop a thought's meaning, we need only determine what conduct it is fitted to produce: that conduct is for us its sole significance.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 46]<br />
|}<br />
<br />
Figures&nbsp;33-i and 33-ii depict two stages in the form of analysis that will be applied to transformations throughout the remainder of this study. From now on our interest is staked on an operator denoted <math>{}^{\backprime\backprime} \mathsf{E} {}^{\prime\prime},\!</math> which receives the principal investment of analytic attention, and on the constituent parts of <math>\mathsf{E},\!</math> which derive their shares of significance as developed by the analysis. In the sequel, we refer to <math>\mathsf{E}\!</math> as the ''secant operator'', taking it for granted that a context has been chosen that defines its type. The secant operator has the component description <math>\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}),\!</math> and its active ingredient <math>\mathrm{E}\!</math> is known as the ''enlargement operator''. (Here, we name <math>\mathrm{E}\!</math> after the literal ancestor of the shift operator in the calculus of finite differences, defined so that <math>\mathrm{E}f(x) = f(x+1)\!</math> for any suitable function <math>f,\!</math> though of course the logical analogue that we take up here must have a rather different definition.)<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
U% $E$ $E$U% $E$U% $E$U%<br />
o------------------>o============o============o<br />
| | | |<br />
| | | |<br />
| | | |<br />
| | | |<br />
F | | $E$F = | $d$^0.F + | $r$^0.F<br />
| | | |<br />
| | | |<br />
| | | |<br />
v v v v<br />
o------------------>o============o============o<br />
X% $E$ $E$X% $E$X% $E$X%<br />
<br />
Figure 33-i. Analytic Diagram (1)<br />
</pre><br />
|}<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
U% $E$ $E$U% $E$U% $E$U% $E$U%<br />
o------------------>o============o============o============o<br />
| | | | |<br />
| | | | |<br />
| | | | |<br />
| | | | |<br />
F | | $E$F = | $d$^0.F + | $d$^1.F + | $r$^1.F<br />
| | | | |<br />
| | | | |<br />
| | | | |<br />
v v v v v<br />
o------------------>o============o============o============o<br />
X% $E$ $E$X% $E$X% $E$X% $E$X%<br />
<br />
Figure 33-ii. Analytic Diagram (2)<br />
</pre><br />
|}<br />
<br />
In its action on universes <math>\mathsf{E}\!</math> yields the same result as <math>\mathrm{E},\!</math> a fact that can be expressed in equational form by writing <math>\mathsf{E}U^\bullet = \mathrm{E}U^\bullet\!</math> for any universe <math>U^\bullet.\!</math> Notice that the extended universes across the top and bottom of the diagram are indicated to be strictly identical, rather than requiring a corresponding decomposition for them. In a certain sense, the functional parts of <math>\mathsf{E}F\!</math> are partitioned into separate contexts that have to be re-integrated again, but the best image to use is that of making transparent copies of each universe and then overlapping their functional contents once more at the conclusion of the analysis, as suggested by the graphic conventions that are used at the top of Figure&nbsp;30.<br />
<br />
Acting on a transformation <math>F\!</math> from universe <math>U^\bullet\!</math> to universe <math>X^\bullet,\!</math> the operator <math>\mathsf{E}\!</math> determines a transformation <math>\mathsf{E}F\!</math> from <math>\mathsf{E}U^\bullet\!</math> to <math>\mathsf{E}X^\bullet.\!</math> The map <math>\mathsf{E}F\!</math> forms the main body of evidence to be investigated in performing a differential analysis of <math>F.\!</math> Because we shall frequently be focusing on small pieces of this map for considerable lengths of time, and consequently lose sight of the &ldquo;big picture&rdquo;, it is critically important to emphasize that the map <math>\mathsf{E}F\!</math> is a transformation that determines a relation from one extended universe into another. This means that we should not be satisfied with our understanding of a transformation <math>F\!</math> until we can lay out the full &ldquo;parts diagram&rdquo; of <math>\mathsf{E}F\!</math> along the lines of the generic frame in Figure&nbsp;30.<br />
<br />
Working within the confines of propositional calculus, it is possible to give an elementary definition of <math>\mathsf{E}F\!</math> by means of a system of propositional equations, as we now describe.<br />
<br />
Given a transformation<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>F = (F_1, \ldots, F_k) : \mathbb{B}^n \to \mathbb{B}^k\!</math><br />
|}<br />
<br />
of concrete type<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>F : [u_1, \ldots, u_n] \to [x_1, \ldots, x_k],\!</math><br />
|}<br />
<br />
the transformation<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\mathsf{E}F = (F_1, \ldots, F_k, \mathrm{E}F_1, \ldots, \mathrm{E}F_k) : \mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B}^k \times \mathbb{D}^k\!</math><br />
|}<br />
<br />
of concrete type<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\mathsf{E}F : [u_1, \dots, u_n, \mathrm{d}u_1, \dots, \mathrm{d}u_n] \to [x_1, \ldots, x_k, \mathrm{d}x_1, \ldots, \mathrm{d}x_k]\!</math><br />
|}<br />
<br />
is defined by means of the following system of logical equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\\[16pt]<br />
\mathrm{d}x_1<br />
& = & \mathrm{E}F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1 + \mathrm{d}u_1, \ldots, u_n + \mathrm{d}u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
\mathrm{d}x_k<br />
& = & \mathrm{E}F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1 + \mathrm{d}u_1, \ldots, u_n + \mathrm{d}u_n)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
It is important to note that this system of equations can be read as a conjunction of equational propositions, in effect, as a single proposition in the universe of discourse generated by all the named variables. Specifically, this is the universe of discourse over <math>2(n+k)\!</math> variables denoted by:<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
\mathrm{E}[\mathcal{U} \cup \mathcal{X}]<br />
& = &<br />
[u_1, \ldots, u_n, ~ x_1, \ldots, x_k, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n, ~ \mathrm{d}x_1, \ldots, \mathrm{d}x_k].<br />
\end{matrix}</math><br />
|}<br />
<br />
In this light, it should be clear that the system of equations defining <math>\mathsf{E}F\!</math> embodies, in a higher rank and differentially extended version, an analogy with the process of thematization that we treated earlier for propositions of type <math>F : \mathbb{B}^n \to \mathbb{B}.\!</math><br />
<br />
The entire collection of constraints that is represented in the above system of equations may be abbreviated by writing <math>\mathsf{E}F = (\boldsymbol\varepsilon F, \mathrm{E}F),\!</math> for any map <math>F.\!</math> This is tantamount to regarding <math>\mathsf{E}\!</math> as a complex operator, <math>\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}),\!</math> with a form of application that distributes each component of the operator to work on each component of the operand, as follows:<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
\mathsf{E}F<br />
& = &<br />
(\boldsymbol\varepsilon, \mathrm{E})F<br />
& = &<br />
(\boldsymbol\varepsilon F, \mathrm{E}F)<br />
& = &<br />
(\boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k, ~ \mathrm{E}F_1, \ldots, \mathrm{E}F_k).<br />
\end{matrix}</math><br />
|}<br />
<br />
Quite a lot of &ldquo;thematic infrastructure&rdquo; or interpretive information is being swept under the rug in the use of such abbreviations. When confusion arises about the meaning of such constructions, one always has recourse to the defining system of equations, in its totality a purely propositional expression. This means that the parenthesized argument lists, that were used in this context to build an image of multi-component transformations, should not be expected to determine a well-defined product in themselves but only to serve as reminders of the prior thematic decisions (choices of variable names, etc.) that have to be made in order to determine one. Accordingly, the argument list notation can be regarded as a kind of ''thematic frame'', an interpretive storage device that preserves the proper associations of concrete logical features between the extended universes at the source and target of <math>\mathsf{E}F.\!</math><br />
<br />
The generic notations <math>\mathsf{d}^0\!F, \mathsf{d}^1\!F, \ldots, \mathsf{d}^m\!F\!</math> in Figure&nbsp;33 refer to the increasing orders of differentials that are extracted in the course of analyzing <math>F.\!</math> When the analysis is halted at a partial stage of development, notations like <math>\mathsf{r}^0\!F, \mathsf{r}^1\!F, \ldots, \mathsf{r}^m\!F\!</math> may be used to summarize the contributions to <math>\mathsf{E}F\!</math> that remain to be analyzed. The Figure illustrates a convention that makes <math>\mathsf{r}^m\!F,\!</math> in effect, the sum of all differentials of order strictly greater than <math>m.\!</math><br />
<br />
We next discuss the operators that figure into this form of analysis, describing their effects on transformations. In simplified or specialized contexts these operators tend to take on a variety of different names and notations, some of whose number we introduce along the way.<br />
<br />
=====The Radius Operator : '''e'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
And the tangible fact at the root of all our thought-distinctions, however subtle, is that there is no one of them so fine as to consist in anything but a possible difference of practice.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 46]<br />
|}<br />
<br />
The operator identified as <math>\mathrm{d}^0\!</math> in the analytic diagram (Figure&nbsp;33) has the sole purpose of creating a proxy for <math>F\!</math> in the appropriately extended context. Construed in terms of its broadest components, <math>\mathrm{d}^0\!</math> is equivalent to the doubly tacit extension operator <math>(\boldsymbol\varepsilon, \boldsymbol\varepsilon),\!</math> in recognition of which let us redub it as <math>{}^{\backprime\backprime} \mathsf{e} {}^{\prime\prime}.\!</math> Pursuing a geometric analogy, we may refer to <math>\mathsf{e} =(\boldsymbol\varepsilon, \boldsymbol\varepsilon) = \mathrm{d}^0\!</math> as the ''radius operator''. The operation intended by all of these forms is defined by the following equation:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathsf{e}F<br />
& = &<br />
(\boldsymbol\varepsilon, \boldsymbol\varepsilon)F<br />
\\[4pt]<br />
& = &<br />
(\boldsymbol\varepsilon F, ~ \boldsymbol\varepsilon F)<br />
\\[4pt]<br />
& = &<br />
(\boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k, ~ \boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k).<br />
\end{array}</math><br />
|}<br />
<br />
which is tantamount to the system of equations below.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\\[16pt]<br />
\mathrm{d}x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
\mathrm{d}x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
=====The Phantom of the Operators : '''&eta;'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
I was wondering what the reason could be, when I myself raised my head and everything within me seemed drawn towards the Unseen, ''which was playing the most perfect music''!<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 81]<br />
|}<br />
<br />
We now describe an operator whose persistent but elusive action behind the scenes, whose slightly twisted and ambivalent character, and whose fugitive disposition, caught somewhere in flight between the arrantly negative and the positive but errant intent, has cost us some painstaking trouble to detect. In the end we shall place it among the other extensions and projections, as a shade among shadows, of muted tones and motley hue, that adumbrates its own thematic frame and paradoxically lights the way toward a whole new spectrum of values.<br />
<br />
Given a transformation <math>F : [u_1, \ldots, u_n] \to [x_1, \dots, x_k],\!</math> we often have call to consider a family of related transformations, all having the form:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>F^\dagger : [u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n] \to [\mathrm{d}x_1, \dots, \mathrm{d}x_k].\!</math><br />
|}<br />
<br />
The operator <math>\eta</math> is introduced to deal with the simplest one of these maps:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\eta F : [u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n] \to [\mathrm{d}x_1, \ldots \mathrm{d}x_k],\!</math><br />
|}<br />
<br />
which is defined by the following equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
\mathrm{d}x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
In effect, the operator <math>\eta\!</math> is nothing but the stand-alone version of a procedure that is otherwise invoked subordinate to the work of the radius operator <math>\mathsf{e}.\!</math> Operating independently, <math>\eta\!</math> achieves precisely the same results that the second <math>\boldsymbol\varepsilon\!</math> in <math>(\boldsymbol\varepsilon, \boldsymbol\varepsilon)\!</math> accomplishes by working within the context of its ordered pair thematic frame. From this point on, because the use of <math>\boldsymbol\varepsilon\!</math> and <math>\eta\!</math> in this setting combines the aims of both the tacit and the thematic extensions, and because <math>\eta\!</math> reflects in regard to <math>\boldsymbol\varepsilon\!</math> little more than the application of a differential twist, a mere turn of phrase, we refer to <math>\eta\!</math> as the ''trope extension'' operator.<br />
<br />
=====The Chord Operator : '''D'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
What difference would it practically make to any one if this notion rather than that notion were true? If no practical difference whatever can be traced, then the alternatives mean practically the same thing, and all dispute is idle.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 45]<br />
|}<br />
<br />
Next we discuss an operator that is always immanent in this form of analysis, and remains implicitly present in the entire proceeding. It may appear once as a record: a relic or revenant that reprises the reminders of an earlier stage of development. Or it may appear always as a resource: a reserve or redoubt that caches in advance an echo of what remains to be played out, cleared up, and requited in full at a future stage. And all of this remains true whether or not we recall the key at any time, and whether or not the subtending theme is recited explicitly at any stage of play.<br />
<br />
This is the operator that is referred to as <math>\mathsf{r}^0\!</math> in the initial stage of analysis (Figure&nbsp;33-i) and that is expanded as <math>\mathsf{d}^1 + \mathsf{r}^1\!</math> in the subsequent step (Figure&nbsp;33-ii). In congruence, but not quite harmony with our allusions of analogy that are not quite geometry, we call this the ''chord operator'' and denote it <math>\mathsf{D}.\!</math> In the more casual terms that are here introduced, <math>\mathsf{D}</math> is defined as the remainder of <math>\mathsf{E}\!</math> and <math>\mathsf{e}\!</math> and it assigns a due measure to each undertone of accord or discord that is struck between the note of enterprise <math>\mathsf{E}\!</math> and the bar of exigency <math>\mathsf{e}.\!</math><br />
<br />
The tension between these counterposed notions, in balance transient but regular in stridence, may be refracted along familiar lines, though never by any such fraction resolved. In this style we write <math>\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}),\!</math> calling <math>\mathrm{D}\!</math> the ''difference operator'' and noting that it plays a role in this realm of mutable and diverse discourse that is analogous to the part taken by the discrete difference operator in the ordinary difference calculus. Finally, we should note that the chord <math>\mathsf{D}\!</math> is not one that need be lost at any stage of development. At the <math>m^\text{th}\!</math> stage of play it can always be reconstituted in the following form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathsf{D}<br />
& = & \mathsf{E} - \mathsf{e}<br />
\\[6pt]<br />
& = & \mathsf{r}^0<br />
\\[6pt]<br />
& = & \mathsf{d}^1 + \mathsf{r}^1<br />
\\[6pt]<br />
& = & \mathsf{d}^1 + \ldots + \mathsf{d}^m + \mathsf{r}^m<br />
\\[6pt]<br />
& = & \displaystyle \sum_{i=1}^m \mathsf{d}^i + \mathsf{r}^m<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====The Tangent Operator : '''T'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
They take part in scenes of whose significance they have no inkling. They are merely tangent to curves of history the beginnings and ends and forms of which pass wholly beyond their ken. So we are tangent to the wider life of things.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 300]<br />
|}<br />
<br />
The operator tagged as <math>\mathsf{d}^1\!</math> in the analytic diagram (Figure&nbsp;33) is called the ''tangent operator'' and is usually denoted in this text as <math>\mathsf{d}\!</math> or <math>\mathsf{T}.\!</math> Because it has the properties required to qualify as a functor, namely, preserving the identity element of the composition operation and the articulated form of every composition of transformations, it also earns the title of a ''tangent functor''. According to the custom adopted here, we dissect it as <math>\mathsf{T} = \mathsf{d} = (\boldsymbol\varepsilon, \mathrm{d}),\!</math> where <math>\mathrm{d}\!</math> is the operator that yields the first order differential <math>\mathrm{d}F\!</math> when applied to a transformation <math>F,\!</math> and whose name is legion.<br />
<br />
Figure&nbsp;34 illustrates a stage of analysis where we ignore everything but the tangent functor <math>\mathsf{T}\!</math> and attend to it chiefly as it bears on the first order differential <math>\mathrm{d}F\!</math> in the analytic expansion of <math>F.\!</math> In this situation we often refer to the extended universes <math>\mathrm{E}U^\bullet\!</math> and <math>\mathrm{E}X^\bullet\!</math> under the equivalent designations <math>\mathsf{T}U^\bullet\!</math> and <math>\mathsf{T}X^\bullet,\!</math> respectively. The purpose of the tangent functor <math>\mathsf{T}\!</math> is to extract the tangent map <math>\mathsf{T}F\!</math> at each point of <math>U^\bullet,\!</math> and the tangent map <math>\mathsf{T}F = (\boldsymbol\varepsilon, \mathrm{d})F\!</math> tells us not only what the transformation <math>F\!</math> is doing at each point of the universe <math>U^\bullet\!</math> but also what <math>F\!</math> is doing to states in the neighborhood of that point, approximately, linearly, and relatively speaking.<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
U% $T$ $T$U% $T$U%<br />
o------------------>o============o<br />
| | |<br />
| | |<br />
| | |<br />
| | |<br />
F | | $T$F = | <!e!, d> F<br />
| | |<br />
| | |<br />
| | |<br />
v v v<br />
o------------------>o============o<br />
X% $T$ $T$X% $T$X%<br />
<br />
Figure 34. Tangent Functor Diagram<br />
</pre><br />
|}<br />
<br />
* '''NB.''' There is one aspect of the preceding construction that remains especially problematic. Why did we define the operators <math>\mathrm{W}\!</math> in <math>\{ \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}\!</math> so that the ranges of their resulting maps all fall within the realms of differential quality, even fabricating a variant of the tacit extension operator to have that character? Clearly, not all of the operator maps <math>\mathrm{W}F\!</math> have equally good reasons for placing their values in differential stocks. The reason for it appears to be that, without doing this, we cannot justify the comparison and combination of their functional values in the various analytic steps. By default, only those values in the same functional component can be brought into algebraic modes of interaction. Up till now the only mechanism provided for their broader association has been a purely logical one, their common placement in a target universe of discourse, but the task of converting this logical circumstance into algebraic forms of application has not yet been taken up.<br />
<br />
===Transformations of Type '''B'''<sup>2</sup> &rarr; '''B'''<sup>1</sup>===<br />
<br />
To study the effects of these analytic operators in the simplest possible setting, let us revert to a still more primitive case. Consider the singular proposition <math>J(u, v)= u\!\cdot\!v,\!</math> regarded either as the functional product of the maps <math>u\!</math> and <math>v\!</math> or as the logical conjunction of the features <math>u\!</math> and <math>v,\!</math> a map whose fiber of truth <math>J^{-1}(1)\!</math> picks out the single cell of that logical description in the universe of discourse <math>U^\bullet.\!</math> Thus <math>J,\!</math> or <math>u\!\cdot\!v,\!</math> may be treated as another name for the point whose coordinates are <math>(1, 1)\!</math> in <math>U^\bullet.\!</math><br />
<br />
====Analytic Expansion of Conjunction====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
<p>In her sufferings she read a great deal and discovered that she had lost something, the possession of which she had previously not been much aware of: a&nbsp;soul.</p><br />
<br />
<p>What is that? It is easily defined negatively: it is simply what curls up and hides when there is any mention of algebraic series.</p><br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 118]<br />
|}<br />
<br />
Figure&nbsp;35 pictures the form of conjunction <math>J : \mathbb{B}^2 \to \mathbb{B}\!</math> as a transformation from the <math>2\!</math>-dimensional universe <math>[u, v]\!</math> to the <math>1\!</math>-dimensional universe <math>[x].\!</math> This is a subtle but significant change of viewpoint on the proposition, attaching an arbitrary but concrete quality to its functional value. Using the language introduced earlier, we can express this change by saying that the proposition <math>J : \langle u, v \rangle \to \mathbb{B}\!</math> is being recast into the thematized role of a transformation <math>J : [u, v] \to [x],\!</math> where the new variable <math>x\!</math> takes the part of a thematic variable <math>\check{J}.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 35 -- A Conjunction Viewed as a Transformation.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 35.} ~~ \text{Conjunction as Transformation}\!</math><br />
|}<br />
<br />
=====Tacit Extension of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
I teach straying from me, yet who can stray from me?<br><br />
I follow you whoever you are from the present hour;<br><br />
My words itch at your ears till you understand them.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 83]<br />
|}<br />
<br />
Earlier we defined the tacit extension operators <math>\boldsymbol\varepsilon : X^\bullet \to Y^\bullet\!</math> as maps embedding each proposition of a given universe <math>X^\bullet~\!</math> in a more generously given universe <math>Y^\bullet \supset X^\bullet.\!</math> Of immediate interest are the tacit extensions <math>\boldsymbol\varepsilon : U^\bullet \to \mathrm{E}U^\bullet,\!</math> that locate each proposition of <math>U^\bullet\!</math> in the enlarged context of <math>\mathrm{E}U^\bullet.\!</math> In its application to the propositional conjunction <math>J = u\!\cdot\!v</math> in <math>[u, v],\!</math> the tacit extension operator <math>\boldsymbol\varepsilon\!</math> yields the proposition <math>\boldsymbol\varepsilon J\!</math> in <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v].\!</math> The extended proposition <math>\boldsymbol\varepsilon J\!</math> may be computed according to the scheme in Table&nbsp;36, in effect doing nothing more that conjoining a tautology of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> to <math>J\!</math> in <math>U^\bullet.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 36.} ~~ \text{Computation of}~ \boldsymbol\varepsilon J\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\boldsymbol\varepsilon J & = & J {}_{^\langle} u, v {}_{^\rangle}<br />
\\[4pt]<br />
& = & u \cdot v<br />
\\[4pt]<br />
& = & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{ } \mathrm{d}v \texttt{ }<br />
& + & u \cdot v \cdot \texttt{ } \mathrm{d}u \texttt{ } \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \cdot v \cdot \texttt{ } \mathrm{d}u \texttt{ } \cdot \texttt{ } \mathrm{d}v \texttt{ }<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{array}{*{4}{l}}<br />
\boldsymbol\varepsilon J<br />
& = && u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & u \cdot v \cdot \texttt{~} \mathrm{d}u \texttt{~} \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & u \cdot v \cdot \texttt{~} \mathrm{d}u \texttt{~} \cdot \texttt{~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
The lower portion of the Table contains the dispositional features of <math>\boldsymbol\varepsilon J\!</math> arranged in such a way that the variety of ordinary features spreads across the rows and the variety of differential features runs through the columns. This organization serves to facilitate pattern matching in the remainder of our computations. Again, the tacit extension is usually so trivial a concern that we do not always bother to make an explicit note of it, taking it for granted that any function <math>F\!</math> being employed in a differential context is equivalent to <math>\boldsymbol\varepsilon F\!</math> for a suitable <math>\boldsymbol\varepsilon.\!</math><br />
<br />
Figures&nbsp;37-a through 37-d present several pictures of the proposition <math>J\!</math> and its tacit extension <math>\boldsymbol\varepsilon J.\!</math> Notice in these Figures how <math>\boldsymbol\varepsilon J\!</math> in <math>\mathrm{E}U^\bullet\!</math> visibly extends <math>J\!</math> in <math>U^\bullet\!</math> by annexing to the indicated cells of <math>J\!</math> all the arcs that exit from or flow out of them. In effect, this extension attaches to these cells all the dispositions that spring from them, in other words, it attributes to these cells all the conceivable changes that are their issue.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-a -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-a.} ~~ \text{Tacit Extension of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-b -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-b.} ~~ \text{Tacit Extension of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-c -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-c.} ~~ \text{Tacit Extension of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-d -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-d.} ~~ \text{Tacit Extension of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
The computational scheme shown in Table&nbsp;36 treated <math>J\!</math> as a proposition in <math>U^\bullet\!</math> and formed <math>\boldsymbol\varepsilon J\!</math> as a proposition in <math>\mathrm{E}U^\bullet.\!</math> When <math>J\!</math> is regarded as a mapping <math>J : U^\bullet \to X^\bullet\!</math> then <math>\boldsymbol\varepsilon J\!</math> must be obtained as a mapping <math>\boldsymbol\varepsilon J : \mathrm{E}U^\bullet \to X^\bullet.\!</math> By default, the tacit extension of the map <math>J : [u, v] \to [x]\!</math> is naturally taken to be a particular map,<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\boldsymbol\varepsilon J : [u, v, \mathrm{d}u, \mathrm{d}v] \to [x] \subseteq [x, \mathrm{d}x],\!</math><br />
|}<br />
<br />
namely, the one that looks like <math>J\!</math> when painted in the frame of the extended source universe and that takes the same thematic variable in the extended target universe as the one that <math>J\!</math> already takes.<br />
<br />
But the choice of a particular thematic variable, for example <math>x\!</math> for <math>\check{J},\!</math> is a shade more arbitrary than the choice of original variable names <math>\{ u, v \},\!</math> so the map we are calling the ''trope extension'',<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\eta J : [u, v, \mathrm{d}u, \mathrm{d}v] \to [\mathrm{d}x] \subseteq [x, \mathrm{d}x],\!</math><br />
|}<br />
<br />
since it looks just the same as <math>\boldsymbol\varepsilon J\!</math> in the way its fibers paint the source domain, belongs just as fully to the family of tacit extensions, generically considered.<br />
<br />
These considerations have the practical consequence that all of our computations and illustrations of <math>\boldsymbol\varepsilon J\!</math> perform the double duty of capturing <math>\eta J\!</math> as well. In other words, we are saved the work of carrying out calculations and drawing figures for the trope extension <math>\eta J,\!</math> because it would be identical to the work already done for <math>\boldsymbol\varepsilon J.\!</math> Since the computations given for <math>\boldsymbol\varepsilon J\!</math> are expressed solely in terms of the variables <math>\{ u, v, \mathrm{d}u, \mathrm{d}v \},\!</math> they work equally well for finding <math>\eta J.\!</math> Further, since each of the above Figures shows only how the level sets of <math>\boldsymbol\varepsilon J\!</math> partition the extended source universe <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v],\!</math> all of them serve equally well as portraits of <math>\eta J.\!</math><br />
<br />
=====Enlargement Map of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
No one could have established the existence of any details that might not just as well have existed in earlier times too; but all the relations between things had shifted slightly. Ideas that had once been of lean account grew fat.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 62]<br />
|}<br />
<br />
The enlargement map <math>\mathrm{E}J\!</math> is computed from the proposition <math>J\!</math> by making a particular class of formal substitutions for its variables, in this case <math>u + \mathrm{d}u\!</math> for <math>u\!</math> and <math>v + \mathrm{d}v\!</math> for <math>v,\!</math> and afterwards expanding the result in whatever way is found convenient.<br />
<br />
Table&nbsp;38 shows a typical scheme of computation, following a systematic method of exploiting boolean expansions over selected variables and ultimately developing <math>\mathrm{E}J\!</math> over the cells of <math>[u, v].\!</math> The critical step of this procedure uses the facts that <math>\texttt{(} 0, x \texttt{)} = 0 + x = x\!</math> and <math>\texttt{(} 1, x \texttt{)} = 1 + x = \texttt{(} x \texttt{)}\!</math> for any boolean variable <math>x.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 38.} ~~ \text{Computation of}~ \mathrm{E}J ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{E}J & = & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}<br />
\\[4pt]<br />
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
& = & \texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot J_{(1 + \mathrm{d}u, 1 + \mathrm{d}v)}<br />
& + & \texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot J_{(1 + \mathrm{d}u, \mathrm{d}v)}<br />
& + & \texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot J_{(\mathrm{d}u, 1 + \mathrm{d}v)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot J_{(\mathrm{d}u, \mathrm{d}v)}<br />
\\[4pt]<br />
& = &<br />
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot J_{(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})}<br />
& + &<br />
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot J_{(\texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v)}<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot J_{(\mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)})}<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot J_{(\mathrm{d}u, \mathrm{d}v)}<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{E}J<br />
& = &<br />
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&& + &<br />
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{ } \mathrm{d}v \texttt{ }<br />
\\[4pt]<br />
&&&&& + &<br />
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot \texttt{ } \mathrm{d}u \texttt{ } \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&&&&&& + &<br />
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \texttt{ } \mathrm{d}u \texttt{ }~\texttt{ } \mathrm{d}v \texttt{ }<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;39 exhibits another method that happens to work quickly in this particular case, using distributive laws to multiply things out in an algebraic manner, arranging the notations of feature and fluxion according to a scale of simple character and degree. Proceeding this way leads through an intermediate step which, in chiming the changes of ordinary calculus, should take on a familiar ring. Consequential properties of exclusive disjunction then carry us on to the concluding line.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 39.} ~~ \text{Computation of}~ \mathrm{E}J ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{c}}<br />
\mathrm{E}J<br />
& = & (u + \mathrm{d}u) \cdot (v + \mathrm{d}v)<br />
\\[6pt]<br />
& = & u \cdot v<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = &<br />
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{ } \mathrm{d}v \texttt{ }<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot \texttt{ } \mathrm{d}u \texttt{ } \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \texttt{ } \mathrm{d}u \texttt{ }~\texttt{ } \mathrm{d}v \texttt{ }<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;40-a through 40-d present several views of the enlarged proposition <math>\mathrm{E}J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-a -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-a.} ~~ \text{Enlargement of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-b -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-b.} ~~ \text{Enlargement of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-c -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-c.} ~~ \text{Enlargement of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-d -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-d.} ~~ \text{Enlargement of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
An intuitive reading of the proposition <math>\mathrm{E}J\!</math> becomes available at this point. Recall that propositions in the extended universe <math>\mathrm{E}U^\bullet\!</math> express the ''dispositions'' of a system and the constraints that are placed on them. In other words, a differential proposition in <math>\mathrm{E}U^\bullet\!</math> can be read as referring to various changes that a system might undergo in and from its various states. In particular, we can understand <math>\mathrm{E}J\!</math> as a statement that tells us what changes need to be made with regard to each state in the universe of discourse in order to reach the ''truth'' of <math>J,\!</math> that is, the region of the universe where <math>J\!</math> is true. This interpretation is visibly clear in the Figures above and appeals to the imagination in a satisfying way but it has the added benefit of giving fresh meaning to the original name of the shift operator <math>\mathrm{E}.\!</math> Namely, <math>\mathrm{E}J\!</math> can be read as a proposition that ''enlarges'' on the meaning of <math>J,\!</math> in the sense of explaining its practical bearings and clarifying what it means in terms of actions and effects &mdash; the available options for differential action and the consequential effects that result from each choice.<br />
<br />
Read this way, the enlargement <math>\mathrm{E}J\!</math> has strong ties to the normal use of <math>J,\!</math> no matter whether it is understood as a proposition or a function, namely, to act as a figurative device for indicating the models of <math>J,\!</math> in effect, pointing to the interpretive elements in its fiber of truth <math>J^{-1}(1).\!</math> It is this kind of &ldquo;use&rdquo; that is often contrasted with the &ldquo;mention&rdquo; of a proposition, and thereby hangs a tale.<br />
<br />
=====Digression : Reflection on Use and Mention=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Reflection is turning a topic over in various aspects and in various lights so that nothing significant about it shall be overlooked &mdash; almost as one might turn a stone over to see what its hidden side is like or what is covered by it.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 57]<br />
|}<br />
<br />
The contrast drawn in logic between the ''use'' and the ''mention'' of a proposition corresponds to the difference that we observe in functional terms between using <math>{}^{\backprime\backprime} J \, {}^{\prime\prime}\!</math> to indicate the region <math>J^{-1}(1)\!</math> and using <math>{}^{\backprime\backprime} J \, {}^{\prime\prime}\!</math> to indicate the function <math>J.\!</math> You may think that one of these uses ought to be proscribed, and logicians are quick to prescribe against their confusion. But there seems to be no likelihood in practice that their interactions can be avoided. If the name <math>{}^{\backprime\backprime} J \, {}^{\prime\prime}\!</math> is used as a sign of the function <math>J,\!</math> and if the function <math>J\!</math> has its use in signifying something else, as would constantly be the case when some future theory of signs has given a functional meaning to every sign whatsoever, then is not <math>J,\!</math> by transitivity a sign of the thing itself? There are, of course, two answers to this question. Not every act of signifying or referring need be transitive. Not every warrant or guarantee or certificate is automatically transferable, indeed, not many. Not every feature of a feature is a feature of the featuree. Otherwise, if a buffalo is white, and white is a color, then a buffalo would ''be'' a color.<br />
<br />
The logical or pragmatic distinction between use and mention is cogent and necessary, and so is the analogous functional distinction between determining a value and determining what determines that value, but so are the normal techniques that we use to make these distinctions apply flexibly in practice. The way that the hue and cry about use and mention is raised in logical discussions, you might be led to think that this single dimension of choices embraces the only kinds of use worth mentioning and the only kinds of mention worth using. It will constitute the expeditionary and taxonomic tasks of that future theory of signs to explore and to classify the many other constellations and dimensions of use and mention that are yet to be opened up by the generative potential of full-fledged sign relations.<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
The well-known capacity that thoughts have &mdash; as doctors have discovered &mdash; for dissolving and dispersing those hard lumps of deep, ingrowing, morbidly entangled conflict that arise out of gloomy regions of the self probably rests on nothing other than their social and worldly nature, which links the individual being with other people and things; but unfortunately what gives them their power of healing seems to be the same as what diminishes the quality of personal experience in them.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 130]<br />
|}<br />
<br />
=====Difference Map of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
&ldquo;It doesn't matter what one does,&rdquo; the Man Without Qualities said to himself, shrugging his shoulders. &ldquo;In a tangle of forces like this it doesn't make a scrap of difference.&rdquo; He turned away like a man who has learned renunciation, almost indeed like a sick man who shrinks from any intensity of contact. And then, striding through his adjacent dressing-room, he passed a punching-ball that hung there; he gave it a blow far swifter and harder than is usual in moods of resignation or states of weakness.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 8]<br />
|}<br />
<br />
With the tacit extension map <math>\boldsymbol\varepsilon J\!</math> and the enlargement map <math>\mathrm{E}J\!</math> well in place, the difference map <math>\mathrm{D}J\!</math> can be computed along the lines displayed in Table&nbsp;41, ending up with an expansion of <math>\mathrm{D}J\!</math> over the cells of <math>[u, v].\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 41.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & \mathrm{E}J<br />
& + & \boldsymbol\varepsilon J<br />
\\[6pt]<br />
& = & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}<br />
& + & J_{(u, v)}<br />
\\[6pt]<br />
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \cdot v<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = &<br />
u \cdot v \cdot \qquad 0<br />
\\[6pt]<br />
& + &<br />
u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v<br />
& + &<br />
u \cdot \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v<br />
\\[6pt]<br />
& + &<br />
u \cdot v \cdot \texttt{~} \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
&&& + &<br />
\texttt{(} u \texttt{)} \cdot v \cdot \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
& + &<br />
u \cdot v \cdot \texttt{~} \mathrm{d}u \;\cdot\; \mathrm{d}v \texttt{~}<br />
&&&&& + &<br />
\texttt{(} u \texttt{)} \cdot \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = &<br />
u \cdot v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
u \cdot \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v<br />
& + &<br />
\texttt{(} u \texttt{)} \cdot v \cdot \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)} \cdot \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Alternatively, the difference map <math>\mathrm{D}J\!</math> can be expanded over the cells of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> to arrive at the formulation shown in Table&nbsp;42. The same development would be obtained from the previous Table by collecting terms in an alternate manner, along the rows rather than the columns in the middle portion of the Table.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 42.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & \boldsymbol\varepsilon J<br />
& + & \mathrm{E}J<br />
\\[6pt]<br />
& = & J_{(u, v)}<br />
& + & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}<br />
\\[6pt]<br />
& = & u \cdot v<br />
& + & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
& = & 0<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}J<br />
& = & 0<br />
& + & u \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Even more simply, the same result is reached by matching up the propositional coefficients of <math>\boldsymbol\varepsilon J</math> and <math>\mathrm{E}J\!</math> along the cells of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> and adding the pairs under boolean addition, that is, &ldquo;mod 2&rdquo;, where 1&nbsp;+&nbsp;1&nbsp;=&nbsp;0, as shown in Table&nbsp;43.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 43.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 3)}\!</math><br />
|<br />
<math>\begin{array}{*{5}{l}}<br />
\mathrm{D}J & = & \boldsymbol\varepsilon J & + & \mathrm{E}J<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\boldsymbol\varepsilon J<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ u \,\cdot\, v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~ u \;\cdot\; v \;\cdot\; \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u ~ \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} ~ v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot\, \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & ~~ 0 ~~ \,\cdot\, ~ \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~~ u ~ \,\cdot\, ~~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ ~ v ~~ \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
The difference map <math>\mathrm{D}J\!</math> can also be given a ''dispositional'' interpretation. First, recall that <math>\boldsymbol\varepsilon J\!</math> exhibits the dispositions to change from anywhere in <math>J\!</math> to anywhere at all in the universe of discourse and <math>\mathrm{E}J\!</math> exhibits the dispositions to change from anywhere in the universe to anywhere in <math>J.\!</math> Next, observe that each of these classes of dispositions may be divided in accordance with the case of <math>J\!</math> versus <math>\texttt{(} J \texttt{)}\!</math> that applies to their points of departure and destination, as shown below. Then, since the dispositions corresponding to <math>\boldsymbol\varepsilon J</math> and <math>\mathrm{E}J\!</math> have in common the dispositions to preserve <math>J,\!</math> their symmetric difference <math>\texttt{(} \boldsymbol\varepsilon J, \mathrm{E}J \texttt{)}\!</math> is made up of all the remaining dispositions, which are in fact disposed to cross the boundary of <math>J\!</math> in one direction or the other. In other words, we may conclude that <math>\mathrm{D}J\!</math> expresses the collective disposition to make a definite change with respect to <math>J,\!</math> no matter what value it holds in the current state of affairs.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
\boldsymbol\varepsilon J<br />
& = & \{ \text{Dispositions from}~ J ~\text{to}~ J \}<br />
& + & \{ \text{Dispositions from}~ J ~\text{to}~ \texttt{(} J \texttt{)} \}<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = & \{ \text{Dispositions from}~ J ~\text{to}~ J \}<br />
& + & \{ \text{Dispositions from}~ \texttt{(} J \texttt{)} ~\text{to}~ J \}<br />
\\[6pt]<br />
\mathrm{D}J<br />
& = & \{ \text{Dispositions from}~ J ~\text{to}~ \texttt{(} J \texttt{)} \}<br />
& + & \{ \text{Dispositions from}~ \texttt{(} J \texttt{)} ~\text{to}~ J \}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;44-a through 44-d illustrate the difference proposition <math>\mathrm{D}J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-a -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-a.} ~~ \text{Difference Map of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-b -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-b.} ~~ \text{Difference Map of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-c -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-c.} ~~ \text{Difference Map of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-d -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-d.} ~~ \text{Difference Map of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
=====Differential of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
By deploying discourse throughout a calendar, and by giving a date to each of its elements, one does not obtain a definitive hierarchy of precessions and originalities; this hierarchy is never more than relative to the systems of discourse that it sets out to evaluate.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Archaeology of Knowledge'', [Fou, 143]<br />
|}<br />
<br />
Finally, at long last, the differential proposition <math>\mathrm{d}J\!</math> can be gleaned from the difference proposition <math>\mathrm{D}J\!</math> by ranging over the cells of <math>[u, v]\!</math> and picking out the linear proposition of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> that is &ldquo;closest&rdquo; to the portion of <math>\mathrm{D}J\!</math> that touches on each point. The idea of distance that would give this definition unequivocal sense has been referred to in cautionary quotes, the kind we use to distance ourselves from taking a final position. There are obvious notions of approximation that suggest themselves, but finding one that can be justified as ultimately correct is not as straightforward as it seems.<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
He had drifted into the very heart of the world. From him to the distant beloved was as far as to the next tree.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 144]<br />
|}<br />
<br />
Let us venture a guess as to where these developments might be heading. From the present vantage point it appears that the ultimate answer to the quandary of distances and the question of a fitting measure may be that, rather than having the constitution of an analytic series depend on our familiar notions of approach, proximity, and approximation, it will be found preferable, and perhaps unavoidable, to turn the tables and let the orders of approximation be defined in terms of our favored and operative notions of formal analysis. Only the aftermath of this conversion, if it does converge, could be hoped to prove whether this hortatory form of analysis and the cohort idea of an analytic form &mdash; the limitary concept of a self-corrective process and the coefficient concept of a completable product &mdash; are truly (in practical reality) the more inceptive and persistent of principles and really (for all practical purposes) the more effective and regulative of ideas.<br />
<br />
Awaiting that determination, I proceed with what seems like the obvious course, and compute <math>\mathrm{d}J\!</math> according to the pattern in Table&nbsp;45.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 45.} ~~ \text{Computation of}~ \mathrm{d}J\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}J<br />
& = &<br />
u\!\cdot\!v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
u \, \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \, \mathrm{d}v<br />
& + &<br />
\texttt{(} u \texttt{)} \, v \cdot \mathrm{d}u \, \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \!\cdot\! \mathrm{d}v \texttt{~}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}J<br />
& = &<br />
u\!\cdot\!v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + &<br />
u \, \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + &<br />
\texttt{(} u \texttt{)} \, v \cdot \mathrm{d}u<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;46-a through 46-d illustrate the proposition <math>{\mathrm{d}J},\!</math> rounded out in our usual array of prospects. This proposition of <math>\mathrm{E}U^\bullet\!</math> is what we refer to as the (first order) differential of <math>J,\!</math> and normally regard as ''the'' differential proposition corresponding to <math>J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-a -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-a.} ~~ \text{Differential of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-b -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-b.} ~~ \text{Differential of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-c -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-c.} ~~ \text{Differential of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-d -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-d.} ~~ \text{Differential of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
=====Remainder of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
<p>I bequeath myself to the dirt to grow from the grass I love,<br><br />
If you want me again look for me under your bootsoles.</p><br />
<br />
<p>You will hardly know who I am or what I mean,<br><br />
But I shall be good health to you nevertheless,<br><br />
And filter and fibre your blood.</p><br />
<br />
<p>Failing to fetch me at first keep encouraged,<br><br />
Missing me one place search another,<br><br />
I stop some where waiting for you</p><br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]<br />
|}<br />
<br />
<br><br />
<br />
Let us recapitulate the story so far. We have in effect been carrying out a decomposition of the enlarged proposition <math>\mathrm{E}J\!</math> in a series of stages. First, we considered the equation <math>\mathrm{E}J = \boldsymbol\varepsilon J + \mathrm{D}J,\!</math> which was involved in the definition of <math>\mathrm{D}J\!</math> as the difference <math>\mathrm{E}J - \boldsymbol\varepsilon J.\!</math> Next, we contemplated the equation <math>\mathrm{D}J = \mathrm{d}J + \mathrm{r}J,\!</math> which expresses <math>\mathrm{D}J\!</math> in terms of two components, the differential <math>\mathrm{d}J\!</math> that was just extracted and the residual component <math>\mathrm{r}J = \mathrm{D}J - \mathrm{d}J.~\!</math> This remaining proposition <math>\mathrm{r}J\!</math> can be computed as shown in Table&nbsp;47.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 47.} ~~ \text{Computation of}~ \mathrm{r}J\!</math><br />
|<br />
<math>\begin{array}{*{5}{l}}<br />
\mathrm{r}J & = & \mathrm{D}J & + & \mathrm{d}J<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}J<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{r}J ~<br />
& = & u \!\cdot\! v \cdot ~ \mathrm{d}u \cdot \mathrm{d}v ~ ~ ~ ~ ~<br />
& + & u \texttt{(} v \texttt{)} \cdot \, \mathrm{d}u \cdot \mathrm{d}v \,<br />
& + & \texttt{(} u \texttt{)} v \cdot \, \mathrm{d}u \cdot \mathrm{d}v \,<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \, \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
As it happens, the remainder <math>\mathrm{r}J\!</math> falls under the description of a second order differential <math>\mathrm{r}J = \mathrm{d}^2 J.\!</math> This means that the expansion of <math>\mathrm{E}J\!</math> in the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|<br />
<math>\begin{array}{*{7}{l}}<br />
\mathrm{E}J<br />
& = & \boldsymbol\varepsilon J<br />
& + & \mathrm{D}J<br />
\\[6pt]<br />
& = & \boldsymbol\varepsilon J<br />
& + & \mathrm{d}J<br />
& + & \mathrm{r}J<br />
\\[6pt]<br />
& = & \mathrm{d}^0 J<br />
& + & \mathrm{d}^1 J<br />
& + & \mathrm{d}^2 J<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
which is nothing other than the propositional analogue of a Taylor series, is a decomposition that terminates in a finite number of steps.<br />
<br />
Figures&nbsp;48-a through 48-d illustrate the proposition <math>\mathrm{r}J = \mathrm{d}^2 J,\!</math> which forms the remainder map of <math>J\!</math> and also, in this instance, the second order differential of <math>J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-a -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-a.} ~~ \text{Remainder of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-b -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-b.} ~~ \text{Remainder of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-c -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-c.} ~~ \text{Remainder of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-d -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-d.} ~~ \text{Remainder of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
=====Summary of Conjunction=====<br />
<br />
To establish a convenient reference point for further discussion, Table&nbsp;49 summarizes the operator actions that have been computed for the form of conjunction, as exemplified by the proposition <math>J.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 49.} ~~ \text{Computation Summary for}~ J~\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon J<br />
& = & u \!\cdot\! v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}J<br />
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}J<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{r}J<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Analytic Series : Coordinate Method====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
And if he is told that something ''is'' the way it is, then he thinks: Well, it could probably just as easily be some other way. So the sense of possibility might be defined outright as the capacity to think how everything could &ldquo;just as easily&rdquo; be, and to attach no more importance to what is than to what is not.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 12]<br />
|}<br />
<br />
Table&nbsp;50 exhibits a truth table method for computing the analytic series (or the differential expansion) of a proposition in terms of coordinates.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:70%"<br />
|+ style="height:30px" | <math>\text{Table 50.} ~~ \text{Computation of an Analytic Series in Terms of Coordinates}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:8%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:8%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:8%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}u\!</math><br />
| style="width:8%; border-bottom:1px solid black" | <math>\mathrm{d}v\!</math><br />
| style="width:8%; border-bottom:1px solid black; border-left:1px solid black" | <math>u'\!</math><br />
| style="width:8%; border-bottom:1px solid black" | <math>v'\!</math><br />
| style="width:10%; border-bottom:1px solid black; border-left:4px double black" | <math>\boldsymbol\varepsilon J\!</math><br />
| style="width:10%; border-bottom:1px solid black" | <math>\mathrm{E}J\!</math><br />
| style="width:12%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{D}J\!</math><br />
| style="width:10%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}J\!</math><br />
| style="width:10%; border-bottom:1px solid black" | <math>\mathrm{d}^2\!J\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
The first six columns of the Table, taken as a whole, represent the variables of a construct called the ''contingent universe'' <math>[u, v, \mathrm{d}u, \mathrm{d}v, u', v'],\!</math> or the bundle of ''contingency spaces'' <math>[\mathrm{d}u, \mathrm{d}v, u', v']\!</math> over the universe <math>[u, v].\!</math> Their placement to the left of the double bar indicates that all of them amount to independent variables, but there is a co-dependency among them, as described by the following equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
u' & = & u + \mathrm{d}u & = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\\[8pt]<br />
v' & = & v + \mathrm{d}v & = & \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
These relations correspond to the formal substitutions that are made in defining <math>\mathrm{E}J\!</math> and <math>\mathrm{D}J.\!</math> For now, the whole rigamarole of contingency spaces can be regarded as a technical device for achieving the effect of these substitutions, adapted to a setting where functional compositions and other symbolic manipulations are difficult to contemplate and execute.<br />
<br />
The five columns to the right of the double bar in Table&nbsp;50 contain the values of the dependent variables <math>\{ \boldsymbol\varepsilon J, ~\mathrm{E}J, ~\mathrm{D}J, ~\mathrm{d}J, ~\mathrm{d}^2\!J \}.\!</math> These are normally interpreted as values of functions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B}\!</math> or as values of propositions in the extended universe <math>[u, v, \mathrm{d}u, \mathrm{d}v]\!</math> but the dependencies prevailing in the contingent universe make it possible to regard these same final values as arising via functions on alternative lists of arguments, for example, the set <math>\{ u, v, u', v' \}.\!</math><br />
<br />
The column for <math>\boldsymbol\varepsilon J\!</math> is computed as <math>J(u, v) = uv\!</math> and together with the columns for <math>u\!</math> and <math>v\!</math> illustrates how we &ldquo;share structure&rdquo; in the Table by listing only the first entries of each constant block.<br />
<br />
The column for <math>\mathrm{E}J\!</math> is computed by means of the following chain of identities, where the contingent variables <math>u'\!</math> and <math>v'\!</math> are defined as <math>u' = u + \mathrm{d}u\!</math> and <math>v' = v + \mathrm{d}v.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathrm{E}J(u, v, \mathrm{d}u, \mathrm{d}v)<br />
& = &<br />
J(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& = &<br />
J(u', v')<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
This makes it easy to determine <math>\mathrm{E}J\!</math> by inspection, computing the conjunction <math>J(u', v') = u'v'\!</math> from the columns headed <math>u'\!</math> and <math>v'.\!</math> Since each of these forms expresses the same proposition <math>\mathrm{E}J\!</math> in <math>\mathrm{E}U^\bullet,\!</math> the dependence on <math>\mathrm{d}u\!</math> and <math>\mathrm{d}v\!</math> is still present but merely left implicit in the final variant <math>J(u', v').\!</math><br />
<br />
* '''Note.''' On occasion, it is tempting to use the further notation <math>J'(u, v) = J(u', v'),\!</math> especially to suggest a transformation that acts on whole propositions, for example, taking the proposition <math>J\!</math> into the proposition <math>J' = \mathrm{E}J.\!</math> The prime <math>( {}^{\prime} )\!</math> then signifies an action that is mediated by a field of choices, namely, the values that are picked out for the contingent variables in sweeping through the initial universe. But this heaps an unwieldy lot of construed intentions on a rather slight character and puts too high a premium on the constant correctness of its interpretation. In practice, therefore, it is best to avoid this usage.<br />
<br />
Given the values of <math>\boldsymbol\varepsilon J\!</math> and <math>\mathrm{E}J,\!</math> the columns for the remaining functions can be filled in quickly. The difference map is computed according to the relation <math>\mathrm{D}J = \boldsymbol\varepsilon J + \mathrm{E}J.\!</math> The first order differential <math>\mathrm{d}J\!</math> is found by looking in each block of constant argument pairs <math>u, v\!</math> and choosing the linear function of <math>\mathrm{d}u, \mathrm{d}v\!</math> that best approximates <math>\mathrm{D}J\!</math> in that block. Finally, the remainder is computed as <math>\mathrm{r}J = \mathrm{D}J + \mathrm{d}J,\!</math> in this case yielding the second order differential <math>\mathrm{d}^2\!J.\!</math><br />
<br />
====Analytic Series : Recap====<br />
<br />
Let us now summarize the results of Table&nbsp;50 by writing down for each column and for each block of constant argument pairs <math>u, v\!</math> a reasonably canonical symbolic expression for the function of <math>\mathrm{d}u, \mathrm{d}v\!</math> that appears there. The synopsis formed in this way is presented in Table&nbsp;51. As one has a right to expect, it confirms the results that were obtained previously by operating solely in terms of the formal calculus.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:70%"<br />
|+ style="height:30px" | <math>\text{Table 51.} ~~ \text{Computation of an Analytic Series in Symbolic Terms}\!</math><br />
|- style="height:35px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black" | <math>J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{E}J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{D}J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{d}J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{d}^2\!J\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}u \!\;\cdot\;\! \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}u \!\;\cdot\;\! \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
\mathrm{d}u<br />
\\[4pt]<br />
\mathrm{d}v<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;52 and 53 provide a quick overview of the analysis performed so far, giving the successive decompositions of <math>\mathrm{E}J = J + \mathrm{D}J\!</math> and <math>\mathrm{D}J = \mathrm{d}J + \mathrm{r}J\!</math> in two different styles of diagram.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 52 -- Decomposition of EJ.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 52.} ~~ \text{Decomposition of}~ \mathrm{E}J\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 53 -- Decomposition of DJ.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 53.} ~~ \text{Decomposition of}~ \mathrm{D}J\!</math><br />
|}<br />
<br />
====Terminological Interlude====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Lastly, my attention was especially attracted, not so much to the scene, as to the mirrors that produced it. These mirrors were broken in parts. Yes, they were marked and scratched; they had been &ldquo;starred&rdquo;, in spite of their solidity &hellip;<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 230]<br />
|}<br />
<br />
At this point several issues of terminology have accrued enough substance to intrude on our discussion. The remarks of this Subsection are intended to accomplish two goals. First, we call attention to significant aspects of the previous series of Figures, translating into literal terms what they depict in iconic forms, and we re-stress the most important structural elements they indicate. Next, we prepare the way for taking on more complex examples of transformations, those whose target universes have more than one dimension.<br />
<br />
In talking about the actions of operators it is important to keep in mind the distinctions between the operators per&nbsp;se, their operands, and their results. Furthermore, in working with composite forms of operators <math>\mathrm{W} = (\mathrm{W}_1, \ldots, \mathrm{W}_n),\!</math> transformations <math>\mathrm{F} = (\mathrm{F}_1, \ldots, \mathrm{F}_n),\!</math> and target domains <math>X^\bullet = [x_1, \ldots, x_n],\!</math> we need to preserve a clear distinction between the compound entity of each given type and any one of its separate components. It is curious, given the usefulness of the concepts ''operator'' and ''operand'', that we seem to lack a generic term, formed on the same root, for the corresponding result of an operation. Following the obvious paradigm would lead to words like ''opus'', ''opera'', and ''operant'', but these words are too affected with clang associations to work well at present, though they might be adapted in time. One current usage gets around this problem by using the substantive ''map'' as a systematic epithet to express the result of each operator's action. We will follow this practice as far as possible, for example, using the phrase ''tangent map'' to denote the end product of the tangent functor acting on its operand map.<br />
<br />
* '''Scholium.''' See [JGH, 6-9] for a good account of tangent functors and tangent maps in ordinary analysis and for examples of their use in mechanics. This work as a whole is a model of clarity in applying functorial principles to problems in physical dynamics.<br />
<br />
Whenever we focus on isolated propositions, on single components of composite operators, or on the portions of transformations that have <math>1\!</math>-dimensional ranges, we are free to shift between the native form of a proposition <math>J : U \to \mathbb{B}\!</math> and the thematized form of a mapping <math>J : U^\bullet \to [x]\!</math> without much trouble. In these cases we are able to tolerate a higher degree of ambiguity about the precise nature of an operator's input and output domains than we otherwise might. For example, in the preceding treatment of the example <math>J,\!</math> and for each operator <math>\mathrm{W}\!</math> in the set <math>\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \},\!</math> both the operand <math>J\!</math> and the result <math>\mathrm{W}J\!</math> could be viewed in either one of two ways. On one hand we may treat them as propositions <math>J : U \to \mathbb{B}\!</math> and <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B},\!</math> ignoring the distinction between the range <math>[x] \cong \mathbb{B}\!</math> of <math>\boldsymbol\varepsilon J\!</math> and the range <math>[\mathrm{d}x] \cong \mathbb{D}\!</math> of the other types of <math>\mathrm{W}J.\!</math> This is what we usually do when we content ourselves with simply coloring in regions of venn diagrams. On the other hand we may view these entities as maps <math>J : U^\bullet \to [x] = X^\bullet\!</math> and <math>\boldsymbol\varepsilon J : \mathrm{E}U^\bullet \to [x] \subseteq \mathrm{E}X^\bullet\!</math> or <math>\mathrm{W}J : \mathrm{E}U^\bullet \to [\mathrm{d}x] \subseteq \mathrm{E}X^\bullet,\!</math> in which case the qualitative characters of the output features are not ignored.<br />
<br />
At the beginning of this Section we recast the natural form of a proposition <math>J : U \to \mathbb{B}\!</math> into the thematic role of a transformation <math>J : U^\bullet \to [x],\!</math> where <math>x\!</math> was a variable recruited to express the newly independent <math>\check{J}.\!</math> However, in our computations and representations of operator actions we immediately lapsed back to viewing the results as native elements of the extended universe <math>\mathrm{E}U^\bullet,\!</math> in other words, as propositions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B},\!</math> where <math>\mathrm{W}\!</math> ranged over the set <math>\{ \boldsymbol\varepsilon, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}.\!</math> That is as it should be. We have worked hard to devise a language that gives us these advantages &mdash; the flexibility to exchange terms and types of equal information value and the capacity to reflect as quickly and as wittingly as a controlled reflex on the fibers of our propositions, independently of whether they express amusements, beliefs, or conjectures.<br />
<br />
As we take on target spaces of increasing dimension, however, these types of confusions (and confusions of types) become less and less permissible. For this reason, Tables&nbsp;54 and 55 present a rather detailed summary of the notation and the terminology we are using, as applied to the case <math>J = uv.\!</math> The rationale of these Tables is not so much to train more elephant guns on this poor drosophila of a concrete example but to invest our paradigm with enough solidity to bear the weight of abstraction to come.<br />
<br />
Table&nbsp;54 provides basic notation and descriptive information for the objects and operators used in this Example, giving the generic type (or broadest defined type) for each entity. Here, the sans&nbsp;serif operators <math>\mathsf{W} \in \{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{d}, \mathsf{r} \}\!</math> and their components <math>\mathrm{W} \in \{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}\!</math> both have the same broad type <math>\mathsf{W}, \mathrm{W} : (U^\bullet \to X^\bullet) \to (\mathrm{E}U^\bullet \to \mathrm{E}X^\bullet),\!</math> as appropriate to operators that map transformations <math>J : U^\bullet \to X^\bullet\!</math> to extended transformations <math>\mathsf{W}J, \mathrm{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 54.} ~~ \text{Cast of Characters : Expansive Subtypes of Objects and Operators}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| align="center" | <math>\text{Symbol}\!</math><br />
| align="center" | <math>\text{Notation}\!</math><br />
| align="center" | <math>\text{Description}\!</math><br />
| align="center" | <math>\text{Type}\!</math><br />
|-<br />
| align="center" | <math>U^\bullet\!</math><br />
| <math>= [u, v]\!</math><br />
| <math>\text{Source universe}\!</math><br />
| <math>[\mathbb{B}^2]\!</math><br />
|-<br />
| align="center" | <math>X^\bullet~\!</math><br />
| <math>= [x]\!</math><br />
| <math>\text{Target universe}\!</math><br />
| <math>[\mathbb{B}^1]~\!</math><br />
|-<br />
| align="center" | <math>\mathrm{E}U^\bullet\!</math><br />
| <math>= [u, v, \mathrm{d}u, \mathrm{d}v]\!</math><br />
| <math>\text{Extended source universe}\!</math><br />
| <math>[\mathbb{B}^2 \!\times\! \mathbb{D}^2]</math><br />
|-<br />
| align="center" | <math>\mathrm{E}X^\bullet\!</math><br />
| <math>= [x, \mathrm{d}x]~\!</math><br />
| <math>\text{Extended target universe}\!</math><br />
| <math>[\mathbb{B}^1 \!\times\! \mathbb{D}^1]</math><br />
|-<br />
| align="center" | <math>J\!</math><br />
| <math>J : U \!\to\! \mathbb{B}\!</math><br />
| <math>\text{Proposition}\!</math><br />
| <math>(\mathbb{B}^2 \!\to\! \mathbb{B}) \in [\mathbb{B}^2]\!</math><br />
|-<br />
| align="center" | <math>J\!</math><br />
| <math>J : U^\bullet \!\to\! X^\bullet\!</math><br />
| <math>\text{Transformation or Map}\!</math><br />
| <math>[\mathbb{B}^2] \!\to\! [\mathbb{B}^1]\!</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\boldsymbol\varepsilon<br />
\\<br />
\eta<br />
\\<br />
\mathrm{E}<br />
\\<br />
\mathrm{D}<br />
\\<br />
\mathrm{d}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{W} : U^\bullet \!\to\! \mathrm{E}U^\bullet,<br />
\\<br />
\mathrm{W} : X^\bullet \!\to\! \mathrm{E}X^\bullet,<br />
\\<br />
\mathrm{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\\<br />
\text{for each}~ \mathrm{W} ~\text{in the set:}<br />
\\<br />
\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{ll}<br />
\text{Tacit extension operator} & \boldsymbol\varepsilon<br />
\\<br />
\text{Trope extension operator} & \eta<br />
\\<br />
\text{Enlargement operator} & \mathrm{E}<br />
\\<br />
\text{Difference operator} & \mathrm{D}<br />
\\<br />
\text{Differential operator} & \mathrm{d}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^2] \!\to\! [\mathbb{B}^2 \!\times\! \mathbb{D}^2]},<br />
\\<br />
{[\mathbb{B}^1] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]},<br />
\\\\<br />
([\mathbb{B}^2] \!\to\! [\mathbb{B}^1]) \!\to\!<br />
\\<br />
([\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1])<br />
\end{array}</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\mathsf{e}<br />
\\<br />
\mathsf{E}<br />
\\<br />
\mathsf{D}<br />
\\<br />
\mathsf{T}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{W} : U^\bullet \!\to\! \mathsf{T}U^\bullet = \mathrm{E}U^\bullet,<br />
\\<br />
\mathsf{W} : X^\bullet \!\to\! \mathsf{T}X^\bullet = \mathrm{E}X^\bullet,<br />
\\<br />
\mathsf{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathsf{T}U^\bullet \!\to\! \mathsf{T}X^\bullet)<br />
\\<br />
\text{for each}~ \mathsf{W} ~\text{in the set:}<br />
\\<br />
\{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{T} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{lll}<br />
\text{Radius operator} & \mathsf{e} & = (\boldsymbol\varepsilon, \eta)<br />
\\<br />
\text{Secant operator} & \mathsf{E} & = (\boldsymbol\varepsilon, \mathrm{E})<br />
\\<br />
\text{Chord operator} & \mathsf{D} & = (\boldsymbol\varepsilon, \mathrm{D})<br />
\\<br />
\text{Tangent functor} & \mathsf{T} & = (\boldsymbol\varepsilon, \mathrm{d})<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^2] \!\to\! [\mathbb{B}^2 \!\times\! \mathbb{D}^2]},<br />
\\<br />
{[\mathbb{B}^1] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]},<br />
\\\\<br />
([\mathbb{B}^2] \!\to\! [\mathbb{B}^1]) \!\to\!<br />
\\<br />
([\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1])<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;55 supplies a more detailed outline of terminology for operators and their results. Here, we list the restrictive subtype (or narrowest defined subtype) that applies to each entity and we indicate across the span of the Table the whole spectrum of alternative types that color the interpretation of each symbol. For example, all the component operator maps <math>\mathrm{W}J\!</math> have <math>1\!</math>-dimensional ranges, either <math>\mathbb{B}^1\!</math> or <math>\mathbb{D}^1,\!</math> and so they can be viewed either as propositions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B}\!</math> or as logical transformations <math>\mathrm{W}J : \mathrm{E}U^\bullet \to X^\bullet.\!</math> As a rule, the plan of the Table allows us to name each entry by detaching the underlined adjective at the left of its row and prefixing it to the generic noun at the top of its column. In one case, however, it is customary to depart from this scheme. Because the phrase ''differential proposition'', applied to the result <math>\mathrm{d}J : \mathrm{E}U \to \mathbb{D},\!</math> does not distinguish it from the general run of differential propositions <math>\mathrm{G}: \mathrm{E}U \to \mathbb{B},\!</math> it is usual to single out <math>\mathrm{d}J\!</math> as the ''tangent proposition'' of <math>J.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 55.} ~~ \text{Synopsis of Terminology : Restrictive and Alternative Subtypes}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| &nbsp;<br />
| align="center" | <math>\text{Operator}\!</math><br />
| align="center" | <math>\text{Proposition}\!</math><br />
| align="center" | <math>\text{Map}\!</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tacit}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\boldsymbol\varepsilon : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\boldsymbol\varepsilon : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{B} \\<br />
\boldsymbol\varepsilon J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{B}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x] \\<br />
\boldsymbol\varepsilon J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Trope}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\eta : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\eta : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\eta J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\eta J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Enlargement}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{E} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{E}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{E}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Difference}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{D} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{D}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{D}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Differential}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{d} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{d} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{d}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}\!</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Remainder}}\\\text{operator}\end{matrix}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{r} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{r} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{r}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{r}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Radius}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e} = (\boldsymbol\varepsilon, \eta) \\<br />
\mathsf{e} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{e}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Secant}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}) \\<br />
\mathsf{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{E}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Chord}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}) \\<br />
\mathsf{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{D}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tangent}}\\\text{functor}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T} = (\boldsymbol\varepsilon, \mathrm{d}) \\<br />
\mathsf{T} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{T}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====End of Perfunctory Chatter : Time to Roll the Clip!====<br />
<br />
Two steps remain to finish the analysis of <math>J\!</math> that we began so long ago. First, we need to paste our accumulated heap of flat pictures into the frames of transformations, filling out the shapes of the operator maps <math>\mathsf{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet.~\!</math> This scheme is executed in two styles, using the ''areal views'' in Figures&nbsp;56-a and the ''box views'' in Figures&nbsp;56-b. Finally, in Figures&nbsp;57-1 to 57-4 we put all the pieces together to construct the full operator diagrams for <math>\mathsf{W} : J \to \mathsf{W}J.\!</math> There is a considerable amount of redundancy among the following three series of Figures but that will hopefully provide a fuller picture of the operations under review, enabling these snapshots to serve as successive frames in the animation of logic they are meant to become.<br />
<br />
=====Operator Maps : Areal Views=====<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a1 -- Radius Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a1.} ~~ \text{Radius Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a2 -- Secant Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a2.} ~~ \text{Secant Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a3 -- Chord Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a3.} ~~ \text{Chord Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a4 -- Tangent Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a4.} ~~ \text{Tangent Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
=====Operator Maps : Box Views=====<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b1 -- Radius Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b1.} ~~ \text{Radius Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b2 -- Secant Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b2.} ~~ \text{Secant Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b3 -- Chord Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b3.} ~~ \text{Chord Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b4 -- Tangent Map of J ISW.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b4.} ~~ \text{Tangent Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
=====Operator Diagrams for the Conjunction J = uv=====<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-1 -- Radius Operator Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-1.} ~~ \text{Radius Operator Diagram for the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-2 -- Secant Operator Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-2.} ~~ \text{Secant Operator Diagram for the Conjunction}~ J = uv~\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-3 -- Chord Operator Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-3.} ~~ \text{Chord Operator Diagram for the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-4 -- Tangent Functor Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-4.} ~~ \text{Tangent Functor Diagram for the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
===Taking Aim at Higher Dimensional Targets===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
The past and present wilt . . . . I have filled them and<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;emptied them,<br><br />
And proceed to fill my next fold of the future.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 87]<br />
|}<br />
<br />
In the next Section we consider a transformation <math>F\!</math> of concrete type <math>F : [u, v] \to [x, y]\!</math> and abstract type <math>F : [\mathbb{B}^2] \to [\mathbb{B}^2].\!</math> From the standpoint of propositional calculus we naturally approach the task of understanding such a transformation by parsing it into component maps with <math>1\!</math>-dimensional ranges, as follows:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{ccccccl}<br />
F & = & (F_1, F_2) & = & (f, g) & : & [u, v] \to [x, y],<br />
\\[6pt]<br />
&& F_1 & = & f & : & [u, v] \to [x],<br />
\\[6pt]<br />
&& F_2 & = & g & : & [u, v] \to [y].<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Then we tackle the separate components, now viewed as propositions <math>F_i : U \to \mathbb{B},\!</math> one at a time. At the completion of this analytic phase, we return to the task of synthesizing these partial and transient impressions into an agile form of integrity, a solidly coordinated and deeply integrated comprehension of the ongoing transformation. (Very often, of course, in tangling with refractory cases, we never get as far as the beginning again.)<br />
<br />
Let us now refer to the dimension of the target space or codomain as the ''toll'' (or ''tole'') of a transformation, as distinguished from the dimension of the range or image that is customarily called the ''rank''. When we keep to transformations with a toll of <math>1,\!</math> as <math>J : [u, v] \to [x],\!</math> we tend to get lazy about distinguishing a logical transformation from its component propositions. However, if we deal with transformations of a higher toll, this form of indolence can no longer be tolerated.<br />
<br />
Well, perhaps we can carry it a little further. After all, the operator result <math>\mathrm{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet\!</math> is a map of toll <math>2,\!</math> and cannot be unfolded in one piece as a proposition. But when a map has rank <math>1,\!</math> like <math>\boldsymbol\varepsilon J : \mathrm{E}U \to X \subseteq \mathrm{E}X\!</math> or <math>\mathrm{d}J : \mathrm{E}U \to \mathrm{d}X \subseteq \mathrm{E}X,\!</math> we naturally choose to concentrate on the <math>1\!</math>-dimensional range of the operator result <math>\mathrm{W}J,\!</math> ignoring the final difference in quality between the spaces <math>X\!</math> and <math>\mathrm{d}X,\!</math> and view <math>\mathrm{W}J\!</math> as a proposition about <math>\mathrm{E}U.\!</math><br />
<br />
In this way, an initial ambivalence about the role of the operand <math>J\!</math> conveys a double duty to the result <math>\mathrm{W}J.\!</math> The pivot that is formed by our focus of attention is essential to the linkage that transfers this double moment, as the whole process takes its bearing and wheels around the precise measure of a narrow bead that we can draw on the range of <math>\mathrm{W}J.\!</math> This is the escapement that it takes to get away with what may otherwise seem to be a simple duplicity, and this is the tolerance that is needed to counterbalance a certain arrogance of equivocation, by all of which machinations we make ourselves free to indicate the operator results <math>\mathrm{W}J\!</math> as propositions or as transformations, indifferently.<br />
<br />
But that's it, and no further. Neglect of these distinctions in range and target universes of higher dimensions is bound to cause a hopeless confusion. To guard against these adverse prospects, Tables&nbsp;58 and 59 lay the groundwork for discussing a typical map <math>F : [\mathbb{B}^2] \to [\mathbb{B}^2],\!</math> and begin to pave the way to some extent for discussing any transformation of the form <math>F : [\mathbb{B}^n] \to [\mathbb{B}^k].\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 58.} ~~ \text{Cast of Characters : Expansive Subtypes of Objects and Operators}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| align="center" | <math>\text{Symbol}\!</math><br />
| align="center" | <math>\text{Notation}\!</math><br />
| align="center" | <math>\text{Description}\!</math><br />
| align="center" | <math>\text{Type}\!</math><br />
|-<br />
| align="center" | <math>U^\bullet\!</math><br />
| <math>= [u, v]\!</math><br />
| <math>\text{Source universe}\!</math><br />
| <math>[\mathbb{B}^n]\!</math><br />
|-<br />
| align="center" | <math>X^\bullet~\!</math><br />
| <math>\begin{array}{l}<br />
= [x, y] \\<br />
= [f, g]<br />
\end{array}</math><br />
| <math>\text{Target universe}\!</math><br />
| <math>[\mathbb{B}^k]\!</math><br />
|-<br />
| align="center" | <math>\mathrm{E}U^\bullet\!</math><br />
| <math>= [u, v, \mathrm{d}u, \mathrm{d}v]\!</math><br />
| <math>\text{Extended source universe}\!</math><br />
| <math>[\mathbb{B}^n \!\times\! \mathbb{D}^n]\!</math><br />
|-<br />
| align="center" | <math>\mathrm{E}X^\bullet\!</math><br />
| <math>\begin{array}{l}<br />
= [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
= [f, g, \mathrm{d}f, \mathrm{d}g]<br />
\end{array}</math><br />
| <math>\text{Extended target universe}\!</math><br />
| <math>[\mathbb{B}^k \!\times\! \mathbb{D}^k]\!</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
f \\ g<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{ll}<br />
f : U \!\to\! [x] \cong \mathbb{B} \\<br />
g : U \!\to\! [y] \cong \mathbb{B}<br />
\end{array}</math><br />
| <math>\text{Proposition}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathbb{B}^n \!\to\! \mathbb{B} \\<br />
\in (\mathbb{B}^n, \mathbb{B}^n \!\to\! \mathbb{B}) = [\mathbb{B}^n]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>F\!</math><br />
| <math>F = (f, g) : U^\bullet \!\to\! X^\bullet\!</math><br />
| <math>\text{Transformation of Map}\!</math><br />
| <math>[\mathbb{B}^n] \!\to\! [\mathbb{B}^k]</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\boldsymbol\varepsilon<br />
\\<br />
\eta<br />
\\<br />
\mathrm{E}<br />
\\<br />
\mathrm{D}<br />
\\<br />
\mathrm{d}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{W} : U^\bullet \!\to\! \mathrm{E}U^\bullet,<br />
\\<br />
\mathrm{W} : X^\bullet \!\to\! \mathrm{E}X^\bullet,<br />
\\<br />
\mathrm{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\\<br />
\text{for each}~ \mathrm{W} ~\text{in the set:}<br />
\\<br />
\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{ll}<br />
\text{Tacit extension operator} & \boldsymbol\varepsilon<br />
\\<br />
\text{Trope extension operator} & \eta<br />
\\<br />
\text{Enlargement operator} & \mathrm{E}<br />
\\<br />
\text{Difference operator} & \mathrm{D}<br />
\\<br />
\text{Differential operator} & \mathrm{d}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^n] \!\to\! [\mathbb{B}^n \!\times\! \mathbb{D}^n]},<br />
\\<br />
{[\mathbb{B}^k] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]},<br />
\\\\<br />
([\mathbb{B}^n] \!\to\! [\mathbb{B}^k]) \!\to\!<br />
\\<br />
([\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k])<br />
\end{array}</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\mathsf{e}<br />
\\<br />
\mathsf{E}<br />
\\<br />
\mathsf{D}<br />
\\<br />
\mathsf{T}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{W} : U^\bullet \!\to\! \mathsf{T}U^\bullet = \mathrm{E}U^\bullet,<br />
\\<br />
\mathsf{W} : X^\bullet \!\to\! \mathsf{T}X^\bullet = \mathrm{E}X^\bullet,<br />
\\<br />
\mathsf{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathsf{T}U^\bullet \!\to\! \mathsf{T}X^\bullet)<br />
\\<br />
\text{for each}~ \mathsf{W} ~\text{in the set:}<br />
\\<br />
\{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{T} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{lll}<br />
\text{Radius operator} & \mathsf{e} & = (\boldsymbol\varepsilon, \eta)<br />
\\<br />
\text{Secant operator} & \mathsf{E} & = (\boldsymbol\varepsilon, \mathrm{E})<br />
\\<br />
\text{Chord operator} & \mathsf{D} & = (\boldsymbol\varepsilon, \mathrm{D})<br />
\\<br />
\text{Tangent functor} & \mathsf{T} & = (\boldsymbol\varepsilon, \mathrm{d})<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^n] \!\to\! [\mathbb{B}^n \!\times\! \mathbb{D}^n]},<br />
\\<br />
{[\mathbb{B}^k] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]},<br />
\\\\<br />
([\mathbb{B}^n] \!\to\! [\mathbb{B}^k]) \!\to\!<br />
\\<br />
([\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k])<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 59.} ~~ \text{Synopsis of Terminology : Restrictive and Alternative Subtypes}~\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| &nbsp;<br />
| align="center" | <math>\begin{matrix}\text{Operator}\\\text{or}\\\text{Operand}\end{matrix}</math><br />
| align="center" | <math>\begin{matrix}\text{Proposition}\\\text{or}\\\text{Component}\end{matrix}</math><br />
| align="center" | <math>\begin{matrix}\text{Transformation}\\\text{or}\\\text{Map}\end{matrix}</math><br />
|-<br />
| align="center" | <math>\underline{\text{Operand}}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
F = (F_1, F_2) \\<br />
F = (f, g) : U \!\to\! X<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
F_i : \langle u, v \rangle \!\to\! \mathbb{B} \\<br />
F_i : \mathbb{B}^n \!\to\! \mathbb{B}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
F : [u, v] \!\to\! [x, y] \\<br />
F : [\mathbb{B}^n] \!\to\! [\mathbb{B}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tacit}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\boldsymbol\varepsilon : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\boldsymbol\varepsilon : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{B} \\<br />
\boldsymbol\varepsilon F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{B}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y] \\<br />
\boldsymbol\varepsilon F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Trope}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\eta : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\eta : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\eta F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\eta F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Enlargement}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{E} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{E}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{E}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Difference}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{D} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{D}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{D}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Differential}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{d} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{d} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{d}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}\!</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Remainder}}\\\text{operator}\end{matrix}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{r} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{r} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{r}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{r}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Radius}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e} = (\boldsymbol\varepsilon, \eta) \\<br />
\mathsf{e} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{e}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Secant}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}) \\<br />
\mathsf{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{E}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Chord}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}) \\<br />
\mathsf{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{D}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tangent}}\\\text{functor}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T} = (\boldsymbol\varepsilon, \mathrm{d}) \\<br />
\mathsf{T} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{T}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
===Transformations of Type '''B'''<sup>2</sup> &rarr; '''B'''<sup>2</sup>===<br />
<br />
To take up a slightly more complex example, but one that remains simple enough to pursue through a complete series of developments, consider the transformation from <math>U^\bullet = [u, v]\!</math> to <math>X^\bullet = [x, y]\!</math> that is defined by the following system of equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
x<br />
& = & f(u, v)<br />
& = & \texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[8pt]<br />
y<br />
& = & g(u, v)<br />
& = & \texttt{((} u \texttt{,} v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
The component notation <math>F = (F_1, F_2) = (f, g) : U^\bullet \to X^\bullet\!</math> allows us to give a name and a type to this transformation and permits defining it by the compact description that follows:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
(x, y)<br />
& = & F(u, v)<br />
& = & (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Logical Transformations====<br />
<br />
The information that defines the logical transformation <math>F\!</math> can be represented in the form of a truth table, as shown in Table&nbsp;60. To cut down on subscripts in this example we continue to use plain letter equivalents for all components of spaces and maps.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:60%"<br />
|+ style="height:30px" | <math>\text{Table 60.} ~~ \text{A Propositional Transformation}\!</math><br />
|- style="height:40px; background:ghostwhite; width:100%"<br />
| style="width:25%" | <math>u\!</math><br />
| style="width:25%" | <math>v\!</math><br />
| style="width:25%; border-left:1px solid black" | <math>f\!</math><br />
| style="width:25%" | <math>g\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="border-top:1px solid black" | <math>u\!</math><br />
| style="border-top:1px solid black" | <math>v\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;61 shows how we might paint a picture of the transformation <math>F\!</math> in the manner of Figure&nbsp;30.<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 61 -- Propositional Transformation.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 61.} ~~ \text{A Propositional Transformation}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;62 extracts the gist of Figure&nbsp;61, exhibiting a style of diagram that is adequate for most purposes.<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 62 -- Propositional Transformation (Short Form).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 62.} ~~ \text{A Propositional Transformation (Short Form)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Local Transformations====<br />
<br />
Figure&nbsp;63 gives a more complete picture of the transformation <math>F,\!</math> showing how the points of <math>U^\bullet\!</math> are transformed into points of <math>X^\bullet.\!</math> The bold lines crossing from one universe to the other trace the action that <math>F\!</math> induces on points, in other words, they show how the transformation acts as a mapping from points to points and chart its effects on the elements that are variously called cells, points, positions, or singular propositions.<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 63 -- Transformation of Positions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 63.} ~~ \text{A Transformation of Positions}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;64 shows how the action of <math>F\!</math> on cells or points can be computed in terms of coordinates.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 64.} ~~ \text{A Transformation of Positions}\!</math><br />
|- style="height:40px; background:ghostwhite; width:100%"<br />
| style="width:8%" | <math>u\!</math><br />
| style="width:8%" | <math>v\!</math><br />
| style="width:12%; border-left:1px solid black" | <math>x\!</math><br />
| style="width:12%" | <math>y\!</math><br />
| style="width:10%; border-left:1px solid black" | <math>x~y\!</math><br />
| style="width:10%" | <math>x \texttt{(} y \texttt{)}\!</math><br />
| style="width:10%" | <math>\texttt{(} x \texttt{)} y\!</math><br />
| style="width:10%" | <math>\texttt{(} x \texttt{)(} y \texttt{)}\!</math><br />
| style="width:20%; border-left:1px solid black" | <math>X^\bullet = [x, y]\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\uparrow<br />
\\[4pt]<br />
F =<br />
\\[4pt]<br />
(f, g)<br />
\\[4pt]<br />
\uparrow<br />
\end{matrix}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="border-top:1px solid black" | <math>u\!</math><br />
| style="border-top:1px solid black" | <math>v\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
| style="border-top:1px solid black" | <math>\texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>u~v\!</math><br />
| style="border-top:1px solid black" | <math>\texttt{(} u \texttt{,} v \texttt{)}\!</math><br />
| style="border-top:1px solid black" | <math>\texttt{(} u \texttt{)(} v \texttt{)}\!</math><br />
| style="border-top:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>U^\bullet = [u, v]\!</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;65 extends this scheme from single cells to arbitrary regions, showing how we might compute the action of a logical transformation on arbitrary propositions in the universe of discourse. The effect of a point-transformation on arbitrary propositions, or any other structures erected on points, is referred to as the ''induced action'' of the transformation on the structures in question.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 65-a.} ~~ \text{An Induced Transformation on Propositions}\!</math><br />
|- style="height:50px; background:ghostwhite"<br />
| style="width:20%" | <math>X^\bullet~\!</math><br />
| style="width:20%; border-right:none" | <math>\longleftarrow\!</math><br />
| style="width:20%; border-left:none; border-right:none" | <math>F = (f, g)\!</math><br />
| style="width:20%; border-left:none" | <math>\longleftarrow\!</math><br />
| style="width:20%" | <math>U^\bullet~\!</math><br />
|- style="background:ghostwhite"<br />
| rowspan="2" | <math>f_i (x, y)\!</math><br />
| align="right" | <math>\begin{matrix}u = \\ v =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~0~0\\1~0~1~0\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= u \\ = v\end{matrix}</math><br />
| rowspan="2" style="width:20%" | <math>f_j (u, v)\!</math><br />
|- style="background:ghostwhite"<br />
| align="right" | <math>\begin{matrix}x = \\ y =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~1~0\\1~0~0~1\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= f(u, v) \\ = g(u, v)\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{2}<br />
\\[2pt]<br />
f_{3}<br />
\\[2pt]<br />
f_{4}<br />
\\[2pt]<br />
f_{5}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{7}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)~ ~}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{~ ~(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{,~} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{~~} y \texttt{)}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~0<br />
\\[2pt]<br />
0~0~0~0<br />
\\[2pt]<br />
0~0~0~1<br />
\\[2pt]<br />
0~0~0~1<br />
\\[2pt]<br />
0~1~1~0<br />
\\[2pt]<br />
0~1~1~0<br />
\\[2pt]<br />
0~1~1~1<br />
\\[2pt]<br />
0~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{,~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{,~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{~~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{~~} v \texttt{)}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[2pt]<br />
f_{0}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{7}<br />
\\[2pt]<br />
f_{7}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{8}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{10}<br />
\\[2pt]<br />
f_{11}<br />
\\[2pt]<br />
f_{12}<br />
\\[2pt]<br />
f_{13}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{15}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[2pt]<br />
\texttt{~ ~ ~ ~} y \texttt{~~}<br />
\\[2pt]<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[2pt]<br />
\texttt{~~} x \texttt{~ ~ ~ ~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
1~0~0~0<br />
\\[2pt]<br />
1~0~0~0<br />
\\[2pt]<br />
1~0~0~1<br />
\\[2pt]<br />
1~0~0~1<br />
\\[2pt]<br />
1~1~1~0<br />
\\[2pt]<br />
1~1~1~0<br />
\\[2pt]<br />
1~1~1~1<br />
\\[2pt]<br />
1~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~~} u \texttt{~~} v \texttt{~~}<br />
\\[2pt]<br />
\texttt{~~} u \texttt{~~} v \texttt{~~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{8}<br />
\\[2pt]<br />
f_{8}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{15}<br />
\\[2pt]<br />
f_{15}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 65-b.} ~~ \text{An Induced Transformation on Propositions}\!</math><br />
|- style="height:50px; background:ghostwhite"<br />
| style="width:20%" | <math>X^\bullet~\!</math><br />
| style="width:20%; border-right:none" | <math>\longleftarrow\!</math><br />
| style="width:20%; border-left:none; border-right:none" | <math>F = (f, g)\!</math><br />
| style="width:20%; border-left:none" | <math>\longleftarrow\!</math><br />
| style="width:20%" | <math>U^\bullet~\!</math><br />
|- style="background:ghostwhite"<br />
| rowspan="2" | <math>f_i (x, y)\!</math><br />
| align="right" | <math>\begin{matrix}u = \\ v =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~0~0\\1~0~1~0\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= u \\ = v\end{matrix}</math><br />
| rowspan="2" style="width:20%" | <math>f_j (u, v)\!</math><br />
|- style="background:ghostwhite"<br />
| align="right" | <math>\begin{matrix}x = \\ y =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~1~0\\1~0~0~1\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= f(u, v) \\ = g(u, v)\end{matrix}</math><br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>f_{0}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[2pt]<br />
f_{2}<br />
\\[2pt]<br />
f_{4}<br />
\\[2pt]<br />
f_{8}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~0<br />
\\[2pt]<br />
0~0~0~1<br />
\\[2pt]<br />
0~1~1~0<br />
\\[2pt]<br />
1~0~0~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{,~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{~} u \texttt{~~} v \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{8}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{3}<br />
\\[2pt]<br />
f_{12}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[2pt]<br />
1~1~1~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} u \texttt{)(} v \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[2pt]<br />
f_{14}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[2pt]<br />
f_{9}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[2pt]<br />
1~0~0~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{~~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{~} u \texttt{~~} v \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[2pt]<br />
f_{8}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{5}<br />
\\[2pt]<br />
f_{10}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[2pt]<br />
1~0~0~1<br />
\end{matrix}~\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} u \texttt{,~} v \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[2pt]<br />
f_{9}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[2pt]<br />
f_{11}<br />
\\[2pt]<br />
f_{13}<br />
\\[2pt]<br />
f_{14}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[2pt]<br />
1~0~0~1<br />
\\[2pt]<br />
1~1~1~0<br />
\\[2pt]<br />
1~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} u \texttt{~~} v \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{15}<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>f_{15}\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Difference Operators and Tangent Functors====<br />
<br />
Given the alphabets <math>\mathcal{U} = \{ u, v \}\!</math> and <math>\mathcal{X} = \{ x, y \},\!</math> along with the corresponding universes of discourse <math>U^\bullet, X^\bullet \cong [\mathbb{B}^2],\!</math> how many logical transformations of the general form <math>G = (G_1, G_2) : U^\bullet \to X^\bullet\!</math> are there? Since <math>G_1\!</math> and <math>G_2\!</math> can be any propositions of the type <math>\mathbb{B}^2 \to \mathbb{B},\!</math> there are <math>2^4 = 16\!</math> choices for each of the maps <math>G_1\!</math> and <math>G_2\!</math> and thus there are <math>2^4 \cdot 2^4 = 2^8 = 256\!</math> different mappings altogether of the form <math>G : U^\bullet \to X^\bullet.\!</math> The set of functions of a given type is denoted by placing its type indicator in parentheses, in the present instance writing <math>(U^\bullet \to X^\bullet) = \{ G : U^\bullet \to X^\bullet \},\!</math> and so the cardinality of the ''function space'' <math>(U^\bullet \to X^\bullet)\!</math> is summed up by writing <math>|(U^\bullet \to X^\bullet)| = |(\mathbb{B}^2 \to \mathbb{B}^2)| = 4^4 = 256.\!</math><br />
<br />
Given a transformation <math>G = (G_1, G_2) : U^\bullet \to X^\bullet\!</math> of this type, we proceed to define a pair of further transformations, related to <math>G,\!</math> that operate between the extended universes, <math>\mathrm{E}U^\bullet\!</math> and <math>\mathrm{E}X^\bullet,\!</math> of its source and target domains.<br />
<br />
First, the ''enlargement map'' (or ''secant transformation'') <math>\mathrm{E}G = (\mathrm{E}G_1, \mathrm{E}G_2) : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet\!</math> is defined by the following set of component equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}G_i<br />
& = & G_i (u + \mathrm{d}u, v + \mathrm{d}v)<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Next, the ''difference map'' (or ''chordal transformation'') <math>\mathrm{D}G = (\mathrm{D}G_1, \mathrm{D}G_2) : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet~\!</math> is defined in component-wise fashion as the boolean sum of the initial proposition <math>G_i\!</math> and the enlarged proposition <math>\mathrm{E}G_i,\!</math> for <math>i = 1, 2,\!</math> according to the following set of equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
\mathrm{D}G_i<br />
& = & G_i (u, v)<br />
& + & \mathrm{E}G_i (u, v, \mathrm{d}u, \mathrm{d}v)<br />
\\[8pt]<br />
& = & G_i (u, v)<br />
& + & G_i (u + \mathrm{d}u, v + \mathrm{d}v)<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Maintaining a strict analogy with ordinary difference calculus would perhaps have us write <math>\mathrm{D}G_i = \mathrm{E}G_i - G_i,\!</math> but the sum and difference operations are the same thing in boolean arithmetic. It is more often natural in the logical context to consider an initial proposition <math>q,\!</math> then to compute the enlargement <math>\mathrm{E}q,\!</math> and finally to determine the difference <math>\mathrm{D}q = q + \mathrm{E}q,\!</math> so we let the variant order of terms reflect this sequence of considerations.<br />
<br />
Viewed in this light the difference operator <math>\mathrm{D}\!</math> is imagined to be a function of very wide scope and polymorphic application, one that is able to realize the association between each transformation <math>G\!</math> and its difference map <math>\mathrm{D}G,\!</math> for example, taking the function space <math>(U^\bullet \to X^\bullet)\!</math> into <math>(\mathrm{E}U^\bullet \to \mathrm{E}X^\bullet).\!</math> When we consider the variety of interpretations permitted to propositions over the contexts in which we put them to use, it should be clear that an operator of this scope is not at all a trivial matter to define in general and that it may take some trouble to work out. For the moment we content ourselves with returning to particular cases.<br />
<br />
Acting on the logical transformation <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;),\!</math> the operators <math>\mathrm{E}\!</math> and <math>\mathrm{D}\!</math> yield the enlarged map <math>\mathrm{E}F = (\mathrm{E}f, \mathrm{E}g)\!</math> and the difference map <math>\mathrm{D}F = (\mathrm{D}f, \mathrm{D}g),\!</math> respectively, whose components are given as follows.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}f<br />
& = & \texttt{((} u + \mathrm{d}u \texttt{)(} v + \mathrm{d}v \texttt{))}<br />
\\[8pt]<br />
\mathrm{E}g<br />
& = & \texttt{((} u + \mathrm{d}u \texttt{,~} v + \mathrm{d}v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
\mathrm{D}f<br />
& = & \texttt{((} u \texttt{)(} v \texttt{))}<br />
& + & \texttt{((} u + \mathrm{d}u \texttt{)(} v + \mathrm{d}v \texttt{))}<br />
\\[8pt]<br />
\mathrm{D}g<br />
& = & \texttt{((} u \texttt{,~} v \texttt{))}<br />
& + & \texttt{((} u + \mathrm{d}u \texttt{,~} v + \mathrm{d}v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
But these initial formulas are purely definitional, and help us little in understanding either the purpose of the operators or the meaning of their results. Working symbolically, let us apply the same method to the separate components <math>f\!</math> and <math>g\!</math> that we earlier used on <math>J.\!</math> This work is recorded in Appendix&nbsp;3 and a summary of the results is presented in Tables&nbsp;66-i and 66-ii.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 66-i.} ~~ \text{Computation Summary for}~ f(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f<br />
& = & u \!\cdot\! v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 1<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}f<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \cdot \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{d}f<br />
& = & u \!\cdot\! v \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}f<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 66-ii.} ~~ \text{Computation Summary for}~ g(u, v) = \texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon g<br />
& = & u \!\cdot\! v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}g<br />
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}g<br />
& = & u \!\cdot\! v \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;67 shows how to compute the analytic series for <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math> in terms of coordinates, and Table&nbsp;68 recaps these results in symbolic terms, agreeing with earlier derivations.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 67.} ~~ \text{Computation of an Analytic Series in Terms of Coordinates}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:6%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}u\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>\mathrm{d}v\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>u'\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>v'\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:4px double black" | <math>f\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>g\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{E}f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{E}g}\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{D}f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{D}g}\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{d}f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{d}g}\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{d}^2\!f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{d}^2\!g}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 68.} ~~ \text{Computation of an Analytic Series in Symbolic Terms}\!</math><br />
|- style="height:40px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black" | <math>f\!</math><br />
| <math>g\!</math><br />
| style="border-left:1px solid black" | <math>{\mathrm{D}f}\!</math><br />
| <math>{\mathrm{D}g}\!</math><br />
| style="border-left:1px solid black" | <math>{\mathrm{d}f}\!</math><br />
| <math>{\mathrm{d}g}\!</math><br />
| style="border-left:1px solid black" | <math>{\mathrm{d}^2\!f}\!</math><br />
| <math>{\mathrm{d}^2\!g}\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[4pt]<br />
\texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
\\[4pt]<br />
\texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
\\[4pt]<br />
\texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;69 gives a graphical picture of the difference map <math>\mathrm{D}F = (\mathrm{D}f, \mathrm{D}g)\!</math> for the transformation <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math> This represents the same information about <math>\mathrm{D}f~\!</math> and <math>\mathrm{D}g~\!</math> that was given in the corresponding rows of Tables&nbsp;66-i and 66-ii, for ease of reference repeated below.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[8pt]<br />
\mathrm{D}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 69 -- Difference Map (Short Form).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 69.} ~~ \text{Difference Map of}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;70-a shows a way of visualizing the tangent functor map <math>\mathrm{d}F = (\mathrm{d}f, \mathrm{d}g)\!</math> for the transformation <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math> This amounts to the same information about <math>\mathrm{d}f~\!</math> and <math>\mathrm{d}g~\!</math> that was given in Tables&nbsp;66-i and 66-ii, the corresponding rows of which are repeated below.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{d}f<br />
& = & u \!\cdot\! v \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[8pt]<br />
\mathrm{d}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 70-a -- Tangent Functor Diagram.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 70-a.} ~~ \text{Tangent Functor Diagram for}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math><br />
|}<br />
<br />
<br><br />
<br />
Figure 70-b shows another way to picture the action of the tangent functor on the logical transformation <math>F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 70-b -- Tangent Functor Diagram.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 70-b.} ~~ \text{Tangent Functor Ferris Wheel for}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math><br />
|}<br />
<br />
<br><br />
<br />
* '''Note.''' The original Figure&nbsp;70-b lost some of its labeling in a succession of platform metamorphoses over the years, so we have included an ASCII version below to indicate where the missing labels go.<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| /////\ /////\ | | /XXXX\ /XXXX\ | | /\\\\\ /\\\\\ |<br />
| ///////o//////\ | | /XXXXXXoXXXXXX\ | | /\\\\\\o\\\\\\\ |<br />
| //////// \//////\ | | /XXXXXX/ \XXXXXX\ | | /\\\\\\/ \\\\\\\\ |<br />
| o/////// \//////o | | oXXXXXX/ \XXXXXXo | | o\\\\\\/ \\\\\\\o |<br />
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |<br />
| |/du//| |//dv/| | | |XXXXX| |XXXXX| | | |\du\\| |\\dv\| |<br />
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |<br />
| o//////\ ///////o | | oXXXXXX\ /XXXXXXo | | o\\\\\\\ /\\\\\\o |<br />
| \//////\ //////// | | \XXXXXX\ /XXXXXX/ | | \\\\\\\\ /\\\\\\/ |<br />
| \//////o/////// | | \XXXXXXoXXXXXX/ | | \\\\\\\o\\\\\\/ |<br />
| \///// \///// | | \XXXX/ \XXXX/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ (u)(v) o-----------------------o dv' @ (u)(v) =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| / \ /////\ | | /\\\\\ /XXXX\ | | /\\\\\ /\\\\\ |<br />
| / o//////\ | | /\\\\\\oXXXXXX\ | | /\\\\\\o\\\\\\\ |<br />
| / //\//////\ | | /\\\\\\//\XXXXXX\ | | /\\\\\\/ \\\\\\\\ |<br />
| o ////\//////o | | o\\\\\\////\XXXXXXo | | o\\\\\\/ \\\\\\\o |<br />
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |<br />
| | du |/////|//dv/| | | |\\\\\|/////|XXXXX| | | |\du\\| |\\dv\| |<br />
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |<br />
| o \//////////o | | o\\\\\\\////XXXXXXo | | o\\\\\\\ /\\\\\\o |<br />
| \ \///////// | | \\\\\\\\//XXXXXX/ | | \\\\\\\\ /\\\\\\/ |<br />
| \ o/////// | | \\\\\\\oXXXXXX/ | | \\\\\\\o\\\\\\/ |<br />
| \ / \///// | | \\\\\/ \XXXX/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ (u) v o-----------------------o dv' @ (u) v =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| /////\ / \ | | /XXXX\ /\\\\\ | | /\\\\\ /\\\\\ |<br />
| ///////o \ | | /XXXXXXo\\\\\\\ | | /\\\\\\o\\\\\\\ |<br />
| /////////\ \ | | /XXXXXX//\\\\\\\\ | | /\\\\\\/ \\\\\\\\ |<br />
| o//////////\ o | | oXXXXXX////\\\\\\\o | | o\\\\\\/ \\\\\\\o |<br />
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |<br />
| |/du//|/////| dv | | | |XXXXX|/////|\\\\\| | | |\du\\| |\\dv\| |<br />
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |<br />
| o//////\//// o | | oXXXXXX\////\\\\\\o | | o\\\\\\\ /\\\\\\o |<br />
| \//////\// / | | \XXXXXX\//\\\\\\/ | | \\\\\\\\ /\\\\\\/ |<br />
| \//////o / | | \XXXXXXo\\\\\\/ | | \\\\\\\o\\\\\\/ |<br />
| \///// \ / | | \XXXX/ \\\\\/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ u (v) o-----------------------o dv' @ u (v) =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| / \ / \ | | /\\\\\ /\\\\\ | | /\\\\\ /\\\\\ |<br />
| / o \ | | /\\\\\\o\\\\\\\ | | /\\\\\\o\\\\\\\ |<br />
| / / \ \ | | /\\\\\\/ \\\\\\\\ | | /\\\\\\/ \\\\\\\\ |<br />
| o / \ o | | o\\\\\\/ \\\\\\\o | | o\\\\\\/ \\\\\\\o |<br />
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |<br />
| | du | | dv | | | |\\\\\| |\\\\\| | | |\du\\| |\\dv\| |<br />
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |<br />
| o \ / o | | o\\\\\\\ /\\\\\\o | | o\\\\\\\ /\\\\\\o |<br />
| \ \ / / | | \\\\\\\\ /\\\\\\/ | | \\\\\\\\ /\\\\\\/ |<br />
| \ o / | | \\\\\\\o\\\\\\/ | | \\\\\\\o\\\\\\/ |<br />
| \ / \ / | | \\\\\/ \\\\\/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ u v o-----------------------o dv' @ u v =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| U | |\U\\\\\\\\\\\\\\\\\\\\\| |\U\\\\\\\\\\\\\\\\\\\\\|<br />
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|<br />
| /////\ /////\ | |\\\\\/////\\/////\\\\\\| |\\\\\/ \\/ \\\\\\|<br />
| ///////o//////\ | |\\\\///////o//////\\\\\| |\\\\/ o \\\\\|<br />
| /////////\//////\ | |\\\////////X\//////\\\\| |\\\/ /\\ \\\\|<br />
| o//////////\//////o | |\\o///////XXX\//////o\\| |\\o /\\\\ o\\|<br />
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|<br />
| |//u//|/////|//v//| | |\\|//u//|XXXXX|//v//|\\| |\\| u |\\\\\| v |\\|<br />
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|<br />
| o//////\//////////o | |\\o//////\XXX///////o\\| |\\o \\\\/ o\\|<br />
| \//////\///////// | |\\\\//////\X////////\\\| |\\\\ \\/ /\\\|<br />
| \//////o/////// | |\\\\\//////o///////\\\\| |\\\\\ o /\\\\|<br />
| \///// \///// | |\\\\\\/////\\/////\\\\\| |\\\\\\ /\\ /\\\\\|<br />
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|<br />
| | |\\\\\\\\\\\\\\\\\\\\\\\| |\\\\\\\\\\\\\\\\\\\\\\\|<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= u' o-----------------------o v' =<br />
= | U' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/u'//|XXXXX|\\v'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
Figure 70-b. Tangent Functor Ferris Wheel for F<u, v> = <((u)(v)), ((u, v))><br />
</pre><br />
|}<br />
<br />
==Epilogue, Enchoiry, Exodus==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
It is time to explain myself . . . . let us stand up.<br />
| width="4%" | &nbsp;<br />
|- <br />
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 79]<br />
|}<br />
<br />
==Appendices==<br />
<br />
===Appendix 1. Propositional Forms and Differential Expansions===<br />
<br />
====Table A1. Propositional Forms on Two Variables====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A1.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="background:ghostwhite"<br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}</math><br />
| width="25%" | <math>\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{0}\\f_{1}\\f_{2}\\f_{3}\\f_{4}\\f_{5}\\f_{6}\\f_{7}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0000}\\f_{0001}\\f_{0010}\\f_{0011}\\f_{0100}\\f_{0101}\\f_{0110}\\f_{0111}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~0\\0~0~0~1\\0~0~1~0\\0~0~1~1\\0~1~0~0\\0~1~0~1\\0~1~1~0\\0~1~1~1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)~ ~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~ ~(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{,~} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{~~} y \texttt{)}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{false}<br />
\\<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\<br />
y ~\text{without}~ x<br />
\\<br />
\text{not}~ x<br />
\\<br />
x ~\text{without}~ y<br />
\\<br />
\text{not}~ y<br />
\\<br />
x ~\text{not equal to}~ y<br />
\\<br />
\text{not both}~ x ~\text{and}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\<br />
\lnot x \land \lnot y<br />
\\<br />
\lnot x \land y<br />
\\<br />
\lnot x<br />
\\<br />
x \land \lnot y<br />
\\<br />
\lnot y<br />
\\<br />
x \ne y<br />
\\<br />
\lnot x \lor \lnot y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{8}\\f_{9}\\f_{10}\\f_{11}\\f_{12}\\f_{13}\\f_{14}\\f_{15}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{1000}\\f_{1001}\\f_{1010}\\f_{1011}\\f_{1100}\\f_{1101}\\f_{1110}\\f_{1111}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1~0~0~0\\1~0~0~1\\1~0~1~0\\1~0~1~1\\1~1~0~0\\1~1~0~1\\1~1~1~0\\1~1~1~1<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~ ~ ~ ~} y \texttt{~~}<br />
\\<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{~~} x \texttt{~ ~ ~ ~}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{((~))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{and}~ y<br />
\\<br />
x ~\text{equal to}~ y<br />
\\<br />
y<br />
\\<br />
\text{not}~ x ~\text{without}~ y<br />
\\<br />
x<br />
\\<br />
\text{not}~ y ~\text{without}~ x<br />
\\<br />
x ~\text{or}~ y<br />
\\<br />
\text{true}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x \land y<br />
\\<br />
x = y<br />
\\<br />
y<br />
\\<br />
x \Rightarrow y<br />
\\<br />
x<br />
\\<br />
x \Leftarrow y<br />
\\<br />
x \lor y<br />
\\<br />
1<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A2. Propositional Forms on Two Variables====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A2.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="background:ghostwhite"<br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}</math><br />
| width="25%" | <math>\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>f_{0000}\!</math><br />
| <math>0~0~0~0</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0001}\\f_{0010}\\f_{0100}\\f_{1000}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~1\\0~0~1~0\\0~1~0~0\\1~0~0~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\<br />
y ~\text{without}~ x<br />
\\<br />
x ~\text{without}~ y<br />
\\<br />
x ~\text{and}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot x \land \lnot y<br />
\\<br />
\lnot x \land y<br />
\\<br />
x \land \lnot y<br />
\\<br />
x \land y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{3}\\f_{12}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0011}\\f_{1100}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~1~1\\1~1~0~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ x<br />
\\<br />
x<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot x<br />
\\<br />
x<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{6}\\f_{9}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0110}\\f_{1001}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~0\\1~0~0~1<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{not equal to}~ y<br />
\\<br />
x ~\text{equal to}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x \ne y<br />
\\<br />
x = y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{5}\\f_{10}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0101}\\f_{1010}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~0~1\\1~0~1~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ y<br />
\\<br />
y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot y<br />
\\<br />
y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{7}\\f_{11}\\f_{13}\\f_{14}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0111}\\f_{1011}\\f_{1101}\\f_{1110}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~1\\1~0~1~1\\1~1~0~1\\1~1~1~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not both}~ x ~\text{and}~ y<br />
\\<br />
\text{not}~ x ~\text{without}~ y<br />
\\<br />
\text{not}~ y ~\text{without}~ x<br />
\\<br />
x ~\text{or}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot x \lor \lnot y<br />
\\<br />
x \Rightarrow y<br />
\\<br />
x \Leftarrow y<br />
\\<br />
x \lor y<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>f_{1111}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A3. E''f'' Expanded Over Differential Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A3.} ~~ \mathrm{E}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{T}_{11}f\\\mathrm{E}f|_{\mathrm{d}x ~ \mathrm{d}y}\end{matrix}</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{T}_{10}f\\\mathrm{E}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}\end{matrix}</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{T}_{01}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}\end{matrix}</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{T}_{00}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{3}\\f_{12}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{6}\\f_{9}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{5}\\f_{10}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{7}\\f_{11}\\f_{13}\\f_{14}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
|- style="background:ghostwhite"<br />
| style="border-top:1px solid black" colspan="2" | <math>\text{Fixed Point Total}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>4\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>4\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>4\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>16\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A4. D''f'' Expanded Over Differential Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A4.} ~~ \mathrm{D}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\mathrm{D}f|_{\mathrm{d}x ~ \mathrm{d}y}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}~\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
y<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
y<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
x<br />
\\<br />
x<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
x<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
y<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
y<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <br />
<math>\begin{matrix}<br />
x<br />
\\<br />
x<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A5. E''f'' Expanded Over Ordinary Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A5.} ~~ \mathrm{E}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\mathrm{E}f|_{xy}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{E}f|_{x \texttt{(} y \texttt{)}}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{E}f|_{\texttt{(} x \texttt{)} y}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{E}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{3}\\f_{12}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{6}\\f_{9}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{5}\\f_{10}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{7}\\f_{11}\\f_{13}\\f_{14}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A6. D''f'' Expanded Over Ordinary Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A6.} ~~ \mathrm{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\mathrm{D}f|_{xy}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{x \texttt{(} y \texttt{)}}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} x \texttt{)} y}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\texttt{(} x \texttt{)}\\\texttt{~} x \texttt{~}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Appendix 2. Differential Forms===<br />
<br />
The actions of the difference operator <math>\mathrm{D}\!</math> and the tangent operator <math>\mathrm{d}\!</math> on the 16 bivariate propositions are shown in Tables&nbsp;A7 and A8.<br />
<br />
Table A7 expands the differential forms that result over a ''logical basis'':<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
|<br />
<math>\{~ \texttt{(}\mathrm{d}x\texttt{)(}\mathrm{d}y\texttt{)}, ~\mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}, ~\texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.\!</math><br />
|}<br />
<br />
This set consists of the singular propositions in the first order differential variables, indicating mutually exclusive and exhaustive ''cells'' of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the cell-basis, point-basis, or singular differential basis. In this setting it is frequently convenient to use the following abbreviations:<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
|<br />
<math>\partial x ~=~ \mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}\!</math> &nbsp; &nbsp; and &nbsp; &nbsp; <math>\partial y ~=~ \texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y.\!</math><br />
|}<br />
<br />
Table A8 expands the differential forms that result over an ''algebraic basis'':<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
| <math>\{~ 1, ~\mathrm{d}x, ~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.\!</math><br />
|}<br />
<br />
This set consists of the ''positive propositions'' in the first order differential variables, indicating overlapping positive regions of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the ''positive differential basis''.<br />
<br />
====Table A7. Differential Forms Expanded on a Logical Basis====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A7.} ~~ \text{Differential Forms Expanded on a Logical Basis}\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| &nbsp;<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" | <math>\mathrm{D}f~\!</math><br />
| <math>\mathrm{d}f~\!</math><br />
|-<br />
| <math>f_{0}\!</math><br />
| style="border-right:none" | <math>\texttt{(~)}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x) ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\\<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\\<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\\<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\partial x<br />
\\<br />
\partial x<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
\\<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\partial x & + & \partial y<br />
\\<br />
\partial x & + & \partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\partial y<br />
\\<br />
\partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x) ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\\<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\\<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\\<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| style="border-right:none" | <math>\texttt{((~))}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A8. Differential Forms Expanded on an Algebraic Basis====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A8.} ~~ \text{Differential Forms Expanded on an Algebraic Basis}\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| &nbsp;<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" | <math>\mathrm{D}f~\!</math><br />
| <math>\mathrm{d}f~\!</math><br />
|-<br />
| <math>f_{0}\!</math><br />
| style="border-right:none" | <math>\texttt{(~)}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x<br />
\\<br />
\mathrm{d}x<br />
\end{matrix}\!</math><br />
| <math>\begin{matrix}<br />
\mathrm{d}x<br />
\\<br />
\mathrm{d}x<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\\<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\\<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}y<br />
\\<br />
\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y<br />
\\<br />
\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| style="border-right:none" | <math>\texttt{((~))}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A9. Tangent Proposition as Pointwise Linear Approximation====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A9.} ~~ \text{Tangent Proposition}~ \mathrm{d}f = \text{Pointwise Linear Approximation to the Difference Map}~ \mathrm{D}f\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}f =<br />
\\[2pt]<br />
\partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}^2\!f =<br />
\\[2pt]<br />
\partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\mathrm{d}f|_{x \, y}</math><br />
| <math>\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math><br />
|-<br />
| style="border-right:none" | <math>f_0\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{5}\\f_{10}\end{matrix}\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\end{matrix}\!</math><br />
| <math>\begin{matrix}<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>f_{15}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A10. Taylor Series Expansion Df = d''f'' + d<sup>2</sup>''f''====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" |<br />
<math>\text{Table A10.} ~~ \text{Taylor Series Expansion}~ {\mathrm{D}f = \mathrm{d}f + \mathrm{d}^2\!f}\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{D}f<br />
\\<br />
= & \mathrm{d}f & + & \mathrm{d}^2\!f<br />
\\<br />
= & \partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y & + & \partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\mathrm{d}f|_{x \, y}</math><br />
| <math>\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math><br />
|-<br />
| style="border-right:none" | <math>f_0\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>f_{15}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A11. Partial Differentials and Relative Differentials====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A11.} ~~ \text{Partial Differentials and Relative Differentials}\!</math><br />
|- style="background:ghostwhite; height:50px"<br />
| &nbsp;<br />
| <math>f\!</math><br />
| <math>\frac{\partial f}{\partial x}\!</math><br />
| <math>\frac{\partial f}{\partial y}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}f =<br />
\\[2pt]<br />
\partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\left. \frac{\partial x}{\partial y} \right| f\!</math><br />
| <math>\left. \frac{\partial y}{\partial x} \right| f\!</math><br />
|-<br />
| <math>f_0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A12. Detail of Calculation for the Difference Map====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:4px double black; border-left:4px double black; border-right:4px double black; border-top:4px double black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A12.} ~~ \text{Detail of Calculation for}~ {\mathrm{E}f + f = \mathrm{D}f}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:6%" | &nbsp;<br />
| style="width:14%; border-left:1px solid black" | <math>f\!</math><br />
| style="width:20%; border-left:4px double black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\mathrm{d}x ~ \mathrm{d}y}<br />
\\[4pt]<br />
+ & f|_{\mathrm{d}x ~ \mathrm{d}y}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\mathrm{d}x ~ \mathrm{d}y}<br />
\end{array}</math><br />
| style="width:20%; border-left:1px solid black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}<br />
\\[4pt]<br />
+ & f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}<br />
\end{array}</math><br />
| style="width:20%; border-left:1px solid black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
+ & f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}<br />
\end{array}</math><br />
| style="width:20%; border-left:1px solid black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
+ & f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}<br />
\end{array}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{0}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:4px double black; border-left:4px double black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{1}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{)(} y \texttt{)~}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{2}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{)~} y \texttt{~~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{4}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~~} x \texttt{~(} y \texttt{)~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{8}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~~} x \texttt{~~} y \texttt{~~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{3}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{(} x \texttt{)}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{12}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>x\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{6}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{,~} y \texttt{)~}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{9}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} x \texttt{,~} y \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{5}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{(} y \texttt{)}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{10}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>y\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{7}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{~~} y \texttt{)~}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{11}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{~(} y \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{13}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} x \texttt{)~} y \texttt{)~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{14}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} x \texttt{)(} y \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{15}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:4px double black; border-left:4px double black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Appendix 3. Computational Details===<br />
<br />
====Operator Maps for the Logical Conjunction ''f''<sub>8</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{8}~\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{8}<br />
& = && f_{8}(u, v)<br />
\\[4pt]<br />
& = && uv<br />
\\[4pt]<br />
& = && uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & uv \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & uv \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{8}<br />
& = && uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & uv \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & uv \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & uv \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.2-i} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{E}f_{8}<br />
& = & f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{8}(\mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{8}(\mathrm{d}u, \mathrm{d}v)<br />
\\[4pt]<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[20pt]<br />
\mathrm{E}f_{8}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
\\[4pt]<br />
&&&&& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&&&&&& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.2-ii} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{c}}<br />
\mathrm{E}f_{8}<br />
& = & (u + \mathrm{d}u) \cdot (v + \mathrm{d}v)<br />
\\[6pt]<br />
& = & u \cdot v<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}f_{8}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{8}<br />
& = && \mathrm{E}f_{8}<br />
& + & \boldsymbol\varepsilon f_{8}<br />
\\[4pt]<br />
& = && f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{8}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & uv<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{~}<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}f_{8}<br />
& = & \boldsymbol\varepsilon f_{8}<br />
& + & \mathrm{E}f_{8}<br />
\\[6pt]<br />
& = & f_{8}(u, v)<br />
& + & f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[6pt]<br />
& = & uv<br />
& + & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
& = & 0<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}f_{8}<br />
& = & 0<br />
& + & u \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.3-iii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 3)}\!</math><br />
|<br />
<math>\begin{array}{c*{9}{l}}<br />
\mathrm{D}f_{8}<br />
& = & \boldsymbol\varepsilon f_{8} ~+~ \mathrm{E}f_{8}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{8}<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ u \,\cdot\, v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~ u \;\cdot\; v \;\cdot\; \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}f_{8}<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u ~ \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} ~ v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot\, \mathrm{d}u ~ \mathrm{d}v<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = & ~ ~ 0 ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~ ~ u ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ ~ ~ v ~~ \cdot ~ \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
=====Computation of d''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.4} ~~ \text{Computation of}~ \mathrm{d}f_{8}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{8}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.5} ~~ \text{Computation of}~ \mathrm{r}f_{8}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{8} & = & \mathrm{D}f_{8} ~+~ \mathrm{d}f_{8}<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[20pt]<br />
\mathrm{r}f_{8}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Conjunction=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.6} ~~ \text{Computation Summary for}~ f_{8}(u, v) = uv\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{8}<br />
& = & uv \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}f_{8}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{r}f_{8}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Operator Maps for the Logical Equality ''f''<sub>9</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{9}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{9}<br />
& = && f_{9}(u, v)<br />
\\[4pt]<br />
& = && \texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{9}(1, 1)<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{9}(1, 0)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{9}(0, 1)<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{9}(0, 0)<br />
\\[4pt]<br />
& = && u v & + & 0 & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{9}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.2} ~~ \text{Computation of}~ \mathrm{E}f_{9}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{E}f_{9}<br />
& = && f_{9}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! f_{9}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! f_{9}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{ } \mathrm{d}v \texttt{ })<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! f_{9}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! f_{9}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{ } \mathrm{d}v \texttt{ })<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{ (} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{) }<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{ (} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{) }<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{E}f_{9}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{9}<br />
& = && \mathrm{E}f_{9}<br />
& + & \boldsymbol\varepsilon f_{9}<br />
\\[4pt]<br />
& = && f_{9}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{9}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{9}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[20pt]<br />
\mathrm{D}f_{9}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}f_{9}<br />
& = & 0 \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & 1 \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & 1 \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & 0 \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of d''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.4} ~~ \text{Computation of}~ \mathrm{d}f_{9}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.5} ~~ \text{Computation of}~ \mathrm{r}f_{9}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{9} & = & \mathrm{D}f_{9} ~+~ \mathrm{d}f_{9}<br />
\\[20pt]<br />
\mathrm{D}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[20pt]<br />
\mathrm{r}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Equality=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.6} ~~ \text{Computation Summary for}~ f_{9}(u, v) = \texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{9}<br />
& = & uv \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}f_{9}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f_{9}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{9}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}f_{9}<br />
& = & uv \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Operator Maps for the Logical Implication ''f''<sub>11</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{11}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{11}<br />
& = && f_{11}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(} u \texttt{(} v \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{11}(1, 1)<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{11}(1, 0)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{11}(0, 1)<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{11}(0, 0)<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ }<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ }<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{11}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.2} ~~ \text{Computation of}~ \mathrm{E}f_{11}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{E}f_{11}<br />
& = && f_{11}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = &&<br />
\texttt{(}<br />
\\<br />
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\\<br />
&&& \texttt{(}<br />
\\<br />
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\<br />
&&& \texttt{))}<br />
\\[4pt]<br />
& = &&<br />
u v<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)((} \mathrm{d}v \texttt{)))}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u \texttt{((} \mathrm{d}v \texttt{)))}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[4pt]<br />
& = &&<br />
u v<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{E}f_{11}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{11}<br />
& = && \mathrm{E}f_{11}<br />
& + & \boldsymbol\varepsilon f_{11}<br />
\\[4pt]<br />
& = && f_{11}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{11}(u, v)<br />
\\[4pt]<br />
& = &&<br />
\texttt{(} \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\texttt{(} \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{(} v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & u v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & 0<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & 0<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = && u v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{c*{9}{l}}<br />
\mathrm{D}f_{11}<br />
& = & \boldsymbol\varepsilon f_{11} ~+~ \mathrm{E}f_{11}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{11}<br />
& = & u v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}f_{11}<br />
& = &<br />
u v<br />
\cdot<br />
\texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\cdot<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\cdot<br />
\texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\cdot<br />
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = &<br />
u v<br />
\cdot<br />
\texttt{~(} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{~}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\cdot<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\cdot<br />
\texttt{~} \mathrm{d}u ~ \mathrm{d}v \texttt{~}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\cdot<br />
\texttt{~} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of d''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.4} ~~ \text{Computation of}~ \mathrm{d}f_{11}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{11}<br />
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{11}<br />
& = & u v \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.5} ~~ \text{Computation of}~ \mathrm{r}f_{11}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{11} & = & \mathrm{D}f_{11} ~+~ \mathrm{d}f_{11}<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{11}<br />
& = & u v \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u<br />
\\[20pt]<br />
\mathrm{r}f_{11}<br />
& = & u v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Implication=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.6} ~~ \text{Computation Summary for}~ f_{11}(u, v) = \texttt{(} u \texttt{(} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{11}<br />
& = & u v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}f_{11}<br />
& = & u v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f_{11}<br />
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{11}<br />
& = & u v \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u<br />
\\[6pt]<br />
\mathrm{r}f_{11}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Operator Maps for the Logical Disjunction ''f''<sub>14</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{14}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{14}<br />
& = && f_{14}(u, v)<br />
\\[4pt]<br />
& = && \texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{14}(1, 1)<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{14}(1, 0)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{14}(0, 1)<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{14}(0, 0)<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ }<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ }<br />
& + & 0<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{14}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.2} ~~ \text{Computation of}~ \mathrm{E}f_{14}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{E}f_{14}<br />
& = && f_{14}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = &&<br />
\texttt{((}<br />
\\<br />
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\\<br />
&&& \texttt{)(}<br />
\\<br />
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\<br />
&&& \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! f_{14}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! f_{14}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{ } \mathrm{d}v \texttt{ })<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! f_{14}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! f_{14}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{ } \mathrm{d}v \texttt{ })<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{E}f_{14}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{14}<br />
& = && \mathrm{E}f_{14}<br />
& + & \boldsymbol\varepsilon f_{14}<br />
\\[4pt]<br />
& = && f_{14}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{14}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{))((} v \texttt{,} \mathrm{d}v \texttt{)))}<br />
& + & \texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{14}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
\\[4pt]<br />
&& + & 0<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
\\[4pt]<br />
&& + & uv \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
\\[20pt]<br />
\mathrm{D}f_{14}<br />
& = && uv \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}f_{14}<br />
& = & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & 0 \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of d''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.4} ~~ \text{Computation of}~ \mathrm{d}f_{14}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.5} ~~ \text{Computation of}~ \mathrm{r}f_{14}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{14} & = & \mathrm{D}f_{14} ~+~ \mathrm{d}f_{14}<br />
\\[20pt]<br />
\mathrm{D}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{d}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[20pt]<br />
\mathrm{r}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Disjunction=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.6} ~~ \text{Computation Summary for}~ f_{14}(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{14}<br />
& = & uv \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 1<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}f_{14}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f_{14}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{d}f_{14}<br />
& = & uv \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}f_{14}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
===Appendix 4. Source Materials===<br />
<br />
===Appendix 5. Various Definitions of the Tangent Vector===<br />
<br />
==References==<br />
<br />
===Works Cited===<br />
<br />
{| cellpadding=3<br />
| valign=top | [AuM]<br />
| Auslander, L., and MacKenzie, R.E., ''Introduction to Differentiable Manifolds'', McGraw-Hill, 1963. Reprinted, Dover, New York, NY, 1977.<br />
|-<br />
| valign=top | [BiG]<br />
| Bishop, R.L., and Goldberg, S.I., ''Tensor Analysis on Manifolds'', Macmillan, 1968. Reprinted, Dover, New York, NY, 1980.<br />
|-<br />
| valign=top | [Boo]<br />
| Boole, G., ''An Investigation of The Laws of Thought'', Macmillan, 1854. Reprinted, Dover, New York, NY, 1958.<br />
|-<br />
| valign=top | [BoT]<br />
| Bott, R., and Tu, L.W., ''Differential Forms in Algebraic Topology'', Springer-Verlag, New York, NY, 1982.<br />
|-<br />
| valign=top | [dCa]<br />
| do Carmo, M.P., ''Riemannian Geometry''. Originally published in Portuguese, 1st editiom 1979, 2nd edition 1988. Translated by F. Flaherty, Birkhäuser, Boston, MA, 1992.<br />
|-<br />
| valign=top | [Che46]<br />
| Chevalley, C., ''Theory of Lie Groups'', Princeton University Press, Princeton, NJ, 1946.<br />
|-<br />
| valign=top | [Che56]<br />
| Chevalley, C., ''Fundamental Concepts of Algebra'', Academic Press, 1956.<br />
|-<br />
| valign=top | [Cho86]<br />
| Chomsky, N., ''Knowledge of Language : Its Nature, Origin, and Use'', Praeger, New York, NY, 1986.<br />
|-<br />
| valign=top | [Cho93]<br />
| Chomsky, N., ''Language and Thought'', Moyer Bell, Wakefield, RI, 1993.<br />
|-<br />
| valign=top | [DoM]<br />
| Doolin, B.F., and Martin, C.F., ''Introduction to Differential Geometry for Engineers'', Marcel Dekker, New York, NY, 1990.<br />
|-<br />
| valign=top | [Fuji]<br />
| Fujiwara, H., ''Logic Testing and Design for Testability'', MIT Press, Cambridge, MA, 1985.<br />
|-<br />
| valign=top | [Hic]<br />
| Hicks, N.J., ''Notes on Differential Geometry'', Van Nostrand, Princeton, NJ, 1965.<br />
|-<br />
| valign=top | [Hir]<br />
| Hirsch, M.W., ''Differential Topology'', Springer-Verlag, New York, NY, 1976.<br />
|-<br />
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| Howard, W.A., "The Formulae-as-Types Notion of Construction", Notes circulated from 1969. Reprinted in [SeH, 479-490].<br />
|-<br />
| valign=top | [JGH]<br />
| Jones, A., Gray, A., and Hutton, R., ''Manifolds and Mechanics'', Cambridge University Press, Cambridge, UK, 1987.<br />
|-<br />
| valign=top | [KoA]<br />
| Kosinski, A.A., ''Differential Manifolds'', Academic Press, San Diego, CA, 1993.<br />
|-<br />
| valign=top | [Koh]<br />
| Kohavi, Z., ''Switching and Finite Automata Theory'', 2nd edition, McGraw-Hill, New York, NY, 1978.<br />
|-<br />
| valign=top | [LaS]<br />
| Lambek, J., and Scott, P.J., ''Introduction to Higher Order Categorical Logic'', Cambridge University Press, Cambridge, UK, 1986.<br />
|-<br />
| valign=top | [La83]<br />
| Lang, S., ''Real Analysis'', 2nd edition, Addison-Wesley, Reading, MA, 1983.<br />
|-<br />
| valign=top | [La84]<br />
| Lang, S., ''Algebra'', 2nd edition, Addison-Wesley, Menlo Park, CA, 1984.<br />
|-<br />
| valign=top | [La85]<br />
| Lang, S., ''Differential Manifolds'', Springer-Verlag, New York, NY, 1985.<br />
|-<br />
| valign=top | [La93]<br />
| Lang, S., ''Real and Functional Analysis'', 3rd edition, Springer-Verlag, New York, NY, 1993.<br />
|-<br />
| valign=top | [Lie80]<br />
| Lie, S., "Sophus Lie's 1880 Transformation Group Paper", in ''Lie Groups : History, Frontiers, and Applications, Volume 1'', translated by M. Ackerman, comments by R. Hermann, Math Sci Press, Brookline, MA, 1975. Original paper 1880.<br />
|-<br />
| valign=top | [Lie84]<br />
| Lie, S., "Sophus Lie's 1884 Differential Invariant Paper", in ''Lie Groups : History, Frontiers, and Applications, Volume 3'', translated by M. Ackerman, comments by R. Hermann, Math Sci Press, Brookline, MA, 1976. Original paper 1884.<br />
|-<br />
| valign=top | [LoS]<br />
| Loomis, L.H., and Sternberg, S., ''Advanced Calculus'', Addison-Wesley, Reading, MA, 1968.<br />
|-<br />
| valign=top | [Mel]<br />
| Melzak, Z.A., ''Companion to Concrete Mathematics, Volume 2 : Mathematical Ideas, Modeling, and Applications'', John Wiley amd Sons, New York, NY, 1976.<br />
|-<br />
| valign=top | [Men]<br />
| Menabrea, L.F., "Sketch of the Analytical Engine Invented by Charles Babbage" with Notes by the Translator, Ada Augusta (Byron), Countess of Lovelace'', in [M&M, 225–297]. Originally published 1842.<br />
|-<br />
| valign=top | [M&M]<br />
| Morrison, P., and Morrison, E. (eds.), ''Charles Babbage on the Principles and Development of the Calculator, and Other Seminal Writings by Charles Babbage and Others, With an Introduction by the Editors'', Dover, Mineola, NY, 1961.<br />
|-<br />
| valign=top | [P1]<br />
| Peirce, C.S., ''Collected Papers of Charles Sanders Peirce'', vols. 1–8, C. Hartshorne, P. Weiss, and A.W. Burks (eds.), Harvard University Press, Cambridge, MA, 1931–1960. Cited as CP [volume].[paragraph].<br />
|-<br />
| valign=top | [P2]<br />
| Peirce, C.S., "Qualitative Logic", in ''The New Elements of Mathematics, Volume 4'', C. Eisele (ed.), Mouton, The Hague, 1976. Cited as NE [volume], [page].<br />
|-<br />
| valign=top | [Rob]<br />
| Roberts, D.D., ''The Existential Graphs of Charles S. Peirce'', Mouton, The Hague, 1973.<br />
|-<br />
| valign=top | [SeH]<br />
| Seldin, J.P., and Hindley, J.R. (eds.), ''To H.B. Curry : Essays on Combinatory Logic, Lambda Calculus, and Formalism'', Academic Press, London, UK, 1980.<br />
|-<br />
| valign=top | [SpB]<br />
| Spencer-Brown, G., ''Laws of Form'', George Allen and Unwin, London, UK, 1969.<br />
|-<br />
| valign=top | [Sp65]<br />
| Spivak, M., ''Calculus on Manifolds : A Modern Approach to Classical Theorems of Advanced Calculus'', W.A. Benjamin, New York, NY, 1965.<br />
|-<br />
| valign=top | [Sp79]<br />
| Spivak, M., ''A Comprehensive Introduction to Differential Geometry'', vols. 1–2. 1st edition 1970. 2nd edition, Publish or Perish Inc., Houston, TX, 1979.<br />
|-<br />
| valign=top | [Sty]<br />
| Styazhkin, N.I., ''History of Mathematical Logic from Leibniz to Peano'', 1st published in Russian, Nauka, Moscow, 1964. MIT Press, Cambridge, MA, 1969.<br />
|-<br />
| valign=top | [Wie]<br />
| Wiener, N., ''Cybernetics : or Control and Communication in the Animal and the Machine'', 1st edition 1948. 2nd edition, MIT Press, Cambridge, MA, 1961.<br />
|}<br />
<br />
===Works Consulted===<br />
<br />
{| cellpadding=3<br />
| valign=top | [Ami]<br />
| Amit, D.J., ''Modeling Brain Function : The World of Attractor Neural Networks'', Cambridge University Press, Cambridge, UK, 1989.<br />
|-<br />
| valign=top | [Ed87]<br />
| Edelman, G.M., ''Neural Darwinism : The Theory of Neuronal Group Selection'', Basic Books, New York, NY, 1987.<br />
|-<br />
| valign=top | [Ed88]<br />
| Edelman, G.M., ''Topobiology : An Introduction to Molecular Embryology'', Basic Books, New York, NY, 1988.<br />
|-<br />
| valign=top | [Fla]<br />
| Flanders, H., ''Differential Forms with Applications to the Physical Sciences'', Academic Press, 1963. Reprinted, Dover, Mineola, NY, 1989. <br />
|-<br />
| valign=top | [Has]<br />
| Hassoun, M.H. (ed.), ''Associative Neural Memories : Theory and Implementation'', Oxford University Press, New York, NY, 1993.<br />
|-<br />
| valign=top | [KoB]<br />
| Kosko, B., ''Neural Networks and Fuzzy Systems : A Dynamical Systems Approach to Machine Intelligence'', Prentice-Hall, Englewood Cliffs, NJ, 1992.<br />
|-<br />
| valign=top | [MaB]<br />
| Mac Lane, S., and Birkhoff, G., ''Algebra'', 3rd edition, Chelsea, New York, NY, 1993.<br />
|-<br />
| valign=top | [Mac]<br />
| Mac Lane, S., ''Categories for the Working Mathematician'', Springer-Verlag, New York, NY, 1971.<br />
|-<br />
| valign=top | [McC]<br />
| McCulloch, W.S., ''Embodiments of Mind'', MIT Press, Cambridge, MA, 1965.<br />
|-<br />
| valign=top | [Mc1]<br />
| McCulloch, W.S., "A Heterarchy of Values Determined by the Topology of Nervous Nets", Bulletin of Mathematical Biophysics, vol. 7 (1945), pp. 89–93. Reprinted in [McC].<br />
|-<br />
| valign=top | [MiP]<br />
| Minsky, M.L., and Papert, S.A., ''Perceptrons : An Introduction to Computational Geometry'', MIT Press, Cambridge, MA, 1969. 2nd printing 1972. Expanded edition 1988.<br />
|-<br />
| valign=top | [Rum]<br />
| Rumelhart, D.E., Hinton, G.E., and McClelland, J.L., "A General Framework for Parallel Distributed Processing" = Chapter 2 in Rumelhart, McClelland, and the PDP Research Group, ''Parallel Distributed Processing, Explorations in the Microstructure of Cognition, Volume 1 : Foundations'', MIT Press, Cambridge, MA, 1986.<br />
|}<br />
<br />
===Incidental Works===<br />
<br />
{| cellpadding=3<br />
| valign=top | [Dew]<br />
| Dewey, John, ''How We Think'', D.C. Heath, Lexington, MA, 1910. Reprinted, Prometheus Books, Buffalo, NY, 1991.<br />
|-<br />
| valign=top | [Fou]<br />
| Foucault, Michel, ''The Archaeology of Knowledge and The Discourse on Language'', A.M. Sheridan-Smith and Rupert Swyer (trans.), Pantheon, New York, NY, 1972. Originally published as ''L´Archéologie du Savoir et L´ordre du discours'', Editions Gallimard, 1969 & 1971.<br />
|-<br />
| valign=top | [Hom]<br />
| Homer, ''The Odyssey'', with an English translation by A.T. Murray, Loeb Classical Library, Harvard University Press, Cambridge, MA, 1980. First printed 1919.<br />
|-<br />
| valign=top | [Jam]<br />
| James, William, ''Pragmatism : A New Name for Some Old Ways of Thinking'', Longmans, Green, and Company, New York, NY, 1907.<br />
|-<br />
| valign=top | [Ler]<br />
| Leroux, Gaston, ''The Phantom of the Opera'', foreword by P. Haining, Dorset Press, New York, NY, 1988. Originally published in French, 1911.<br />
|-<br />
| valign=top | [Mus]<br />
| Musil, Robert, ''The Man Without Qualities'', 3 volumes, translated with a foreword by Eithne Wilkins and Ernst Kaiser, Pan Books, London, UK, 1979. English edition first published by Secker and Warburg, 1954. Originally published in German, ''Der Mann ohne Eigenschaften'', 1930 & 1932.<br />
|-<br />
| valign=top | [PlaR]<br />
| Plato, ''The Republic'', with an English translation by Paul Shorey, Loeb Classical Library, Harvard University Press, Cambridge, MA, 1980. First printed 1930 & 1935.<br />
|-<br />
| valign=top | [PlaS]<br />
| Plato, ''The Sophist'', Loeb Classical Library, William Heinemann, London, 1921, 1987.<br />
|-<br />
| valign=top | [Qui]<br />
| Quine, W.V., ''Mathematical Logic'', 1st edition, 1940. Revised edition, 1951. Harvard University Press, Cambridge, MA, 1981.<br />
|-<br />
| valign=top | [SaD]<br />
| de Santillana, Giorgio, and von Dechend, Hertha, ''Hamlet's Mill : An Essay on Myth and the Frame of Time'', David R. Godine, Publisher, Boston, MA, 1977. 1st published 1969.<br />
|-<br />
| valign=top | [Sha]<br />
| Shakespeare, William, '' William Shakespeare : The Complete Works'', Compact Edition, S. Wells and G. Taylor (eds.), Oxford University Press, Oxford, UK, 1988.<br />
|-<br />
| valign=top | [Sh1]<br />
| Shakespeare, William, ''A Midsummer Night's Dream'', Washington Square Press, New York, NY, 1958.<br />
|-<br />
| valign=top | [Sh2]<br />
| Shakespeare, William, ''The Tragedy of Hamlet, Prince of Denmark'', In [Sha], pp. 654&ndash;690.<br />
|-<br />
| valign=top | [Sh3]<br />
| Shakespeare, William, ''Measure for Measure'', Washington Square Press, New York, NY, 1965.<br />
|-<br />
| valign=top | [Web]<br />
| ''Webster's Ninth New Collegiate Dictionary'', Merriam-Webster, Springfield, MA, 1983.<br />
|-<br />
| valign=top | [Whi]<br />
| Whitman, Walt, ''Leaves of Grass'', Vintage Books / The Library of America, New York, NY, 1992. Originally published in numerous editions, 1855&ndash;1892.<br />
|-<br />
| valign=top | [Wil]<br />
| Wilhelm, R., and Baynes, C.F. (trans.), ''The I Ching, or Book of Changes'', foreword by C.G. Jung, preface by H. Wilhelm, 3rd edition, Bollingen Series XIX, Princeton University Press, Princeton, NJ, 1967.<br />
|}<br />
<br />
==Document History==<br />
<br />
<pre><br />
Author: Jon Awbrey<br />
Created: 16 Dec 1993<br />
Relayed: 31 Oct 1994<br />
Revised: 03 Jun 2003<br />
Recoded: 03 Jun 2007<br />
</pre><br />
<br />
[[Category:Adaptive Systems]]<br />
[[Category:Artificial Intelligence]]<br />
[[Category:Boolean Algebra]]<br />
[[Category:Boolean Functions]]<br />
[[Category:Charles Sanders Peirce]]<br />
[[Category:Combinatorics]]<br />
[[Category:Computer Science]]<br />
[[Category:Cybernetics]]<br />
[[Category:Differential Logic]]<br />
[[Category:Discrete Systems]]<br />
[[Category:Dynamical Systems]]<br />
[[Category:Formal Languages]]<br />
[[Category:Formal Sciences]]<br />
[[Category:Formal Systems]]<br />
[[Category:Functional Logic]]<br />
[[Category:Graph Theory]]<br />
[[Category:Group Theory]]<br />
[[Category:Inquiry]]<br />
[[Category:Knowledge Representation]]<br />
[[Category:Linguistics]]<br />
[[Category:Logic]]<br />
[[Category:Logical Graphs]]<br />
[[Category:Mathematics]]<br />
[[Category:Mathematical Systems Theory]]<br />
[[Category:Philosophy]]<br />
[[Category:Science]]<br />
[[Category:Semiotics]]<br />
[[Category:Systems Science]]<br />
[[Category:Visualization]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Differential_Logic_and_Dynamic_Systems&diff=469882Differential Logic and Dynamic Systems2021-01-14T15:06:21Z<p>Jon Awbrey: remove redirect</p>
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<div></div>Jon Awbreyhttps://mywikibiz.com/index.php?title=User:Jon_Awbrey&diff=469881User:Jon Awbrey2021-01-14T14:20:03Z<p>Jon Awbrey: /* Presently &hellip; */ update</p>
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<p><font face="lucida calligraphy" size="7">Jon Awbrey</font></p><br />
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__NOTOC__<br />
==Presently &hellip;==<br />
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[https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Overview Theme One Program]<br />
<br />
[https://oeis.org/wiki/User:Jon_Awbrey/Exploratory_Qualitative_Analysis_of_Sequential_Observation_Data Sequential Observations]<br />
<br />
[https://oeis.org/wiki/Futures_Of_Logical_Graphs Futures Of Logical Graphs]<br />
<br />
[https://oeis.org/wiki/Pragmatic_Theory_Of_Truth Pragmatic Theory Of Truth]<br />
<br />
[https://oeis.org/wiki/User:Jon_Awbrey/Peirce%27s_Logic_Of_Information Peirce's Logic Of Information]<br />
<br />
[https://oeis.org/wiki/Logical_Graphs Logical Graphs] [https://inquiryintoinquiry.com/2008/07/29/logical-graphs-1/ One] [https://inquiryintoinquiry.com/2008/09/19/logical-graphs-2/ Two]<br />
<br />
[https://oeis.org/wiki/User:Jon_Awbrey/EXCERPTS Collection Of Source Materials]<br />
<br />
[https://oeis.org/wiki/Propositions_As_Types_Analogy Propositions As Types Analogy]<br />
<br />
[https://oeis.org/wiki/Precursors_Of_Category_Theory Precursors Of Category Theory]<br />
<br />
Functional Logic [https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy (1)] [https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Higher_Order_Propositions (2)] [https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Quantification_Theory (3)]<br />
<br />
[https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Overview Peirce's 1870 Logic Of Relatives]<br />
<br />
[https://oeis.org/wiki/Differential_Propositional_Calculus Differential Propositional Calculus]<br />
<br />
[https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Overview Differential Logic] [https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1 (1)] [https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_2 (2)] [https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3 (3)]<br />
<br />
[https://oeis.org/wiki/Differential_Analytic_Turing_Automata Differential Analytic Turing Automata]<br />
<br />
[https://oeis.org/wiki/User:Jon_Awbrey/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]<br />
<br />
[https://oeis.org/wiki/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]<br />
<br />
[https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overview Differential Logic and Dynamic Systems]<br />
<br />
[https://oeis.org/wiki/Information_%3D_Comprehension_%C3%97_Extension Information = Comprehension &times; Extension]<br />
<br />
[https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]<br />
<br />
[https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Overview Inquiry Driven Systems &bull; Inquiry Into Inquiry]<br />
<br />
</center><br />
<br />
==Recent Sightings==<br />
<br />
{| align="center" style="text-align:center" width="100%"<br />
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| [http://inquiryintoinquiry.com/ Inquiry Into Inquiry]<br />
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| [https://web.archive.org/web/20191124081501/http://intersci.ss.uci.edu/wiki/index.php/User:Jon_Awbrey InterSciWiki User Page]<br />
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| [http://mywikibiz.com/User_talk:Jon_Awbrey MyWikiBiz Talk Page]<br />
|-<br />
| [http://planetmath.org/ PlanetMath Project]<br />
| [http://planetmath.org/users/Jon-Awbrey PlanetMath Profile]<br />
|-<br />
| [http://forum.wolframscience.com/ NKS Forum]<br />
| [http://forum.wolframscience.com/member.php?s=&action=getinfo&userid=336 NKS Profile]<br />
|-<br />
| [https://oeis.org/wiki/Welcome OEIS Land]<br />
| [https://oeis.org/search?q=Awbrey Bolgia Mia]<br />
|-<br />
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|-<br />
| [http://list.seqfan.eu/cgi-bin/mailman/listinfo/seqfan Fantasia Sequentia]<br />
| [http://list.seqfan.eu/pipermail/seqfan/ SeqFan Archive]<br />
|-<br />
| [http://mathforum.org/kb/ Math Forum Project]<br />
| [http://mathforum.org/kb/accountView.jspa?userID=99854 Math Forum Profile]<br />
|-<br />
| [http://www.mathweb.org/wiki/User:Jon_Awbrey MathWeb Page]<br />
| [http://www.mathweb.org/wiki/User_talk:Jon_Awbrey MathWeb Talk]<br />
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| [http://www.proofwiki.org/wiki/User:Jon_Awbrey ProofWiki Page]<br />
| [http://www.proofwiki.org/wiki/User_talk:Jon_Awbrey ProofWiki Talk]<br />
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|-<br />
| [http://p2pfoundation.net/User:JonAwbrey P2P Wiki Page]<br />
| [http://p2pfoundation.net/User_talk:JonAwbrey P2P Wiki Talk]<br />
|-<br />
| [http://vectors.usc.edu/ Vectors Project]<br />
| [http://vectors.usc.edu/thoughtmesh/ ThoughtMesh]<br />
|-<br />
| [http://ontolog.cim3.net/ OntoLog Project]<br />
| [http://ontolog.cim3.net/cgi-bin/wiki.pl?JonAwbrey OntoLog Profile]<br />
|-<br />
| [http://web.archive.org/web/20150127223035/http://semanticweb.org/wiki/User:Jon_Awbrey SemanticWeb Page]<br />
| [http://web.archive.org/web/20100716202618/http://semanticweb.org/wiki/User_talk:Jon_Awbrey SemanticWeb Talk]<br />
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| [https://zh.wikipedia.org/wiki/User_talk:Jon_Awbrey~zhwiki 维基百科 : 讨论]<br />
|}<br />
<br />
==Contributions==<br />
<br />
===Articles===<br />
<br />
[[Ampheck]]<br />
[[Boolean domain]]<br />
[[Boolean function]]<br />
[[Boolean-valued function]]<br />
[[Charles Sanders Peirce]]<br />
[[Charles Sanders Peirce (Bibliography)]]<br />
[[Comprehension (logic)]]<br />
[[Continuous predicate]]<br />
[[Correspondence theory of truth]]<br />
[[Cybernetics]]<br />
[[Descriptive science]]<br />
[[Differential logic]]<br />
[[Dynamics of inquiry]]<br />
[[Entitative graph]]<br />
[[Exclusive disjunction]]<br />
[[Formal science]]<br />
[[Graph (mathematics)]]<br />
[[Graph theory]]<br />
[[Grounded relation]]<br />
[[Inquiry]]<br />
[[Inquiry driven system]]<br />
[[Integer sequence]]<br />
[[Hypostatic abstraction]]<br />
[[Hypostatic object]]<br />
[[Kaina Stoicheia]]<br />
[[Logic]]<br />
[[Logic of information]]<br />
[[Logic of relatives]]<br />
[[Logic of Relatives (1870)]]<br />
[[Logic of Relatives (1883)]]<br />
[[Logical conjunction]]<br />
[[Logical disjunction]]<br />
[[Logical equality]]<br />
[[Logical graph]]<br />
[[Logical implication]]<br />
[[Logical matrix]]<br />
[[Logical NAND]]<br />
[[Logical negation]]<br />
[[Logical NNOR]]<br />
[[Minimal negation operator]]<br />
[[Multigrade operator]]<br />
[[Normative science]]<br />
[[Null graph]]<br />
[[On a New List of Categories]]<br />
[[Parametric operator]]<br />
[[Peirce's law]]<br />
[[Philosophy of mathematics]]<br />
[[Pragmatic information]]<br />
[[Pragmatic maxim]]<br />
[[Pragmatic theory of truth]]<br />
[[Pragmaticism]] <br />
[[Pragmatism]]<br />
[[Prescisive abstraction]]<br />
[[Propositional calculus]]<br />
[[Relation (mathematics)]]<br />
[[Relation composition]]<br />
[[Relation construction]]<br />
[[Relation reduction]]<br />
[[Relation theory]]<br />
[[Relation type]]<br />
[[Relative term]]<br />
[[Semeiotic]]<br />
[[Semiotic information]]<br />
[[Semiotics]]<br />
[[Sign relation]]<br />
[[Sign relational complex]]<br />
[[Sole sufficient operator]]<br />
[[Tacit extension]]<br />
[[The Simplest Mathematics]]<br />
[[Triadic relation]]<br />
[[Truth table]]<br />
[[Truth theory]]<br />
[[Universe of discourse]]<br />
[[What we've got here is (a) failure to communicate]]<br />
[[Zeroth order logic]]<br />
<br />
===Notes===<br />
<br />
* [[Directory:Jon Awbrey/Notes/Factorization Issues|Factorization Issues]]<br />
<br />
* [[Directory:Jon Awbrey/Notes/Factorization And Reification|Factorization And Reification]]<br />
<br />
===Papers===<br />
<br />
====Functional Logic====<br />
<br />
* [[Directory:Jon Awbrey/Papers/Functional Logic : Higher Order Propositions|Functional Logic : Higher Order Propositions]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Functional Logic : Inquiry and Analogy|Functional Logic : Inquiry and Analogy]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Functional Logic : Quantification Theory|Functional Logic : Quantification Theory]]<br />
<br />
====Differential Logic====<br />
<br />
* [[Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction|Differential Logic : Introduction]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Differential Propositional Calculus|Differential Propositional Calculus]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems|Differential Logic and Dynamic Systems 1.0]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0|Differential Logic and Dynamic Systems 2.0]]<br />
<br />
====Logic and Semiotics====<br />
<br />
* [[Directory:Jon Awbrey/Papers/Futures Of Logical Graphs|Futures Of Logical Graphs]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives|Peirce's 1870 Logic Of Relatives]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Peirce's Logic Of Information|Peirce's Logic Of Information]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Propositional Equation Reasoning Systems|Propositional Equation Reasoning Systems]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Semiotic Information|Semiotic Information]]<br />
<br />
====Inquiry Driven Systems====<br />
<br />
* [[Directory:Jon Awbrey/Essays/Prospects For Inquiry Driven Systems|Prospects for Inquiry Driven Systems]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Introduction to Inquiry Driven Systems|Introduction to Inquiry Driven Systems]]<br />
<br />
* [[Directory:Jon Awbrey/Essays/Inquiry Driven Systems : Fields Of Inquiry|Inquiry Driven Systems : Fields Of Inquiry]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems|Inquiry Driven Systems : Inquiry Into Inquiry]]<br />
<br />
===Projects===<br />
<br />
* [[Directory:Jon Awbrey/Projects/Cactus Language|Cactus Language]]<br />
<br />
* [[Directory:Jon Awbrey/Projects/Differential Logic|Differential Logic]]<br />
<br />
* [[Directory:Jon Awbrey/Projects/Inquiry|Inquiry]]<br />
** [[Directory:Jon Awbrey/Projects/Architecture For Inquiry|Architecture For Inquiry]]<br />
** [[Directory:Jon Awbrey/Projects/Inquiry Driven Systems|Inquiry Driven Systems]]<br />
<br />
* [[Directory:Jon Awbrey/Projects/Logic Of Information|Logic Of Information]]<br />
** [[Directory:Jon Awbrey/Projects/Pragmatic Theory Of Information|Pragmatic Theory Of Information]]<br />
** [[Directory:Jon Awbrey/Projects/Semiotic Theory Of Information|Semiotic Theory Of Information]]<br />
<br />
* [[Directory:Jon Awbrey/Projects/Notes And Queries|Notes And Queries]]<br />
<br />
* [[Directory:Jon Awbrey/Projects/Peircean Pragmata|Peircean Pragmata]]<br />
<br />
* [[Directory:Jon Awbrey/Projects/Theme One Program|Theme One Program]]<br />
<br />
* [[Directory:Jon Awbrey/Projects/Theory Of Relations|Theory Of Relations]]<br />
<br />
===Poetry===<br />
<br />
* [[Directory:Jon Awbrey/Poetry/Past All Reckoning|Past All Reckoning]]<br />
<br />
* [[Directory:Jon Awbrey/Poetry/Poems Of Emediate Moment|Poems Of Emediate Moment]]<br />
<br />
* [[Directory:Jon Awbrey/Poetry/Questionable Verses|Questionable Verses]]<br />
<br />
* [[Directory:Jon_Awbrey/Poetry/Iconoclast|Iconoclast]]<br />
<br />
===User Pages===<br />
<br />
* [[Directory:Jon Awbrey/EXCERPTS|Collection Of Source Materials]]<br />
* [[User:Jon Awbrey/Examples Of Inquiry|Examples Of Inquiry]]<br />
* [[User:Jon Awbrey/Mathematical Notes|Mathematical Notes]]<br />
* [[User:Jon Awbrey/Philosophical Notes|Philosophical Notes]]<br />
<br />
* [http://mywikibiz.com/index.php?title=Special%3APrefixIndex&prefix=Jon+Awbrey&namespace=2 MyWikiBiz User Pages]<br />
* [http://intersci.ss.uci.edu/wiki/index.php?title=Special%3APrefixIndex&prefix=Jon+Awbrey&namespace=2 InterSciWiki User Pages]<br />
* [http://mywikibiz.com/index.php?title=Special%3APrefixIndex&prefix=Jon+Awbrey&namespace=110 MyWikiBiz Directory Pages]<br />
<br />
==Presentations and Publications==<br />
<br />
* Awbrey, S.M., and Awbrey, J.L. (May 2001), &ldquo;Conceptual Barriers to Creating Integrative Universities&rdquo;, ''Organization : The Interdisciplinary Journal of Organization, Theory, and Society'' 8(2), Sage Publications, London, UK, pp. 269&ndash;284. [http://org.sagepub.com/cgi/content/abstract/8/2/269 Abstract].<br />
<br />
* Awbrey, S.M., and Awbrey, J.L. (September 1999), &ldquo;Organizations of Learning or Learning Organizations : The Challenge of Creating Integrative Universities for the Next Century&rdquo;, ''Second International Conference of the Journal &lsquo;Organization&rsquo;'', ''Re-Organizing Knowledge, Trans-Forming Institutions : Knowing, Knowledge, and the University in the 21st Century'', University of Massachusetts, Amherst, MA. [http://cspeirce.com/menu/library/aboutcsp/awbrey/integrat.htm Online].<br />
<br />
* Awbrey, J.L., and Awbrey, S.M. (Autumn 1995), &ldquo;Interpretation as Action : The Risk of Inquiry&rdquo;, ''Inquiry : Critical Thinking Across the Disciplines'' 15(1), pp. 40&ndash;52. [https://web.archive.org/web/20001210162300/http://chss.montclair.edu/inquiry/fall95/awbrey.html Archive]. [https://www.pdcnet.org/inquiryct/content/inquiryct_1995_0015_0001_0040_0052 Journal]. [https://independent.academia.edu/JonAwbrey/Papers/1302117/Interpretation_as_Action_The_Risk_of_Inquiry Online].<br />
<br />
* Awbrey, J.L., and Awbrey, S.M. (June 1992), &ldquo;Interpretation as Action : The Risk of Inquiry&rdquo;, ''The Eleventh International Human Science Research Conference'', Oakland University, Rochester, Michigan.<br />
<br />
* Awbrey, S.M., and Awbrey, J.L. (May 1991), &ldquo;An Architecture for Inquiry : Building Computer Platforms for Discovery&rdquo;, ''Proceedings of the Eighth International Conference on Technology and Education'', Toronto, Canada, pp. 874&ndash;875. [http://www.academia.edu/1270327/An_Architecture_for_Inquiry_Building_Computer_Platforms_for_Discovery Online].<br />
<br />
* Awbrey, J.L., and Awbrey, S.M. (January 1991), &ldquo;Exploring Research Data Interactively : Developing a Computer Architecture for Inquiry&rdquo;, Poster presented at the ''Annual Sigma Xi Research Forum'', University of Texas Medical Branch, Galveston, TX.<br />
<br />
* Awbrey, J.L., and Awbrey, S.M. (August 1990), &ldquo;Exploring Research Data Interactively. Theme One : A Program of Inquiry&rdquo;, ''Proceedings of the Sixth Annual Conference on Applications of Artificial Intelligence and CD-ROM in Education and Training'', Society for Applied Learning Technology, Washington, DC, pp. 9&ndash;15. [http://academia.edu/1272839/Exploring_Research_Data_Interactively._Theme_One_A_Program_of_Inquiry Online].<br />
<br />
==Education==<br />
<br />
* 1993&ndash;2003. [http://web.archive.org/web/20120202222443/http://www2.oakland.edu/secs/dispprofile.asp?Fname=Fatma&Lname=Mili Graduate Study], [http://web.archive.org/web/20120203004703/http://www2.oakland.edu/secs/dispprofile.asp?Fname=Mohamed&Lname=Zohdy Systems Engineering], [http://meadowbrookhall.org/explore/history/meadowbrookhall Oakland University].<br />
<br />
* '''1989. [http://web.archive.org/web/20001206101800/http://www.msu.edu/dig/msumap/psychology.html M.A. Psychology]''', [http://web.archive.org/web/20000902103838/http://www.msu.edu/dig/msumap/beaumont.html Michigan State University].<br />
<br />
* 1985&ndash;1986. [http://quod.lib.umich.edu/cgi/i/image/image-idx?id=S-BHL-X-BL001808%5DBL001808 Graduate Study, Mathematics], [http://www.umich.edu/ University of Michigan].<br />
<br />
* 1985. [http://web.archive.org/web/20061230174652/http://www.uiuc.edu/navigation/buildings/altgeld.top.html Graduate Study, Mathematics], [http://www.uiuc.edu/ University of Illinois at Urbana&ndash;Champaign].<br />
<br />
* 1984. [http://www.psych.uiuc.edu/graduate/ Graduate Study, Psychology], [http://www.uiuc.edu/ University of Illinois at Champaign&ndash;Urbana].<br />
<br />
* '''1980. [http://www.mth.msu.edu/images/wells_medium.jpg M.A. Mathematics]''', [http://www.msu.edu/~hvac/survey/BeaumontTower.html Michigan State University].<br />
<br />
* '''1976. [http://web.archive.org/web/20001206050600/http://www.msu.edu/dig/msumap/phillips.html B.A. Mathematical and Philosophical Method]''', <br> [http://www.enolagaia.com/JMC.html Justin Morrill College], [http://www.msu.edu/ Michigan State University].<br />
<br />
==Category and Subject Interests==<br />
<br />
[[Category:Artificial Intelligence]]<br />
[[Category:Automata Theory]]<br />
[[Category:Category Theory]]<br />
[[Category:Charles Sanders Peirce]]<br />
[[Category:Cognitive Science]] <br />
[[Category:Combinatorics]]<br />
[[Category:Computer Science]]<br />
[[Category:Critical Thinking]]<br />
[[Category:Cybernetics]]<br />
[[Category:Differential Logic]]<br />
[[Category:Education]]<br />
[[Category:Formal Languages]]<br />
[[Category:Formal Sciences]]<br />
[[Category:Graph Theory]]<br />
[[Category:Group Theory]]<br />
[[Category:Hermeneutics]]<br />
[[Category:Information Systems]]<br />
[[Category:Information Theory]]<br />
[[Category:Inquiry]]<br />
[[Category:Inquiry Driven Systems]]<br />
[[Category:Integer Sequences]]<br />
[[Category:Intelligence Amplification]]<br />
[[Category:Learning Organizations]]<br />
[[Category:Linguistics]]<br />
[[Category:Knowledge Representation]]<br />
[[Category:Logic]]<br />
[[Category:Mathematics]]<br />
[[Category:Natural Languages]]<br />
[[Category:Philosophy]]<br />
[[Category:Pragmatics]]<br />
[[Category:Psychology]]<br />
[[Category:Science]]<br />
[[Category:Semantics]]<br />
[[Category:Semiotics]]<br />
[[Category:Statistics]]<br />
[[Category:Systems Science]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Directory:Jon_Awbrey/Differential_Logic_and_Dynamic_Systems_2.0&diff=469880Directory:Jon Awbrey/Differential Logic and Dynamic Systems 2.02021-01-13T18:36:34Z<p>Jon Awbrey: check</p>
<hr />
<div>{{DISPLAYTITLE:Differential Logic and Dynamic Systems 2.0}}<br />
'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''<br />
<br />
{| align="center" cellpadding="10"<br />
| [[Image:Tangent_Functor_Ferris_Wheel.gif]]<br />
|}<br />
<br />
{| style="height:36px; width:100%"<br />
| align="left" | ''Stand and unfold yourself.''<br />
| align="right" | Hamlet: Francsico&mdash;1.1.2<br />
|}<br />
<br />
This article develops a differential extension of propositional calculus and applies it to a context of problems arising in dynamic systems. The work pursued here is coordinated with a parallel application that focuses on neural network systems, but the dependencies are arranged to make the present article the main and the more self-contained work, to serve as a conceptual frame and a technical background for the network project.<br />
<br />
==Review and Transition==<br />
<br />
This note continues a previous discussion on the problem of dealing with change and diversity in logic-based intelligent systems. It is useful to begin by summarizing essential material from previous reports.<br />
<br />
Table 1 outlines a notation for propositional calculus based on two types of logical connectives, both of variable <math>k\!</math>-ary scope.<br />
<br />
* A bracketed list of propositional expressions in the form <math>\texttt{(} e_1, e_2, \ldots, e_{k-1}, e_k \texttt{)}\!</math> indicates that exactly one of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> is false.<br />
<br />
* A concatenation of propositional expressions in the form <math>e_1 ~ e_2 ~ \ldots ~ e_{k-1} ~ e_k</math> indicates that all of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> are true, in other words, that their [[logical conjunction]] is true.<br />
<br />
All other propositional connectives can be obtained in a very efficient style of representation through combinations of these two forms. Strictly speaking, the concatenation form is dispensable in light of the bracketed form, but it is convenient to maintain it as an abbreviation of more complicated bracket expressions.<br />
<br />
This treatment of propositional logic is derived from the work of C.S. Peirce [P1, P2], who gave this approach an extensive development in his graphical systems of predicate, relational, and modal logic [Rob]. More recently, these ideas were revived and supplemented in an alternative interpretation by George Spencer-Brown [SpB]. Both of these authors used other forms of enclosure where I use parentheses, but the structural topologies of expression and the functional varieties of interpretation are fundamentally the same.<br />
<br />
While working with expressions solely in propositional calculus, it is easiest to use plain parentheses for logical connectives. In contexts where parentheses are needed for other purposes &ldquo;teletype&rdquo; parentheses <math>\texttt{(} \ldots \texttt{)}\!</math> or barred parentheses <math>(\!| \ldots |\!)</math> may be used for logical operators.<br />
<br />
The briefest expression for logical truth is the empty word, usually denoted by <math>{}^{\backprime\backprime} \boldsymbol\varepsilon {}^{\prime\prime}\!</math> or <math>{}^{\backprime\backprime} \boldsymbol\lambda {}^{\prime\prime}\!</math> in formal languages, where it forms the identity element for concatenation. To make it visible in this text, it may be denoted by the equivalent expression <math>{}^{\backprime\backprime} \texttt{((~))} {}^{\prime\prime},\!</math> or, especially if operating in an algebraic context, by a simple <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}.\!</math> Also when working in an algebraic mode, the plus sign <math>{}^{\backprime\backprime} + {}^{\prime\prime}\!</math> may be used for [[exclusive disjunction]]. For example, we have the following paraphrases of algebraic expressions by bracket expressions:<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
|<br />
<math>\begin{matrix}<br />
x + y ~=~ \texttt{(} x, y \texttt{)}<br />
\\[6pt]<br />
x + y + z ~=~ \texttt{((} x, y \texttt{)}, z \texttt{)} ~=~ \texttt{(} x, \texttt{(} y, z \texttt{))}<br />
\end{matrix}</math><br />
|}<br />
<br />
It is important to note that the last expressions are not equivalent to the triple bracket <math>\texttt{(} x, y, z \texttt{)}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 1.} ~~ \text{Syntax and Semantics of a Calculus for Propositional Logic}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Expression}~\!</math><br />
| <math>\text{Interpretation}\!</math><br />
| <math>\text{Other Notations}\!</math><br />
|-<br />
| &nbsp;<br />
| <math>\text{True}\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{False}\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>x\!</math><br />
| <math>x\!</math><br />
| <math>x\!</math><br />
|-<br />
| <math>\texttt{(} x \texttt{)}\!</math><br />
| <math>\text{Not}~ x\!</math><br />
|<br />
<math>\begin{matrix}<br />
x'<br />
\\<br />
\tilde{x}<br />
\\<br />
\lnot x<br />
\end{matrix}\!</math><br />
|-<br />
| <math>x~y~z\!</math><br />
| <math>x ~\text{and}~ y ~\text{and}~ z\!</math><br />
| <math>x \land y \land z\!</math><br />
|-<br />
| <math>\texttt{((} x \texttt{)(} y \texttt{)(} z \texttt{))}\!</math><br />
| <math>x ~\text{or}~ y ~\text{or}~ z\!</math><br />
| <math>x \lor y \lor z\!</math><br />
|-<br />
| <math>\texttt{(} x ~ \texttt{(} y \texttt{))}\!</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{implies}~ y<br />
\\<br />
\mathrm{If}~ x ~\text{then}~ y<br />
\end{matrix}</math><br />
| <math>x \Rightarrow y\!</math><br />
|-<br />
| <math>\texttt{(} x \texttt{,} y \texttt{)}\!</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{not equal to}~ y<br />
\\<br />
x ~\text{exclusive or}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x \ne y<br />
\\<br />
x + y<br />
\end{matrix}</math><br />
|-<br />
| <math>\texttt{((} x \texttt{,} y \texttt{))}\!</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{is equal to}~ y<br />
\\<br />
x ~\text{if and only if}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x = y<br />
\\<br />
x \Leftrightarrow y<br />
\end{matrix}</math><br />
|-<br />
| <math>\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Just one of}<br />
\\<br />
x, y, z<br />
\\<br />
\text{is false}.<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x'y~z~ & \lor<br />
\\<br />
x~y'z~ & \lor<br />
\\<br />
x~y~z' &<br />
\end{matrix}</math><br />
|-<br />
| <math>\texttt{((} x \texttt{),(} y \texttt{),(} z \texttt{))}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Just one of}<br />
\\<br />
x, y, z<br />
\\<br />
\text{is true}.<br />
\\<br />
&<br />
\\<br />
\text{Partition all}<br />
\\<br />
\text{into}~ x, y, z.<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x~y'z' & \lor<br />
\\<br />
x'y~z' & \lor<br />
\\<br />
x'y'z~ &<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,} y \texttt{),} z \texttt{)}<br />
\\<br />
&<br />
\\<br />
\texttt{(} x \texttt{,(} y \texttt{,} z \texttt{))}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Oddly many of}<br />
\\<br />
x, y, z<br />
\\<br />
\text{are true}.<br />
\end{matrix}\!</math><br />
|<br />
<p><math>x + y + z\!</math></p><br />
<br><br />
<p><math>\begin{matrix}<br />
x~y~z~ & \lor<br />
\\<br />
x~y'z' & \lor<br />
\\<br />
x'y~z' & \lor<br />
\\<br />
x'y'z~ &<br />
\end{matrix}\!</math></p><br />
|-<br />
| <math>\texttt{(} w \texttt{,(} x \texttt{),(} y \texttt{),(} z \texttt{))}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Partition}~ w<br />
\\<br />
\text{into}~ x, y, z.<br />
\\<br />
&<br />
\\<br />
\text{Genus}~ w ~\text{comprises}<br />
\\<br />
\text{species}~ x, y, z.<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
w'x'y'z' & \lor<br />
\\<br />
w~x~y'z' & \lor<br />
\\<br />
w~x'y~z' & \lor<br />
\\<br />
w~x'y'z~ &<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
'''Note.''' The usage that one often sees, of a plus sign "<math>+\!</math>" to represent inclusive disjunction, and the reference to this operation as ''boolean addition'', is a misnomer on at least two counts. Boole used the plus sign to represent exclusive disjunction (at any rate, an operation of aggregation restricted in its logical interpretation to cases where the represented sets are disjoint (Boole, 32)), as any mathematician with a sensitivity to the ring and field properties of algebra would do:<br />
<br />
<blockquote><br />
The expression <math>x + y\!</math> seems indeed uninterpretable, unless it be assumed that the things represented by <math>x\!</math> and the things represented by <math>y\!</math> are entirely separate; that they embrace no individuals in common. (Boole, 66).<br />
</blockquote><br />
<br />
It was only later that Peirce and Jevons treated inclusive disjunction as a fundamental operation, but these authors, with a respect for the algebraic properties that were already associated with the plus sign, used a variety of other symbols for inclusive disjunction (Sty, 177, 189). It seems to have been Schröder who later reassigned the plus sign to inclusive disjunction (Sty, 208). Additional information, discussion, and references can be found in (Boole) and (Sty, 177&ndash;263). Aside from these historical points, which never really count against a current practice that has gained a life of its own, this usage does have a further disadvantage of cutting or confounding the lines of communication between algebra and logic. For this reason, it will be avoided here.<br />
<br />
==A Functional Conception of Propositional Calculus==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Out of the dimness opposite equals advance . . . .<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Always substance and increase,<br><br />
Always a knit of identity . . . . always distinction . . . .<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;always a breed of life.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]<br />
|}<br />
<br />
In the general case, we start with a set of logical features <math>\{a_1, \ldots, a_n\}</math> that represent properties of objects or propositions about the world. In concrete examples the features <math>\{a_i\!\}</math> commonly appear as capital letters from an ''alphabet'' like <math>\{A, B, C, \ldots\}</math> or as meaningful words from a linguistic ''vocabulary'' of codes. This language can be drawn from any sources, whether natural, technical, or artificial in character and interpretation. In the application to dynamic systems we tend to use the letters <math>\{x_1, \ldots, x_n\}</math> as our coordinate propositions, and to interpret them as denoting properties of a system's ''state'', that is, as propositions about its location in configuration space. Because I have to consider non-deterministic systems from the outset, I often use the word ''state'' in a loose sense, to denote the position or configuration component of a contemplated state vector, whether or not it ever gets a deterministic completion.<br />
<br />
The set of logical features <math>\{a_1, \ldots, a_n\}</math> provides a basis for generating an <math>n\!</math>-dimensional ''[[universe of discourse]]'' that I denote as <math>[a_1, \ldots, a_n].</math> It is useful to consider each universe of discourse as a unified categorical object that incorporates both the set of points <math>\langle a_1, \ldots, a_n \rangle</math> and the set of propositions <math>f : \langle a_1, \ldots, a_n \rangle \to \mathbb{B}</math> that are implicit with the ordinary picture of a venn diagram on <math>n\!</math> features. Thus, we may regard the universe of discourse <math>[a_1, \ldots, a_n]</math> as an ordered pair having the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}),</math> and we may abbreviate this last type designation as <math>\mathbb{B}^n\ +\!\to \mathbb{B},</math> or even more succinctly as <math>[\mathbb{B}^n].</math> (Used this way, the angle brackets <math>\langle\ldots\rangle</math> are referred to as ''generator brackets''.)<br />
<br />
Table 2 exhibits the scheme of notation I use to formalize the domain of propositional calculus, corresponding to the logical content of truth tables and venn diagrams. Although it overworks the square brackets a bit, I also use either one of the equivalent notations <math>[n]\!</math> or <math>\mathbf{n}</math> to denote the data type of a finite set on <math>n\!</math> elements.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 2.} ~~ \text{Propositional Calculus : Basic Notation}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Symbol}\!</math><br />
| <math>\text{Notation}\!</math><br />
| <math>\text{Description}\!</math><br />
| <math>\text{Type}\!</math><br />
|-<br />
| <math>\mathfrak{A}\!</math><br />
| <math>\{ {}^{\backprime\backprime} a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} a_n {}^{\prime\prime} \}\!</math><br />
| <math>\text{Alphabet}\!</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>\mathcal{A}\!</math><br />
| <math>\{ a_1, \ldots, a_n \}\!</math><br />
| <math>\text{Basis}\!</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>A_i\!</math><br />
| <math>\{ \texttt{(} a_i \texttt{)}, a_i \}\!</math><br />
| <math>\text{Dimension}~ i\!</math><br />
| <math>\mathbb{B}\!</math><br />
|-<br />
| <math>A\!</math><br />
|<br />
<math>\begin{matrix}<br />
\langle \mathcal{A} \rangle<br />
\\[2pt]<br />
\langle a_1, \ldots, a_n \rangle<br />
\\[2pt]<br />
\{ (a_1, \ldots, a_n) \}<br />
\\[2pt]<br />
A_1 \times \ldots \times A_n<br />
\\[2pt]<br />
\textstyle \prod_{i=1}^n A_i<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Set of cells},<br />
\\[2pt]<br />
\text{coordinate tuples},<br />
\\[2pt]<br />
\text{points, or vectors}<br />
\\[2pt]<br />
\text{in the universe}<br />
\\[2pt]<br />
\text{of discourse}<br />
\end{matrix}</math><br />
| <math>\mathbb{B}^n\!</math><br />
|-<br />
| <math>A^*\!</math><br />
| <math>(\mathrm{hom} : A \to \mathbb{B})\!</math><br />
| <math>\text{Linear functions}\!</math><br />
| <math>(\mathbb{B}^n)^* \cong \mathbb{B}^n\!</math><br />
|-<br />
| <math>A^\uparrow\!</math><br />
| <math>(A \to \mathbb{B})\!</math><br />
| <math>\text{Boolean functions}\!</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}\!</math><br />
|-<br />
| <math>A^\bullet\!</math><br />
|<br />
<math>\begin{matrix}<br />
[\mathcal{A}]<br />
\\[2pt]<br />
(A, A^\uparrow)<br />
\\[2pt]<br />
(A ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
(A, (A \to \mathbb{B}))<br />
\\[2pt]<br />
[a_1, \ldots, a_n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Universe of discourse}<br />
\\[2pt]<br />
\text{based on the features}<br />
\\[2pt]<br />
\{ a_1, \ldots, a_n \}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))<br />
\\[2pt]<br />
(\mathbb{B}^n ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
[\mathbb{B}^n]<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
===Qualitative Logic and Quantitative Analogy===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
''Logical'', however, is used in a third sense, which is at once more vital and more practical; to denote, namely, the systematic care, negative and positive, taken to safeguard reflection so that it may yield the best results under the given conditions.<br />
| width="4%" | &nbsp;<br />
|- <br />
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 56]<br />
|}<br />
<br />
These concepts and notations may now be explained in greater detail. In order to begin as simply as possible, let us distinguish two levels of analysis and set out initially on the easier path. On the first level of analysis we take spaces like <math>\mathbb{B},</math> <math>\mathbb{B}^n,</math> and <math>(\mathbb{B}^n \to \mathbb{B})</math> at face value and treat them as the primary objects of interest. On the second level of analysis we use these spaces as coordinate charts for talking about points and functions in more fundamental spaces.<br />
<br />
A pair of spaces, of types <math>\mathbb{B}^n</math> and <math>(\mathbb{B}^n \to \mathbb{B}),</math> give typical expression to everything that we commonly associate with the ordinary picture of a venn diagram. The dimension, <math>n,\!</math> counts the number of "circles" or simple closed curves that are inscribed in the universe of discourse, corresponding to its relevant logical features or basic propositions. Elements of type <math>\mathbb{B}^n</math> correspond to what are often called propositional ''interpretations'' in logic, that is, the different assignments of truth values to sentence letters. Relative to a given universe of discourse, these interpretations are visualized as its ''cells'', in other words, the smallest enclosed areas or undivided regions of the venn diagram. The functions <math>f : \mathbb{B}^n \to \mathbb{B}</math> correspond to the different ways of shading the venn diagram to indicate arbitrary propositions, regions, or sets. Regions included under a shading indicate the ''models'', and regions excluded represent the ''non-models'' of a proposition. To recognize and formalize the natural cohesion of these two layers of concepts into a single universe of discourse, we introduce the type notations <math>[\mathbb{B}^n] = \mathbb{B}^n\ +\!\to \mathbb{B}</math> to stand for the pair of types <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})).</math> The resulting "stereotype" serves to frame the universe of discourse as a unified categorical object, and makes it subject to prescribed sets of evaluations and transformations (categorical morphisms or ''arrows'') that affect the universe of discourse as an integrated whole.<br />
<br />
Most of the time we can serve the algebraic, geometric, and logical interests of our study without worrying about their occasional conflicts and incidental divergences. The conventions and definitions already set down will continue to cover most of the algebraic and functional aspects of our discussion, but to handle the logical and qualitative aspects we will need to add a few more. In general, abstract sets may be denoted by gothic, greek, or script capital variants of <math>A, B, C,\!</math> and so on, with their elements being denoted by a corresponding set of subscripted letters in plain lower case, for example, <math>\mathcal{A} = \{a_i\}.</math> Most of the time, a set such as <math>\mathcal{A} = \{a_i\}\!</math> will be employed as the ''alphabet'' of a [[formal language]]. These alphabet letters serve to name the logical features (properties or propositions) that generate a particular universe of discourse. When we want to discuss the particular features of a universe of discourse, beyond the abstract designation of a type like <math>(\mathbb{B}^n\ +\!\to \mathbb{B}),</math> then we may use the following notations. If <math>\mathcal{A} = \{a_1, \ldots, a_n\}</math> is an alphabet of logical features, then <math>A = \langle \mathcal{A} \rangle = \langle a_1, \ldots, a_n \rangle</math> is the set of interpretations, <math>A^\uparrow = (A \to \mathbb{B})</math> is the set of propositions, and <math>A^\bullet = [\mathcal{A}] = [a_1, \ldots, a_n]</math> is the combination of these interpretations and propositions into the universe of discourse that is based on the features <math>\{a_1, \ldots, a_n\}.</math><br />
<br />
As always, especially in concrete examples, these rules may be dropped whenever necessary, reverting to a free assortment of feature labels. However, when we need to talk about the logical aspects of a space that is already named as a vector space, it will be necessary to make special provisions. At any rate, these elaborations can be deferred until actually needed.<br />
<br />
===Philosophy of Notation : Formal Terms and Flexible Types===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Where number is irrelevant, regimented mathematical technique has hitherto tended to be lacking. Thus it is that the progress of natural science has depended so largely upon the discernment of measurable quantity of one sort or another.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]<br />
|}<br />
<br />
For much of our discussion propositions and boolean functions are treated as the same formal objects, or as different interpretations of the same formal calculus. This rule of interpretation has exceptions, though. There is a distinctively logical interest in the use of propositional calculus that is not exhausted by its functional interpretation. It is part of our task in this study to deal with these uniquely logical characteristics as they present themselves both in our subject matter and in our formal calculus. Just to provide a hint of what's at stake: In logic, as opposed to the more imaginative realms of mathematics, we consider it a good thing to always know what we are talking about. Where mathematics encourages tolerance for uninterpreted symbols as intermediate terms, logic exerts a keener effort to interpret directly each oblique carrier of meaning, no matter how slight, and to unfold the complicities of every indirection in the flow of information. Translated into functional terms, this means that we want to maintain a continual, immediate, and persistent sense of both the converse relation <math>f^{-1} \subseteq \mathbb{B} \times \mathbb{B}^n,</math> or what is the same thing, <math>f^{-1} : \mathbb{B} \to \mathcal{P}(\mathbb{B}^n),</math> and the ''fibers'' or inverse images <math>f^{-1}(0)\!</math> and <math>f^{-1}(1),\!</math> associated with each boolean function <math>f : \mathbb{B}^n \to \mathbb{B}</math> that we use. In practical terms, the desired implementation of a propositional interpreter should incorporate our intuitive recognition that the induced partition of the functional domain into level sets <math>f^{-1}(b),\!</math> for <math>b \in \mathbb{B},</math> is part and parcel of understanding the denotative uses of each propositional function <math>f.\!</math><br />
<br />
===Special Classes of Propositions===<br />
<br />
It is important to remember that the coordinate propositions <math>\{a_i\},\!</math> besides being projection maps <math>a_i : \mathbb{B}^n \to \mathbb{B},</math> are propositions on an equal footing with all others, even though employed as a basis in a particular moment. This set of <math>n\!</math> propositions may sometimes be referred to as the ''basic propositions'', the ''coordinate propositions'', or the ''simple propositions'' that found a universe of discourse. Either one of the equivalent notations, <math>\{a_i : \mathbb{B}^n \to \mathbb{B}\}</math> or <math>(\mathbb{B}^n \xrightarrow{i} \mathbb{B}),</math> may be used to indicate the adoption of the propositions <math>a_i\!</math> as a basis for describing a universe of discourse.<br />
<br />
Among the <math>2^{2^n}</math> propositions in <math>[a_1, \ldots, a_n]</math> are several families of <math>2^n\!</math> propositions each that take on special forms with respect to the basis <math>\{ a_1, \ldots, a_n \}.</math> Three of these families are especially prominent in the present context, the ''linear'', the ''positive'', and the ''singular'' propositions. Each family is naturally parameterized by the coordinate <math>n\!</math>-tuples in <math>\mathbb{B}^n</math> and falls into <math>n + 1\!</math> ranks, with a binomial coefficient <math>\tbinom{n}{k}</math> giving the number of propositions that have rank or weight <math>k.\!</math><br />
<br />
<ul><br />
<br />
<li><br />
<p>The ''linear propositions'', <math>\{ \ell : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{\ell} \mathbb{B}),\!</math> may be written as sums:</p><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\sum_{i=1}^n e_i ~=~ e_1 + \ldots + e_n<br />
~\text{where}~<br />
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 0 \end{matrix}\right\}<br />
~\text{for}~ i = 1 ~\text{to}~ n.\!</math><br />
|}<br />
</li><br />
<br />
<li><br />
<p>The ''positive propositions'', <math>\{ p : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{p} \mathbb{B}),\!</math> may be written as products:</p><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\prod_{i=1}^n e_i ~=~ e_1 \cdot \ldots \cdot e_n<br />
~\text{where}~<br />
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 1 \end{matrix}\right\}<br />
~\text{for}~ i = 1 ~\text{to}~ n.\!</math><br />
|}<br />
</li><br />
<br />
<li><br />
<p>The ''singular propositions'', <math>\{ \mathbf{x} : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{s} \mathbb{B}),\!</math> may be written as products:</p><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\prod_{i=1}^n e_i ~=~ e_1 \cdot \ldots \cdot e_n<br />
~\text{where}~<br />
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = \texttt{(} a_i \texttt{)} \end{matrix}\right\}<br />
~\text{for}~ i = 1 ~\text{to}~ n.\!</math><br />
|}<br />
</li><br />
<br />
</ul><br />
<br />
In each case the rank <math>k\!</math> ranges from <math>0\!</math> to <math>n\!</math> and counts the number of positive appearances of the coordinate propositions <math>a_1, \ldots, a_n\!</math> in the resulting expression. For example, for <math>{n = 3},\!</math> the linear proposition of rank <math>0\!</math> is <math>0,\!</math> the positive proposition of rank <math>0\!</math> is <math>1,\!</math> and the singular proposition of rank <math>0\!</math> is <math>\texttt{(} a_1 \texttt{)(} a_2 \texttt{)(} a_3\texttt{)}.\!</math><br />
<br />
The basic propositions <math>a_i : \mathbb{B}^n \to \mathbb{B}</math> are both linear and positive. So these two kinds of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions.<br />
<br />
Linear propositions and positive propositions are generated by taking boolean sums and products, respectively, over selected subsets of basic propositions, so both families of propositions are parameterized by the powerset <math>\mathcal{P}(\mathcal{I}),</math> that is, the set of all subsets <math>J\!</math> of the basic index set <math>\mathcal{I} = \{1, \ldots, n\}.\!</math><br />
<br />
Let us define <math>\mathcal{A}_J</math> as the subset of <math>\mathcal{A}</math> that is given by <math>\{a_i : i \in J\}.\!</math> Then we may comprehend the action of the linear and the positive propositions in the following terms:<br />
<br />
* The linear proposition <math>\ell_J : \mathbb{B}^n \to \mathbb{B}</math> evaluates each cell <math>\mathbf{x}</math> of <math>\mathbb{B}^n</math> by looking at the coefficients of <math>\mathbf{x}</math> with respect to the features that <math>\ell_J</math> "likes", namely those in <math>\mathcal{A}_J,</math> and then adds them up in <math>\mathbb{B}.</math> Thus, <math>\ell_J(\mathbf{x})</math> computes the parity of the number of features that <math>\mathbf{x}</math> has in <math>\mathcal{A}_J,</math> yielding one for odd and zero for even. Expressed in this idiom, <math>\ell_J(\mathbf{x}) = 1</math> says that <math>\mathbf{x}</math> seems ''odd'' (or ''oddly true'') to <math>\mathcal{A}_J,</math> whereas <math>\ell_J(\mathbf{x}) = 0</math> says that <math>\mathbf{x}</math> seems ''even'' (or ''evenly true'') to <math>\mathcal{A}_J,</math> so long as we recall that ''zero times'' is evenly often, too.<br />
<br />
* The positive proposition <math>p_J : \mathbb{B}^n \to \mathbb{B}</math> evaluates each cell <math>\mathbf{x}</math> of <math>\mathbb{B}^n</math> by looking at the coefficients of <math>\mathbf{x}</math> with regard to the features that <math>p_J\!</math> "likes", namely those in <math>\mathcal{A}_J,</math> and then takes their product in <math>\mathbb{B}.</math> Thus, <math>p_J(\mathbf{x})</math> assesses the unanimity of the multitude of features that <math>\mathbf{x}</math> has in <math>\mathcal{A}_J,</math> yielding one for all and aught for else. In these consensual or contractual terms, <math>p_J(\mathbf{x}) = 1</math> means that <math>\mathbf{x}</math> is ''AOK'' or congruent with all of the conditions of <math>\mathcal{A}_J,</math> while <math>p_J(\mathbf{x}) = 0</math> means that <math>\mathbf{x}</math> defaults or dissents from some condition of <math>\mathcal{A}_J.</math><br />
<br />
===Basis Relativity and Type Ambiguity===<br />
<br />
Finally, two things are important to keep in mind with regard to the simplicity, linearity, positivity, and singularity of propositions.<br />
<br />
First, all of these properties are relative to a particular basis. For example, a singular proposition with respect to a basis <math>\mathcal{A}</math> will not remain singular if <math>\mathcal{A}</math> is extended by a number of new and independent features. Even if we stick to the original set of pairwise options <math>\{a_i\} \cup \{ \texttt{(} a_i \texttt{)} \}\!</math> to select a new basis, the sets of linear and positive propositions are determined by the choice of simple propositions, and this determination is tantamount to the conventional choice of a cell as origin.<br />
<br />
Second, the singular propositions <math>\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B},</math> picking out as they do a single cell or a coordinate tuple <math>\mathbf{x}</math> of <math>\mathbb{B}^n,</math> become the carriers or the vehicles of a certain type-ambiguity that vacillates between the dual forms <math>\mathbb{B}^n</math> and <math>(\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B})</math> and infects the whole hierarchy of types built on them. In other words, the terms that signify the interpretations <math>\mathbf{x} : \mathbb{B}^n</math> and the singular propositions <math>\mathbf{x} : \mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B}</math> are fully equivalent in information, and this means that every token of the type <math>\mathbb{B}^n</math> can be reinterpreted as an appearance of the subtype <math>\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B}.</math> And vice versa, the two types can be exchanged with each other everywhere that they turn up. In practical terms, this allows the use of singular propositions as a way of denoting points, forming an alternative to coordinate tuples.<br />
<br />
For example, relative to the universe of discourse <math>[a_1, a_2, a_3]\!</math> the singular proposition <math>a_1 a_2 a_3 : \mathbb{B}^3 \xrightarrow{s} \mathbb{B}</math> could be explicitly retyped as <math>a_1 a_2 a_3 : \mathbb{B}^3</math> to indicate the point <math>(1, 1, 1)\!</math> but in most cases the proper interpretation could be gathered from context. Both notations remain dependent on a particular basis, but the code that is generated under the singular option has the advantage in its self-commenting features, in other words, it constantly reminds us of its basis in the process of denoting points. When the time comes to put a multiplicity of different bases into play, and to search for objects and properties that remain invariant under the transformations between them, this infinitesimal potential advantage may well evolve into an overwhelming practical necessity.<br />
<br />
===The Analogy Between Real and Boolean Types===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Measurement consists in correlating our subject matter with the series of real numbers; and such correlations are desirable because, once they are set up, all the well-worked theory of numerical mathematics lies ready at hand as a tool for our further reasoning.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]<br />
|}<br />
<br />
There are two further reasons why it useful to spend time on a careful treatment of types, and they both have to do with our being able to take full computational advantage of certain dimensions of flexibility in the types that apply to terms. First, the domains of differential geometry and logic programming are connected by analogies between real and boolean types of the same pattern. Second, the types involved in these patterns have important isomorphisms connecting them that apply on both the real and the boolean sides of the picture.<br />
<br />
Amazingly enough, these isomorphisms are themselves schematized by the axioms and theorems of propositional logic. This fact is known as the ''propositions as types'' analogy or the Curry&ndash;Howard isomorphism [How]. In another formulation it says that terms are to types as proofs are to propositions. See [LaS, 42&ndash;46] and [SeH] for a good discussion and further references. To anticipate the bearing of these issues on our immediate topic, Table&nbsp;3 sketches a partial overview of the Real to Boolean analogy that may serve to illustrate the paradigm in question.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 3.} ~~ \text{Analogy Between Real and Boolean Types}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Real Domain} ~ \mathbb{R}\!</math><br />
| <math>\longleftrightarrow\!</math><br />
| <math>\text{Boolean Domain} ~ \mathbb{B}\!</math><br />
|-<br />
| <math>\mathbb{R}^n\!</math><br />
| <math>\text{Basic Space}\!</math><br />
| <math>\mathbb{B}^n\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}\!</math><br />
| <math>\text{Function Space}\!</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R}\!</math><br />
| <math>\text{Tangent Vector}\!</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to ((\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R})\!</math><br />
| <math>\text{Vector Field}\!</math><br />
| <math>\mathbb{B}^n \to ((\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B})\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \times (\mathbb{R}^n \to \mathbb{R})) \to \mathbb{R}\!</math><br />
| '''<font size="4">"</font>'''<br />
| <math>(\mathbb{B}^n \times (\mathbb{B}^n \to \mathbb{B})) \to \mathbb{B}\!</math><br />
|-<br />
| <math>((\mathbb{R}^n \to \mathbb{R}) \times \mathbb{R}^n) \to \mathbb{R}\!</math><br />
| '''<font size="4">"</font>'''<br />
| <math>((\mathbb{B}^n \to \mathbb{B}) \times \mathbb{B}^n) \to \mathbb{B}\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^n \to \mathbb{R})~\!</math><br />
| <math>\text{Derivation}\!</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^n \to \mathbb{B})\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}^m\!</math><br />
| <math>\begin{matrix}\text{Basic}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}^m\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^m \to \mathbb{R})\!</math><br />
| <math>\begin{matrix}\text{Function}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^m \to \mathbb{B})\!</math><br />
|}<br />
<br />
<br><br />
<br />
The Table exhibits a sample of likely parallels between the real and boolean domains. The central column gives a selection of terminology that is borrowed from differential geometry and extended in its meaning to the logical side of the Table. These are the varieties of spaces that come up when we turn to analyzing the dynamics of processes that pursue their courses through the states of an arbitrary space <math>X.\!</math> Moreover, when it becomes necessary to approach situations of overwhelming dynamic complexity in a succession of qualitative reaches, then the methods of logic that are afforded by the boolean domains, with their declarative means of synthesis and deductive modes of analysis, supply a natural battery of tools for the task.<br />
<br />
It is usually expedient to take these spaces two at a time, in dual pairs of the form <math>X\!</math> and <math>(X \to \mathbb{K}).</math> In general, one creates pairs of type schemas by replacing any space <math>X\!</math> with its dual <math>(X \to \mathbb{K}),</math> for example, pairing the type <math>X \to Y</math> with the type <math>(X \to \mathbb{K}) \to (Y \to \mathbb{K}),</math> and <math>X \times Y</math> with <math>(X \to \mathbb{K}) \times (Y \to \mathbb{K}).</math> The word ''dual'' is used here in its broader sense to mean all of the functionals, not just the linear ones. Given any function <math>f : X \to \mathbb{K},</math> the ''converse'' or inverse relation corresponding to <math>f\!</math> is denoted <math>f^{-1},\!</math> and the subsets of <math>X\!</math> that are defined by <math>f^{-1}(k),\!</math> taken over <math>k\!</math> in <math>\mathbb{K},</math> are called the ''fibers'' or the ''level sets'' of the function <math>f.\!</math><br />
<br />
===Theory of Control and Control of Theory===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
You will hardly know who I am or what I mean,<br><br />
But I shall be good health to you nevertheless,<br><br />
And filter and fibre your blood.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]<br />
|}<br />
<br />
In the boolean context a function <math>f : X \to \mathbb{B}</math> is tantamount to a ''proposition'' about elements of <math>X,\!</math> and the elements of <math>X\!</math> constitute the ''interpretations'' of that proposition. The fiber <math>f^{-1}(1)\!</math> comprises the set of ''models'' of <math>f,\!</math> or examples of elements in <math>X\!</math> satisfying the proposition <math>f.\!</math> The fiber <math>f^{-1}(0)\!</math> collects the complementary set of ''anti-models'', or the exceptions to the proposition <math>f\!</math> that exist in <math>X.\!</math> Of course, the space of functions <math>(X \to \mathbb{B})\!</math> is isomorphic to the set of all subsets of <math>X,\!</math> called the ''power set'' of <math>X,\!</math> and often denoted <math>\mathcal{P}(X)</math> or <math>2^X.\!</math><br />
<br />
The operation of replacing <math>X\!</math> by <math>(X \to \mathbb{B})\!</math> in a type schema corresponds to a certain shift of attitude towards the space <math>X,\!</math> in which one passes from a focus on the ostensibly individual elements of <math>X\!</math> to a concern with the states of information and uncertainty that one possesses about objects and situations in <math>X.\!</math> The conceptual obstacles in the path of this transition can be smoothed over by using singular functions <math>(\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B})</math> as stepping stones. First of all, it's an easy step from an element <math>\mathbf{x}</math> of type <math>\mathbb{B}^n</math> to the equivalent information of a singular proposition <math>\mathbf{x} : X \xrightarrow{s} \mathbb{B}, </math> and then only a small jump of generalization remains to reach the type of an arbitrary proposition <math>f : X \to \mathbb{B},</math> perhaps understood to indicate a relaxed constraint on the singularity of points or a neighborhood circumscribing the original <math>\mathbf{x}.</math> This is frequently a useful transformation, communicating between the ''objective'' and the ''intentional'' perspectives, in spite perhaps of the open objection that this distinction is transient in the mean time and ultimately superficial.<br />
<br />
It is hoped that this measure of flexibility, allowing us to stretch a point into a proposition, can be useful in the examination of inquiry driven systems, where the differences between empirical, intentional, and theoretical propositions constitute the discrepancies and the distributions that drive experimental activity. I can give this model of inquiry a cybernetic cast by realizing that theory change and theory evolution, as well as the choice and the evaluation of experiments, are actions that are taken by a system or its agent in response to the differences that are detected between observational contents and theoretical coverage.<br />
<br />
All of the above notwithstanding, there are several points that distinguish these two tasks, namely, the ''theory of control'' and the ''control of theory'', features that are often obscured by too much precipitation in the quickness with which we understand their similarities. In the control of uncertainty through inquiry, some of the actuators that we need to be concerned with are axiom changers and theory modifiers, operators with the power to compile and to revise the theories that generate expectations and predictions, effectors that form and edit our grammars for the languages of observational data, and agencies that rework the proposed model to fit the actual sequences of events and the realized relationships of values that are observed in the environment. Moreover, when steps must be taken to carry out an experimental action, there must be something about the particular shape of our uncertainty that guides us in choosing what directions to explore, and this impression is more than likely influenced by previous accumulations of experience. Thus it must be anticipated that much of what goes into scientific progress, or any sustainable effort toward a goal of knowledge, is necessarily predicated on long term observation and modal expectations, not only on the more local or short term prediction and correction.<br />
<br />
===Propositions as Types and Higher Order Types===<br />
<br />
The types collected in Table&nbsp;3 (repeated below) serve to illustrate the themes of ''higher order propositional expressions'' and the ''propositions as types'' (PAT) analogy.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 3.} ~~ \text{Analogy Between Real and Boolean Types}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Real Domain} ~ \mathbb{R}\!</math><br />
| <math>\longleftrightarrow\!</math><br />
| <math>\text{Boolean Domain} ~ \mathbb{B}\!</math><br />
|-<br />
| <math>\mathbb{R}^n\!</math><br />
| <math>\text{Basic Space}\!</math><br />
| <math>\mathbb{B}^n\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}\!</math><br />
| <math>\text{Function Space}\!</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R}\!</math><br />
| <math>\text{Tangent Vector}\!</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to ((\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R})\!</math><br />
| <math>\text{Vector Field}\!</math><br />
| <math>\mathbb{B}^n \to ((\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B})\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \times (\mathbb{R}^n \to \mathbb{R})) \to \mathbb{R}\!</math><br />
| '''<font size="4">"</font>'''<br />
| <math>(\mathbb{B}^n \times (\mathbb{B}^n \to \mathbb{B})) \to \mathbb{B}\!</math><br />
|-<br />
| <math>((\mathbb{R}^n \to \mathbb{R}) \times \mathbb{R}^n) \to \mathbb{R}\!</math><br />
| '''<font size="4">"</font>'''<br />
| <math>((\mathbb{B}^n \to \mathbb{B}) \times \mathbb{B}^n) \to \mathbb{B}\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^n \to \mathbb{R})~\!</math><br />
| <math>\text{Derivation}\!</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^n \to \mathbb{B})\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}^m\!</math><br />
| <math>\begin{matrix}\text{Basic}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}^m\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^m \to \mathbb{R})\!</math><br />
| <math>\begin{matrix}\text{Function}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^m \to \mathbb{B})\!</math><br />
|}<br />
<br />
<br><br />
<br />
First, observe that the type of a ''tangent vector at a point'', also known as a ''directional derivative'' at that point, has the form <math>(\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K},</math> where <math>\mathbb{K}</math> is the chosen ground field, in the present case either <math>\mathbb{R}</math> or <math>\mathbb{B}.</math> At a point in a space of type <math>\mathbb{K}^n,</math> a directional derivative operator <math>\vartheta\!</math> takes a function on that space, an <math>f\!</math> of type <math>(\mathbb{K}^n \to \mathbb{K}),</math> and maps it to a ground field value of type <math>\mathbb{K}.</math> This value is known as the ''derivative'' of <math>f\!</math> in the direction <math>\vartheta\!</math> [Che46, 76&ndash;77]. In the boolean case <math>\vartheta : (\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math> has the form of a proposition about propositions, in other words, a proposition of the next higher type.<br />
<br />
Next, by way of illustrating the propositions as types idea, consider a proposition of the form <math>X \Rightarrow (Y \Rightarrow Z).</math> One knows from propositional calculus that this is logically equivalent to a proposition of the form <math>(X \land Y) \Rightarrow Z.</math> But this equivalence should remind us of the functional isomorphism that exists between a construction of the type <math>X \to (Y \to Z)</math> and a construction of the type <math>(X \times Y) \to Z.</math> The propositions as types analogy permits us to take a functional type like this and, under the right conditions, replace the functional arrows &ldquo;<math>\to~\!</math>&rdquo; and products &ldquo;<math>\times\!</math>&rdquo; with the respective logical arrows &ldquo;<math>\Rightarrow\!</math>&rdquo; and products &ldquo;<math>\land\!</math>&rdquo;. Accordingly, viewing the result as a proposition, we can employ axioms and theorems of propositional calculus to suggest appropriate isomorphisms among the categorical and functional constructions.<br />
<br />
Finally, examine the middle four rows of Table&nbsp;3. These display a series of isomorphic types that stretch from the categories that are labeled ''Vector Field'' to those that are labeled ''Derivation''. A ''vector field'', also known as an ''infinitesimal transformation'', associates a tangent vector at a point with each point of a space. In symbols, a vector field is a function of the form <math>\textstyle \xi : X \to \bigcup_{x \in X} \xi_x\!</math> that assigns to each point <math>x\!</math> of the space <math>X\!</math> a tangent vector to <math>X\!</math> at that point, namely, the tangent vector <math>\xi_x\!</math> [Che46, 82&ndash;83]. If <math>X\!</math> is of the type <math>\mathbb{K}^n,</math> then <math>\xi\!</math> is of the type <math>\mathbb{K}^n \to ((\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K}).</math> This has the pattern <math>X \to (Y \to Z),</math> with <math>X = \mathbb{K}^n,</math> <math>Y = (\mathbb{K}^n \to \mathbb{K}),</math> and <math>Z = \mathbb{K}.</math><br />
<br />
Applying the propositions as types analogy, one can follow this pattern through a series of metamorphoses from the type of a vector field to the type of a derivation, as traced out in Table&nbsp;4. Observe how the function <math>f : X \to \mathbb{K},</math> associated with the place of <math>Y\!</math> in the pattern, moves through its paces from the second to the first position. In this way, the vector field <math>\xi,\!</math> initially viewed as attaching each tangent vector <math>\xi_x\!</math> to the site <math>x\!</math> where it acts in <math>X,\!</math> now comes to be seen as acting on each scalar potential <math>f : X \to \mathbb{K}</math> like a generalized species of differentiation, producing another function <math>\xi f : X \to \mathbb{K}</math> of the same type.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 4.} ~~ \text{An Equivalence Based on the Propositions as Types Analogy}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Pattern}\!</math><br />
| <math>\text{Construct}\!</math><br />
| <math>\text{Instance}\!</math><br />
|-<br />
| <math>X \to (Y \to Z)\!</math><br />
| <math>\text{Vector Field}\!</math><br />
| <math>\mathbb{K}^n \to ((\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K})\!</math><br />
|-<br />
| <math>(X \times Y) \to Z\!</math><br />
| <math>\Uparrow\!</math><br />
| <math>(\mathbb{K}^n \times (\mathbb{K}^n \to \mathbb{K})) \to \mathbb{K}\!</math><br />
|-<br />
| <math>(Y \times X) \to Z\!</math><br />
| <math>\Downarrow\!</math><br />
| <math>((\mathbb{K}^n \to \mathbb{K}) \times \mathbb{K}^n) \to \mathbb{K}\!</math><br />
|-<br />
| <math>Y \to (X \to Z)\!</math><br />
| <math>\text{Derivation}\!</math><br />
| <math>(\mathbb{K}^n \to \mathbb{K}) \to (\mathbb{K}^n \to \mathbb{K})\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Reality at the Threshold of Logic===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
But no science can rest entirely on measurement, and many scientific investigations are quite out of reach of that device. To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]<br />
|}<br />
<br />
Table 5 accumulates an array of notation that I hope will not be too distracting. Some of it is rarely needed, but has been filled in for the sake of completeness. Its purpose is simple, to give literal expression to the visual intuitions that come with venn diagrams, and to help build a bridge between our qualitative and quantitative outlooks on dynamic systems.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 5.} ~~ \text{A Bridge Over Troubled Waters}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| align="center" | <math>\text{Linear Space}\!</math><br />
| align="center" | <math>\text{Liminal Space}\!</math><br />
| align="center" | <math>\text{Logical Space}\!</math><br />
|-<br />
| <math>\begin{matrix}\mathcal{X} & = & \{ x_1, \ldots, x_n \}\end{matrix}</math><br />
| <math>\begin{matrix}\underline{\mathcal{X}} & = & \{ \underline{x}_1, \ldots, \underline{x}_n \}\end{matrix}</math><br />
| <math>\begin{matrix}\mathcal{A} & = & \{ a_1, \ldots, a_n \}\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X_i & = & \langle x_i \rangle<br />
\\<br />
& \cong & \mathbb{K}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}_i & = & \{ \texttt{(} \underline{x}_i \texttt{)}, \underline{x}_i \}<br />
\\<br />
& \cong & \mathbb{B}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A_i & = & \{ \texttt{(} a_i \texttt{)}, a_i \}<br />
\\<br />
& \cong & \mathbb{B}<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X<br />
\\<br />
= & \langle \mathcal{X} \rangle<br />
\\<br />
= & \langle x_1, \ldots, x_n \rangle<br />
\\<br />
= & X_1 \times \ldots \times X_n<br />
\\<br />
= & \prod_{i=1}^n X_i<br />
\\<br />
\cong & \mathbb{K}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}<br />
\\<br />
= & \langle \underline{\mathcal{X}} \rangle<br />
\\<br />
= & \langle \underline{x}_1, \ldots, \underline{x}_n \rangle<br />
\\<br />
= & \underline{X}_1 \times \ldots \times \underline{X}_n<br />
\\<br />
= & \prod_{i=1}^n \underline{X}_i<br />
\\<br />
\cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A<br />
\\<br />
= & \langle \mathcal{A} \rangle<br />
\\<br />
= & \langle a_1, \ldots, a_n \rangle<br />
\\<br />
= & A_1 \times \ldots \times A_n<br />
\\<br />
= & \prod_{i=1}^n A_i<br />
\\<br />
\cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X^* & = & (\ell : X \to \mathbb{K})<br />
\\<br />
& \cong & \mathbb{K}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}^* & = & (\ell : \underline{X} \to \mathbb{B})<br />
\\<br />
& \cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A^* & = & (\ell : A \to \mathbb{B})<br />
\\<br />
& \cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X^\uparrow & = & (X \to \mathbb{K})<br />
\\<br />
& \cong & (\mathbb{K}^n \to \mathbb{K})<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}^\uparrow & = & (\underline{X} \to \mathbb{B})<br />
\\<br />
& \cong & (\mathbb{B}^n \to \mathbb{B})<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A^\uparrow & = & (A \to \mathbb{B})<br />
\\<br />
& \cong & (\mathbb{B}^n \to \mathbb{B})<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X^\bullet<br />
\\<br />
= & [\mathcal{X}]<br />
\\<br />
= & [x_1, \ldots, x_n]<br />
\\<br />
= & (X, X^\uparrow)<br />
\\<br />
= & (X ~+\!\to \mathbb{K})<br />
\\<br />
= & (X, (X \to \mathbb{K}))<br />
\\<br />
\cong & (\mathbb{K}^n, (\mathbb{K}^n \to \mathbb{K}))<br />
\\<br />
= & (\mathbb{K}^n ~+\!\to \mathbb{K})<br />
\\<br />
= & [\mathbb{K}^n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}^\bullet<br />
\\<br />
= & [\underline{\mathcal{X}}]<br />
\\<br />
= & [\underline{x}_1, \ldots, \underline{x}_n]<br />
\\<br />
= & (\underline{X}, \underline{X}^\uparrow)<br />
\\<br />
= & (\underline{X} ~+\!\to \mathbb{B})<br />
\\<br />
= & (\underline{X}, (\underline{X} \to \mathbb{B}))<br />
\\<br />
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))<br />
\\<br />
= & (\mathbb{B}^n ~+\!\to \mathbb{B})<br />
\\<br />
= & [\mathbb{B}^n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A^\bullet<br />
\\<br />
= & [\mathcal{A}]<br />
\\<br />
= & [a_1, \ldots, a_n]<br />
\\<br />
= & (A, A^\uparrow)<br />
\\<br />
= & (A ~+\!\to \mathbb{B})<br />
\\<br />
= & (A, (A \to \mathbb{B}))<br />
\\<br />
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))<br />
\\<br />
= & (\mathbb{B}^n ~+\!\to \mathbb{B})<br />
\\<br />
= & [\mathbb{B}^n]<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
The left side of the Table collects mostly standard notation for an <math>n\!</math>-dimensional vector space over a field <math>\mathbb{K}.</math> The right side of the table repeats the first elements of a notation that I sketched above, to be used in further developments of propositional calculus. (I plan to use this notation in the logical analysis of neural network systems.) The middle column of the table is designed as a transitional step from the case of an arbitrary field <math>\mathbb{K},</math> with a special interest in the continuous line <math>\mathbb{R},</math> to the qualitative and discrete situations that are instanced and typified by <math>\mathbb{B}.</math><br />
<br />
I now proceed to explain these concepts in more detail. The most important ideas developed in Table&nbsp;5 are these:<br />
<br />
* The idea of a universe of discourse, which includes both a space of ''points'' and a space of ''maps'' on those points.<br />
<br />
* The idea of passing from a more complex universe to a simpler universe by a process of ''thresholding'' each dimension of variation down to a single bit of information.<br />
<br />
For the sake of concreteness, let us suppose that we start with a continuous <math>n\!</math>-dimensional vector space like <math>X = \langle x_1, \ldots, x_n \rangle \cong \mathbb{R}^n.</math> The coordinate system <math>\mathcal{X} = \{x_i\}</math> is a set of maps <math>x_i : \mathbb{R}^n \to \mathbb{R},</math> also known as the ''coordinate projections''. Given a "dataset" of points <math>\mathbf{x}</math> in <math>\mathbb{R}^n,</math> there are numerous ways of sensibly reducing the data down to one bit for each dimension. One strategy that is general enough for our present purposes is as follows. For each <math>i\!</math> we choose an <math>n\!</math>-ary relation <math>L_i\!</math> on <math>\mathbb{R}^n,</math> that is, a subset of <math>\mathbb{R}^n,</math> and then we define the <math>i^\mathrm{th}\!</math> threshold map, or ''limen'' <math>\underline{x}_i</math> as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\underline{x}_i : \mathbb{R}^n \to \mathbb{B}\ \text{such that:}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
\underline{x}_i(\mathbf{x}) = 1 & \text{if} & \mathbf{x} \in L_i,<br />
\\[4pt]<br />
\underline{x}_i(\mathbf{x}) = 0 & \text{if} & \mathbf{x} \not\in L_i.<br />
\end{matrix}</math><br />
|}<br />
<br />
In other notations that are sometimes used, the operator <math>\chi (\ldots)</math> or the corner brackets <math>\lceil\ldots\rceil</math> can be used to denote a ''characteristic function'', that is, a mapping from statements to their truth values in <math>\mathbb{B}.</math> Finally, it is not uncommon to use the name of the relation itself as a predicate that maps <math>n\!</math>-tuples into truth values. Thus we have the following notational variants of the above definition:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
\underline{x}_i (\mathbf{x}) & = & \chi (\mathbf{x} \in L_i) & = & \lceil \mathbf{x} \in L_i \rceil & = & L_i (\mathbf{x}).<br />
\end{matrix}</math><br />
|}<br />
<br />
Notice that, as defined here, there need be no actual relation between the <math>n\!</math>-dimensional subsets <math>\{L_i\}\!</math> and the coordinate axes corresponding to <math>\{x_i\},\!</math> aside from the circumstance that the two sets have the same cardinality. In concrete cases, though, one usually has some reason for associating these "volumes" with these "lines", for instance, <math>L_i\!</math> is bounded by some hyperplane that intersects the <math>i^\text{th}\!</math> axis at a unique threshold value <math>r_i \in \mathbb{R}.\!</math> Often, the hyperplane is chosen normal to the axis. In recognition of this motive, let us make the following convention. When the set <math>L_i\!</math> has points on the <math>i^\text{th}\!</math> axis, that is, points of the form <math>(0, \ldots, 0, r_i, 0, \ldots, 0)</math> where only the <math>x_i\!</math> coordinate is possibly non-zero, we may pick any one of these coordinate values as a parametric index of the relation. In this case we say that the indexing is ''real'', otherwise the indexing is ''imaginary''. For a knowledge based system <math>X,\!</math> this should serve once again to mark the distinction between ''acquaintance'' and ''opinion''.<br />
<br />
States of knowledge about the location of a system or about the distribution of a population of systems in a state space <math>X = \mathbb{R}^n</math> can now be expressed by taking the set <math>\underline{\mathcal{X}} = \{\underline{x}_i\}</math> as a basis of logical features. In picturesque terms, one may think of the underscore and the subscript as combining to form a subtextual spelling for the <math>i^\text{th}\!</math> threshold map. This can help to remind us that the ''threshold operator'' <math>(\underline{~})_i</math> acts on <math>\mathbf{x}</math> by setting up a kind of a &ldquo;hurdle&rdquo; for it. In this interpretation the coordinate proposition <math>\underline{x}_i</math> asserts that the representative point <math>\mathbf{x}</math> resides ''above'' the <math>i^\mathrm{th}\!</math> threshold.<br />
<br />
Primitive assertions of the form <math>\underline{x}_i (\mathbf{x})</math> may then be negated and joined by means of propositional connectives in the usual ways to provide information about the state <math>\mathbf{x}</math> of a contemplated system or a statistical ensemble of systems. Parentheses <math>\texttt{(} \ldots \texttt{)}\!</math> may be used to indicate logical negation. Eventually one discovers the usefulness of the <math>k\!</math>-ary ''just one false'' operators of the form <math>\texttt{(} a_1 \texttt{,} \ldots \texttt{,} a_k \texttt{)},\!</math> as treated in earlier reports. This much tackle generates a space of points (cells, interpretations), <math>\underline{X} \cong \mathbb{B}^n,</math> and a space of functions (regions, propositions), <math>\underline{X}^\uparrow \cong (\mathbb{B}^n \to \mathbb{B}).</math> Together these form a new universe of discourse <math>\underline{X}^\bullet</math> of the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})),</math> which we may abbreviate as <math>\mathbb{B}^n\ +\!\to \mathbb{B}</math> or most succinctly as <math>[\mathbb{B}^n].</math><br />
<br />
The square brackets have been chosen to recall the rectangular frame of a venn diagram. In thinking about a universe of discourse it is a good idea to keep this picture in mind, graphically illustrating the links among the elementary cells <math>\underline{\mathbf{x}},</math> the defining features <math>\underline{x}_i,</math> and the potential shadings <math>f : \underline{X} \to \mathbb{B}</math> all at the same time, not to mention the arbitrariness of the way we choose to inscribe our distinctions in the medium of a continuous space.<br />
<br />
Finally, let <math>X^*\!</math> denote the space of linear functions, <math>(\ell : X \to \mathbb{K}),</math> which has in the finite case the same dimensionality as <math>X,\!</math> and let the same notation be extended across the Table.<br />
<br />
We have just gone through a lot of work, apparently doing nothing more substantial than spinning a complex spell of notational devices through a labyrinth of baffled spaces and baffling maps. The reason for doing this was to bind together and to constitute the intuitive concept of a universe of discourse into a coherent categorical object, the kind of thing, once grasped, that can be turned over in the mind and considered in all its manifold changes and facets. The effort invested in these preliminary measures is intended to pay off later, when we need to consider the state transformations and the time evolution of neural network systems.<br />
<br />
===Tables of Propositional Forms===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope. It provides explicit techniques for manipulating the most basic ingredients of discourse.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7&ndash;8]<br />
|}<br />
<br />
To prepare for the next phase of discussion, Tables&nbsp;6 and 7 collect and summarize all of the propositional forms on one and two variables. These propositional forms are represented over bases of boolean variables as complete sets of boolean-valued functions. Adjacent to their names and specifications are listed what are roughly the simplest expressions in the ''[[cactus language]]'', the particular syntax for propositional calculus that I use in formal and computational contexts. For the sake of orientation, the English paraphrases and the more common notations are listed in the last two columns. As simple and circumscribed as these low-dimensional universes may appear to be, a careful exploration of their differential extensions will involve us in complexities sufficient to demand our attention for some time to come.<br />
<br />
Propositional forms on one variable correspond to boolean functions <math>f : \mathbb{B}^1 \to \mathbb{B}.</math> In Table&nbsp;6 these functions are listed in a variant form of [[truth table]], one in which the axes of the usual arrangement are rotated through a right angle. Each function <math>f_i\!</math> is indexed by the string of values that it takes on the points of the universe <math>X^\bullet = [x] \cong \mathbb{B}^1.</math> The binary index generated in this way is converted to its decimal equivalent and these are used as conventional names for the <math>f_i,\!</math> as shown in the first column of the Table. In their own right the <math>2^1\!</math> points of the universe <math>X^\bullet</math> are coordinated as a space of type <math>\mathbb{B}^1,</math> this in light of the universe <math>X^\bullet</math> being a functional domain where the coordinate projection <math>x\!</math> takes on its values in <math>\mathbb{B}.</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 6.} ~~ \text{Propositional Forms on One Variable}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_0\!</math><br />
| <math>f_{00}\!</math><br />
| <math>0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>f_1\!</math><br />
| <math>f_{01}\!</math><br />
| <math>0~1\!</math><br />
| <math>\texttt{(} x \texttt{)}\!</math><br />
| <math>\text{not}~ x\!</math><br />
| <math>\lnot x\!</math><br />
|-<br />
| <math>f_2\!</math><br />
| <math>f_{10}\!</math><br />
| <math>1~0\!</math><br />
| <math>x\!</math><br />
| <math>x\!</math><br />
| <math>x\!</math><br />
|-<br />
| <math>f_3\!</math><br />
| <math>f_{11}\!</math><br />
| <math>1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
Propositional forms on two variables correspond to boolean functions <math>f : \mathbb{B}^2 \to \mathbb{B}.</math> In Table&nbsp;7 each function <math>f_i\!</math> is indexed by the values that it takes on the points of the universe <math>X^\bullet = [x, y] \cong \mathbb{B}^2.</math> Converting the binary index thus generated to a decimal equivalent, we obtain the functional nicknames that are listed in the first column. The <math>2^2\!</math> points of the universe <math>X^\bullet</math> are coordinated as a space of type <math>\mathbb{B}^2,</math> as indicated under the heading of the Table, where the coordinate projections <math>x\!</math> and <math>y\!</math> run through the various combinations of their values in <math>\mathbb{B}.</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 7-a.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}\!</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}\!</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}\!</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}\!</math><br />
| style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}\!</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}\!</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[4pt]<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\\[4pt]<br />
f_{3}<br />
\\[4pt]<br />
f_{4}<br />
\\[4pt]<br />
f_{5}<br />
\\[4pt]<br />
f_{6}<br />
\\[4pt]<br />
f_{7}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0000}<br />
\\[4pt]<br />
f_{0001}<br />
\\[4pt]<br />
f_{0010}<br />
\\[4pt]<br />
f_{0011}<br />
\\[4pt]<br />
f_{0100}<br />
\\[4pt]<br />
f_{0101}<br />
\\[4pt]<br />
f_{0110}<br />
\\[4pt]<br />
f_{0111}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~0<br />
\\[4pt]<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~0~1~1<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
0~1~0~1<br />
\\[4pt]<br />
0~1~1~0<br />
\\[4pt]<br />
0~1~1~1<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{)} ~ y ~<br />
\\[4pt]<br />
\texttt{(} x \texttt{)}<br />
\\[4pt]<br />
~ x ~ \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{,} ~ y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x ~ y \texttt{)}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{false}<br />
\\[4pt]<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\[4pt]<br />
y ~\text{without}~ x<br />
\\[4pt]<br />
\text{not}~ x<br />
\\[4pt]<br />
x ~\text{without}~ y<br />
\\[4pt]<br />
\text{not}~ y<br />
\\[4pt]<br />
x ~\text{not equal to}~ y<br />
\\[4pt]<br />
\text{not both}~ x ~\text{and}~ y<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
\lnot x \land \lnot y<br />
\\[4pt]<br />
\lnot x \land y<br />
\\[4pt]<br />
\lnot x<br />
\\[4pt]<br />
x \land \lnot y<br />
\\[4pt]<br />
\lnot y<br />
\\[4pt]<br />
x \ne y<br />
\\[4pt]<br />
\lnot x \lor \lnot y<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{8}<br />
\\[4pt]<br />
f_{9}<br />
\\[4pt]<br />
f_{10}<br />
\\[4pt]<br />
f_{11}<br />
\\[4pt]<br />
f_{12}<br />
\\[4pt]<br />
f_{13}<br />
\\[4pt]<br />
f_{14}<br />
\\[4pt]<br />
f_{15}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1000}<br />
\\[4pt]<br />
f_{1001}<br />
\\[4pt]<br />
f_{1010}<br />
\\[4pt]<br />
f_{1011}<br />
\\[4pt]<br />
f_{1100}<br />
\\[4pt]<br />
f_{1101}<br />
\\[4pt]<br />
f_{1110}<br />
\\[4pt]<br />
f_{1111}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
1~0~0~0<br />
\\[4pt]<br />
1~0~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\\[4pt]<br />
1~1~1~1<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x ~ y<br />
\\[4pt]<br />
\texttt{((} x \texttt{,} ~ y \texttt{))}<br />
\\[4pt]<br />
y<br />
\\[4pt]<br />
\texttt{(} x ~ \texttt{(} y \texttt{))}<br />
\\[4pt]<br />
x<br />
\\[4pt]<br />
\texttt{((} x \texttt{)} ~ y \texttt{)}<br />
\\[4pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
\texttt{((~))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x ~\text{and}~ y<br />
\\[4pt]<br />
x ~\text{equal to}~ y<br />
\\[4pt]<br />
y<br />
\\[4pt]<br />
\text{not}~ x ~\text{without}~ y<br />
\\[4pt]<br />
x<br />
\\[4pt]<br />
\text{not}~ y ~\text{without}~ x<br />
\\[4pt]<br />
x ~\text{or}~ y<br />
\\[4pt]<br />
\text{true}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x \land y<br />
\\[4pt]<br />
x = y<br />
\\[4pt]<br />
y<br />
\\[4pt]<br />
x \Rightarrow y<br />
\\[4pt]<br />
x<br />
\\[4pt]<br />
x \Leftarrow y<br />
\\[4pt]<br />
x \lor y<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 7-b.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}</math><br />
| style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>f_{0000}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\\[4pt]<br />
f_{4}<br />
\\[4pt]<br />
f_{8}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0001}<br />
\\[4pt]<br />
f_{0010}<br />
\\[4pt]<br />
f_{0100}<br />
\\[4pt]<br />
f_{1000}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
1~0~0~0<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{)} ~ y ~<br />
\\[4pt]<br />
~ x ~ \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
~ x ~~ y ~<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\[4pt]<br />
y ~\text{without}~ x<br />
\\[4pt]<br />
x ~\text{without}~ y<br />
\\[4pt]<br />
x ~\text{and}~ y<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot x \land \lnot y<br />
\\[4pt]<br />
\lnot x \land y<br />
\\[4pt]<br />
x \land \lnot y<br />
\\[4pt]<br />
x \land y<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{3}<br />
\\[4pt]<br />
f_{12}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0011}<br />
\\[4pt]<br />
f_{1100}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\[4pt]<br />
x<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{not}~ x<br />
\\[4pt]<br />
x<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot x<br />
\\[4pt]<br />
x<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[4pt]<br />
f_{9}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0110}<br />
\\[4pt]<br />
f_{1001}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[4pt]<br />
1~0~0~1<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{,} y \texttt{)}<br />
\\[4pt]<br />
\texttt{((} x \texttt{,} y \texttt{))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x ~\text{not equal to}~ y<br />
\\[4pt]<br />
x ~\text{equal to}~ y<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x \ne y<br />
\\[4pt]<br />
x = y<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{5}<br />
\\[4pt]<br />
f_{10}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0101}<br />
\\[4pt]<br />
f_{1010}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\[4pt]<br />
y<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{not}~ y<br />
\\[4pt]<br />
y<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot y<br />
\\[4pt]<br />
y<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[4pt]<br />
f_{11}<br />
\\[4pt]<br />
f_{13}<br />
\\[4pt]<br />
f_{14}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0111}<br />
\\[4pt]<br />
f_{1011}<br />
\\[4pt]<br />
f_{1101}<br />
\\[4pt]<br />
f_{1110}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} ~ x ~~ y ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{(} ~ x ~ \texttt{(} y \texttt{))}<br />
\\[4pt]<br />
\texttt{((} x \texttt{)} ~ y ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{not both}~ x ~\text{and}~ y<br />
\\[4pt]<br />
\text{not}~ x ~\text{without}~ y<br />
\\[4pt]<br />
\text{not}~ y ~\text{without}~ x<br />
\\[4pt]<br />
x ~\text{or}~ y<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot x \lor \lnot y<br />
\\[4pt]<br />
x \Rightarrow y<br />
\\[4pt]<br />
x \Leftarrow y<br />
\\[4pt]<br />
x \lor y<br />
\end{matrix}\!</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>f_{1111}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
==A Differential Extension of Propositional Calculus==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Fire over water:<br><br />
The image of the condition before transition.<br><br />
Thus the superior man is careful<br><br />
In the differentiation of things,<br><br />
So that each finds its place.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; ''I Ching'', Hexagram 64, [Wil, 249]<br />
|}<br />
<br />
This much preparation is enough to begin introducing my subject, if I excuse myself from giving full arguments for my definitional choices until some later stage. I am trying to develop a ''differential theory of qualitative equations'' that parallels the application of differential geometry to dynamic systems. The idea of a tangent vector is key to this work and a major goal is to find the right logical analogues of tangent spaces, bundles, and functors. The strategy is taken of looking for the simplest versions of these constructions that can be discovered within the realm of propositional calculus, so long as they serve to fill out the general theme.<br />
<br />
===Differential Propositions : Qualitative Analogues of Differential Equations===<br />
<br />
In order to define the differential extension of a universe of discourse <math>[\mathcal{A}],</math> the initial alphabet <math>\mathcal{A}</math> must be extended to include a collection of symbols for ''differential features'', or basic ''changes'' that are capable of occurring in <math>[\mathcal{A}].</math> Intuitively, these symbols may be construed as denoting primitive features of change, qualitative attributes of motion, or propositions about how things or points in the universe may be changing or moving with respect to the features that are noted in the initial alphabet.<br />
<br />
Therefore, let us define the corresponding ''differential alphabet'' or ''tangent alphabet'' as <math>\mathrm{d}\mathcal{A}\!</math> <math>=\!</math> <math>\{\mathrm{d}a_1, \ldots, \mathrm{d}a_n\},</math> in principle just an arbitrary alphabet of symbols, disjoint from the initial alphabet <math>\mathcal{A}\!</math> <math>=\!</math> <math>\{ a_1, \ldots, a_n \},\!</math> that is intended to be interpreted in the way just indicated. It only remains to be understood that the precise interpretation of the symbols in <math>\mathrm{d}\mathcal{A}\!</math> is often conceived to be changeable from point to point of the underlying space <math>A.\!</math> Indeed, for all we know, the state space <math>A\!</math> might well be the state space of a language interpreter, one that is concerned, among other things, with the idiomatic meanings of the dialect generated by <math>\mathcal{A}</math> and <math>\mathrm{d}\mathcal{A}.\!</math><br />
<br />
The ''tangent space'' to <math>A\!</math> at one of its points <math>x,\!</math> sometimes written <math>\mathrm{T}_x(A),</math> takes the form <math>\mathrm{d}A</math> <math>=\!</math> <math>\langle \mathrm{d}\mathcal{A} \rangle</math> <math>=\!</math> <math>\langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle.\!</math> Strictly speaking, the name ''cotangent space'' is probably more correct for this construction, but the fact that we take up spaces and their duals in pairs to form our universes of discourse allows our language to be pliable here.<br />
<br />
Proceeding as we did with the base space <math>A,\!</math> the tangent space <math>\mathrm{d}A</math> at a point of <math>A\!</math> can be analyzed as a product of distinct and independent factors:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\mathrm{d}A ~=~ \prod_{i=1}^n \mathrm{d}A_i ~=~ \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n.\!</math><br />
|}<br />
<br />
Here, <math>\mathrm{d}A_i\!</math> is a set of two differential propositions, <math>\mathrm{d}A_i = \{ (\mathrm{d}a_i), \mathrm{d}a_i \},\!</math> where <math>\texttt{(} \mathrm{d}a_i \texttt{)}\!</math> is a proposition with the logical value of <math>\text{not} ~ \mathrm{d}a_i.\!</math> Each component <math>\mathrm{d}A_i\!</math> has the type <math>\mathbb{B},\!</math> operating under the ordered correspondence <math>\{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \} \cong \{ 0, 1 \}.\!</math> However, clarity is often served by acknowledging this differential usage with a superficially distinct type <math>\mathbb{D},\!</math> whose intension may be indicated as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\mathbb{D} = \{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \} = \{ \text{same}, \text{different} \} = \{ \text{stay}, \text{change} \} = \{ \text{stop}, \text{step} \}.\!</math><br />
|}<br />
<br />
Viewed within a coordinate representation, spaces of type <math>\mathbb{B}^n\!</math> and <math>\mathbb{D}^n\!</math> may appear to be identical sets of binary vectors, but taking a view at this level of abstraction would be like ignoring the qualitative units and the diverse dimensions that distinguish position and momentum, or the different roles of quantity and impulse.<br />
<br />
===An Interlude on the Path===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
There would have been no beginnings: instead, speech would proceed from me, while I stood in its path &ndash; a slender gap &ndash; the point of its possible disappearance.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
A sense of the relation between <math>\mathbb{B}</math> and <math>\mathbb{D}</math> may be obtained by considering the ''path classifier'' (or the ''equivalence class of curves'') approach to tangent vectors. Consider a universe <math>[\mathcal{X}].\!</math> Given the boolean value system, a path in the space <math>X = \langle \mathcal{X} \rangle</math> is a map <math>q : \mathbb{B} \to X.</math> In this context the set of paths <math>(\mathbb{B} \to X)</math> is isomorphic to the cartesian square <math>X^2 = X \times X,</math> or the set of ordered pairs chosen from <math>X.\!</math><br />
<br />
We may analyze <math>X^2 = \{ (u, v) : u, v \in X \}</math> into two parts, specifically, the ordered pairs <math>(u, v)\!</math> that lie on and off the diagonal:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}X^2 & = & \{ (u, v) : u = v \} & \cup & \{ (u, v) : u \ne v \}.\end{matrix}</math><br />
|}<br />
<br />
This partition may also be expressed in the following symbolic form:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}X^2 & \cong & \operatorname{diag} (X) & + & 2 \binom{X}{2}.\end{matrix}</math><br />
|}<br />
<br />
The separate terms of this formula are defined as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}\operatorname{diag} (X) & = & \{ (x, x) : x \in X \}.\end{matrix}\!</math><br />
|}<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}\binom{X}{k} & = & X ~\text{choose}~ k & = & \{ k\text{-sets from}~ X \}.\end{matrix}\!</math><br />
|}<br />
<br />
Thus we have:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}\binom{X}{2} & = & \{ \{ u, v \} : u, v \in X \}.\end{matrix}</math><br />
|}<br />
<br />
We may now use the features in <math>\mathrm{d}\mathcal{X} = \{ \mathrm{d}x_i \} = \{ \mathrm{d}x_1, \ldots, \mathrm{d}x_n \}</math> to classify the paths of <math>(\mathbb{B} \to X)</math> by way of the pairs in <math>X^2.\!</math> If <math>X \cong \mathbb{B}^n,</math> then a path <math>q\!</math> in <math>X\!</math> has the following form:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
q : (\mathbb{B} \to \mathbb{B}^n) & \cong & \mathbb{B}^n \times \mathbb{B}^n & \cong & \mathbb{B}^{2n} & \cong & (\mathbb{B}^2)^n.<br />
\end{matrix}</math><br />
|}<br />
<br />
Intuitively, we want to map this <math>(\mathbb{B}^2)^n</math> onto <math>\mathbb{D}^n</math> by mapping each component <math>\mathbb{B}^2</math> onto a copy of <math>\mathbb{D}.</math> But in the presenting context <math>{}^{\backprime\backprime} \mathbb{D} {}^{\prime\prime}</math> is just a name associated with, or an incidental quality attributed to, coefficient values in <math>\mathbb{B}</math> when they are attached to features in <math>\mathrm{d}\mathcal{X}.</math><br />
<br />
Taking these intentions into account, define <math>\mathrm{d}x_i : X^2 \to \mathbb{B}</math> in the following manner:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcrcl}<br />
\mathrm{d}x_i(u, v)<br />
& = & \texttt{(} ~ x_i(u) & \texttt{,} & x_i(v) ~ \texttt{)}<br />
\\<br />
& = & x_i(u) & + & x_i(v)<br />
\\<br />
& = & x_i(v) & - & x_i(u).<br />
\end{array}</math><br />
|}<br />
<br />
In the above transcription, the operator bracket of the form <math>\texttt{(} \ldots \texttt{,} \ldots \texttt{)}\!</math> is a ''cactus lobe'', in general signifying that just one of the arguments listed is false. In the case of two arguments this is the same thing as saying that the arguments are not equal. The plus sign signifies boolean addition, in the sense of addition in <math>\mathrm{GF}(2),</math> and thus means the same thing in this context as the minus sign, in the sense of adding the additive inverse.<br />
<br />
The above definition of <math>\mathrm{d}x_i : X^2 \to \mathbb{B}</math> is equivalent to defining <math>\mathrm{d}x_i : (\mathbb{B} \to X) \to \mathbb{B}\!</math> in the following way:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcrcl}<br />
\mathrm{d}x_i (q)<br />
& = & \texttt{(} ~ x_i(q_0) & \texttt{,} & x_i(q_1) ~ \texttt{)}<br />
\\<br />
& = & x_i(q_0) & + & x_i(q_1)<br />
\\<br />
& = & x_i(q_1) & - & x_i(q_0).<br />
\end{array}</math><br />
|}<br />
<br />
In this definition <math>q_b = q(b),\!</math> for each <math>b\!</math> in <math>\mathbb{B}.</math> Thus, the proposition <math>\mathrm{d}x_i</math> is true of the path <math>q = (u, v)\!</math> exactly if the terms of <math>q,\!</math> the endpoints <math>u\!</math> and <math>v,\!</math> lie on different sides of the question <math>x_i.\!</math><br />
<br />
The language of features in <math>\langle \mathrm{d}\mathcal{X} \rangle,</math> indeed the whole calculus of propositions in <math>[\mathrm{d}\mathcal{X}],</math> may now be used to classify paths and sets of paths. In other words, the paths can be taken as models of the propositions <math>g : \mathrm{d}X \to \mathbb{B}.</math> For example, the paths corresponding to <math>\mathrm{diag}(X)</math> fall under the description <math>\texttt{(} \mathrm{d}x_1 \texttt{)} \cdots \texttt{(} \mathrm{d}x_n \texttt{)},\!</math> which says that nothing changes against the backdrop of the coordinate frame <math>\{ x_1, \ldots, x_n \}.\!</math><br />
<br />
Finally, a few words of explanation may be in order. If this concept of a path appears to be described in a roundabout fashion, it is because I am trying to avoid using any assumption of vector space properties for the space <math>X\!</math> that contains its range. In many ways the treatment is still unsatisfactory, but improvements will have to wait for the introduction of substitution operators acting on singular propositions.<br />
<br />
===The Extended Universe of Discourse===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
At the moment of speaking, I would like to have perceived a nameless voice, long preceding me, leaving me merely to enmesh myself in it, taking up its cadence, and to lodge myself, when no one was looking, in its interstices as if it had paused an instant, in suspense, to beckon to me.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
Next we define the ''extended alphabet'' or ''bundled alphabet'' <math>\mathrm{E}\mathcal{A}</math> as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lclcl}<br />
\mathrm{E}\mathcal{A}<br />
& = & \mathcal{A} \cup \mathrm{d}\mathcal{A}<br />
& = & \{ a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}.<br />
\end{array}</math><br />
|}<br />
<br />
This supplies enough material to construct the ''differential extension'' <math>\mathrm{E}A,</math> or the ''tangent bundle'' over the initial space <math>A,\!</math> in the following fashion:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcl}<br />
\mathrm{E}A<br />
& = & \langle \mathrm{E}\mathcal{A} \rangle<br />
\\[4pt]<br />
& = & \langle \mathcal{A} \cup \mathrm{d}\mathcal{A} \rangle<br />
\\[4pt]<br />
& = & \langle a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle,<br />
\end{array}</math><br />
|}<br />
<br />
and also:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcl}<br />
\mathrm{E}A<br />
& = & A \times \mathrm{d}A<br />
\\[4pt]<br />
& = & A_1 \times \ldots \times A_n \times \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n.<br />
\end{array}</math><br />
|}<br />
<br />
This gives <math>\mathrm{E}A</math> the type <math>\mathbb{B}^n \times \mathbb{D}^n.</math><br />
<br />
Finally, the tangent universe <math>\mathrm{E}A^\bullet = [\mathrm{E}\mathcal{A}]\!</math> is constituted from the totality of points and maps, or interpretations and propositions, that are based on the extended set of features <math>\mathrm{E}\mathcal{A},</math> and this fact is summed up in the following notation:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lclcl}<br />
\mathrm{E}A^\bullet<br />
& = & [\mathrm{E}\mathcal{A}]<br />
& = & [a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n].<br />
\end{array}</math><br />
|}<br />
<br />
This gives the tangent universe <math>\mathrm{E}A^\bullet\!</math> the type:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcl}<br />
(\mathbb{B}^n \times \mathbb{D}^n\ +\!\to \mathbb{B})<br />
& = & (\mathbb{B}^n \times \mathbb{D}^n, (\mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B})).<br />
\end{array}</math><br />
|}<br />
<br />
A proposition in the tangent universe <math>[\mathrm{E}\mathcal{A}]</math> is called a ''differential proposition'' and forms the analogue of a system of differential equations, constraints, or relations in ordinary calculus.<br />
<br />
With these constructions, the differential extension <math>\mathrm{E}A</math> and the space of differential propositions <math>(\mathrm{E}A \to \mathbb{B}),\!</math> we have arrived, in main outline, at one of the major subgoals of this study. Table&nbsp;8 summarizes the concepts that have been introduced for working with differentially extended universes of discourse.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 8.} ~~ \text{Differential Extension : Basic Notation}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Symbol}\!</math><br />
| <math>\text{Notation}\!</math><br />
| <math>\text{Description}\!</math><br />
| <math>\text{Type}\!</math><br />
|-<br />
| <math>\mathrm{d}\mathfrak{A}\!</math><br />
| <math>\{ {}^{\backprime\backprime} \mathrm{d}a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} \mathrm{d}a_n {}^{\prime\prime} \}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Alphabet of}<br />
\\[2pt]<br />
\text{differential symbols}<br />
\end{matrix}</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>\mathrm{d}\mathcal{A}\!</math><br />
| <math>\{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Basis of}<br />
\\[2pt]<br />
\text{differential features}<br />
\end{matrix}</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>\mathrm{d}A_i\!</math><br />
| <math>\{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \}\!</math><br />
| <math>\text{Differential dimension}~ i\!</math><br />
| <math>\mathbb{D}\!</math><br />
|-<br />
| <math>\mathrm{d}A\!</math><br />
|<br />
<math>\begin{matrix}<br />
\langle \mathrm{d}\mathcal{A} \rangle<br />
\\[2pt]<br />
\langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle<br />
\\[2pt]<br />
\{ (\mathrm{d}a_1, \ldots, \mathrm{d}a_n) \}<br />
\\[2pt]<br />
\mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n<br />
\\[2pt]<br />
\textstyle \prod_i \mathrm{d}A_i<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Tangent space at a point:}<br />
\\[2pt]<br />
\text{Set of changes, motions,}<br />
\\[2pt]<br />
\text{steps, tangent vectors}<br />
\\[2pt]<br />
\text{at a point}<br />
\end{matrix}</math><br />
| <math>\mathbb{D}^n\!</math><br />
|-<br />
| <math>\mathrm{d}A^*\!</math><br />
| <math>(\mathrm{hom} : \mathrm{d}A \to \mathbb{B})\!</math><br />
| <math>\text{Linear functions on}~ \mathrm{d}A\!</math><br />
| <math>(\mathbb{D}^n)^* \cong \mathbb{D}^n\!</math><br />
|-<br />
| <math>\mathrm{d}A^\uparrow\!</math><br />
| <math>(\mathrm{d}A \to \mathbb{B})\!</math><br />
| <math>\text{Boolean functions on}~ \mathrm{d}A\!</math><br />
| <math>\mathbb{D}^n \to \mathbb{B}\!</math><br />
|-<br />
| <math>\mathrm{d}A^\bullet\!</math><br />
|<br />
<math>\begin{matrix}<br />
[\mathrm{d}\mathcal{A}]<br />
\\[2pt]<br />
(\mathrm{d}A, \mathrm{d}A^\uparrow)<br />
\\[2pt]<br />
(\mathrm{d}A ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
(\mathrm{d}A, (\mathrm{d}A \to \mathbb{B}))<br />
\\[2pt]<br />
[\mathrm{d}a_1, \ldots, \mathrm{d}a_n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Tangent universe at a point of}~ A^\bullet,<br />
\\[2pt]<br />
\text{based on the tangent features}<br />
\\[2pt]<br />
\{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
(\mathbb{D}^n, (\mathbb{D}^n \to \mathbb{B}))<br />
\\[2pt]<br />
(\mathbb{D}^n ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
[\mathbb{D}^n]<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
The adjectives ''differential'' or ''tangent'' are systematically attached to every construct based on the differential alphabet <math>\mathrm{d}\mathfrak{A},</math> taken by itself. Strictly speaking, we probably ought to call <math>\mathrm{d}\mathcal{A}</math> the set of ''cotangent'' features derived from <math>\mathcal{A},</math> but the only time this distinction really seems to matter is when we need to distinguish the tangent vectors as maps of type <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math> from cotangent vectors as elements of type <math>\mathbb{D}^n.</math> In like fashion, having defined <math>\mathrm{E}\mathcal{A} = \mathcal{A} \cup \mathrm{d}\mathcal{A},</math> we can systematically attach the adjective ''extended'' or the substantive ''bundle'' to all of the constructs associated with this full complement of <math>{2n}\!</math> features.<br />
<br />
It eventually becomes necessary to extend the initial alphabet even further, to allow for the discussion of higher order differential expressions. Table&nbsp;9 provides a suggestion of how these further extensions can be carried out.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 9.} ~~ \text{Higher Order Differential Features}\!</math><br />
|<br />
<p><math>\begin{array}{lllll}<br />
\mathrm{d}^0 \mathcal{A} & = & \{ a_1, \ldots, a_n \} & = & \mathcal{A}<br />
\\<br />
\mathrm{d}^1 \mathcal{A} & = & \{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \} & = & \mathrm{d}\mathcal{A}<br />
\end{array}</math></p><br />
<br />
<p><math>\begin{array}{lll}<br />
\mathrm{d}^k \mathcal{A} & = & \{ \mathrm{d}^k a_1, \ldots, \mathrm{d}^k a_n \}<br />
\\<br />
\mathrm{d}^* \mathcal{A} & = & \{ \mathrm{d}^0 \mathcal{A}, \ldots, \mathrm{d}^k \mathcal{A}, \ldots \}<br />
\end{array}</math></p><br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}^0 \mathcal{A} & = & \mathrm{d}^0 \mathcal{A}<br />
\\<br />
\mathrm{E}^1 \mathcal{A} & = & \mathrm{d}^0 \mathcal{A} ~\cup~ \mathrm{d}^1 \mathcal{A}<br />
\\<br />
\mathrm{E}^k \mathcal{A} & = & \mathrm{d}^0 \mathcal{A} ~\cup~ \ldots ~\cup~ \mathrm{d}^k \mathcal{A}<br />
\\<br />
\mathrm{E}^\infty \mathcal{A} & = & \bigcup~ \mathrm{d}^* \mathcal{A}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
===Intentional Propositions===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Do you guess I have some intricate purpose?<br><br />
Well I have . . . . for the April rain has, and the mica on<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the side of a rock has.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 45]<br />
|}<br />
<br />
In order to analyze the behavior of a system at successive moments in time, while staying within the limitations of propositional logic, it is necessary to create independent alphabets of logical features for each moment of time that we contemplate using in our discussion. These moments have reference to typical instances and relative intervals, not actual or absolute times. For example, to discuss ''velocities'' (first order rates of change) we need to consider points of time in pairs. There are a number of natural ways of doing this. Given an initial alphabet, we could use its symbols as a lexical basis to generate successive alphabets of compound symbols, say, with temporal markers appended as suffixes.<br />
<br />
As a standard way of dealing with these situations, the following scheme of notation suggests a way of extending any alphabet of logical features through as many temporal moments as a particular order of analysis may demand. The lexical operators <math>\mathrm{p}^k</math> and <math>\mathrm{Q}^k</math> are convenient in many contexts where the accumulation of prime symbols and union symbols would otherwise be cumbersome.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 10.} ~~ \text{A Realm of Intentional Features}\!</math><br />
|<br />
<p><math>\begin{array}{lllll}<br />
\mathrm{p}^0 \mathcal{A} & = & \{ a_1, \ldots, a_n \} & = & \mathcal{A}<br />
\\<br />
\mathrm{p}^1 \mathcal{A} & = & \{ a_1^\prime, \ldots, a_n^\prime \} & = & \mathcal{A}^\prime<br />
\\<br />
\mathrm{p}^2 \mathcal{A} & = & \{ a_1^{\prime\prime}, \ldots, a_n^{\prime\prime} \} & = & \mathcal{A}^{\prime\prime}<br />
\\<br />
\cdots & & \cdots &<br />
\end{array}</math></p><br />
<br />
<p><math>\begin{array}{lll}<br />
\mathrm{p}^k \mathcal{A} & = & \{ \mathrm{p}^k a_1, \ldots, \mathrm{p}^k a_n \}<br />
\end{array}</math></p><br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{Q}^0 \mathcal{A} & = & \mathcal{A}<br />
\\<br />
\mathrm{Q}^1 \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}'<br />
\\<br />
\mathrm{Q}^2 \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}' \cup \mathcal{A}''<br />
\\<br />
\cdots & & \cdots<br />
\\<br />
\mathrm{Q}^k \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}' \cup \ldots \cup \mathrm{p}^k \mathcal{A}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
The resulting augmentations of our logical basis determine a series of discursive universes that may be called the ''intentional extension'' of propositional calculus. This extension follows a pattern analogous to the differential extension, which was developed in terms of the operators <math>\mathrm{d}^k</math> and <math>\mathrm{E}^k,</math> and there is a natural relation between these two extensions that bears further examination. In contexts displaying this pattern, where a sequence of domains stretches from an anchoring domain <math>X\!</math> through an indefinite number of higher reaches, a particular collection of domains based on <math>X\!</math> will be referred to as a ''realm'' of <math>X,\!</math> and when the succession exhibits a temporal aspect, as a ''reign'' of <math>X.\!</math><br />
<br />
For the purposes of this discussion, an ''intentional proposition'' is defined as a proposition in the universe of discourse <math>\mathrm{Q}X^\bullet = [\mathrm{Q}\mathcal{X}],</math> in other words, a map <math>q : \mathrm{Q}X \to \mathbb{B}.</math> The sense of this definition may be seen if we consider the following facts. First, the equivalence <math>\mathrm{Q}X = X \times X'</math> motivates the following chain of isomorphisms between spaces:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lllcl}<br />
(\mathrm{Q}X \to \mathbb{B})<br />
& \cong & (X & \times & ~X' \to \mathbb{B})<br />
\\[4pt]<br />
& \cong & (X & \to & (X' \to \mathbb{B}))<br />
\\[4pt]<br />
& \cong & (X' & \to & (X~ \to \mathbb{B})).<br />
\end{array}</math><br />
|}<br />
<br />
Viewed in this light, an intentional proposition <math>q\!</math> may be rephrased as a map <math>q : X \times X' \to \mathbb{B},</math> which judges the juxtaposition of states in <math>X\!</math> from one moment to the next. Alternatively, <math>q\!</math> may be parsed in two stages in two different ways, as <math>q : X \to (X' \to \mathbb{B})</math> and as <math>q : X' \to (X \to \mathbb{B}),</math> which associate to each point of <math>X\!</math> or <math>X'\!</math> a proposition about states in <math>X'\!</math> or <math>X,\!</math> respectively. In this way, an intentional proposition embodies a type of value system, in effect, a proposal that places a value on a collection of ends-in-view, or a project that evaluates a set of goals as regarded from each point of view in the state space of a system.<br />
<br />
In sum, the intentional proposition <math>q\!</math> indicates a method for the systematic selection of local goals. As a general form of description, a map of the type <math>q : \mathrm{Q}^i X \to \mathbb{B}\!</math> may be referred to as an "<math>i^\text{th}</math> order intentional proposition". Naturally, when we speak of intentional propositions without qualification, we usually mean first order intentions.<br />
<br />
Many different realms of discourse have the same structure as the extensions that have been indicated here. From a strictly logical point of view, each new layer of terms is composed of independent logical variables that are no different in kind from those that go before, and each further course of logical atoms is treated like so many individual, but otherwise indifferent bricks by the prototype computer program that I use as a propositional interpreter. Thus, the names that I use to single out the differential and the intentional extensions, and the lexical paradigms that I follow to construct them, are meant to suggest the interpretations that I have in mind, but they can only hint at the extra meanings that human communicators may pack into their terms and inflections.<br />
<br />
As applied here, the word ''intentional'' is drawn from common use and may have little bearing on its technical use in other, more properly philosophical, contexts. I am merely using the complex of intentional concepts &mdash; aims, ends, goals, objectives, purposes, and so on &mdash; metaphorically to flesh out and vividly to represent any situation where one needs to contemplate a system in multiple aspects of state and destination, that is, its being in certain states and at the same time acting as if headed through certain states. If confusion arises, more neutral words like ''conative'', ''contingent'', ''discretionary'', ''experimental'', ''kinetic'', ''progressive'', ''tentative'', or ''trial'' would probably serve as well.<br />
<br />
===Life on Easy Street===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Failing to fetch me at first keep encouraged,<br><br />
Missing me one place search another,<br><br />
I stop some where waiting for you<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]<br />
|}<br />
<br />
The finite character of the extended universe <math>[\mathrm{E}\mathcal{A}]</math> makes the problem of solving differential propositions relatively straightforward, at least, in principle. The solution set of the differential proposition <math>q : \mathrm{E}A \to \mathbb{B}</math> is the set of models <math>q^{-1}(1)\!</math> in <math>\mathrm{E}A.</math> Finding all the models of <math>q,\!</math> the extended interpretations in <math>\mathrm{E}A</math> that satisfy <math>q,\!</math> can be carried out by a finite search. Being in possession of complete algorithms for propositional calculus modeling, theorem checking, or theorem proving makes the analytic task fairly simple in principle, though the question of efficiency in the face of arbitrary complexity may always remain another matter entirely. While the fact that propositional satisfiability is NP-complete may be discouraging for the prospects of a single efficient algorithm that covers the whole space <math>[\mathrm{E}\mathcal{A}]</math> with equal facility, there appears to be much room for improvement in classifying special forms and in developing algorithms that are tailored to their practical processing.<br />
<br />
In view of these constraints and contingencies, our focus shifts to the tasks of approximation and interpretation that support intuition, especially in dealing with the natural kinds of differential propositions that arise in applications, and in the effort to understand, in succinct and adaptive forms, their dynamic implications. In the absence of direct insights, these tasks are partially carried out by forging analogies with the familiar situations and customary routines of ordinary calculus. But the indirect approach, going by way of specious analogy and intuitive habit, forces us to remain on guard against the circumstance that occurs when the word ''forging'' takes on its shadier nuance, indicting the constant risk of a counterfeit in the proportion.<br />
<br />
==Back to the Beginning : Exemplary Universes==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
I would have preferred to be enveloped in words, borne way beyond all possible beginnings.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
To anchor our understanding of differential logic, let us look at how the various concepts apply in the simplest possible concrete cases, where the initial dimension is only 1 or 2. In spite of the obvious simplicity of these cases, it is possible to observe how central difficulties of the subject begin to arise already at this stage.<br />
<br />
===A One-Dimensional Universe===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
There was never any more inception than there is now,<br><br />
Nor any more youth or age than there is now;<br><br />
And will never be any more perfection than there is now,<br><br />
Nor any more heaven or hell than there is now.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]<br />
|}<br />
<br />
Let <math>\mathcal{X} = \{ x_1 \} = \{ A \}</math> be an alphabet that represents one boolean variable or a single logical feature. In this example the capital letter <math>{}^{\backprime\backprime} A {}^{\prime\prime}\!</math> is used usual informally, to name a feature and not a space, in departure from our formerly stated formal conventions. At any rate, the basis element <math>A = x_1\!</math> may be interpreted as a simple proposition or a coordinate projection <math>A = x_1 : \mathbb{B} \xrightarrow{i} \mathbb{B}.</math> The space <math>X = \langle A \rangle = \{ \texttt{(} A \texttt{)}, A \}</math> of points (cells, vectors, interpretations) has cardinality <math>2^n = 2^1 = 2\!</math> and is isomorphic to <math>\mathbb{B} = \{ 0, 1 \}.</math> Moreover, <math>X\!</math> may be identified with the set of singular propositions <math>\{ x : \mathbb{B} \xrightarrow{s} \mathbb{B} \}.</math> The space of linear propositions <math>X^* = \{ \mathrm{hom} : \mathbb{B} \xrightarrow{\ell} \mathbb{B} \} = \{ 0, A \}</math> is algebraically dual to <math>X\!</math> and also has cardinality <math>2.\!</math> Here, <math>{}^{\backprime\backprime} 0 {}^{\prime\prime}\!</math> is interpreted as denoting the constant function <math>0 : \mathbb{B} \to \mathbb{B},</math> amounting to the linear proposition of rank <math>0,\!</math> while <math>A\!</math> is the linear proposition of rank <math>1.\!</math> Last but not least we have the positive propositions <math>\{ \mathrm{pos} : \mathbb{B} \xrightarrow{p} \mathbb{B} \} = \{ A, 1 \},\!</math> of rank <math>1\!</math> and <math>0,\!</math> respectively, where <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}\!</math> is understood as denoting the constant function <math>1 : \mathbb{B} \to \mathbb{B}.</math> In sum, there are <math>2^{2^n} = 2^{2^1} = 4</math> propositions altogether in the universe of discourse, comprising the set <math>X^\uparrow = \{ f : X \to \mathbb{B} \} = \{ 0, \texttt{(} A \texttt{)}, A, 1 \} \cong (\mathbb{B} \to \mathbb{B}).</math><br />
<br />
The first order differential extension of <math>\mathcal{X}</math> is <math>\mathrm{E}\mathcal{X} = \{ x_1, \mathrm{d}x_1 \} = \{ A, \mathrm{d}A \}.</math> If the feature <math>A\!</math> is understood as applying to some object or state, then the feature <math>\mathrm{d}A</math> may be interpreted as an attribute of the same object or state that says that it is changing ''significantly'' with respect to the property <math>A,\!</math> or that it has an ''escape velocity'' with respect to the state <math>A.\!</math> In practice, differential features acquire their logical meaning through a class of ''temporal inference rules''.<br />
<br />
For example, relative to a frame of observation that is left implicit for now, one is permitted to make the following sorts of inference: From the fact that <math>A\!</math> and <math>\mathrm{d}A</math> are true at a given moment one may infer that <math>\texttt{(} A \texttt{)}\!</math> will be true in the next moment of observation. Altogether in the present instance, there is the fourfold scheme of inference that is shown below:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:50%"<br />
|<br />
<math>\begin{matrix}<br />
\text{From} & \texttt{(} A \texttt{)}<br />
& \text{and} & \texttt{(} \mathrm{d}A \texttt{)}<br />
& \text{infer} & \texttt{(} A \texttt{)} & \text{next.}<br />
\\[8pt]<br />
\text{From} & \texttt{(} A \texttt{)}<br />
& \text{and} & \mathrm{d}A<br />
& \text{infer} & A & \text{next.}<br />
\\[8pt]<br />
\text{From} & A<br />
& \text{and} & \texttt{(} \mathrm{d}A \texttt{)}<br />
& \text{infer} & A & \text{next.}<br />
\\[8pt]<br />
\text{From} & A<br />
& \text{and} & \mathrm{d}A<br />
& \text{infer} & \texttt{(} A \texttt{)} & \text{next.}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
It might be thought that an independent time variable needs to be brought in at this point, but it is an insight of fundamental importance that the idea of process is logically prior to the notion of time. A time variable is a reference to a ''clock'' &mdash; a canonical, conventional process that is accepted or established as a standard of measurement, but in essence no different than any other process. This raises the question of how different subsystems in a more global process can be brought into comparison, and what it means for one process to serve the function of a local standard for others. But these inquiries only wrap up puzzles in further riddles, and are obviously too involved to be handled at our current level of approximation.<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
The clock indicates the moment . . . . but what does<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;eternity indicate?<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 79]<br />
|}<br />
<br />
Observe that the secular inference rules, used by themselves, involve a loss of information, since nothing in them can tell us whether the momenta <math>\{ \texttt{(} \mathrm{d}A \texttt{)}, \mathrm{d}A \}\!</math> are changed or unchanged in the next instance. In order to know this, one would have to determine <math>\mathrm{d}^2 A,\!</math> and so on, pursuing an infinite regress. Ultimately, in order to rest with a finitely determinate system, it is necessary to make an infinite assumption, for example, that <math>\mathrm{d}^k A = 0\!</math> for all <math>k\!</math> greater than some fixed value <math>M.\!</math> Another way to escape the regress is through the provision of a dynamic law, in typical form making higher order differentials dependent on lower degrees and estates.<br />
<br />
===Example 1. A Square Rigging===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Urge and urge and urge,<br><br />
Always the procreant urge of the world.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]<br />
|}<br />
<br />
By way of example, suppose that we are given the initial condition <math>A = \mathrm{d}A\!</math> and the law <math>\mathrm{d}^2 A = \texttt{(} A \texttt{)}.\!</math> Since the equation <math>A = \mathrm{d}A\!</math> is logically equivalent to the disjunction <math>A ~ \mathrm{d}A ~\text{or}~ \texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)},\!</math> we may infer two possible trajectories, as displayed in Table&nbsp;11. In either case the state <math>A ~ \texttt{(} \mathrm{d}A \texttt{)(} \mathrm{d}^2 A \texttt{)}\!</math> is a stable attractor or a terminal condition for both starting points.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 11.} ~~ \text{A Pair of Commodious Trajectories}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Time}\!</math><br />
| <math>\text{Trajectory 1}\!</math><br />
| <math>\text{Trajectory 2}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
2<br />
\\[4pt]<br />
3<br />
\\[4pt]<br />
4<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A & \mathrm{d}A & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
\texttt{(} A \texttt{)} & \mathrm{d}A & \mathrm{d}^2 A<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
{}^{\shortparallel} & {}^{\shortparallel} & {}^{\shortparallel}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{)} & \texttt{(} \mathrm{d}A \texttt{)} & \mathrm{d}^2 A<br />
\\[4pt]<br />
\texttt{(} A \texttt{)} & \mathrm{d}A & \mathrm{d}^2 A<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
{}^{\shortparallel} & {}^{\shortparallel} & {}^{\shortparallel}<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Because the initial space <math>X = \langle A \rangle\!</math> is one-dimensional, we can easily fit the second order extension <math>\mathrm{E}^2 X = \langle A, \mathrm{d}A, \mathrm{d}^2 A \rangle\!</math> within the compass of a single venn diagram, charting the couple of converging trajectories as shown in Figure&nbsp;12.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 12 -- The Anchor.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 12.} ~~ \text{The Anchor}\!</math><br />
|}<br />
<br />
If we eliminate from view the regions of <math>\mathrm{E}^2 X\!</math> that are ruled out by the dynamic law <math>\mathrm{d}^2 A = \texttt{(} A \texttt{)},\!</math> then what remains is the quotient structure that is shown in Figure&nbsp;13. This picture makes it easy to see that the dynamically allowable portion of the universe is partitioned between the properties <math>A\!</math> and <math>\mathrm{d}^2 A\!.</math> As it happens, this fact might have been expressed &ldquo;right off the bat&rdquo; by an equivalent formulation of the differential law, one that uses the exclusive disjunction to state the law as <math>\texttt{(} A \texttt{,} \mathrm{d}^2 A \texttt{)}\!.</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 13 -- The Tiller.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 13.} ~~ \text{The Tiller}\!</math><br />
|}<br />
<br />
What we have achieved in this example is to give a differential description of a simple dynamic process. In effect, we did this by embedding a directed graph, which can be taken to represent the state transitions of a finite automaton, in a dynamically allotted quotient structure that is created from a boolean lattice or an <math>n\!</math>-cube by nullifying all of the regions that the dynamics outlaws. With growth in the dimensions of our contemplated universes, it becomes essential, both for human comprehension and for computer implementation, that the dynamic structures of interest to us be represented not actually, by acquaintance, but virtually, by description. In our present study, we are using the language of propositional calculus to express the relevant descriptions, and to comprehend the structure that is implicit in the subsets of a <math>n\!</math>-cube without necessarily being forced to actualize all of its points.<br />
<br />
One of the reasons for engaging in this kind of extremely reduced, but explicitly controlled case study is to throw light on the general study of languages, formal and natural, in their full array of syntactic, semantic, and pragmatic aspects. Propositional calculus is one of the last points of departure where we can view these three aspects interacting in a non-trivial way without being immediately and totally overwhelmed by the complexity they generate. Often this complexity causes investigators of formal and natural languages to adopt the strategy of focusing on a single aspect and to abandon all hope of understanding the whole, whether it's the still living natural language or the dynamics of inquiry that lies crystallized in formal logic.<br />
<br />
From the perspective that I find most useful here, a language is a syntactic system that is designed or evolved in part to express a set of descriptions. When the explicit symbols of a language have extensions in its object world that are actually infinite, or when the implicit categories and generative devices of a linguistic theory have extensions in its subject matter that are potentially infinite, then the finite characters of terms, statements, arguments, grammars, logics, and rhetorics force an excess of intension to reside in all these symbols and functions, across the spectrum from the object language to the metalinguistic uses. In the aphorism from W. von Humboldt that Chomsky often cites, for example, in [Cho86, 30] and [Cho93, 49], language requires &ldquo;the infinite use of finite means&rdquo;. This is necessarily true when the extensions are infinite, when the referential symbols and grammatical categories of a language possess infinite sets of models and instances. But it also voices a practical truth when the extensions, though finite at every stage, tend to grow at exponential rates.<br />
<br />
This consequence of dealing with extensions that are &ldquo;practically infinite&rdquo; becomes crucial when one tries to build neural network systems that learn, since the learning competence of any intelligent system is limited to the objects and domains that it is able to represent. If we want to design systems that operate intelligently with the full deck of propositions dealt by intact universes of discourse, then we must supply them with succinct representations and efficient transformations in this domain. Furthermore, in the project of constructing inquiry driven systems, we find ourselves forced to contemplate the level of generality that is embodied in propositions, because the dynamic evolution of these systems is driven by the measurable discrepancies that occur among their expectations, intentions, and observations, and because each of these subsystems or components of knowledge constitutes a propositional modality that can take on the fully generic character of an empirical summary or an axiomatic theory.<br />
<br />
A compression scheme by any other name is a symbolic representation, and this is what the differential extension of propositional calculus, through all of its many universes of discourse, is intended to supply. Why is this particular program of mental calisthenics worth carrying out in general? By providing a uniform logical medium for describing dynamic systems we can make the task of understanding complex systems much easier, both in looking for invariant representations of individual cases and in finding points of comparison among diverse structures that would otherwise appear as isolated systems. All of this goes to facilitate the search for compact knowledge and to adapt what is learned from individual cases to the general realm.<br />
<br />
===Back to the Feature===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
I guess it must be the flag of my disposition, out of hopeful<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;green stuff woven.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 31]<br />
|}<br />
<br />
Let us assume that the sense intended for differential features is well enough established in the intuition, for now, that we may continue with outlining the structure of the differential extension <math>[\mathrm{E}\mathcal{X}] = [A, \mathrm{d}A].\!</math> Over the extended alphabet <math>\mathrm{E}\mathcal{X} = \{ x_1, \mathrm{d}x_1 \} = \{ A, \mathrm{d}A \}\!</math> of cardinality <math>2^n = 2\!</math> we generate the set of points <math>\mathrm{E}X\!</math> of cardinality <math>2^{2n} = 4\!</math> that bears the following chain of equivalent descriptions:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}X & = & \langle A, \mathrm{d}A \rangle<br />
\\[4pt]<br />
& = & \{ \texttt{(} A \texttt{)}, A \} ~\times~ \{ \texttt{(} \mathrm{d}A \texttt{)}, \mathrm{d}A \}<br />
\\[4pt]<br />
& = &<br />
\{<br />
\texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)},~<br />
\texttt{(} A \texttt{)} \mathrm{d}A,~<br />
A \texttt{(} \mathrm{d}A \texttt{)},~<br />
A ~ \mathrm{d}A<br />
\}.<br />
\end{array}</math><br />
|}<br />
<br />
The space <math>\mathrm{E}X\!</math> may be assigned the mnemonic type <math>\mathbb{B} \times \mathbb{D},\!</math> which is really no different than <math>\mathbb{B} \times \mathbb{B} = \mathbb{B}^2.\!</math> An individual element of <math>\mathrm{E}X\!</math> may be regarded as a ''disposition at a point'' or a ''situated direction'', in effect, a singular mode of change occurring at a single point in the universe of discourse. In applications, the modality of this change can be interpreted in various ways, for example, as an expectation, an intention, or an observation with respect to the behavior of a system.<br />
<br />
To complete the construction of the extended universe of discourse <math>\mathrm{E}X^\bullet = [x_1, \mathrm{d}x_1] = [A, \mathrm{d}A]\!</math> one must add the set of differential propositions <math>\mathrm{E}X^\uparrow = \{ g : \mathrm{E}X \to \mathbb{B} \} \cong (\mathbb{B} \times \mathbb{D} \to \mathbb{B})\!</math> to the set of dispositions in <math>\mathrm{E}X.\!</math> There are <math>2^{2^{2n}} = 16\!</math> propositions in <math>\mathrm{E}X^\uparrow,\!</math> as detailed in Table&nbsp;14.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 14.} ~~ \text{Differential Propositions}\!</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>A\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>\mathrm{d}A\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>g_{0}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{1}<br />
\\[4pt]<br />
g_{2}<br />
\\[4pt]<br />
g_{4}<br />
\\[4pt]<br />
g_{8}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
1~0~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
\texttt{(} A \texttt{)} ~ \mathrm{d}A ~<br />
\\[4pt]<br />
~ A ~ \texttt{(} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
~ A ~~ \mathrm{d}A ~<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{neither}~ A ~\text{nor}~ \mathrm{d}A<br />
\\[4pt]<br />
\mathrm{d}A ~\text{and not}~ A<br />
\\[4pt]<br />
A ~\text{and not}~ \mathrm{d}A<br />
\\[4pt]<br />
A ~\text{and}~ \mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot A \land \lnot \mathrm{d}A<br />
\\[4pt]<br />
\lnot A \land \mathrm{d}A<br />
\\[4pt]<br />
A \land \lnot \mathrm{d}A<br />
\\[4pt]<br />
A \land \mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
g_{3}<br />
\\[4pt]<br />
g_{12}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{)}<br />
\\[4pt]<br />
A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ A<br />
\\[4pt]<br />
A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot A<br />
\\[4pt]<br />
A<br />
\end{matrix}\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{6}<br />
\\[4pt]<br />
g_{9}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[4pt]<br />
1~0~0~1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{,} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
\texttt{((} A \texttt{,} \mathrm{d}A \texttt{))}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
A ~\text{not equal to}~ \mathrm{d}A<br />
\\[4pt]<br />
A ~\text{equal to}~ \mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
A \ne \mathrm{d}A<br />
\\[4pt]<br />
A = \mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{5}<br />
\\[4pt]<br />
g_{10}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
\mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ \mathrm{d}A<br />
\\[4pt]<br />
\mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot \mathrm{d}A<br />
\\[4pt]<br />
\mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{7}<br />
\\[4pt]<br />
g_{11}<br />
\\[4pt]<br />
g_{13}<br />
\\[4pt]<br />
g_{14}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} ~ A ~~ \mathrm{d}A ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{(} ~ A ~ \texttt{(} \mathrm{d}A \texttt{))}<br />
\\[4pt]<br />
\texttt{((} A \texttt{)} ~ \mathrm{d}A ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{((} A \texttt{)(} \mathrm{d}A \texttt{))}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not both}~ A ~\text{and}~ \mathrm{d}A<br />
\\[4pt]<br />
\text{not}~ A ~\text{without}~ \mathrm{d}A<br />
\\[4pt]<br />
\text{not}~ \mathrm{d}A ~\text{without}~ A<br />
\\[4pt]<br />
A ~\text{or}~ \mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot A \lor \lnot \mathrm{d}A<br />
\\[4pt]<br />
A \Rightarrow \mathrm{d}A<br />
\\[4pt]<br />
A \Leftarrow \mathrm{d}A<br />
\\[4pt]<br />
A \lor \mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
| <math>f_{3}\!</math><br />
| <math>g_{15}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
Aside from changing the names of variables and shuffling the order of rows, this Table follows the format that was used previously for boolean functions of two variables. The rows are grouped to reflect natural similarity classes among the propositions. In a future discussion, these classes will be given additional explanation and motivation as the orbits of a certain transformation group acting on the set of 16 propositions. Notice that four of the propositions, in their logical expressions, resemble those given in the table for <math>X^\uparrow.\!</math> Thus the first set of propositions <math>\{ f_i \}\!</math> is automatically embedded in the present set <math>\{ g_j \}\!</math> and the corresponding inclusions are indicated at the far left margin of the Table.<br />
<br />
===Tacit Extensions===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
I would really like to have slipped imperceptibly into this lecture, as into all the others I shall be delivering, perhaps over the years ahead.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
Strictly speaking, however, there is a subtle distinction in type between the function <math>f_i : X \to \mathbb{B}</math> and the corresponding function <math>g_j : \mathrm{E}X \to \mathbb{B},</math> even though they share the same logical expression. Naturally, we want to maintain the logical equivalence of expressions that represent the same proposition while appreciating the full diversity of that proposition's functional and typical representatives. Both perspectives, and all the levels of abstraction extending through them, have their reasons, as will develop in time.<br />
<br />
Because this special circumstance points up an important general theme, it is a good idea to discuss it more carefully. Whenever there arises a situation like this, where one alphabet <math>\mathcal{X}</math> is a subset of another alphabet <math>\mathcal{Y},</math> then we say that any proposition <math>f : \langle \mathcal{X} \rangle \to \mathbb{B}</math> has a ''tacit extension'' to a proposition <math>\boldsymbol\varepsilon f : \langle \mathcal{Y} \rangle \to \mathbb{B},\!</math> and that the space <math>(\langle \mathcal{X} \rangle \to \mathbb{B})</math> has an ''automatic embedding'' within the space <math>(\langle \mathcal{Y} \rangle \to \mathbb{B}).</math> The extension is defined in such a way that <math>\boldsymbol\varepsilon f\!</math> puts the same constraint on the variables of <math>\mathcal{X}</math> that are contained in <math>\mathcal{Y}</math> as the proposition <math>f\!</math> initially did, while it puts no constraint on the variables of <math>\mathcal{Y}</math> outside of <math>\mathcal{X},</math> in effect, conjoining the two constraints.<br />
<br />
If the variables in question are indexed as <math>\mathcal{X} = \{ x_1, \ldots, x_n \}</math> and <math>\mathcal{Y} = \{ x_1, \ldots, x_n, \ldots, x_{n+k} \},</math> then the definition of the tacit extension from <math>\mathcal{X}</math> to <math>\mathcal{Y}</math> may be expressed in the form of an equation:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\boldsymbol\varepsilon f(x_1, \ldots, x_n, \ldots, x_{n+k}) ~=~ f(x_1, \ldots, x_n).\!</math><br />
|}<br />
<br />
On formal occasions, such as the present context of definition, the tacit extension from <math>\mathcal{X}</math> to <math>\mathcal{Y}</math> is explicitly symbolized by the operator <math>\boldsymbol\varepsilon : (\langle \mathcal{X} \rangle \to \mathbb{B}) \to (\langle \mathcal{Y} \rangle \to \mathbb{B}),</math> where the appropriate alphabets <math>\mathcal{X}</math> and <math>\mathcal{Y}</math> are understood from context, but normally one may leave the "<math>\boldsymbol\varepsilon\!</math>" silent.<br />
<br />
Let's explore what this means for the present Example. Here, <math>\mathcal{X} = \{ A \}</math> and <math>\mathcal{Y} = \mathrm{E}\mathcal{X} = \{ A, \mathrm{d}A \}.</math> For each of the propositions <math>f_i\!</math> over <math>X\!,</math> specifically, those whose expression <math>e_i\!</math> lies in the collection <math>\{ 0, \texttt{(} A \texttt{)}, A, 1 \},\!</math> the tacit extension <math>\boldsymbol\varepsilon f\!</math> of <math>f\!</math> to <math>\mathrm{E}X</math> can be phrased as a logical conjunction of two factors, <math>f_i = e_i \cdot \tau ~ ,\!</math> where <math>\tau\!</math> is a logical tautology that uses all the variables of <math>\mathcal{Y} - \mathcal{X}.</math> Working in these terms, the tacit extensions <math>\boldsymbol\varepsilon f\!</math> of <math>f\!</math> to <math>\mathrm{E}X</math> may be explicated as shown in Table&nbsp;15.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:60%"<br />
|+ style="height:30px" | <math>\text{Table 15.} ~~ \text{Tacit Extension of}~ [A] ~\text{to}~ [A, \mathrm{d}A]\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0<br />
& = & 0 & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))} & = & & 0<br />
\\[8pt]<br />
\texttt{(} A \texttt{)}<br />
& = & \texttt{(} A \texttt{)} & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))}<br />
& = & \texttt{(} A \texttt{)} \, \mathrm{d}A ~ & + & \texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)}<br />
\\[8pt]<br />
A<br />
& = & ~A~ & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))}<br />
& = & ~A~ ~\mathrm{d}A~ & + & ~A~ \texttt{(} \mathrm{d}A \texttt{)}<br />
\\[8pt]<br />
1<br />
& = & 1 & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))} & = & & 1<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
In its effect on the singular propositions over <math>X,\!</math> this analysis has an interesting interpretation. The tacit extension takes us from thinking about a particular state, like <math>A\!</math> or <math>\texttt{(} A \texttt{)},\!</math> to considering the collection of outcomes, the outgoing changes or the singular dispositions, that spring from that state.<br />
<br />
===Example 2. Drives and Their Vicissitudes===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
I open my scuttle at night and see the far-sprinkled systems,<br><br />
And all I see, multiplied as high as I can cipher, edge but<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the rim of the farther systems.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 81]<br />
|}<br />
<br />
Before we leave the one-feature case let's look at a more substantial example, one that illustrates a general class of curves that can be charted through the extended feature spaces and that provides an opportunity to discuss a number of important themes concerning their structure and dynamics.<br />
<br />
Again, let <math>\mathcal{X} = \{ x_1 \} = \{ A \}.\!</math> In the discussion that follows we will consider a class of trajectories having the property that <math>\mathrm{d}^k A = 0\!</math> for all <math>k\!</math> greater than some fixed <math>m\!</math> and we may indulge in the use of some picturesque terms that describe salient classes of such curves. Given the finite order condition, there is a highest order non-zero difference <math>\mathrm{d}^m A\!</math> exhibited at each point in the course of any determinate trajectory that one may wish to consider. With respect to any point of the corresponding orbit or curve let us call this highest order differential feature <math>\mathrm{d}^m A\!</math> the ''drive'' at that point. Curves of constant drive <math>\mathrm{d}^m A\!</math> are then referred to as ''<math>m^\text{th}\!</math>-gear curves''.<br />
<br />
* '''Scholium.''' The fact that a difference calculus can be developed for boolean functions is well known [Fuji], [Koh, &sect; 8-4] and was probably familiar to Boole, who was an expert in difference equations before he turned to logic. And of course there is the strange but true story of how the Turin machines of the 1840s prefigured the Turing machines of the 1940s [Men, 225-297]. At the very outset of general purpose, mechanized computing we find that the motive power driving the Analytical Engine of Babbage, the kernel of an idea behind all of his wheels, was exactly his notion that difference operations, suitably trained, can serve as universal joints for any conceivable computation [M&M], [Mel, ch. 4].<br />
<br />
Given this language, the Example we take up here can be described as the family of <math>4^\text{th}\!</math>-gear curves through <math>\mathrm{E}^4 X\!</math> <math>=\!</math> <math>\langle A, ~\mathrm{d}A, ~\mathrm{d}^2\!A, ~\mathrm{d}^3\!A, ~\mathrm{d}^4\!A \rangle.</math> These are the trajectories generated subject to the dynamic law <math>\mathrm{d}^4 A = 1,\!</math> where it is understood in such a statement that all higher order differences are equal to <math>0.\!</math> Since <math>\mathrm{d}^4 A\!</math> and all higher <math>\mathrm{d}^k A\!</math> are fixed, the temporal or transitional conditions (initial, mediate, terminal &mdash; transient or stable states) vary only with respect to their projections as points of <math>\mathrm{E}^3 X = \langle A, ~\mathrm{d}A, ~\mathrm{d}^2\!A, ~\mathrm{d}^3\!A \rangle.</math> Thus, there is just enough space in a planar venn diagram to plot all of these orbits and to show how they partition the points of <math>\mathrm{E}^3 X.\!</math> It turns out that there are exactly two possible orbits, of eight points each, as illustrated in Figure&nbsp;16.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 16 -- A Couple of Fourth Gear Orbits.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 16.} ~~ \text{A Couple of Fourth Gear Orbits}\!</math><br />
|}<br />
<br />
With a little thought it is possible to devise an indexing scheme for the general run of dynamic states that allows for comparing universes of discourse that weigh in on different scales of observation. With this end in sight, let us index the states <math>q \in \mathrm{E}^m X\!</math> with the dyadic rationals (or the binary fractions) in the half-open interval <math>[0, 2).\!</math> Formally and canonically, a state <math>q_r\!</math> is indexed by a fraction <math>r = \tfrac{s}{t}\!</math> whose denominator is the power of two <math>t = 2^m\!</math> and whose numerator is a binary numeral formed from the coefficients of state in a manner to be described next. The ''differential coefficients'' of the state <math>q\!</math> are just the values <math>\mathrm{d}^k\!A(q)</math> for <math>k = 0 ~\text{to}~ m,\!</math> where <math>\mathrm{d}^0\!A</math> is defined as being identical to <math>A.\!</math> To form the binary index <math>d_0.d_1 \ldots d_m\!</math> of the state <math>q\!</math> the coefficient <math>\mathrm{d}^k\!A(q)</math> is read off as the binary digit <math>d_k\!</math> associated with the place value <math>2^{-k}.\!</math> Expressed by way of algebraic formulas, the rational index <math>r\!</math> of the state <math>q\!</math> can be given by the following equivalent formulations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
r(q)<br />
& = &<br />
\displaystyle\sum_k d_k \cdot 2^{-k}<br />
& = &<br />
\displaystyle\sum_k \text{d}^k A(q) \cdot 2^{-k}<br />
\\[8pt]<br />
=<br />
\\[8pt]<br />
\displaystyle\frac{s(q)}{t}<br />
& = &<br />
\displaystyle\frac{\sum_k d_k \cdot 2^{(m-k)}}{2^m}<br />
& = &<br />
\displaystyle\frac{\sum_k \text{d}^k A(q) \cdot 2^{(m-k)}}{2^m}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Applied to the example of <math>4^\text{th}\!</math>-gear curves, this scheme results in the data of Tables&nbsp;17-a and 17-b, which exhibit one period for each orbit. The states in each orbit are listed as ordered pairs <math>(p_i, q_j),\!</math> where <math>p_i\!</math> may be read as a temporal parameter that indicates the present time of the state and where <math>j\!</math> is the decimal equivalent of the binary numeral <math>s.\!</math> Informally and more casually, the Tables exhibit the states <math>q_s\!</math> as subscripted with the numerators of their rational indices, taking for granted the constant denominators of <math>2^m\! = 2^4 = 16.\!</math> In this set-up the temporal successions of states can be reckoned as given by a kind of ''parallel round-up rule''. That is, if <math>(d_k, d_{k+1})\!</math> is any pair of adjacent digits in the state index <math>r,\!</math> then the value of <math>d_k\!</math> in the next state is <math>{d_k}' = d_k + d_{k+1}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:55%"<br />
|+ style="height:30px" | <math>\text{Table 17-a.} ~~ \text{A Couple of Orbits in Fourth Gear : Orbit 1}\!</math><br />
|- style="background:ghostwhite"<br />
| <math>\text{Time}\!</math><br />
| <math>\text{State}\!</math><br />
| <math>A\!</math><br />
| <math>\mathrm{d}A\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| <math>p_i\!</math><br />
| <math>q_j\!</math><br />
| <math>\mathrm{d}^0\!A</math><br />
| <math>\mathrm{d}^1\!A</math><br />
| <math>\mathrm{d}^2\!A</math><br />
| <math>\mathrm{d}^3\!A</math><br />
| <math>\mathrm{d}^4\!A</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
p_0<br />
\\[4pt]<br />
p_1<br />
\\[4pt]<br />
p_2<br />
\\[4pt]<br />
p_3<br />
\\[4pt]<br />
p_4<br />
\\[4pt]<br />
p_5<br />
\\[4pt]<br />
p_6<br />
\\[4pt]<br />
p_7<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
q_{01}<br />
\\[4pt]<br />
q_{03}<br />
\\[4pt]<br />
q_{05}<br />
\\[4pt]<br />
q_{15}<br />
\\[4pt]<br />
q_{17}<br />
\\[4pt]<br />
q_{19}<br />
\\[4pt]<br />
q_{21}<br />
\\[4pt]<br />
q_{31}<br />
\end{matrix}\!</math><br />
| colspan="5" |<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="text-align:center; width:100%"<br />
|<br />
<math>\begin{matrix}<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
1.<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:55%"<br />
|+ style="height:30px" | <math>\text{Table 17-b.} ~~ \text{A Couple of Orbits in Fourth Gear : Orbit 2}\!</math><br />
|- style="background:ghostwhite"<br />
| <math>\text{Time}\!</math><br />
| <math>\text{State}\!</math><br />
| <math>A\!</math><br />
| <math>\mathrm{d}A\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| <math>p_i\!</math><br />
| <math>q_j\!</math><br />
| <math>\mathrm{d}^0\!A</math><br />
| <math>\mathrm{d}^1\!A</math><br />
| <math>\mathrm{d}^2\!A</math><br />
| <math>\mathrm{d}^3\!A</math><br />
| <math>\mathrm{d}^4\!A</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
p_0<br />
\\[4pt]<br />
p_1<br />
\\[4pt]<br />
p_2<br />
\\[4pt]<br />
p_3<br />
\\[4pt]<br />
p_4<br />
\\[4pt]<br />
p_5<br />
\\[4pt]<br />
p_6<br />
\\[4pt]<br />
p_7<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
q_{25}<br />
\\[4pt]<br />
q_{11}<br />
\\[4pt]<br />
q_{29}<br />
\\[4pt]<br />
q_{07}<br />
\\[4pt]<br />
q_{09}<br />
\\[4pt]<br />
q_{27}<br />
\\[4pt]<br />
q_{13}<br />
\\[4pt]<br />
q_{23}<br />
\end{matrix}\!</math><br />
| colspan="5" |<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="text-align:center; width:100%"<br />
|<br />
<math>\begin{matrix}<br />
1.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
==Transformations of Discourse==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
It is understandable that an engineer should be completely absorbed in his speciality, instead of pouring himself out into the freedom and vastness of the world of thought, even though his machines are being sent off to the ends of the earth; for he no more needs to be capable of applying to his own personal soul what is daring and new in the soul of his subject than a machine is in fact capable of applying to itself the differential calculus on which it is based. The same thing cannot, however, be said about mathematics; for here we have the new method of thought, pure intellect, the very well-spring of the times, the ''fons et origo'' of an unfathomable transformation.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 39]<br />
|}<br />
<br />
In this section we take up the general study of logical transformations, or maps that relate one universe of discourse to another. In many ways, and especially as applied to the subject of intelligent dynamic systems, my argument develops the antithesis of the statement just quoted. Along the way, if incidental to my ends, I hope this essay can pose a fittingly irenic epitaph to the frankly ironic epigraph inscribed at its head.<br />
<br />
My goal in this section is to answer a single question: What is a propositional tangent functor? In other words, my aim is to develop a clear conception of what manner of thing would pass in the logical realm for a genuine analogue of the tangent functor, an object conceived to generalize as far as possible in the abstract terms of category theory the ordinary notions of functional differentiation and the all too familiar operations of taking derivatives.<br />
<br />
As a first step I discuss the kinds of transformations that we already know as ''extensions'' and ''projections'', and I use these special cases to illustrate several different styles of logical and visual representation that will figure heavily in the sequel.<br />
<br />
===Foreshadowing Transformations : Extensions and Projections of Discourse===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
And, despite the care which she took to look behind her at every moment, she failed to see a shadow which followed her like her own shadow, which stopped when she stopped, which started again when she did, and which made no more noise than a well-conducted shadow should.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 126]<br />
|}<br />
<br />
Many times in our discussion we have occasion to place one universe of discourse in the context of a larger universe of discourse. An embedding of the general type <math>[\mathcal{X}] \to [\mathcal{Y}]\!</math> is implied any time that we make use of one alphabet <math>[\mathcal{X}]\!</math> that happens to be included in another alphabet <math>[\mathcal{Y}].\!</math> When we are discussing differential issues we usually have in mind that the extended alphabet <math>[\mathcal{Y}]\!</math> has a special construction or a specific lexical relation with respect to the initial alphabet <math>[\mathcal{X}],\!</math> one that is marked by characteristic types of accents, indices, or inflected forms.<br />
<br />
====Extension from 1 to 2 Dimensions====<br />
<br />
Figure 18-a lays out the ''angular form'' of venn diagram for universes of 1 and 2 dimensions, indicating the embedding map of type <math>\mathbb{B}^1 \to \mathbb{B}^2\!</math> and detailing the coordinates that are associated with individual cells. Because all points, cells, or logical interpretations are represented as connected geometric areas, we can say that these pictures provide us with an ''areal view'' of each universe of discourse.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-a -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-a.} ~~ \text{Extension from 1 to 2 Dimensions : Areal}\!</math><br />
|}<br />
<br />
Figure 18-b shows the differential extension from <math>X^\bullet = [x]\!</math> to <math>\mathrm{E}X^\bullet = [x, \mathrm{d}x]\!</math> in a ''bundle of boxes'' form of venn diagram. As awkward as it may seem at first, this type of picture is often the most natural and the most easily available representation when we want to conceptualize the localized information or momentary knowledge of an intelligent dynamic system. It gives a ready picture of a ''proposition at a point'', in the present instance, of a proposition about changing states which is itself associated with a particular dynamic state of a system. It is easy to see how this application might be extended to conceive of more general types of instantaneous knowledge that are possessed by a system.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-b -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-b.} ~~ \text{Extension from 1 to 2 Dimensions : Bundle}\!</math><br />
|}<br />
<br />
Figure 18-c shows the same extension in a ''compact'' style of venn diagram, where the differential features at each position are represented by arrows extending from that position that cross or do not cross, as the case may be, the corresponding feature boundaries.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-c -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-c.} ~~ \text{Extension from 1 to 2 Dimensions : Compact}\!</math><br />
|}<br />
<br />
Figure 18-d compresses the picture of the differential extension even further, yielding a directed graph or ''digraph'' form of representation. (Notice that my definition of a digraph allows for loops or ''slings'' at individual points, in addition to arcs or ''arrows'' between the points.)<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-d -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-d.} ~~ \text{Extension from 1 to 2 Dimensions : Digraph}\!</math><br />
|}<br />
<br />
====Extension from 2 to 4 Dimensions====<br />
<br />
Figure 19-a lays out the ''areal view'' or the ''angular form'' of venn diagram for universes of 2 and 4 dimensions, indicating the embedding map of type <math>\mathbb{B}^2 \to \mathbb{B}^4.\!</math> In many ways these pictures are the best kind there is, giving full canvass to an ideal vista. Their style allows the clearest, the fairest, and the plainest view that we can form of a universe of discourse, affording equal representation to all dispositions and maintaining a balance with respect to ordinary and differential features. If only we could extend this view! Unluckily, an obvious difficulty beclouds this prospect, and that is how precipitately we run into the limits of our plane and visual intuitions. Even within the scope of the spare few dimensions that we have scanned up to this point subtle discrepancies have crept in already. The circumstances that bind us and the frameworks that block us, the flat distortion of the planar projection and the inevitable ineffability that precludes us from wrapping its rhomb figure into rings around a torus, all of these factors disguise the underlying but true connectivity of the universe of discourse.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-a -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-a.} ~~ \text{Extension from 2 to 4 Dimensions : Areal}\!</math><br />
|}<br />
<br />
Figure 19-b shows the differential extension from <math>U^\bullet = [u, v]\!</math> to <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v]\!</math> in the ''bundle of boxes'' form of venn diagram.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-b -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-b.} ~~ \text{Extension from 2 to 4 Dimensions : Bundle}\!</math><br />
|}<br />
<br />
As dimensions increase, this factorization of the extended universe along the lines that are marked out by the bundle picture begins to look more and more like a practical necessity. But whenever we use a propositional model to address a real situation in the context of nature we need to remain aware that this articulation into factors, affecting our description, may be wholly artificial in nature and cleave to nothing, no joint in nature, nor any juncture in time to be in or out of joint.<br />
<br />
Figure 19-c illustrates the extension from 2 to 4 dimensions in the ''compact'' style of venn diagram. Here, just the changes with respect to the center cell are shown.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-c -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-c.} ~~ \text{Extension from 2 to 4 Dimensions : Compact}\!</math><br />
|}<br />
<br />
Figure 19-d gives the ''digraph'' form of representation for the differential extension <math>U^\bullet \to \mathrm{E}U^\bullet,\!</math> where the 4 nodes marked with a circle <math>{}^{\bigcirc}\!</math> are the cells <math>uv,\, u \texttt{(} v \texttt{)},\, \texttt{(} u \texttt{)} v,\, \texttt{(} u \texttt{)(} v \texttt{)},\!</math> respectively, and where a 2-headed arc counts as 2 arcs of the differential digraph.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-d -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-d.} ~~ \text{Extension from 2 to 4 Dimensions : Digraph}\!</math><br />
|}<br />
<br />
===Thematization of Functions : And a Declaration of Independence for Variables===<br />
<br />
{| width="100%"<br />
| align="left" |<br />
''And as imagination bodies forth''<br><br />
''The forms of things unknown, the poet's pen''<br><br />
''Turns them to shapes, and gives to airy nothing''<br><br />
''A local habitation and a name.''<br />
| align="right" valign="bottom" | A Midsummer Night's Dream, 5.1.18<br />
|}<br />
<br />
In the representation of propositions as functions it is possible to notice different degrees of explicitness in the way their functional character is symbolized. To indicate what I mean by this, the next series of Figures illustrates a set of graphic conventions that will be put to frequent use in the remainder of this discussion, both to mark the relevant distinctions and to help us convert between related expressions at different levels of explicitness in their functionality.<br />
<br />
====Thematization : Venn Diagrams====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
The known universe has one complete lover and that is the greatest poet. He consumes an eternal passion and is indifferent which chance happens and which possible contingency of fortune or misfortune and persuades daily and hourly his delicious pay.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 11&ndash;12]<br />
|}<br />
<br />
Figure 20-i traces the first couple of steps in this order of ''thematic'' progression, that will gradually run the gamut through a complete series of degrees of functional explicitness in the expression of logical propositions. The first venn diagram represents a situation where the function is indicated by a shaded figure and a logical expression. At this stage one may be thinking of the proposition only as expressed by a formula in a particular language and its content only as a subset of the universe of discourse, as when considering the proposition <math>u\!\cdot\!v</math> in the universe <math>[u, v].\!</math> The second venn diagram depicts a situation in which two significant steps have been taken. First, one has taken the trouble to give the proposition <math>u\!\cdot\!v</math> a distinctive functional name <math>{}^{\backprime\backprime} J {}^{\prime\prime}.\!</math> Second, one has come to think explicitly about the target domain that contains the functional values of <math>J,\!</math> as when writing <math>J : \langle u, v \rangle \to \mathbb{B}.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 20-i -- Thematization of Conjunction (Stage 1).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 20-i.} ~~ \text{Thematization of Conjunction (Stage 1)}\!</math><br />
|}<br />
<br />
In Figure 20-ii the proposition <math>J\!</math> is viewed explicitly as a transformation from one universe of discourse to another.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 20-ii -- Thematization of Conjunction (Stage 2).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 20-ii.} ~~ \text{Thematization of Conjunction (Stage 2)}\!</math><br />
|}<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
o-------------------------------o o-------------------------------o<br />
| | | |<br />
| o-----o o-----o | | o-----o o-----o |<br />
| / \ / \ | | / \ / \ |<br />
| / o \ | | / o \ |<br />
| / /`\ \ | | / /`\ \ |<br />
| o o```o o | | o o```o o |<br />
| | u |```| v | | | | u |```| v | |<br />
| o o```o o | | o o```o o |<br />
| \ \`/ / | | \ \`/ / |<br />
| \ o / | | \ o / |<br />
| \ / \ / | | \ / \ / |<br />
| o-----o o-----o | | o-----o o-----o |<br />
| | | |<br />
o-------------------------------o o-------------------------------o<br />
\ / \ /<br />
\ / \ /<br />
\ / \ J /<br />
\ / \ /<br />
\ / \ /<br />
o----------\---------/----------o o----------\---------/----------o<br />
| \ / | | \ / |<br />
| \ / | | \ / |<br />
| o-----@-----o | | o-----@-----o |<br />
| /`````````````\ | | /`````````````\ |<br />
| /```````````````\ | | /```````````````\ |<br />
| /`````````````````\ | | /`````````````````\ |<br />
| o```````````````````o | | o```````````````````o |<br />
| |```````````````````| | | |```````````````````| |<br />
| |```````` J ````````| | | |```````` x ````````| |<br />
| |```````````````````| | | |```````````````````| |<br />
| o```````````````````o | | o```````````````````o |<br />
| \`````````````````/ | | \`````````````````/ |<br />
| \```````````````/ | | \```````````````/ |<br />
| \`````````````/ | | \`````````````/ |<br />
| o-----------o | | o-----------o |<br />
| | | |<br />
| | | |<br />
o-------------------------------o o-------------------------------o<br />
J = u v x = J<u, v><br />
<br />
Figure 20-ii. Thematization of Conjunction (Stage 2)<br />
</pre><br />
|}<br />
<br />
In the first venn diagram the name that is assigned to a composite proposition, function, or region in the source universe is delegated to a simple feature in the target universe. This can result in a single character or term exceeding the responsibilities it can carry off well. Allowing the name of a function <math>J : \langle u, v \rangle \to \mathbb{B}\!</math> to serve as the name of its dependent variable <math>J : \mathbb{B}\!</math> does not mean that one has to confuse a function with any of its values, but it does put one at risk for a number of obvious problems, and we should not be surprised, on numerous and limiting occasions, when quibbling arises from the attempts of a too original syntax to serve these two masters.<br />
<br />
The second venn diagram circumvents these difficulties by introducing a new variable name for each basic feature of the target universe, as when writing <math>J : \langle u, v \rangle \to \langle x \rangle,\!</math> and thereby assigns a concrete type <math>\langle x \rangle</math> to the abstract codomain <math>\mathbb{B}.\!</math> To make this induction of variables more formal one can append subscripts, as in <math>x_J,\!</math> to indicate the origin or derivation of the new characters. Or we may use a lexical modifier to convert function names into variable names, for example, associating the function name <math>J\!</math> with the variable name <math>\check{J}.\!</math> Thus we may think of <math>x = x_J = \check{J}\!</math> as the ''cache variable'' corresponding to the function <math>J\!</math> or the symbol <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> considered as a contingent variable.<br />
<br />
In Figure 20-iii we arrive at a stage where the functional equations <math>J = u\!\cdot\!v</math> and <math>x = u\!\cdot\!v</math> are regarded as propositions in their own right, reigning in and ruling over the 3-feature universes of discourse <math>[u, v, J]~\!</math> and <math>[u, v, x],\!</math> respectively. Subject to the cautions already noted, the function name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> can be reinterpreted as the name of a feature <math>\check{J}</math> and the equation <math>J = u\!\cdot\!v</math> can be read as the logical equivalence <math>\texttt{((} J, u ~ v \texttt{))}.\!</math> To give it a generic name let us call this newly expressed, collateral proposition the ''thematization'' or the ''thematic extension'' of the original proposition <math>J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 20-iii -- Thematization of Conjunction (Stage 3).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 20-iii.} ~~ \text{Thematization of Conjunction (Stage 3)}\!</math><br />
|}<br />
<br />
The first venn diagram represents the thematization of the conjunction <math>J\!</math> with shading in the appropriate regions of the universe <math>[u, v, J].\!</math> Also, it illustrates a quick way of constructing a thematic extension. First, draw a line, in practice or the imagination, that bisects every cell of the original universe, placing half of each cell under the aegis of the thematized proposition and the other half under its antithesis. Next, on the scene where the theme applies leave the shade wherever it lies, and off the stage, where it plays otherwise, stagger the pattern in a harlequin guise.<br />
<br />
In the final venn diagram of this sequence the thematic progression comes full circle and completes one round of its development. The ambiguities that were occasioned by the changing role of the name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> are resolved by introducing a new variable name <math>{}^{\backprime\backprime} x {}^{\prime\prime}</math> to take the place of <math>\check{J},\!</math> and the region that represents this fresh featured <math>x\!</math> is circumscribed in a more conventional symmetry of form and placement. Just as we once gave the name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> to the proposition <math>u\!\cdot\!v,</math> we now give the name <math>{}^{\backprime\backprime} \iota {}^{\prime\prime}</math> to its thematization <math>\texttt{((} x, u ~ v \texttt{))}.\!</math> Already, again, at this culminating stage of reflection, we begin to think of the newly named proposition as a distinctive individual, a particular function <math>\iota : \langle u, v, x \rangle \to \mathbb{B}.\!</math><br />
<br />
From now on, the terms ''thematic extension'' and ''thematization'' will be used to describe both the process and degree of explication that progresses through this series of pictures, both the operation of increasingly explicit symbolization and the dimension of variation that is swept out by it. To speak of this change in general, that takes us in our current example from <math>J\!</math> to <math>\iota,\!</math> we introduce a class of operators symbolized by the Greek letter <math>\theta,\!</math> writing <math>\iota = \theta J\!</math> in the present instance. The operator <math>\theta,\!</math> in the present situation bearing the type <math>\theta : [u, v] \to [u, v, x],\!</math> provides us with a convenient way of recapitulating and summarizing the complete cycle of thematic developments.<br />
<br />
Figure 21 shows how the thematic extension operator <math>\theta\!</math> acts on two further examples, the disjunction <math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math> and the equality <math>\texttt{((} u, v \texttt{))}.\!</math> Referring to the disjunction as <math>f(u, v)\!</math> and the equality as <math>f(u, v),\!</math> we may express the thematic extensions as <math>\varphi = \theta f\!</math> and <math>\gamma = \theta g.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 21 -- Thematization of Disjunction and Equality.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 21.} ~~ \text{Thematization of Disjunction and Equality}\!</math><br />
|}<br />
<br />
====Thematization : Truth Tables====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
That which distorts honest shapes or which creates unearthly beings or places or contingencies is a nuisance and a revolt.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 19]<br />
|}<br />
<br />
Tables 22 through 25 outline a method for computing the thematic extensions of propositions in terms of their coordinate values.<br />
<br />
A preliminary step, as illustrated in Table&nbsp;22, is to write out the truth table representations of the propositional forms whose thematic extensions one wants to compute, in the present instance, the functions <math>f(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math> and <math>g(u, v) = \texttt{((} u, v \texttt{))}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:50%"<br />
|+ style="height:30px" | <math>\text{Table 22.} ~~ \text{Disjunction}~ f ~\text{and Equality}~ g\!</math><br />
|- style="height:35px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black" | <math>f\!</math><br />
| <math>g\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Next, each propositional form is individually represented in the fashion shown in Tables&nbsp;23-i and 23-ii, using <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} g {}^{\prime\prime}\!</math> as function names and creating new variables <math>x\!</math> and <math>y\!</math> to hold the associated functional values. This pair of Tables outlines the first stage in the transition from the <math>2\!</math>-dimensional universes of <math>f\!</math> and <math>g\!</math> to the <math>3\!</math>-dimensional universes of <math>\theta f\!</math> and <math>\theta g.\!</math> The top halves of the Tables replicate the truth table patterns for <math>f\!</math> and <math>g\!</math> in the form <math>f : [u, v] \to [x]\!</math> and <math>g : [u, v] \to [y].\!</math> The bottom halves of the tables print the negatives of these pictures, as it were, and paste the truth tables for <math>\texttt{(} f \texttt{)}\!</math> and <math>\texttt{(} g \texttt{)}\!</math> under the copies for <math>f\!</math> and <math>g.\!</math> At this stage, the columns for <math>\theta f\!</math> and <math>\theta g\!</math> are appended almost as afterthoughts, amounting to indicator functions for the sets of ordered triples that make up the functions <math>f\!</math> and <math>g.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 23-i and 23-ii.} ~~ \text{Thematics of Disjunction and Equality (1)}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 23-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black" | <math>f\!</math><br />
| <math>x\!</math><br />
| <math>\varphi\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |<br />
<math>\begin{matrix}\to\\\to\\\to\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" | &nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 23-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black" | <math>g\!</math><br />
| <math>y\!</math><br />
| <math>\gamma\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |<br />
<math>\begin{matrix}\to\\\to\\\to\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" | &nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
All the data are now in place to give the truth tables for <math>\theta f\!</math> and <math>\theta g.\!</math> All that remains to be done is to permute the rows and change the roles of <math>x\!</math> and <math>y\!</math> from dependent to independent variables. In Tables&nbsp;24-i and 24-ii the rows are arranged in such a way as to put the 3-tuples <math>(u, v, x)\!</math> and <math>(u, v, y)\!</math> in binary numerical order, suitable for viewing as the arguments of the maps <math>\theta f = \varphi : [u, v, x] \to \mathbb{B}\!</math> and <math>\theta g = \gamma : [u, v, y] \to \mathbb{B}.\!</math> Moreover, the structure of the tables is altered slightly, allowing the now vestigial functions <math>\theta f\!</math> and <math>\theta g\!</math> to be passed over without further attention and shifting the heavy vertical bars a notch to the right. In effect, this clinches the fact that the thematic variables <math>x := \check{f}\!</math> and <math>y := \check{g}\!</math> are now treated as independent variables.<br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 24-i and 24-ii.} ~~ \text{Thematics of Disjunction and Equality (2)}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 24-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>f\!</math><br />
| <math>x\!</math><br />
| style="border-left:1px solid black" | <math>\varphi\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <font size="+2">&nbsp;<br></font><math>\begin{matrix}\to\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 24-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>g\!</math><br />
| <math>y\!</math><br />
| style="border-left:1px solid black" | <math>\gamma\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | &nbsp;<br><math>\begin{matrix}\to\\\to\end{matrix}\!</math><br>&nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
An optional reshuffling of the rows brings additional features of the thematic extensions to light. Leaving the columns in place for the sake of comparison, Tables&nbsp;25-i and 25-ii sort the rows in a different order, in effect treating <math>x\!</math> and <math>y\!</math> as the primary variables in their respective 3-tuples. Regarding the thematic extensions in the form <math>\varphi : [x, u, v] \to \mathbb{B}\!</math> and <math>\gamma : [y, u, v] \to \mathbb{B}\!</math> makes it easier to see in this tabular setting a property that was graphically obvious in the venn diagrams above. Specifically, when the thematic variable <math>\check{F}\!</math> is true then <math>\theta F\!</math> exhibits the pattern of the original <math>F,\!</math> and when <math>\check{F}\!</math> is false then <math>\theta F\!</math> exhibits the pattern of its negation <math>\texttt{(} F \texttt{)}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 25-i and 25-ii.} ~~ \text{Thematics of Disjunction and Equality (3)}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 25-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>f\!</math><br />
| <math>x\!</math><br />
| style="border-left:1px solid black" | <math>\varphi\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>{\to}\!</math><br><font size="+2">&nbsp;<br>&nbsp;<br>&nbsp;<br></font><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <font size="+2">&nbsp;<br></font><math>\begin{matrix}\to\\\to\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 25-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>g\!</math><br />
| <math>y\!</math><br />
| style="border-left:1px solid black" | <math>\gamma\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | &nbsp;<br><math>\begin{matrix}\to\\\to\end{matrix}\!</math><br>&nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
Finally, Tables&nbsp;26-i and 26-ii compare the tacit extensions <math>\boldsymbol\varepsilon : [u, v] \to [u, v, x]\!</math> and <math>\boldsymbol\varepsilon : [u, v] \to [u, v, y]\!</math> with the thematic extensions of the same types, as applied to the propositions <math>f\!</math> and <math>g,\!</math> respectively.<br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 26-i and 26-ii.} ~~ \text{Tacit Extension and Thematization}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 26-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>x\!</math><br />
| style="border-left:1px solid black" | <math>\boldsymbol\varepsilon f\!</math><br />
| <math>\theta f\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 26-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>y\!</math><br />
| style="border-left:1px solid black" | <math>\boldsymbol\varepsilon g\!</math><br />
| <math>\theta g\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
Table 27 summarizes the thematic extensions of all propositions on two variables. Column&nbsp;4 lists the equations of form <math>\texttt{((} \check{f_i}, f_i (u, v) \texttt{))}\!</math> and Column&nbsp;5 simplifies these equations into the form of algebraic expressions. As always, <math>{}^{\backprime\backprime} + {}^{\prime\prime}\!</math> refers to exclusive disjunction and each <math>{}^{\backprime\backprime} \check{f} {}^{\prime\prime}\!</math> appearing in the last two Columns refers to the corresponding variable name <math>{}^{\backprime\backprime} \check{f_i} {}^{\prime\prime}.~\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 27.} ~~ \text{Thematization of Bivariate Propositions}\!</math><br />
|- style="height:30px; background:ghostwhite"<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>{f}\!</math><br />
| <math>\theta f\!</math><br />
| <math>\theta f\!</math><br />
|- style="background:ghostwhite"<br />
| align="right" | <math>u\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| align="right" | <math>v\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| align="left" | <math>\texttt{((} \check{f} \texttt{,~(~)~))}\!</math><br />
| align="left" | <math>\check{f} + 1\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\\[4pt]<br />
f_{4}<br />
\\[4pt]<br />
f_{8}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
1~0~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} u \texttt{)~} v \texttt{~}<br />
\\[4pt]<br />
\texttt{~} u \texttt{~(} v \texttt{)}<br />
\\[4pt]<br />
\texttt{~} u \texttt{~~} v \texttt{~}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(u)(v)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~(u)~v~~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~u~(v)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~u~~v~~))}<br />
\end{array}</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + u + v + uv<br />
\\[4pt]<br />
\check{f} + v + uv + 1<br />
\\[4pt]<br />
\check{f} + u + uv + 1<br />
\\[4pt]<br />
\check{f} + uv + 1<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{3}<br />
\\[4pt]<br />
f_{12}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{)}<br />
\\[4pt]<br />
\texttt{~} u \texttt{~}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(u)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~u~~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + u<br />
\\[4pt]<br />
\check{f} + u + 1<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[4pt]<br />
f_{9}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[4pt]<br />
1~0~0~1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{,} v \texttt{)}<br />
\\[4pt]<br />
\texttt{((} u \texttt{,} v \texttt{))}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~~(} u \texttt{,} v \texttt{)~~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~((} u \texttt{,} v \texttt{))~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + u + v + 1<br />
\\[4pt]<br />
\check{f} + u + v<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{5}<br />
\\[4pt]<br />
f_{10}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} v \texttt{)}<br />
\\[4pt]<br />
\texttt{~} v \texttt{~}<br />
\end{matrix}</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(} v \texttt{)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~} v \texttt{~~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + v<br />
\\[4pt]<br />
\check{f} + v + 1<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[4pt]<br />
f_{11}<br />
\\[4pt]<br />
f_{13}<br />
\\[4pt]<br />
f_{14}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(~} u \texttt{~~} v \texttt{~)}<br />
\\[4pt]<br />
\texttt{(~} u \texttt{~(} v \texttt{))}<br />
\\[4pt]<br />
\texttt{((} u \texttt{)~} v \texttt{~)}<br />
\\[4pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(~} u \texttt{~~} v \texttt{~)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~(~} u \texttt{~(} v \texttt{))~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~((} u \texttt{)~} v \texttt{~)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~((} u \texttt{)(} v \texttt{))~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + uv<br />
\\[4pt]<br />
\check{f} + u + uv<br />
\\[4pt]<br />
\check{f} + v + uv<br />
\\[4pt]<br />
\check{f} + u + v + uv + 1<br />
\end{array}\!</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| align="left" | <math>\texttt{((} \check{f} \texttt{,~((~))~))}\!</math><br />
| align="left" | <math>\check{f}\!</math><br />
|}<br />
<br />
<br><br />
<br />
In order to show what all of the thematic extensions from two dimensions to three dimensions look like in terms of coordinates, Tables&nbsp;28 and 29 present ordinary truth tables for the functions <math>f_i : \mathbb{B}^2 \to \mathbb{B}\!</math> and for the corresponding thematizations <math>\theta f_i = \varphi_i : \mathbb{B}^3 \to \mathbb{B}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 28.} ~~ \text{Propositions on Two Variables}\!</math><br />
|- style="height:35px; background:ghostwhite"<br />
| style="width:5%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:5%; border-bottom:1px solid black; border-left:1px solid black" | <math>f_{0}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{1}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{2}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{3}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{4}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{5}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{6}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{7}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{8}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{9}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{10}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{11}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{12}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{13}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{14}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{15}\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 29.} ~~ \text{Thematic Extensions of Bivariate Propositions}\!</math><br />
|- style="height:35px; background:ghostwhite"<br />
| style="width:5%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\check{f}\!</math><br />
| style="width:5%; border-bottom:1px solid black; border-left:1px solid black" | <math>\varphi_{0}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{1}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{2}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{3}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{4}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{5}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{6}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{7}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{8}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{9}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{10}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{11}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{12}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{13}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{14}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{15}\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
|-<br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Propositional Transformations===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
If only the word &lsquo;artificial&rsquo; were associated with the idea of ''art'', or expert skill gained through voluntary apprenticeship (instead of suggesting the factitious and unreal), we might say that ''logical'' refers to artificial thought.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 56&ndash;57]<br />
|}<br />
<br />
In this section we develop a comprehensive set of concepts for dealing with transformations between universes of discourse. In this most general setting the source and target universes of a transformation are allowed to be different, but may be the same. When we apply these concepts to dynamic systems we focus on the important special case of transformations that map a universe into itself, regarding them as the state transitions of a discrete dynamical process and placing them among the myriad ways that a universe of discourse might change, and by that change turn into itself.<br />
<br />
====Alias and Alibi Transformations====<br />
<br />
There are customarily two modes of understanding a transformation, at least, when we try to interpret its relevance to something in reality. A transformation always refers to a changing prospect, to say it in a unified but equivocal way, but this can be taken to mean either a subjective change in the interpreting observer's point of view or an objective change in the systematic subject of discussion. In practice these variant uses of the transformation concept are distinguished in the following terms:<br />
<br />
# A ''perspectival'' or ''alias'' transformation refers to a shift in perspective or a change in language that takes place in the observer's frame of reference.<br />
# A ''transitional'' or ''alibi'' transformation refers to a change of position or an alteration of state that occurs in the object system as it falls under study.<br />
<br />
(For a recent discussion of the alias vs. alibi issue, as it relates to linear transformations in vector spaces and to other issues of an algebraic nature, see [MaB, 256, 582-4].)<br />
<br />
Naturally, when we are concerned with the dynamical properties of a system, the transitional aspect of transformation is the factor that comes to the fore, and this involves us in contemplating all of the ways of changing a universe into itself while remaining under the rule of established dynamical laws. In the prospective application to dynamic systems, and to neural networks viewed in this light, our interest lies chiefly with the transformations of a state space into itself that constitute the state transitions of a discrete dynamic process. Nevertheless, many important properties of these transformations, and some constructions that we need to see most clearly, are independent of the transitional interpretation and are likely to be confounded with irrelevant features if presented first and only in that association.<br />
<br />
In addition, and in partial contrast, intelligent systems are exactly that species of dynamic agents that have the capacity to have a point of view, and we cannot do justice to their peculiar properties without examining their ability to form and transform their own frames of reference in exposure to the elements of their own experience. In this setting, the perspectival aspect of transformation is the facet that shines most brightly, perhaps too often leaving us fascinated with mere glimmerings of its actual potential. It needs to be emphasized that nothing of the ordinary sort needs be moved in carrying out a transformation under the alias interpretation, that it may only involve a change in the forms of address, an amendment of the terms which are customed to approach and fashioned to describe the very same things in the very same world. But again, working within a discipline of realistic computation, we know how formidably complex and resource-consuming such transformations of perspective can be to implement in practice, much less to endow in the self-governed form of a nascently intelligent dynamical system.<br />
<br />
====Transformations of General Type====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
''Es ist passiert'', &ldquo;it just sort of happened&rdquo;, people said there when other people in other places thought heaven knows what had occurred. It was a peculiar phrase, not known in this sense to the Germans and with no equivalent in other languages, the very breath of it transforming facts and the bludgeonings of fate into something light as eiderdown, as thought itself.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 34]<br />
|}<br />
<br />
Consider the situation illustrated in Figure&nbsp;30, where the alphabets <math>\mathcal{U} = \{ u, v \}\!</math> and <math>\mathcal{X} = \{ x, y, z \}\!</math> are used to label basic features in two different logical universes, <math>U^\bullet = [u, v]\!</math> and <math>X^\bullet = [x, y, z].\!</math><br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
o-------------------------------------------------------o<br />
| U |<br />
| |<br />
| o-----------o o-----------o |<br />
| / \ / \ |<br />
| / o \ |<br />
| / / \ \ |<br />
| / / \ \ |<br />
| o o o o |<br />
| | | | | |<br />
| | u | | v | |<br />
| | | | | |<br />
| o o o o |<br />
| \ \ / / |<br />
| \ \ / / |<br />
| \ o / |<br />
| \ / \ / |<br />
| o-----------o o-----------o |<br />
| |<br />
| |<br />
o---------------------------o---------------------------o<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
o-------------------------o o-------------------------o o-------------------------o<br />
| U | | U | | U |<br />
| o---o o---o | | o---o o---o | | o---o o---o |<br />
| / \ / \ | | / \ / \ | | / \ / \ |<br />
| / o \ | | / o \ | | / o \ |<br />
| / / \ \ | | / / \ \ | | / / \ \ |<br />
| o o o o | | o o o o | | o o o o |<br />
| | u | | v | | | | u | | v | | | | u | | v | |<br />
| o o o o | | o o o o | | o o o o |<br />
| \ \ / / | | \ \ / / | | \ \ / / |<br />
| \ o / | | \ o / | | \ o / |<br />
| \ / \ / | | \ / \ / | | \ / \ / |<br />
| o---o o---o | | o---o o---o | | o---o o---o |<br />
| | | | | |<br />
o-------------------------o o-------------------------o o-------------------------o<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ g | \ f / | h /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ o----------|-----------\-----/-----------|----------o /<br />
\ | X | \ / | | /<br />
\ | | \ / | | /<br />
\ | | o-----o-----o | | /<br />
\| | / \ | |/<br />
\ | / \ | /<br />
|\ | / \ | /|<br />
| \ | / \ | / |<br />
| \ | / \ | / |<br />
| \ | o x o | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \| | | |/ |<br />
| o--o--------o o--------o--o |<br />
| / \ \ / / \ |<br />
| / \ \ / / \ |<br />
| / \ o / \ |<br />
| / \ / \ / \ |<br />
| / \ / \ / \ |<br />
| o o--o-----o--o o |<br />
| | | | | |<br />
| | | | | |<br />
| | | | | |<br />
| | y | | z | |<br />
| | | | | |<br />
| | | | | |<br />
| o o o o |<br />
| \ \ / / |<br />
| \ \ / / |<br />
| \ o / |<br />
| \ / \ / |<br />
| \ / \ / |<br />
| o-----------o o-----------o |<br />
| |<br />
| |<br />
o---------------------------------------------------o<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ p , q /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
o<br />
<br />
Figure 30. Generic Frame of a Logical Transformation<br />
</pre><br />
|}<br />
<br />
Enter the picture, as we usually do, in the middle of things, with features like <math>x, y , z\!</math> that present themselves to be simple enough in their own right and that form a satisfactory, if temporary foundation to provide a basis for discussion. In this universe and on these terms we find expression for various propositions and questions of principal interest to ourselves, as indicated by the maps <math>p, q : X \to \mathbb{B}.\!</math> Then we discover that the simple features <math>\{ x, y, z \}\!</math> are really more complex than we thought at first, and it becomes useful to regard them as functions <math>\{ f, g, h \}\!</math> of other features <math>\{ u, v \}\!</math> that we place in a preface to our original discourse, or suppose as topics of a preliminary universe of discourse <math>U^\bullet = [u, v].\!</math> It may happen that these late-blooming but pre-ambling features are found to lie closer, in a sense that may be our job to determine, to the central nature of the situation of interest, in which case they earn our regard as being more fundamental, but these functions and features are only required to supply a critical stance on the universe of discourse or an alternate perspective on the nature of things in order to be preserved as useful.<br />
<br />
A particular transformation <math>F : [u, v] \to [x, y, z]\!</math> may be expressed by a system of equations, as shown below. Here, <math>F\!</math> is defined by its component maps <math>F = (F_1, F_2, F_3) = (f, g, h),\!</math> where each component map in <math>\{ f, g, h \}\!</math> is a proposition of type <math>\mathbb{B}^n \to \mathbb{B}^1.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:50%"<br />
|<br />
<math>\begin{matrix}<br />
x & = & f(u, v)<br />
\\[10pt]<br />
y & = & g(u, v)<br />
\\[10pt]<br />
z & = & h(u, v)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Regarded as a logical statement, this system of equations expresses a relation between a collection of freely chosen propositions <math>\{ f, g, h \}\!</math> in one universe of discourse and the special collection of simple propositions <math>\{ x, y, z \}\!</math> on which is founded another universe of discourse. Growing familiarity with a particular transformation of discourse, and the desire to achieve a ready understanding of its implications, requires that we be able to convert this information about generals and simples into information about all the main subtypes of propositions, including the linear and singular propositions.<br />
<br />
===Analytic Expansions : Operators and Functors===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Consider what effects that might ''conceivably'' have practical bearings you ''conceive'' the objects of your ''conception'' to have. Then, your ''conception'' of those effects is the whole of your ''conception'' of the object.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; C.S. Peirce, &ldquo;The Maxim of Pragmatism&rdquo;, CP 5.438<br />
|}<br />
<br />
Given the barest idea of a logical transformation, as suggested by the sketch in Figure&nbsp;30, and having conceptualized the universe of discourse, with all of its points and propositions, as a beginning object of discussion, we are ready to enter the next phase of our investigation.<br />
<br />
====Operators on Propositions and Transformations====<br />
<br />
The next step is naturally inclined toward objects of the next higher order, namely, with operators that take in argument lists of logical transformations and that give back specified types of logical transformations as their results. For our present aims, we do not need to consider the most general class of such operators, nor any one of them for its own sake. Rather, we are interested in the special sorts of operators that arise in the study and analysis of logical transformations. Figuratively speaking, these operators serve as instruments for the live tomography (and hopefully not the vivisection) of the forms of change under view. Beyond that, they open up ways to implement the changes of view that we need to grasp all the variations on a transformational theme, or to appreciate enough of its significant features to &ldquo;get the drift&rdquo; of the change occurring, to form a passing acquaintance or a synthetic comprehension of its general character and disposition.<br />
<br />
The simplest type of operator is one that takes a single transformation as an argument and returns a single transformation as a result, and most of the operators explicitly considered in our discussion will be of this kind. Figure&nbsp;31 illustrates the typical situation.<br />
<br />
{| align="center" border="0" cellpadding="20"<br />
|<br />
<pre><br />
o---------------------------------------o<br />
| |<br />
| |<br />
| U% F X% |<br />
| o------------------>o |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| !W! | | !W! |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| v v |<br />
| o------------------>o |<br />
| !W!U% !W!F !W!X% |<br />
| |<br />
| |<br />
o---------------------------------------o<br />
Figure 31. Operator Diagram (1)<br />
</pre><br />
|}<br />
<br />
In this Figure <math>{}^{\backprime\backprime} \mathsf{W} {}^{\prime\prime}\!</math> stands for a generic operator <math>\mathsf{W},\!</math> in this case one that takes a logical transformation <math>F\!</math> of type <math>(U^\bullet \to X^\bullet)\!</math> into a logical transformation <math>\mathsf{W}F\!</math> of type <math>(\mathsf{W}U^\bullet \to \mathsf{W}X^\bullet).\!</math> Thus, the operator <math>\mathsf{W}\!</math> must be viewed as making assignments for both families of objects we have previously considered, that is, for universes of discourse like <math>{U^\bullet}\!</math> and <math>{X^\bullet}\!</math> and for logical transformations like <math>F.\!</math><br />
<br />
'''Note.''' Strictly speaking, an operator like <math>\mathsf{W}\!</math> works between two whole categories of universes and transformations, which we call the ''source'' and the ''target'' categories of <math>\mathsf{W}.\!</math> Given this setting, <math>\mathsf{W}\!</math> specifies for each universe <math>U^\bullet\!</math> in its source category a definite universe <math>\mathsf{W}U^\bullet\!</math> in its target category, and to each transformation <math>F\!</math> in its source category it assigns a unique transformation <math>\mathsf{W}F\!</math> in its target category. Naturally, this only works if <math>\mathsf{W}\!</math> takes the source <math>U^\bullet</math> and the target <math>X^\bullet</math> of the map <math>F\!</math> over to the source <math>\mathsf{W}U^\bullet\!</math> and the target <math>\mathsf{W}X^\bullet\!</math> of the map <math>\mathsf{W}F.\!</math> With luck or care enough, we can avoid ever having to put anything like that in words again, letting diagrams do the work. In the situations of present concern we are usually focused on a single transformation <math>F,\!</math> and thus we can take it for granted that the assignment of universes under <math>\mathsf{W}\!</math> is defined appropriately at the source and target ends of <math>F.\!</math> It is not always the case, though, that we need to use the particular names (like <math>{}^{\backprime\backprime} \mathsf{W}U^\bullet {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \mathsf{W}X^\bullet {}^{\prime\prime}\!</math>) that <math>\mathsf{W}\!</math> assigns by default to its operative image universes. In most contexts we will usually have a prior acquaintance with these universes under other names and it is necessary only that we can tell from the information associated with an operator <math>\mathsf{W}\!</math> what universes they are.<br />
<br />
In Figure&nbsp;31 the maps <math>F\!</math> and <math>\mathsf{W}F\!</math> are displayed horizontally, the way one normally orients functional arrows in a written text, and <math>\mathsf{W}\!</math> rolls the map <math>F\!</math> downward into the images that are associated with <math>\mathsf{W}F.\!</math> In Figure&nbsp;32 the same information is redrawn so that the maps <math>F\!</math> and <math>\mathsf{W}F\!</math> flow down the page, and <math>\mathsf{W}\!</math> unfurls the map <math>F\!</math> rightward into domains that are the eminent purview of <math>\mathsf{W}F.\!</math><br />
<br />
{| align="center" border="0" cellpadding="20"<br />
|<br />
<pre><br />
o---------------------------------------o<br />
| |<br />
| |<br />
| U% !W! !W!U% |<br />
| o------------------>o |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| F | | !W!F |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| v v |<br />
| o------------------>o |<br />
| X% !W! !W!X% |<br />
| |<br />
| |<br />
o---------------------------------------o<br />
Figure 32. Operator Diagram (2)<br />
</pre><br />
|}<br />
<br />
The latter arrangement, as exhibited in Figure&nbsp;32, is more congruent with the thinking about operators that we shall do in the rest of this discussion, since all logical transformations from here on out will be pictured vertically, after the fashion of Figure&nbsp;30.<br />
<br />
====Differential Analysis of Propositions and Transformations====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" | The resultant metaphysical problem now is this: ''Does the man go round the squirrel or not?''<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 43]<br />
|}<br />
<br />
The approach to the differential analysis of logical propositions and transformations of discourse to be pursued here is carried out in terms of particular operators <math>\mathsf{W}\!</math> that act on propositions <math>F\!</math> or on transformations <math>F\!</math> to yield the corresponding operator maps <math>\mathsf{W}F.\!</math> The operator results then become the subject of a series of further stages of analysis, which take them apart into their propositional components, rendering them as a set of purely logical constituents. After this is done, all the parts are then re-integrated to reconstruct the original object in the light of a more complete understanding, at least in ways that enable one to appreciate certain aspects of it with fresh insight.<br />
<br />
* '''Remark on Strategy.''' At this point we run into a set of conceptual difficulties that force us to make a strategic choice in how we proceed. Part of the problem can be remedied by extending our discussion of tacit extensions to the transformational context. But the troubles that remain are much more obstinate and lead us to try two different types of solution. The approach that we develop first makes use of a variant type of extension operator, the ''trope extension'', to be defined below. This method is more conservative and requires less preparation, but has features which make it seem unsatisfactory in the long run. A more radical approach, but one with a better hope of long term success, makes use of the notion of ''contingency spaces''. These are an even more generous type of extended universe than the kind we currently use, but are defined subject to certain internal constraints. The extra work needed to set up this method forces us to put it off to a later stage. However, as a compromise, and to prepare the ground for the next pass, we call attention to the various conceptual difficulties as they arise along the way and try to give an honest estimate of how well our first approach deals with them.<br />
<br />
We now describe in general terms the particular operators that are instrumental to this form of analysis. The main series of operators all have the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathsf{W}<br />
& : &<br />
( U^\bullet \to X^\bullet )<br />
& \to &<br />
( \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet )<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
If we assume that the source universe <math>U^\bullet</math> and the target universe <math>X^\bullet</math> have finite dimensions <math>n\!</math> and <math>k,\!</math> respectively, then each operator <math>\mathsf{W}\!</math> is encompassed by the same abstract type:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathsf{W}<br />
& : &<br />
( [\mathbb{B}^n] \to [\mathbb{B}^k] )<br />
& \to &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k \times \mathbb{D}^k] )<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Since the range features of the operator result <math>\mathsf{W}F : [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k \times \mathbb{D}^k]</math> can be sorted by their ordinary versus differential qualities and the component maps can be examined independently, the complete operator <math>\mathsf{W}\!</math> can be separated accordingly into two components, in the form <math>\mathsf{W} = (\boldsymbol\varepsilon, \mathrm{W}).\!</math> Given a fixed context of source and target universes, <math>\boldsymbol\varepsilon\!</math> is always the same type of operator, a multiple component version of the tacit extension operators that were described earlier. In this context <math>\boldsymbol\varepsilon\!</math> has the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{array}{lccccc}<br />
\text{Concrete type}<br />
& \boldsymbol\varepsilon<br />
& : &<br />
( U^\bullet \to X^\bullet )<br />
& \to &<br />
( \mathrm{E}U^\bullet \to X^\bullet )<br />
\\[10pt]<br />
\text{Abstract type}<br />
& \boldsymbol\varepsilon<br />
& : &<br />
( [\mathbb{B}^n] \to [\mathbb{B}^k] )<br />
& \to &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k] )<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
On the other hand, the operator <math>\mathrm{W}\!</math> is specific to each <math>\mathsf{W}.\!</math> In this context <math>\mathrm{W}\!</math> always has the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{array}{lccccc}<br />
\text{Concrete type}<br />
& W<br />
& : &<br />
( U^\bullet \to X^\bullet )<br />
& \to &<br />
( \mathrm{E}U^\bullet \to \mathrm{d}X^\bullet )<br />
\\[10pt]<br />
\text{Abstract type}<br />
& W<br />
& : &<br />
( [\mathbb{B}^n] \to [\mathbb{B}^k] )<br />
& \to &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{D}^k] )<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
In the types just assigned to <math>\boldsymbol\varepsilon\!</math> and <math>\mathrm{W}\!</math> and by implication to their results <math>\boldsymbol\varepsilon F\!</math> and <math>\mathrm{W}F,\!</math> we have listed the most restrictive ranges defined for them rather than the more expansive target spaces that subsume these ranges. When there is need to recognize both, we may use type indications like the following:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\boldsymbol\varepsilon F<br />
& : &<br />
( \mathrm{E}U^\bullet \to X^\bullet \subseteq \mathrm{E}X^\bullet )<br />
& \cong &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k] \subseteq [\mathbb{B}^k \times \mathbb{D}^k] )<br />
\\[10pt]<br />
WF<br />
& : &<br />
( \mathrm{E}U^\bullet \to \mathrm{d}X^\bullet \subseteq \mathrm{E}X^\bullet )<br />
& \cong &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{D}^k] \subseteq [\mathbb{B}^k \times \mathbb{D}^k] )<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Hopefully, though, a general appreciation of these subsumptions will prevent us from having to make such declarations more often than absolutely necessary.<br />
<br />
In giving names to these operators we try to preserve as much of the traditional nomenclature and as many of the classical associations as possible. The chief difficulty in doing this is occasioned by the distinction between the &ldquo;sans&nbsp;serif&rdquo; operators <math>\mathsf{W}\!</math> and their &ldquo;serified&rdquo; components <math>\mathrm{W},\!</math> which forces us to find two distinct but parallel sets of terminology. Here is a plan to that purpose. First, the component operators <math>\mathrm{W}\!</math> are named by analogy with the corresponding operators in the classical difference calculus. Next, the complete operators <math>\mathsf{W} = (\boldsymbol\varepsilon, \mathrm{W})</math> are assigned titles according to their roles in a geometric or trigonometric allegory, if only to ensure that the tangent functor, that belongs to this family and whose exposition we are still working toward, comes out fit with its customary name. Finally, the operator results <math>\mathsf{W}F\!</math> and <math>\mathrm{W}F\!</math> can be fixed in our frame of reference by tethering the operative adjective for <math>\mathsf{W}\!</math> or <math>\mathrm{W}\!</math> to the anchoring epithet &ldquo;map&rdquo;, in conformity with an already standard practice.<br />
<br />
=====The Secant Operator : '''E'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Mr. Peirce, after pointing out that our beliefs are really rules for action, said that, to develop a thought's meaning, we need only determine what conduct it is fitted to produce: that conduct is for us its sole significance.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 46]<br />
|}<br />
<br />
Figures&nbsp;33-i and 33-ii depict two stages in the form of analysis that will be applied to transformations throughout the remainder of this study. From now on our interest is staked on an operator denoted <math>{}^{\backprime\backprime} \mathsf{E} {}^{\prime\prime},\!</math> which receives the principal investment of analytic attention, and on the constituent parts of <math>\mathsf{E},\!</math> which derive their shares of significance as developed by the analysis. In the sequel, we refer to <math>\mathsf{E}\!</math> as the ''secant operator'', taking it for granted that a context has been chosen that defines its type. The secant operator has the component description <math>\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}),\!</math> and its active ingredient <math>\mathrm{E}\!</math> is known as the ''enlargement operator''. (Here, we name <math>\mathrm{E}\!</math> after the literal ancestor of the shift operator in the calculus of finite differences, defined so that <math>\mathrm{E}f(x) = f(x+1)\!</math> for any suitable function <math>f,\!</math> though of course the logical analogue that we take up here must have a rather different definition.)<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
U% $E$ $E$U% $E$U% $E$U%<br />
o------------------>o============o============o<br />
| | | |<br />
| | | |<br />
| | | |<br />
| | | |<br />
F | | $E$F = | $d$^0.F + | $r$^0.F<br />
| | | |<br />
| | | |<br />
| | | |<br />
v v v v<br />
o------------------>o============o============o<br />
X% $E$ $E$X% $E$X% $E$X%<br />
<br />
Figure 33-i. Analytic Diagram (1)<br />
</pre><br />
|}<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
U% $E$ $E$U% $E$U% $E$U% $E$U%<br />
o------------------>o============o============o============o<br />
| | | | |<br />
| | | | |<br />
| | | | |<br />
| | | | |<br />
F | | $E$F = | $d$^0.F + | $d$^1.F + | $r$^1.F<br />
| | | | |<br />
| | | | |<br />
| | | | |<br />
v v v v v<br />
o------------------>o============o============o============o<br />
X% $E$ $E$X% $E$X% $E$X% $E$X%<br />
<br />
Figure 33-ii. Analytic Diagram (2)<br />
</pre><br />
|}<br />
<br />
In its action on universes <math>\mathsf{E}\!</math> yields the same result as <math>\mathrm{E},\!</math> a fact that can be expressed in equational form by writing <math>\mathsf{E}U^\bullet = \mathrm{E}U^\bullet\!</math> for any universe <math>U^\bullet.\!</math> Notice that the extended universes across the top and bottom of the diagram are indicated to be strictly identical, rather than requiring a corresponding decomposition for them. In a certain sense, the functional parts of <math>\mathsf{E}F\!</math> are partitioned into separate contexts that have to be re-integrated again, but the best image to use is that of making transparent copies of each universe and then overlapping their functional contents once more at the conclusion of the analysis, as suggested by the graphic conventions that are used at the top of Figure&nbsp;30.<br />
<br />
Acting on a transformation <math>F\!</math> from universe <math>U^\bullet\!</math> to universe <math>X^\bullet,\!</math> the operator <math>\mathsf{E}\!</math> determines a transformation <math>\mathsf{E}F\!</math> from <math>\mathsf{E}U^\bullet\!</math> to <math>\mathsf{E}X^\bullet.\!</math> The map <math>\mathsf{E}F\!</math> forms the main body of evidence to be investigated in performing a differential analysis of <math>F.\!</math> Because we shall frequently be focusing on small pieces of this map for considerable lengths of time, and consequently lose sight of the &ldquo;big picture&rdquo;, it is critically important to emphasize that the map <math>\mathsf{E}F\!</math> is a transformation that determines a relation from one extended universe into another. This means that we should not be satisfied with our understanding of a transformation <math>F\!</math> until we can lay out the full &ldquo;parts diagram&rdquo; of <math>\mathsf{E}F\!</math> along the lines of the generic frame in Figure&nbsp;30.<br />
<br />
Working within the confines of propositional calculus, it is possible to give an elementary definition of <math>\mathsf{E}F\!</math> by means of a system of propositional equations, as we now describe.<br />
<br />
Given a transformation<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>F = (F_1, \ldots, F_k) : \mathbb{B}^n \to \mathbb{B}^k\!</math><br />
|}<br />
<br />
of concrete type<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>F : [u_1, \ldots, u_n] \to [x_1, \ldots, x_k],\!</math><br />
|}<br />
<br />
the transformation<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\mathsf{E}F = (F_1, \ldots, F_k, \mathrm{E}F_1, \ldots, \mathrm{E}F_k) : \mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B}^k \times \mathbb{D}^k\!</math><br />
|}<br />
<br />
of concrete type<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\mathsf{E}F : [u_1, \dots, u_n, \mathrm{d}u_1, \dots, \mathrm{d}u_n] \to [x_1, \ldots, x_k, \mathrm{d}x_1, \ldots, \mathrm{d}x_k]\!</math><br />
|}<br />
<br />
is defined by means of the following system of logical equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\\[16pt]<br />
\mathrm{d}x_1<br />
& = & \mathrm{E}F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1 + \mathrm{d}u_1, \ldots, u_n + \mathrm{d}u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
\mathrm{d}x_k<br />
& = & \mathrm{E}F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1 + \mathrm{d}u_1, \ldots, u_n + \mathrm{d}u_n)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
It is important to note that this system of equations can be read as a conjunction of equational propositions, in effect, as a single proposition in the universe of discourse generated by all the named variables. Specifically, this is the universe of discourse over <math>2(n+k)\!</math> variables denoted by:<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
\mathrm{E}[\mathcal{U} \cup \mathcal{X}]<br />
& = &<br />
[u_1, \ldots, u_n, ~ x_1, \ldots, x_k, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n, ~ \mathrm{d}x_1, \ldots, \mathrm{d}x_k].<br />
\end{matrix}</math><br />
|}<br />
<br />
In this light, it should be clear that the system of equations defining <math>\mathsf{E}F\!</math> embodies, in a higher rank and differentially extended version, an analogy with the process of thematization that we treated earlier for propositions of type <math>F : \mathbb{B}^n \to \mathbb{B}.\!</math><br />
<br />
The entire collection of constraints that is represented in the above system of equations may be abbreviated by writing <math>\mathsf{E}F = (\boldsymbol\varepsilon F, \mathrm{E}F),\!</math> for any map <math>F.\!</math> This is tantamount to regarding <math>\mathsf{E}\!</math> as a complex operator, <math>\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}),\!</math> with a form of application that distributes each component of the operator to work on each component of the operand, as follows:<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
\mathsf{E}F<br />
& = &<br />
(\boldsymbol\varepsilon, \mathrm{E})F<br />
& = &<br />
(\boldsymbol\varepsilon F, \mathrm{E}F)<br />
& = &<br />
(\boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k, ~ \mathrm{E}F_1, \ldots, \mathrm{E}F_k).<br />
\end{matrix}</math><br />
|}<br />
<br />
Quite a lot of &ldquo;thematic infrastructure&rdquo; or interpretive information is being swept under the rug in the use of such abbreviations. When confusion arises about the meaning of such constructions, one always has recourse to the defining system of equations, in its totality a purely propositional expression. This means that the parenthesized argument lists, that were used in this context to build an image of multi-component transformations, should not be expected to determine a well-defined product in themselves but only to serve as reminders of the prior thematic decisions (choices of variable names, etc.) that have to be made in order to determine one. Accordingly, the argument list notation can be regarded as a kind of ''thematic frame'', an interpretive storage device that preserves the proper associations of concrete logical features between the extended universes at the source and target of <math>\mathsf{E}F.\!</math><br />
<br />
The generic notations <math>\mathsf{d}^0\!F, \mathsf{d}^1\!F, \ldots, \mathsf{d}^m\!F\!</math> in Figure&nbsp;33 refer to the increasing orders of differentials that are extracted in the course of analyzing <math>F.\!</math> When the analysis is halted at a partial stage of development, notations like <math>\mathsf{r}^0\!F, \mathsf{r}^1\!F, \ldots, \mathsf{r}^m\!F\!</math> may be used to summarize the contributions to <math>\mathsf{E}F\!</math> that remain to be analyzed. The Figure illustrates a convention that makes <math>\mathsf{r}^m\!F,\!</math> in effect, the sum of all differentials of order strictly greater than <math>m.\!</math><br />
<br />
We next discuss the operators that figure into this form of analysis, describing their effects on transformations. In simplified or specialized contexts these operators tend to take on a variety of different names and notations, some of whose number we introduce along the way.<br />
<br />
=====The Radius Operator : '''e'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
And the tangible fact at the root of all our thought-distinctions, however subtle, is that there is no one of them so fine as to consist in anything but a possible difference of practice.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 46]<br />
|}<br />
<br />
The operator identified as <math>\mathrm{d}^0\!</math> in the analytic diagram (Figure&nbsp;33) has the sole purpose of creating a proxy for <math>F\!</math> in the appropriately extended context. Construed in terms of its broadest components, <math>\mathrm{d}^0\!</math> is equivalent to the doubly tacit extension operator <math>(\boldsymbol\varepsilon, \boldsymbol\varepsilon),\!</math> in recognition of which let us redub it as <math>{}^{\backprime\backprime} \mathsf{e} {}^{\prime\prime}.\!</math> Pursuing a geometric analogy, we may refer to <math>\mathsf{e} =(\boldsymbol\varepsilon, \boldsymbol\varepsilon) = \mathrm{d}^0\!</math> as the ''radius operator''. The operation intended by all of these forms is defined by the following equation:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathsf{e}F<br />
& = &<br />
(\boldsymbol\varepsilon, \boldsymbol\varepsilon)F<br />
\\[4pt]<br />
& = &<br />
(\boldsymbol\varepsilon F, ~ \boldsymbol\varepsilon F)<br />
\\[4pt]<br />
& = &<br />
(\boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k, ~ \boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k).<br />
\end{array}</math><br />
|}<br />
<br />
which is tantamount to the system of equations below.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\\[16pt]<br />
\mathrm{d}x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
\mathrm{d}x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
=====The Phantom of the Operators : '''&eta;'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
I was wondering what the reason could be, when I myself raised my head and everything within me seemed drawn towards the Unseen, ''which was playing the most perfect music''!<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 81]<br />
|}<br />
<br />
We now describe an operator whose persistent but elusive action behind the scenes, whose slightly twisted and ambivalent character, and whose fugitive disposition, caught somewhere in flight between the arrantly negative and the positive but errant intent, has cost us some painstaking trouble to detect. In the end we shall place it among the other extensions and projections, as a shade among shadows, of muted tones and motley hue, that adumbrates its own thematic frame and paradoxically lights the way toward a whole new spectrum of values.<br />
<br />
Given a transformation <math>F : [u_1, \ldots, u_n] \to [x_1, \dots, x_k],\!</math> we often have call to consider a family of related transformations, all having the form:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>F^\dagger : [u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n] \to [\mathrm{d}x_1, \dots, \mathrm{d}x_k].\!</math><br />
|}<br />
<br />
The operator <math>\eta</math> is introduced to deal with the simplest one of these maps:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\eta F : [u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n] \to [\mathrm{d}x_1, \ldots \mathrm{d}x_k],\!</math><br />
|}<br />
<br />
which is defined by the following equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
\mathrm{d}x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
In effect, the operator <math>\eta\!</math> is nothing but the stand-alone version of a procedure that is otherwise invoked subordinate to the work of the radius operator <math>\mathsf{e}.\!</math> Operating independently, <math>\eta\!</math> achieves precisely the same results that the second <math>\boldsymbol\varepsilon\!</math> in <math>(\boldsymbol\varepsilon, \boldsymbol\varepsilon)\!</math> accomplishes by working within the context of its ordered pair thematic frame. From this point on, because the use of <math>\boldsymbol\varepsilon\!</math> and <math>\eta\!</math> in this setting combines the aims of both the tacit and the thematic extensions, and because <math>\eta\!</math> reflects in regard to <math>\boldsymbol\varepsilon\!</math> little more than the application of a differential twist, a mere turn of phrase, we refer to <math>\eta\!</math> as the ''trope extension'' operator.<br />
<br />
=====The Chord Operator : '''D'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
What difference would it practically make to any one if this notion rather than that notion were true? If no practical difference whatever can be traced, then the alternatives mean practically the same thing, and all dispute is idle.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 45]<br />
|}<br />
<br />
Next we discuss an operator that is always immanent in this form of analysis, and remains implicitly present in the entire proceeding. It may appear once as a record: a relic or revenant that reprises the reminders of an earlier stage of development. Or it may appear always as a resource: a reserve or redoubt that caches in advance an echo of what remains to be played out, cleared up, and requited in full at a future stage. And all of this remains true whether or not we recall the key at any time, and whether or not the subtending theme is recited explicitly at any stage of play.<br />
<br />
This is the operator that is referred to as <math>\mathsf{r}^0\!</math> in the initial stage of analysis (Figure&nbsp;33-i) and that is expanded as <math>\mathsf{d}^1 + \mathsf{r}^1\!</math> in the subsequent step (Figure&nbsp;33-ii). In congruence, but not quite harmony with our allusions of analogy that are not quite geometry, we call this the ''chord operator'' and denote it <math>\mathsf{D}.\!</math> In the more casual terms that are here introduced, <math>\mathsf{D}</math> is defined as the remainder of <math>\mathsf{E}\!</math> and <math>\mathsf{e}\!</math> and it assigns a due measure to each undertone of accord or discord that is struck between the note of enterprise <math>\mathsf{E}\!</math> and the bar of exigency <math>\mathsf{e}.\!</math><br />
<br />
The tension between these counterposed notions, in balance transient but regular in stridence, may be refracted along familiar lines, though never by any such fraction resolved. In this style we write <math>\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}),\!</math> calling <math>\mathrm{D}\!</math> the ''difference operator'' and noting that it plays a role in this realm of mutable and diverse discourse that is analogous to the part taken by the discrete difference operator in the ordinary difference calculus. Finally, we should note that the chord <math>\mathsf{D}\!</math> is not one that need be lost at any stage of development. At the <math>m^\text{th}\!</math> stage of play it can always be reconstituted in the following form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathsf{D}<br />
& = & \mathsf{E} - \mathsf{e}<br />
\\[6pt]<br />
& = & \mathsf{r}^0<br />
\\[6pt]<br />
& = & \mathsf{d}^1 + \mathsf{r}^1<br />
\\[6pt]<br />
& = & \mathsf{d}^1 + \ldots + \mathsf{d}^m + \mathsf{r}^m<br />
\\[6pt]<br />
& = & \displaystyle \sum_{i=1}^m \mathsf{d}^i + \mathsf{r}^m<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====The Tangent Operator : '''T'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
They take part in scenes of whose significance they have no inkling. They are merely tangent to curves of history the beginnings and ends and forms of which pass wholly beyond their ken. So we are tangent to the wider life of things.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 300]<br />
|}<br />
<br />
The operator tagged as <math>\mathsf{d}^1\!</math> in the analytic diagram (Figure&nbsp;33) is called the ''tangent operator'' and is usually denoted in this text as <math>\mathsf{d}\!</math> or <math>\mathsf{T}.\!</math> Because it has the properties required to qualify as a functor, namely, preserving the identity element of the composition operation and the articulated form of every composition of transformations, it also earns the title of a ''tangent functor''. According to the custom adopted here, we dissect it as <math>\mathsf{T} = \mathsf{d} = (\boldsymbol\varepsilon, \mathrm{d}),\!</math> where <math>\mathrm{d}\!</math> is the operator that yields the first order differential <math>\mathrm{d}F\!</math> when applied to a transformation <math>F,\!</math> and whose name is legion.<br />
<br />
Figure&nbsp;34 illustrates a stage of analysis where we ignore everything but the tangent functor <math>\mathsf{T}\!</math> and attend to it chiefly as it bears on the first order differential <math>\mathrm{d}F\!</math> in the analytic expansion of <math>F.\!</math> In this situation we often refer to the extended universes <math>\mathrm{E}U^\bullet\!</math> and <math>\mathrm{E}X^\bullet\!</math> under the equivalent designations <math>\mathsf{T}U^\bullet\!</math> and <math>\mathsf{T}X^\bullet,\!</math> respectively. The purpose of the tangent functor <math>\mathsf{T}\!</math> is to extract the tangent map <math>\mathsf{T}F\!</math> at each point of <math>U^\bullet,\!</math> and the tangent map <math>\mathsf{T}F = (\boldsymbol\varepsilon, \mathrm{d})F\!</math> tells us not only what the transformation <math>F\!</math> is doing at each point of the universe <math>U^\bullet\!</math> but also what <math>F\!</math> is doing to states in the neighborhood of that point, approximately, linearly, and relatively speaking.<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
U% $T$ $T$U% $T$U%<br />
o------------------>o============o<br />
| | |<br />
| | |<br />
| | |<br />
| | |<br />
F | | $T$F = | <!e!, d> F<br />
| | |<br />
| | |<br />
| | |<br />
v v v<br />
o------------------>o============o<br />
X% $T$ $T$X% $T$X%<br />
<br />
Figure 34. Tangent Functor Diagram<br />
</pre><br />
|}<br />
<br />
* '''NB.''' There is one aspect of the preceding construction that remains especially problematic. Why did we define the operators <math>\mathrm{W}\!</math> in <math>\{ \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}\!</math> so that the ranges of their resulting maps all fall within the realms of differential quality, even fabricating a variant of the tacit extension operator to have that character? Clearly, not all of the operator maps <math>\mathrm{W}F\!</math> have equally good reasons for placing their values in differential stocks. The reason for it appears to be that, without doing this, we cannot justify the comparison and combination of their functional values in the various analytic steps. By default, only those values in the same functional component can be brought into algebraic modes of interaction. Up till now the only mechanism provided for their broader association has been a purely logical one, their common placement in a target universe of discourse, but the task of converting this logical circumstance into algebraic forms of application has not yet been taken up.<br />
<br />
===Transformations of Type '''B'''<sup>2</sup> &rarr; '''B'''<sup>1</sup>===<br />
<br />
To study the effects of these analytic operators in the simplest possible setting, let us revert to a still more primitive case. Consider the singular proposition <math>J(u, v)= u\!\cdot\!v,\!</math> regarded either as the functional product of the maps <math>u\!</math> and <math>v\!</math> or as the logical conjunction of the features <math>u\!</math> and <math>v,\!</math> a map whose fiber of truth <math>J^{-1}(1)\!</math> picks out the single cell of that logical description in the universe of discourse <math>U^\bullet.\!</math> Thus <math>J,\!</math> or <math>u\!\cdot\!v,\!</math> may be treated as another name for the point whose coordinates are <math>(1, 1)\!</math> in <math>U^\bullet.\!</math><br />
<br />
====Analytic Expansion of Conjunction====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
<p>In her sufferings she read a great deal and discovered that she had lost something, the possession of which she had previously not been much aware of: a&nbsp;soul.</p><br />
<br />
<p>What is that? It is easily defined negatively: it is simply what curls up and hides when there is any mention of algebraic series.</p><br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 118]<br />
|}<br />
<br />
Figure&nbsp;35 pictures the form of conjunction <math>J : \mathbb{B}^2 \to \mathbb{B}\!</math> as a transformation from the <math>2\!</math>-dimensional universe <math>[u, v]\!</math> to the <math>1\!</math>-dimensional universe <math>[x].\!</math> This is a subtle but significant change of viewpoint on the proposition, attaching an arbitrary but concrete quality to its functional value. Using the language introduced earlier, we can express this change by saying that the proposition <math>J : \langle u, v \rangle \to \mathbb{B}\!</math> is being recast into the thematized role of a transformation <math>J : [u, v] \to [x],\!</math> where the new variable <math>x\!</math> takes the part of a thematic variable <math>\check{J}.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 35 -- A Conjunction Viewed as a Transformation.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 35.} ~~ \text{Conjunction as Transformation}\!</math><br />
|}<br />
<br />
=====Tacit Extension of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
I teach straying from me, yet who can stray from me?<br><br />
I follow you whoever you are from the present hour;<br><br />
My words itch at your ears till you understand them.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 83]<br />
|}<br />
<br />
Earlier we defined the tacit extension operators <math>\boldsymbol\varepsilon : X^\bullet \to Y^\bullet\!</math> as maps embedding each proposition of a given universe <math>X^\bullet~\!</math> in a more generously given universe <math>Y^\bullet \supset X^\bullet.\!</math> Of immediate interest are the tacit extensions <math>\boldsymbol\varepsilon : U^\bullet \to \mathrm{E}U^\bullet,\!</math> that locate each proposition of <math>U^\bullet\!</math> in the enlarged context of <math>\mathrm{E}U^\bullet.\!</math> In its application to the propositional conjunction <math>J = u\!\cdot\!v</math> in <math>[u, v],\!</math> the tacit extension operator <math>\boldsymbol\varepsilon\!</math> yields the proposition <math>\boldsymbol\varepsilon J\!</math> in <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v].\!</math> The extended proposition <math>\boldsymbol\varepsilon J\!</math> may be computed according to the scheme in Table&nbsp;36, in effect doing nothing more that conjoining a tautology of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> to <math>J\!</math> in <math>U^\bullet.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 36.} ~~ \text{Computation of}~ \boldsymbol\varepsilon J\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\boldsymbol\varepsilon J & = & J {}_{^\langle} u, v {}_{^\rangle}<br />
\\[4pt]<br />
& = & u \cdot v<br />
\\[4pt]<br />
& = & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{ } \mathrm{d}v \texttt{ }<br />
& + & u \cdot v \cdot \texttt{ } \mathrm{d}u \texttt{ } \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \cdot v \cdot \texttt{ } \mathrm{d}u \texttt{ } \cdot \texttt{ } \mathrm{d}v \texttt{ }<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{array}{*{4}{l}}<br />
\boldsymbol\varepsilon J<br />
& = && u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & u \cdot v \cdot \texttt{~} \mathrm{d}u \texttt{~} \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & u \cdot v \cdot \texttt{~} \mathrm{d}u \texttt{~} \cdot \texttt{~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
The lower portion of the Table contains the dispositional features of <math>\boldsymbol\varepsilon J\!</math> arranged in such a way that the variety of ordinary features spreads across the rows and the variety of differential features runs through the columns. This organization serves to facilitate pattern matching in the remainder of our computations. Again, the tacit extension is usually so trivial a concern that we do not always bother to make an explicit note of it, taking it for granted that any function <math>F\!</math> being employed in a differential context is equivalent to <math>\boldsymbol\varepsilon F\!</math> for a suitable <math>\boldsymbol\varepsilon.\!</math><br />
<br />
Figures&nbsp;37-a through 37-d present several pictures of the proposition <math>J\!</math> and its tacit extension <math>\boldsymbol\varepsilon J.\!</math> Notice in these Figures how <math>\boldsymbol\varepsilon J\!</math> in <math>\mathrm{E}U^\bullet\!</math> visibly extends <math>J\!</math> in <math>U^\bullet\!</math> by annexing to the indicated cells of <math>J\!</math> all the arcs that exit from or flow out of them. In effect, this extension attaches to these cells all the dispositions that spring from them, in other words, it attributes to these cells all the conceivable changes that are their issue.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-a -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-a.} ~~ \text{Tacit Extension of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-b -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-b.} ~~ \text{Tacit Extension of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-c -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-c.} ~~ \text{Tacit Extension of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-d -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-d.} ~~ \text{Tacit Extension of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
The computational scheme shown in Table&nbsp;36 treated <math>J\!</math> as a proposition in <math>U^\bullet\!</math> and formed <math>\boldsymbol\varepsilon J\!</math> as a proposition in <math>\mathrm{E}U^\bullet.\!</math> When <math>J\!</math> is regarded as a mapping <math>J : U^\bullet \to X^\bullet\!</math> then <math>\boldsymbol\varepsilon J\!</math> must be obtained as a mapping <math>\boldsymbol\varepsilon J : \mathrm{E}U^\bullet \to X^\bullet.\!</math> By default, the tacit extension of the map <math>J : [u, v] \to [x]\!</math> is naturally taken to be a particular map,<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\boldsymbol\varepsilon J : [u, v, \mathrm{d}u, \mathrm{d}v] \to [x] \subseteq [x, \mathrm{d}x],\!</math><br />
|}<br />
<br />
namely, the one that looks like <math>J\!</math> when painted in the frame of the extended source universe and that takes the same thematic variable in the extended target universe as the one that <math>J\!</math> already takes.<br />
<br />
But the choice of a particular thematic variable, for example <math>x\!</math> for <math>\check{J},\!</math> is a shade more arbitrary than the choice of original variable names <math>\{ u, v \},\!</math> so the map we are calling the ''trope extension'',<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\eta J : [u, v, \mathrm{d}u, \mathrm{d}v] \to [\mathrm{d}x] \subseteq [x, \mathrm{d}x],\!</math><br />
|}<br />
<br />
since it looks just the same as <math>\boldsymbol\varepsilon J\!</math> in the way its fibers paint the source domain, belongs just as fully to the family of tacit extensions, generically considered.<br />
<br />
These considerations have the practical consequence that all of our computations and illustrations of <math>\boldsymbol\varepsilon J\!</math> perform the double duty of capturing <math>\eta J\!</math> as well. In other words, we are saved the work of carrying out calculations and drawing figures for the trope extension <math>\eta J,\!</math> because it would be identical to the work already done for <math>\boldsymbol\varepsilon J.\!</math> Since the computations given for <math>\boldsymbol\varepsilon J\!</math> are expressed solely in terms of the variables <math>\{ u, v, \mathrm{d}u, \mathrm{d}v \},\!</math> they work equally well for finding <math>\eta J.\!</math> Further, since each of the above Figures shows only how the level sets of <math>\boldsymbol\varepsilon J\!</math> partition the extended source universe <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v],\!</math> all of them serve equally well as portraits of <math>\eta J.\!</math><br />
<br />
=====Enlargement Map of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
No one could have established the existence of any details that might not just as well have existed in earlier times too; but all the relations between things had shifted slightly. Ideas that had once been of lean account grew fat.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 62]<br />
|}<br />
<br />
The enlargement map <math>\mathrm{E}J\!</math> is computed from the proposition <math>J\!</math> by making a particular class of formal substitutions for its variables, in this case <math>u + \mathrm{d}u\!</math> for <math>u\!</math> and <math>v + \mathrm{d}v\!</math> for <math>v,\!</math> and afterwards expanding the result in whatever way is found convenient.<br />
<br />
Table&nbsp;38 shows a typical scheme of computation, following a systematic method of exploiting boolean expansions over selected variables and ultimately developing <math>\mathrm{E}J\!</math> over the cells of <math>[u, v].\!</math> The critical step of this procedure uses the facts that <math>\texttt{(} 0, x \texttt{)} = 0 + x = x\!</math> and <math>\texttt{(} 1, x \texttt{)} = 1 + x = \texttt{(} x \texttt{)}\!</math> for any boolean variable <math>x.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 38.} ~~ \text{Computation of}~ \mathrm{E}J ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{E}J & = & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}<br />
\\[4pt]<br />
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
& = & \texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot J_{(1 + \mathrm{d}u, 1 + \mathrm{d}v)}<br />
& + & \texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot J_{(1 + \mathrm{d}u, \mathrm{d}v)}<br />
& + & \texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot J_{(\mathrm{d}u, 1 + \mathrm{d}v)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot J_{(\mathrm{d}u, \mathrm{d}v)}<br />
\\[4pt]<br />
& = &<br />
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot J_{(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})}<br />
& + &<br />
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot J_{(\texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v)}<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot J_{(\mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)})}<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot J_{(\mathrm{d}u, \mathrm{d}v)}<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{E}J<br />
& = &<br />
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&& + &<br />
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{ } \mathrm{d}v \texttt{ }<br />
\\[4pt]<br />
&&&&& + &<br />
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot \texttt{ } \mathrm{d}u \texttt{ } \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&&&&&& + &<br />
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \texttt{ } \mathrm{d}u \texttt{ }~\texttt{ } \mathrm{d}v \texttt{ }<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;39 exhibits another method that happens to work quickly in this particular case, using distributive laws to multiply things out in an algebraic manner, arranging the notations of feature and fluxion according to a scale of simple character and degree. Proceeding this way leads through an intermediate step which, in chiming the changes of ordinary calculus, should take on a familiar ring. Consequential properties of exclusive disjunction then carry us on to the concluding line.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 39.} ~~ \text{Computation of}~ \mathrm{E}J ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{c}}<br />
\mathrm{E}J<br />
& = & (u + \mathrm{d}u) \cdot (v + \mathrm{d}v)<br />
\\[6pt]<br />
& = & u \cdot v<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = &<br />
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{ } \mathrm{d}v \texttt{ }<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot \texttt{ } \mathrm{d}u \texttt{ } \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \texttt{ } \mathrm{d}u \texttt{ }~\texttt{ } \mathrm{d}v \texttt{ }<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;40-a through 40-d present several views of the enlarged proposition <math>\mathrm{E}J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-a -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-a.} ~~ \text{Enlargement of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-b -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-b.} ~~ \text{Enlargement of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-c -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-c.} ~~ \text{Enlargement of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-d -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-d.} ~~ \text{Enlargement of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
An intuitive reading of the proposition <math>\mathrm{E}J\!</math> becomes available at this point. Recall that propositions in the extended universe <math>\mathrm{E}U^\bullet\!</math> express the ''dispositions'' of a system and the constraints that are placed on them. In other words, a differential proposition in <math>\mathrm{E}U^\bullet\!</math> can be read as referring to various changes that a system might undergo in and from its various states. In particular, we can understand <math>\mathrm{E}J\!</math> as a statement that tells us what changes need to be made with regard to each state in the universe of discourse in order to reach the ''truth'' of <math>J,\!</math> that is, the region of the universe where <math>J\!</math> is true. This interpretation is visibly clear in the Figures above and appeals to the imagination in a satisfying way but it has the added benefit of giving fresh meaning to the original name of the shift operator <math>\mathrm{E}.\!</math> Namely, <math>\mathrm{E}J\!</math> can be read as a proposition that ''enlarges'' on the meaning of <math>J,\!</math> in the sense of explaining its practical bearings and clarifying what it means in terms of actions and effects &mdash; the available options for differential action and the consequential effects that result from each choice.<br />
<br />
Read this way, the enlargement <math>\mathrm{E}J\!</math> has strong ties to the normal use of <math>J,\!</math> no matter whether it is understood as a proposition or a function, namely, to act as a figurative device for indicating the models of <math>J,\!</math> in effect, pointing to the interpretive elements in its fiber of truth <math>J^{-1}(1).\!</math> It is this kind of &ldquo;use&rdquo; that is often contrasted with the &ldquo;mention&rdquo; of a proposition, and thereby hangs a tale.<br />
<br />
=====Digression : Reflection on Use and Mention=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Reflection is turning a topic over in various aspects and in various lights so that nothing significant about it shall be overlooked &mdash; almost as one might turn a stone over to see what its hidden side is like or what is covered by it.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 57]<br />
|}<br />
<br />
The contrast drawn in logic between the ''use'' and the ''mention'' of a proposition corresponds to the difference that we observe in functional terms between using <math>{}^{\backprime\backprime} J \, {}^{\prime\prime}\!</math> to indicate the region <math>J^{-1}(1)\!</math> and using <math>{}^{\backprime\backprime} J \, {}^{\prime\prime}\!</math> to indicate the function <math>J.\!</math> You may think that one of these uses ought to be proscribed, and logicians are quick to prescribe against their confusion. But there seems to be no likelihood in practice that their interactions can be avoided. If the name <math>{}^{\backprime\backprime} J \, {}^{\prime\prime}\!</math> is used as a sign of the function <math>J,\!</math> and if the function <math>J\!</math> has its use in signifying something else, as would constantly be the case when some future theory of signs has given a functional meaning to every sign whatsoever, then is not <math>J,\!</math> by transitivity a sign of the thing itself? There are, of course, two answers to this question. Not every act of signifying or referring need be transitive. Not every warrant or guarantee or certificate is automatically transferable, indeed, not many. Not every feature of a feature is a feature of the featuree. Otherwise, if a buffalo is white, and white is a color, then a buffalo would ''be'' a color.<br />
<br />
The logical or pragmatic distinction between use and mention is cogent and necessary, and so is the analogous functional distinction between determining a value and determining what determines that value, but so are the normal techniques that we use to make these distinctions apply flexibly in practice. The way that the hue and cry about use and mention is raised in logical discussions, you might be led to think that this single dimension of choices embraces the only kinds of use worth mentioning and the only kinds of mention worth using. It will constitute the expeditionary and taxonomic tasks of that future theory of signs to explore and to classify the many other constellations and dimensions of use and mention that are yet to be opened up by the generative potential of full-fledged sign relations.<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
The well-known capacity that thoughts have &mdash; as doctors have discovered &mdash; for dissolving and dispersing those hard lumps of deep, ingrowing, morbidly entangled conflict that arise out of gloomy regions of the self probably rests on nothing other than their social and worldly nature, which links the individual being with other people and things; but unfortunately what gives them their power of healing seems to be the same as what diminishes the quality of personal experience in them.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 130]<br />
|}<br />
<br />
=====Difference Map of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
&ldquo;It doesn't matter what one does,&rdquo; the Man Without Qualities said to himself, shrugging his shoulders. &ldquo;In a tangle of forces like this it doesn't make a scrap of difference.&rdquo; He turned away like a man who has learned renunciation, almost indeed like a sick man who shrinks from any intensity of contact. And then, striding through his adjacent dressing-room, he passed a punching-ball that hung there; he gave it a blow far swifter and harder than is usual in moods of resignation or states of weakness.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 8]<br />
|}<br />
<br />
With the tacit extension map <math>\boldsymbol\varepsilon J\!</math> and the enlargement map <math>\mathrm{E}J\!</math> well in place, the difference map <math>\mathrm{D}J\!</math> can be computed along the lines displayed in Table&nbsp;41, ending up with an expansion of <math>\mathrm{D}J\!</math> over the cells of <math>[u, v].\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 41.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & \mathrm{E}J<br />
& + & \boldsymbol\varepsilon J<br />
\\[6pt]<br />
& = & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}<br />
& + & J_{(u, v)}<br />
\\[6pt]<br />
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \cdot v<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = &<br />
u \cdot v \cdot \qquad 0<br />
\\[6pt]<br />
& + &<br />
u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v<br />
& + &<br />
u \cdot \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v<br />
\\[6pt]<br />
& + &<br />
u \cdot v \cdot \texttt{~} \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
&&& + &<br />
\texttt{(} u \texttt{)} \cdot v \cdot \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
& + &<br />
u \cdot v \cdot \texttt{~} \mathrm{d}u \;\cdot\; \mathrm{d}v \texttt{~}<br />
&&&&& + &<br />
\texttt{(} u \texttt{)} \cdot \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = &<br />
u \cdot v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
u \cdot \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v<br />
& + &<br />
\texttt{(} u \texttt{)} \cdot v \cdot \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)} \cdot \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Alternatively, the difference map <math>\mathrm{D}J\!</math> can be expanded over the cells of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> to arrive at the formulation shown in Table&nbsp;42. The same development would be obtained from the previous Table by collecting terms in an alternate manner, along the rows rather than the columns in the middle portion of the Table.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 42.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & \boldsymbol\varepsilon J<br />
& + & \mathrm{E}J<br />
\\[6pt]<br />
& = & J_{(u, v)}<br />
& + & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}<br />
\\[6pt]<br />
& = & u \cdot v<br />
& + & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
& = & 0<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}J<br />
& = & 0<br />
& + & u \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Even more simply, the same result is reached by matching up the propositional coefficients of <math>\boldsymbol\varepsilon J</math> and <math>\mathrm{E}J\!</math> along the cells of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> and adding the pairs under boolean addition, that is, &ldquo;mod 2&rdquo;, where 1&nbsp;+&nbsp;1&nbsp;=&nbsp;0, as shown in Table&nbsp;43.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 43.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 3)}\!</math><br />
|<br />
<math>\begin{array}{*{5}{l}}<br />
\mathrm{D}J & = & \boldsymbol\varepsilon J & + & \mathrm{E}J<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\boldsymbol\varepsilon J<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ u \,\cdot\, v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~ u \;\cdot\; v \;\cdot\; \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u ~ \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} ~ v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot\, \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & ~~ 0 ~~ \,\cdot\, ~ \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~~ u ~ \,\cdot\, ~~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ ~ v ~~ \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
The difference map <math>\mathrm{D}J\!</math> can also be given a ''dispositional'' interpretation. First, recall that <math>\boldsymbol\varepsilon J\!</math> exhibits the dispositions to change from anywhere in <math>J\!</math> to anywhere at all in the universe of discourse and <math>\mathrm{E}J\!</math> exhibits the dispositions to change from anywhere in the universe to anywhere in <math>J.\!</math> Next, observe that each of these classes of dispositions may be divided in accordance with the case of <math>J\!</math> versus <math>\texttt{(} J \texttt{)}\!</math> that applies to their points of departure and destination, as shown below. Then, since the dispositions corresponding to <math>\boldsymbol\varepsilon J</math> and <math>\mathrm{E}J\!</math> have in common the dispositions to preserve <math>J,\!</math> their symmetric difference <math>\texttt{(} \boldsymbol\varepsilon J, \mathrm{E}J \texttt{)}\!</math> is made up of all the remaining dispositions, which are in fact disposed to cross the boundary of <math>J\!</math> in one direction or the other. In other words, we may conclude that <math>\mathrm{D}J\!</math> expresses the collective disposition to make a definite change with respect to <math>J,\!</math> no matter what value it holds in the current state of affairs.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
\boldsymbol\varepsilon J<br />
& = & \{ \text{Dispositions from}~ J ~\text{to}~ J \}<br />
& + & \{ \text{Dispositions from}~ J ~\text{to}~ \texttt{(} J \texttt{)} \}<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = & \{ \text{Dispositions from}~ J ~\text{to}~ J \}<br />
& + & \{ \text{Dispositions from}~ \texttt{(} J \texttt{)} ~\text{to}~ J \}<br />
\\[6pt]<br />
\mathrm{D}J<br />
& = & \{ \text{Dispositions from}~ J ~\text{to}~ \texttt{(} J \texttt{)} \}<br />
& + & \{ \text{Dispositions from}~ \texttt{(} J \texttt{)} ~\text{to}~ J \}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;44-a through 44-d illustrate the difference proposition <math>\mathrm{D}J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-a -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-a.} ~~ \text{Difference Map of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-b -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-b.} ~~ \text{Difference Map of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-c -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-c.} ~~ \text{Difference Map of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-d -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-d.} ~~ \text{Difference Map of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
=====Differential of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
By deploying discourse throughout a calendar, and by giving a date to each of its elements, one does not obtain a definitive hierarchy of precessions and originalities; this hierarchy is never more than relative to the systems of discourse that it sets out to evaluate.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Archaeology of Knowledge'', [Fou, 143]<br />
|}<br />
<br />
Finally, at long last, the differential proposition <math>\mathrm{d}J\!</math> can be gleaned from the difference proposition <math>\mathrm{D}J\!</math> by ranging over the cells of <math>[u, v]\!</math> and picking out the linear proposition of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> that is &ldquo;closest&rdquo; to the portion of <math>\mathrm{D}J\!</math> that touches on each point. The idea of distance that would give this definition unequivocal sense has been referred to in cautionary quotes, the kind we use to distance ourselves from taking a final position. There are obvious notions of approximation that suggest themselves, but finding one that can be justified as ultimately correct is not as straightforward as it seems.<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
He had drifted into the very heart of the world. From him to the distant beloved was as far as to the next tree.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 144]<br />
|}<br />
<br />
Let us venture a guess as to where these developments might be heading. From the present vantage point it appears that the ultimate answer to the quandary of distances and the question of a fitting measure may be that, rather than having the constitution of an analytic series depend on our familiar notions of approach, proximity, and approximation, it will be found preferable, and perhaps unavoidable, to turn the tables and let the orders of approximation be defined in terms of our favored and operative notions of formal analysis. Only the aftermath of this conversion, if it does converge, could be hoped to prove whether this hortatory form of analysis and the cohort idea of an analytic form &mdash; the limitary concept of a self-corrective process and the coefficient concept of a completable product &mdash; are truly (in practical reality) the more inceptive and persistent of principles and really (for all practical purposes) the more effective and regulative of ideas.<br />
<br />
Awaiting that determination, I proceed with what seems like the obvious course, and compute <math>\mathrm{d}J\!</math> according to the pattern in Table&nbsp;45.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 45.} ~~ \text{Computation of}~ \mathrm{d}J\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}J<br />
& = &<br />
u\!\cdot\!v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
u \, \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \, \mathrm{d}v<br />
& + &<br />
\texttt{(} u \texttt{)} \, v \cdot \mathrm{d}u \, \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \!\cdot\! \mathrm{d}v \texttt{~}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}J<br />
& = &<br />
u\!\cdot\!v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + &<br />
u \, \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + &<br />
\texttt{(} u \texttt{)} \, v \cdot \mathrm{d}u<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;46-a through 46-d illustrate the proposition <math>{\mathrm{d}J},\!</math> rounded out in our usual array of prospects. This proposition of <math>\mathrm{E}U^\bullet\!</math> is what we refer to as the (first order) differential of <math>J,\!</math> and normally regard as ''the'' differential proposition corresponding to <math>J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-a -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-a.} ~~ \text{Differential of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-b -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-b.} ~~ \text{Differential of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-c -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-c.} ~~ \text{Differential of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-d -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-d.} ~~ \text{Differential of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
=====Remainder of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
<p>I bequeath myself to the dirt to grow from the grass I love,<br><br />
If you want me again look for me under your bootsoles.</p><br />
<br />
<p>You will hardly know who I am or what I mean,<br><br />
But I shall be good health to you nevertheless,<br><br />
And filter and fibre your blood.</p><br />
<br />
<p>Failing to fetch me at first keep encouraged,<br><br />
Missing me one place search another,<br><br />
I stop some where waiting for you</p><br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]<br />
|}<br />
<br />
<br><br />
<br />
Let us recapitulate the story so far. We have in effect been carrying out a decomposition of the enlarged proposition <math>\mathrm{E}J\!</math> in a series of stages. First, we considered the equation <math>\mathrm{E}J = \boldsymbol\varepsilon J + \mathrm{D}J,\!</math> which was involved in the definition of <math>\mathrm{D}J\!</math> as the difference <math>\mathrm{E}J - \boldsymbol\varepsilon J.\!</math> Next, we contemplated the equation <math>\mathrm{D}J = \mathrm{d}J + \mathrm{r}J,\!</math> which expresses <math>\mathrm{D}J\!</math> in terms of two components, the differential <math>\mathrm{d}J\!</math> that was just extracted and the residual component <math>\mathrm{r}J = \mathrm{D}J - \mathrm{d}J.~\!</math> This remaining proposition <math>\mathrm{r}J\!</math> can be computed as shown in Table&nbsp;47.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 47.} ~~ \text{Computation of}~ \mathrm{r}J\!</math><br />
|<br />
<math>\begin{array}{*{5}{l}}<br />
\mathrm{r}J & = & \mathrm{D}J & + & \mathrm{d}J<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}J<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{r}J ~<br />
& = & u \!\cdot\! v \cdot ~ \mathrm{d}u \cdot \mathrm{d}v ~ ~ ~ ~ ~<br />
& + & u \texttt{(} v \texttt{)} \cdot \, \mathrm{d}u \cdot \mathrm{d}v \,<br />
& + & \texttt{(} u \texttt{)} v \cdot \, \mathrm{d}u \cdot \mathrm{d}v \,<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \, \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
As it happens, the remainder <math>\mathrm{r}J\!</math> falls under the description of a second order differential <math>\mathrm{r}J = \mathrm{d}^2 J.\!</math> This means that the expansion of <math>\mathrm{E}J\!</math> in the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|<br />
<math>\begin{array}{*{7}{l}}<br />
\mathrm{E}J<br />
& = & \boldsymbol\varepsilon J<br />
& + & \mathrm{D}J<br />
\\[6pt]<br />
& = & \boldsymbol\varepsilon J<br />
& + & \mathrm{d}J<br />
& + & \mathrm{r}J<br />
\\[6pt]<br />
& = & \mathrm{d}^0 J<br />
& + & \mathrm{d}^1 J<br />
& + & \mathrm{d}^2 J<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
which is nothing other than the propositional analogue of a Taylor series, is a decomposition that terminates in a finite number of steps.<br />
<br />
Figures&nbsp;48-a through 48-d illustrate the proposition <math>\mathrm{r}J = \mathrm{d}^2 J,\!</math> which forms the remainder map of <math>J\!</math> and also, in this instance, the second order differential of <math>J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-a -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-a.} ~~ \text{Remainder of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-b -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-b.} ~~ \text{Remainder of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-c -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-c.} ~~ \text{Remainder of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-d -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-d.} ~~ \text{Remainder of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
=====Summary of Conjunction=====<br />
<br />
To establish a convenient reference point for further discussion, Table&nbsp;49 summarizes the operator actions that have been computed for the form of conjunction, as exemplified by the proposition <math>J.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 49.} ~~ \text{Computation Summary for}~ J~\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon J<br />
& = & u \!\cdot\! v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}J<br />
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}J<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{r}J<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Analytic Series : Coordinate Method====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
And if he is told that something ''is'' the way it is, then he thinks: Well, it could probably just as easily be some other way. So the sense of possibility might be defined outright as the capacity to think how everything could &ldquo;just as easily&rdquo; be, and to attach no more importance to what is than to what is not.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 12]<br />
|}<br />
<br />
Table&nbsp;50 exhibits a truth table method for computing the analytic series (or the differential expansion) of a proposition in terms of coordinates.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:70%"<br />
|+ style="height:30px" | <math>\text{Table 50.} ~~ \text{Computation of an Analytic Series in Terms of Coordinates}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:8%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:8%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:8%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}u\!</math><br />
| style="width:8%; border-bottom:1px solid black" | <math>\mathrm{d}v\!</math><br />
| style="width:8%; border-bottom:1px solid black; border-left:1px solid black" | <math>u'\!</math><br />
| style="width:8%; border-bottom:1px solid black" | <math>v'\!</math><br />
| style="width:10%; border-bottom:1px solid black; border-left:4px double black" | <math>\boldsymbol\varepsilon J\!</math><br />
| style="width:10%; border-bottom:1px solid black" | <math>\mathrm{E}J\!</math><br />
| style="width:12%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{D}J\!</math><br />
| style="width:10%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}J\!</math><br />
| style="width:10%; border-bottom:1px solid black" | <math>\mathrm{d}^2\!J\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
The first six columns of the Table, taken as a whole, represent the variables of a construct called the ''contingent universe'' <math>[u, v, \mathrm{d}u, \mathrm{d}v, u', v'],\!</math> or the bundle of ''contingency spaces'' <math>[\mathrm{d}u, \mathrm{d}v, u', v']\!</math> over the universe <math>[u, v].\!</math> Their placement to the left of the double bar indicates that all of them amount to independent variables, but there is a co-dependency among them, as described by the following equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
u' & = & u + \mathrm{d}u & = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\\[8pt]<br />
v' & = & v + \mathrm{d}v & = & \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
These relations correspond to the formal substitutions that are made in defining <math>\mathrm{E}J\!</math> and <math>\mathrm{D}J.\!</math> For now, the whole rigamarole of contingency spaces can be regarded as a technical device for achieving the effect of these substitutions, adapted to a setting where functional compositions and other symbolic manipulations are difficult to contemplate and execute.<br />
<br />
The five columns to the right of the double bar in Table&nbsp;50 contain the values of the dependent variables <math>\{ \boldsymbol\varepsilon J, ~\mathrm{E}J, ~\mathrm{D}J, ~\mathrm{d}J, ~\mathrm{d}^2\!J \}.\!</math> These are normally interpreted as values of functions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B}\!</math> or as values of propositions in the extended universe <math>[u, v, \mathrm{d}u, \mathrm{d}v]\!</math> but the dependencies prevailing in the contingent universe make it possible to regard these same final values as arising via functions on alternative lists of arguments, for example, the set <math>\{ u, v, u', v' \}.\!</math><br />
<br />
The column for <math>\boldsymbol\varepsilon J\!</math> is computed as <math>J(u, v) = uv\!</math> and together with the columns for <math>u\!</math> and <math>v\!</math> illustrates how we &ldquo;share structure&rdquo; in the Table by listing only the first entries of each constant block.<br />
<br />
The column for <math>\mathrm{E}J\!</math> is computed by means of the following chain of identities, where the contingent variables <math>u'\!</math> and <math>v'\!</math> are defined as <math>u' = u + \mathrm{d}u\!</math> and <math>v' = v + \mathrm{d}v.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathrm{E}J(u, v, \mathrm{d}u, \mathrm{d}v)<br />
& = &<br />
J(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& = &<br />
J(u', v')<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
This makes it easy to determine <math>\mathrm{E}J\!</math> by inspection, computing the conjunction <math>J(u', v') = u'v'\!</math> from the columns headed <math>u'\!</math> and <math>v'.\!</math> Since each of these forms expresses the same proposition <math>\mathrm{E}J\!</math> in <math>\mathrm{E}U^\bullet,\!</math> the dependence on <math>\mathrm{d}u\!</math> and <math>\mathrm{d}v\!</math> is still present but merely left implicit in the final variant <math>J(u', v').\!</math><br />
<br />
* '''Note.''' On occasion, it is tempting to use the further notation <math>J'(u, v) = J(u', v'),\!</math> especially to suggest a transformation that acts on whole propositions, for example, taking the proposition <math>J\!</math> into the proposition <math>J' = \mathrm{E}J.\!</math> The prime <math>( {}^{\prime} )\!</math> then signifies an action that is mediated by a field of choices, namely, the values that are picked out for the contingent variables in sweeping through the initial universe. But this heaps an unwieldy lot of construed intentions on a rather slight character and puts too high a premium on the constant correctness of its interpretation. In practice, therefore, it is best to avoid this usage.<br />
<br />
Given the values of <math>\boldsymbol\varepsilon J\!</math> and <math>\mathrm{E}J,\!</math> the columns for the remaining functions can be filled in quickly. The difference map is computed according to the relation <math>\mathrm{D}J = \boldsymbol\varepsilon J + \mathrm{E}J.\!</math> The first order differential <math>\mathrm{d}J\!</math> is found by looking in each block of constant argument pairs <math>u, v\!</math> and choosing the linear function of <math>\mathrm{d}u, \mathrm{d}v\!</math> that best approximates <math>\mathrm{D}J\!</math> in that block. Finally, the remainder is computed as <math>\mathrm{r}J = \mathrm{D}J + \mathrm{d}J,\!</math> in this case yielding the second order differential <math>\mathrm{d}^2\!J.\!</math><br />
<br />
====Analytic Series : Recap====<br />
<br />
Let us now summarize the results of Table&nbsp;50 by writing down for each column and for each block of constant argument pairs <math>u, v\!</math> a reasonably canonical symbolic expression for the function of <math>\mathrm{d}u, \mathrm{d}v\!</math> that appears there. The synopsis formed in this way is presented in Table&nbsp;51. As one has a right to expect, it confirms the results that were obtained previously by operating solely in terms of the formal calculus.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:70%"<br />
|+ style="height:30px" | <math>\text{Table 51.} ~~ \text{Computation of an Analytic Series in Symbolic Terms}\!</math><br />
|- style="height:35px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black" | <math>J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{E}J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{D}J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{d}J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{d}^2\!J\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}u \!\;\cdot\;\! \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}u \!\;\cdot\;\! \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
\mathrm{d}u<br />
\\[4pt]<br />
\mathrm{d}v<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;52 and 53 provide a quick overview of the analysis performed so far, giving the successive decompositions of <math>\mathrm{E}J = J + \mathrm{D}J\!</math> and <math>\mathrm{D}J = \mathrm{d}J + \mathrm{r}J\!</math> in two different styles of diagram.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 52 -- Decomposition of EJ.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 52.} ~~ \text{Decomposition of}~ \mathrm{E}J\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 53 -- Decomposition of DJ.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 53.} ~~ \text{Decomposition of}~ \mathrm{D}J\!</math><br />
|}<br />
<br />
====Terminological Interlude====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Lastly, my attention was especially attracted, not so much to the scene, as to the mirrors that produced it. These mirrors were broken in parts. Yes, they were marked and scratched; they had been &ldquo;starred&rdquo;, in spite of their solidity &hellip;<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 230]<br />
|}<br />
<br />
At this point several issues of terminology have accrued enough substance to intrude on our discussion. The remarks of this Subsection are intended to accomplish two goals. First, we call attention to significant aspects of the previous series of Figures, translating into literal terms what they depict in iconic forms, and we re-stress the most important structural elements they indicate. Next, we prepare the way for taking on more complex examples of transformations, those whose target universes have more than one dimension.<br />
<br />
In talking about the actions of operators it is important to keep in mind the distinctions between the operators per&nbsp;se, their operands, and their results. Furthermore, in working with composite forms of operators <math>\mathrm{W} = (\mathrm{W}_1, \ldots, \mathrm{W}_n),\!</math> transformations <math>\mathrm{F} = (\mathrm{F}_1, \ldots, \mathrm{F}_n),\!</math> and target domains <math>X^\bullet = [x_1, \ldots, x_n],\!</math> we need to preserve a clear distinction between the compound entity of each given type and any one of its separate components. It is curious, given the usefulness of the concepts ''operator'' and ''operand'', that we seem to lack a generic term, formed on the same root, for the corresponding result of an operation. Following the obvious paradigm would lead to words like ''opus'', ''opera'', and ''operant'', but these words are too affected with clang associations to work well at present, though they might be adapted in time. One current usage gets around this problem by using the substantive ''map'' as a systematic epithet to express the result of each operator's action. We will follow this practice as far as possible, for example, using the phrase ''tangent map'' to denote the end product of the tangent functor acting on its operand map.<br />
<br />
* '''Scholium.''' See [JGH, 6-9] for a good account of tangent functors and tangent maps in ordinary analysis and for examples of their use in mechanics. This work as a whole is a model of clarity in applying functorial principles to problems in physical dynamics.<br />
<br />
Whenever we focus on isolated propositions, on single components of composite operators, or on the portions of transformations that have <math>1\!</math>-dimensional ranges, we are free to shift between the native form of a proposition <math>J : U \to \mathbb{B}\!</math> and the thematized form of a mapping <math>J : U^\bullet \to [x]\!</math> without much trouble. In these cases we are able to tolerate a higher degree of ambiguity about the precise nature of an operator's input and output domains than we otherwise might. For example, in the preceding treatment of the example <math>J,\!</math> and for each operator <math>\mathrm{W}\!</math> in the set <math>\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \},\!</math> both the operand <math>J\!</math> and the result <math>\mathrm{W}J\!</math> could be viewed in either one of two ways. On one hand we may treat them as propositions <math>J : U \to \mathbb{B}\!</math> and <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B},\!</math> ignoring the distinction between the range <math>[x] \cong \mathbb{B}\!</math> of <math>\boldsymbol\varepsilon J\!</math> and the range <math>[\mathrm{d}x] \cong \mathbb{D}\!</math> of the other types of <math>\mathrm{W}J.\!</math> This is what we usually do when we content ourselves with simply coloring in regions of venn diagrams. On the other hand we may view these entities as maps <math>J : U^\bullet \to [x] = X^\bullet\!</math> and <math>\boldsymbol\varepsilon J : \mathrm{E}U^\bullet \to [x] \subseteq \mathrm{E}X^\bullet\!</math> or <math>\mathrm{W}J : \mathrm{E}U^\bullet \to [\mathrm{d}x] \subseteq \mathrm{E}X^\bullet,\!</math> in which case the qualitative characters of the output features are not ignored.<br />
<br />
At the beginning of this Section we recast the natural form of a proposition <math>J : U \to \mathbb{B}\!</math> into the thematic role of a transformation <math>J : U^\bullet \to [x],\!</math> where <math>x\!</math> was a variable recruited to express the newly independent <math>\check{J}.\!</math> However, in our computations and representations of operator actions we immediately lapsed back to viewing the results as native elements of the extended universe <math>\mathrm{E}U^\bullet,\!</math> in other words, as propositions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B},\!</math> where <math>\mathrm{W}\!</math> ranged over the set <math>\{ \boldsymbol\varepsilon, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}.\!</math> That is as it should be. We have worked hard to devise a language that gives us these advantages &mdash; the flexibility to exchange terms and types of equal information value and the capacity to reflect as quickly and as wittingly as a controlled reflex on the fibers of our propositions, independently of whether they express amusements, beliefs, or conjectures.<br />
<br />
As we take on target spaces of increasing dimension, however, these types of confusions (and confusions of types) become less and less permissible. For this reason, Tables&nbsp;54 and 55 present a rather detailed summary of the notation and the terminology we are using, as applied to the case <math>J = uv.\!</math> The rationale of these Tables is not so much to train more elephant guns on this poor drosophila of a concrete example but to invest our paradigm with enough solidity to bear the weight of abstraction to come.<br />
<br />
Table&nbsp;54 provides basic notation and descriptive information for the objects and operators used in this Example, giving the generic type (or broadest defined type) for each entity. Here, the sans&nbsp;serif operators <math>\mathsf{W} \in \{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{d}, \mathsf{r} \}\!</math> and their components <math>\mathrm{W} \in \{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}\!</math> both have the same broad type <math>\mathsf{W}, \mathrm{W} : (U^\bullet \to X^\bullet) \to (\mathrm{E}U^\bullet \to \mathrm{E}X^\bullet),\!</math> as appropriate to operators that map transformations <math>J : U^\bullet \to X^\bullet\!</math> to extended transformations <math>\mathsf{W}J, \mathrm{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 54.} ~~ \text{Cast of Characters : Expansive Subtypes of Objects and Operators}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| align="center" | <math>\text{Symbol}\!</math><br />
| align="center" | <math>\text{Notation}\!</math><br />
| align="center" | <math>\text{Description}\!</math><br />
| align="center" | <math>\text{Type}\!</math><br />
|-<br />
| align="center" | <math>U^\bullet\!</math><br />
| <math>= [u, v]\!</math><br />
| <math>\text{Source universe}\!</math><br />
| <math>[\mathbb{B}^2]\!</math><br />
|-<br />
| align="center" | <math>X^\bullet~\!</math><br />
| <math>= [x]\!</math><br />
| <math>\text{Target universe}\!</math><br />
| <math>[\mathbb{B}^1]~\!</math><br />
|-<br />
| align="center" | <math>\mathrm{E}U^\bullet\!</math><br />
| <math>= [u, v, \mathrm{d}u, \mathrm{d}v]\!</math><br />
| <math>\text{Extended source universe}\!</math><br />
| <math>[\mathbb{B}^2 \!\times\! \mathbb{D}^2]</math><br />
|-<br />
| align="center" | <math>\mathrm{E}X^\bullet\!</math><br />
| <math>= [x, \mathrm{d}x]~\!</math><br />
| <math>\text{Extended target universe}\!</math><br />
| <math>[\mathbb{B}^1 \!\times\! \mathbb{D}^1]</math><br />
|-<br />
| align="center" | <math>J\!</math><br />
| <math>J : U \!\to\! \mathbb{B}\!</math><br />
| <math>\text{Proposition}\!</math><br />
| <math>(\mathbb{B}^2 \!\to\! \mathbb{B}) \in [\mathbb{B}^2]\!</math><br />
|-<br />
| align="center" | <math>J\!</math><br />
| <math>J : U^\bullet \!\to\! X^\bullet\!</math><br />
| <math>\text{Transformation or Map}\!</math><br />
| <math>[\mathbb{B}^2] \!\to\! [\mathbb{B}^1]\!</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\boldsymbol\varepsilon<br />
\\<br />
\eta<br />
\\<br />
\mathrm{E}<br />
\\<br />
\mathrm{D}<br />
\\<br />
\mathrm{d}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{W} : U^\bullet \!\to\! \mathrm{E}U^\bullet,<br />
\\<br />
\mathrm{W} : X^\bullet \!\to\! \mathrm{E}X^\bullet,<br />
\\<br />
\mathrm{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\\<br />
\text{for each}~ \mathrm{W} ~\text{in the set:}<br />
\\<br />
\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{ll}<br />
\text{Tacit extension operator} & \boldsymbol\varepsilon<br />
\\<br />
\text{Trope extension operator} & \eta<br />
\\<br />
\text{Enlargement operator} & \mathrm{E}<br />
\\<br />
\text{Difference operator} & \mathrm{D}<br />
\\<br />
\text{Differential operator} & \mathrm{d}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^2] \!\to\! [\mathbb{B}^2 \!\times\! \mathbb{D}^2]},<br />
\\<br />
{[\mathbb{B}^1] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]},<br />
\\\\<br />
([\mathbb{B}^2] \!\to\! [\mathbb{B}^1]) \!\to\!<br />
\\<br />
([\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1])<br />
\end{array}</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\mathsf{e}<br />
\\<br />
\mathsf{E}<br />
\\<br />
\mathsf{D}<br />
\\<br />
\mathsf{T}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{W} : U^\bullet \!\to\! \mathsf{T}U^\bullet = \mathrm{E}U^\bullet,<br />
\\<br />
\mathsf{W} : X^\bullet \!\to\! \mathsf{T}X^\bullet = \mathrm{E}X^\bullet,<br />
\\<br />
\mathsf{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathsf{T}U^\bullet \!\to\! \mathsf{T}X^\bullet)<br />
\\<br />
\text{for each}~ \mathsf{W} ~\text{in the set:}<br />
\\<br />
\{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{T} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{lll}<br />
\text{Radius operator} & \mathsf{e} & = (\boldsymbol\varepsilon, \eta)<br />
\\<br />
\text{Secant operator} & \mathsf{E} & = (\boldsymbol\varepsilon, \mathrm{E})<br />
\\<br />
\text{Chord operator} & \mathsf{D} & = (\boldsymbol\varepsilon, \mathrm{D})<br />
\\<br />
\text{Tangent functor} & \mathsf{T} & = (\boldsymbol\varepsilon, \mathrm{d})<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^2] \!\to\! [\mathbb{B}^2 \!\times\! \mathbb{D}^2]},<br />
\\<br />
{[\mathbb{B}^1] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]},<br />
\\\\<br />
([\mathbb{B}^2] \!\to\! [\mathbb{B}^1]) \!\to\!<br />
\\<br />
([\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1])<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;55 supplies a more detailed outline of terminology for operators and their results. Here, we list the restrictive subtype (or narrowest defined subtype) that applies to each entity and we indicate across the span of the Table the whole spectrum of alternative types that color the interpretation of each symbol. For example, all the component operator maps <math>\mathrm{W}J\!</math> have <math>1\!</math>-dimensional ranges, either <math>\mathbb{B}^1\!</math> or <math>\mathbb{D}^1,\!</math> and so they can be viewed either as propositions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B}\!</math> or as logical transformations <math>\mathrm{W}J : \mathrm{E}U^\bullet \to X^\bullet.\!</math> As a rule, the plan of the Table allows us to name each entry by detaching the underlined adjective at the left of its row and prefixing it to the generic noun at the top of its column. In one case, however, it is customary to depart from this scheme. Because the phrase ''differential proposition'', applied to the result <math>\mathrm{d}J : \mathrm{E}U \to \mathbb{D},\!</math> does not distinguish it from the general run of differential propositions <math>\mathrm{G}: \mathrm{E}U \to \mathbb{B},\!</math> it is usual to single out <math>\mathrm{d}J\!</math> as the ''tangent proposition'' of <math>J.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 55.} ~~ \text{Synopsis of Terminology : Restrictive and Alternative Subtypes}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| &nbsp;<br />
| align="center" | <math>\text{Operator}\!</math><br />
| align="center" | <math>\text{Proposition}\!</math><br />
| align="center" | <math>\text{Map}\!</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tacit}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\boldsymbol\varepsilon : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\boldsymbol\varepsilon : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{B} \\<br />
\boldsymbol\varepsilon J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{B}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x] \\<br />
\boldsymbol\varepsilon J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Trope}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\eta : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\eta : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\eta J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\eta J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Enlargement}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{E} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{E}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{E}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Difference}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{D} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{D}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{D}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Differential}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{d} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{d} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{d}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}\!</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Remainder}}\\\text{operator}\end{matrix}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{r} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{r} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{r}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{r}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Radius}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e} = (\boldsymbol\varepsilon, \eta) \\<br />
\mathsf{e} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{e}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Secant}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}) \\<br />
\mathsf{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{E}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Chord}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}) \\<br />
\mathsf{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{D}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tangent}}\\\text{functor}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T} = (\boldsymbol\varepsilon, \mathrm{d}) \\<br />
\mathsf{T} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{T}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====End of Perfunctory Chatter : Time to Roll the Clip!====<br />
<br />
Two steps remain to finish the analysis of <math>J\!</math> that we began so long ago. First, we need to paste our accumulated heap of flat pictures into the frames of transformations, filling out the shapes of the operator maps <math>\mathsf{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet.~\!</math> This scheme is executed in two styles, using the ''areal views'' in Figures&nbsp;56-a and the ''box views'' in Figures&nbsp;56-b. Finally, in Figures&nbsp;57-1 to 57-4 we put all the pieces together to construct the full operator diagrams for <math>\mathsf{W} : J \to \mathsf{W}J.\!</math> There is a considerable amount of redundancy among the following three series of Figures but that will hopefully provide a fuller picture of the operations under review, enabling these snapshots to serve as successive frames in the animation of logic they are meant to become.<br />
<br />
=====Operator Maps : Areal Views=====<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a1 -- Radius Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a1.} ~~ \text{Radius Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a2 -- Secant Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a2.} ~~ \text{Secant Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a3 -- Chord Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a3.} ~~ \text{Chord Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a4 -- Tangent Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a4.} ~~ \text{Tangent Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
=====Operator Maps : Box Views=====<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b1 -- Radius Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b1.} ~~ \text{Radius Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b2 -- Secant Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b2.} ~~ \text{Secant Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b3 -- Chord Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b3.} ~~ \text{Chord Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b4 -- Tangent Map of J ISW.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b4.} ~~ \text{Tangent Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
=====Operator Diagrams for the Conjunction J = uv=====<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-1 -- Radius Operator Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-1.} ~~ \text{Radius Operator Diagram for the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-2 -- Secant Operator Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-2.} ~~ \text{Secant Operator Diagram for the Conjunction}~ J = uv~\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-3 -- Chord Operator Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-3.} ~~ \text{Chord Operator Diagram for the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-4 -- Tangent Functor Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-4.} ~~ \text{Tangent Functor Diagram for the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
===Taking Aim at Higher Dimensional Targets===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
The past and present wilt . . . . I have filled them and<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;emptied them,<br><br />
And proceed to fill my next fold of the future.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 87]<br />
|}<br />
<br />
In the next Section we consider a transformation <math>F\!</math> of concrete type <math>F : [u, v] \to [x, y]\!</math> and abstract type <math>F : [\mathbb{B}^2] \to [\mathbb{B}^2].\!</math> From the standpoint of propositional calculus we naturally approach the task of understanding such a transformation by parsing it into component maps with <math>1\!</math>-dimensional ranges, as follows:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{ccccccl}<br />
F & = & (F_1, F_2) & = & (f, g) & : & [u, v] \to [x, y],<br />
\\[6pt]<br />
&& F_1 & = & f & : & [u, v] \to [x],<br />
\\[6pt]<br />
&& F_2 & = & g & : & [u, v] \to [y].<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Then we tackle the separate components, now viewed as propositions <math>F_i : U \to \mathbb{B},\!</math> one at a time. At the completion of this analytic phase, we return to the task of synthesizing these partial and transient impressions into an agile form of integrity, a solidly coordinated and deeply integrated comprehension of the ongoing transformation. (Very often, of course, in tangling with refractory cases, we never get as far as the beginning again.)<br />
<br />
Let us now refer to the dimension of the target space or codomain as the ''toll'' (or ''tole'') of a transformation, as distinguished from the dimension of the range or image that is customarily called the ''rank''. When we keep to transformations with a toll of <math>1,\!</math> as <math>J : [u, v] \to [x],\!</math> we tend to get lazy about distinguishing a logical transformation from its component propositions. However, if we deal with transformations of a higher toll, this form of indolence can no longer be tolerated.<br />
<br />
Well, perhaps we can carry it a little further. After all, the operator result <math>\mathrm{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet\!</math> is a map of toll <math>2,\!</math> and cannot be unfolded in one piece as a proposition. But when a map has rank <math>1,\!</math> like <math>\boldsymbol\varepsilon J : \mathrm{E}U \to X \subseteq \mathrm{E}X\!</math> or <math>\mathrm{d}J : \mathrm{E}U \to \mathrm{d}X \subseteq \mathrm{E}X,\!</math> we naturally choose to concentrate on the <math>1\!</math>-dimensional range of the operator result <math>\mathrm{W}J,\!</math> ignoring the final difference in quality between the spaces <math>X\!</math> and <math>\mathrm{d}X,\!</math> and view <math>\mathrm{W}J\!</math> as a proposition about <math>\mathrm{E}U.\!</math><br />
<br />
In this way, an initial ambivalence about the role of the operand <math>J\!</math> conveys a double duty to the result <math>\mathrm{W}J.\!</math> The pivot that is formed by our focus of attention is essential to the linkage that transfers this double moment, as the whole process takes its bearing and wheels around the precise measure of a narrow bead that we can draw on the range of <math>\mathrm{W}J.\!</math> This is the escapement that it takes to get away with what may otherwise seem to be a simple duplicity, and this is the tolerance that is needed to counterbalance a certain arrogance of equivocation, by all of which machinations we make ourselves free to indicate the operator results <math>\mathrm{W}J\!</math> as propositions or as transformations, indifferently.<br />
<br />
But that's it, and no further. Neglect of these distinctions in range and target universes of higher dimensions is bound to cause a hopeless confusion. To guard against these adverse prospects, Tables&nbsp;58 and 59 lay the groundwork for discussing a typical map <math>F : [\mathbb{B}^2] \to [\mathbb{B}^2],\!</math> and begin to pave the way to some extent for discussing any transformation of the form <math>F : [\mathbb{B}^n] \to [\mathbb{B}^k].\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 58.} ~~ \text{Cast of Characters : Expansive Subtypes of Objects and Operators}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| align="center" | <math>\text{Symbol}\!</math><br />
| align="center" | <math>\text{Notation}\!</math><br />
| align="center" | <math>\text{Description}\!</math><br />
| align="center" | <math>\text{Type}\!</math><br />
|-<br />
| align="center" | <math>U^\bullet\!</math><br />
| <math>= [u, v]\!</math><br />
| <math>\text{Source universe}\!</math><br />
| <math>[\mathbb{B}^n]\!</math><br />
|-<br />
| align="center" | <math>X^\bullet~\!</math><br />
| <math>\begin{array}{l}<br />
= [x, y] \\<br />
= [f, g]<br />
\end{array}</math><br />
| <math>\text{Target universe}\!</math><br />
| <math>[\mathbb{B}^k]\!</math><br />
|-<br />
| align="center" | <math>\mathrm{E}U^\bullet\!</math><br />
| <math>= [u, v, \mathrm{d}u, \mathrm{d}v]\!</math><br />
| <math>\text{Extended source universe}\!</math><br />
| <math>[\mathbb{B}^n \!\times\! \mathbb{D}^n]\!</math><br />
|-<br />
| align="center" | <math>\mathrm{E}X^\bullet\!</math><br />
| <math>\begin{array}{l}<br />
= [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
= [f, g, \mathrm{d}f, \mathrm{d}g]<br />
\end{array}</math><br />
| <math>\text{Extended target universe}\!</math><br />
| <math>[\mathbb{B}^k \!\times\! \mathbb{D}^k]\!</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
f \\ g<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{ll}<br />
f : U \!\to\! [x] \cong \mathbb{B} \\<br />
g : U \!\to\! [y] \cong \mathbb{B}<br />
\end{array}</math><br />
| <math>\text{Proposition}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathbb{B}^n \!\to\! \mathbb{B} \\<br />
\in (\mathbb{B}^n, \mathbb{B}^n \!\to\! \mathbb{B}) = [\mathbb{B}^n]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>F\!</math><br />
| <math>F = (f, g) : U^\bullet \!\to\! X^\bullet\!</math><br />
| <math>\text{Transformation of Map}\!</math><br />
| <math>[\mathbb{B}^n] \!\to\! [\mathbb{B}^k]</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\boldsymbol\varepsilon<br />
\\<br />
\eta<br />
\\<br />
\mathrm{E}<br />
\\<br />
\mathrm{D}<br />
\\<br />
\mathrm{d}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{W} : U^\bullet \!\to\! \mathrm{E}U^\bullet,<br />
\\<br />
\mathrm{W} : X^\bullet \!\to\! \mathrm{E}X^\bullet,<br />
\\<br />
\mathrm{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\\<br />
\text{for each}~ \mathrm{W} ~\text{in the set:}<br />
\\<br />
\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{ll}<br />
\text{Tacit extension operator} & \boldsymbol\varepsilon<br />
\\<br />
\text{Trope extension operator} & \eta<br />
\\<br />
\text{Enlargement operator} & \mathrm{E}<br />
\\<br />
\text{Difference operator} & \mathrm{D}<br />
\\<br />
\text{Differential operator} & \mathrm{d}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^n] \!\to\! [\mathbb{B}^n \!\times\! \mathbb{D}^n]},<br />
\\<br />
{[\mathbb{B}^k] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]},<br />
\\\\<br />
([\mathbb{B}^n] \!\to\! [\mathbb{B}^k]) \!\to\!<br />
\\<br />
([\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k])<br />
\end{array}</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\mathsf{e}<br />
\\<br />
\mathsf{E}<br />
\\<br />
\mathsf{D}<br />
\\<br />
\mathsf{T}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{W} : U^\bullet \!\to\! \mathsf{T}U^\bullet = \mathrm{E}U^\bullet,<br />
\\<br />
\mathsf{W} : X^\bullet \!\to\! \mathsf{T}X^\bullet = \mathrm{E}X^\bullet,<br />
\\<br />
\mathsf{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathsf{T}U^\bullet \!\to\! \mathsf{T}X^\bullet)<br />
\\<br />
\text{for each}~ \mathsf{W} ~\text{in the set:}<br />
\\<br />
\{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{T} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{lll}<br />
\text{Radius operator} & \mathsf{e} & = (\boldsymbol\varepsilon, \eta)<br />
\\<br />
\text{Secant operator} & \mathsf{E} & = (\boldsymbol\varepsilon, \mathrm{E})<br />
\\<br />
\text{Chord operator} & \mathsf{D} & = (\boldsymbol\varepsilon, \mathrm{D})<br />
\\<br />
\text{Tangent functor} & \mathsf{T} & = (\boldsymbol\varepsilon, \mathrm{d})<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^n] \!\to\! [\mathbb{B}^n \!\times\! \mathbb{D}^n]},<br />
\\<br />
{[\mathbb{B}^k] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]},<br />
\\\\<br />
([\mathbb{B}^n] \!\to\! [\mathbb{B}^k]) \!\to\!<br />
\\<br />
([\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k])<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 59.} ~~ \text{Synopsis of Terminology : Restrictive and Alternative Subtypes}~\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| &nbsp;<br />
| align="center" | <math>\begin{matrix}\text{Operator}\\\text{or}\\\text{Operand}\end{matrix}</math><br />
| align="center" | <math>\begin{matrix}\text{Proposition}\\\text{or}\\\text{Component}\end{matrix}</math><br />
| align="center" | <math>\begin{matrix}\text{Transformation}\\\text{or}\\\text{Map}\end{matrix}</math><br />
|-<br />
| align="center" | <math>\underline{\text{Operand}}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
F = (F_1, F_2) \\<br />
F = (f, g) : U \!\to\! X<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
F_i : \langle u, v \rangle \!\to\! \mathbb{B} \\<br />
F_i : \mathbb{B}^n \!\to\! \mathbb{B}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
F : [u, v] \!\to\! [x, y] \\<br />
F : [\mathbb{B}^n] \!\to\! [\mathbb{B}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tacit}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\boldsymbol\varepsilon : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\boldsymbol\varepsilon : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{B} \\<br />
\boldsymbol\varepsilon F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{B}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y] \\<br />
\boldsymbol\varepsilon F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Trope}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\eta : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\eta : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\eta F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\eta F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Enlargement}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{E} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{E}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{E}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Difference}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{D} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{D}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{D}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Differential}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{d} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{d} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{d}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}\!</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Remainder}}\\\text{operator}\end{matrix}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{r} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{r} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{r}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{r}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Radius}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e} = (\boldsymbol\varepsilon, \eta) \\<br />
\mathsf{e} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{e}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Secant}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}) \\<br />
\mathsf{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{E}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Chord}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}) \\<br />
\mathsf{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{D}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tangent}}\\\text{functor}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T} = (\boldsymbol\varepsilon, \mathrm{d}) \\<br />
\mathsf{T} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{T}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
===Transformations of Type '''B'''<sup>2</sup> &rarr; '''B'''<sup>2</sup>===<br />
<br />
To take up a slightly more complex example, but one that remains simple enough to pursue through a complete series of developments, consider the transformation from <math>U^\bullet = [u, v]\!</math> to <math>X^\bullet = [x, y]\!</math> that is defined by the following system of equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
x<br />
& = & f(u, v)<br />
& = & \texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[8pt]<br />
y<br />
& = & g(u, v)<br />
& = & \texttt{((} u \texttt{,} v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
The component notation <math>F = (F_1, F_2) = (f, g) : U^\bullet \to X^\bullet\!</math> allows us to give a name and a type to this transformation and permits defining it by the compact description that follows:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
(x, y)<br />
& = & F(u, v)<br />
& = & (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Logical Transformations====<br />
<br />
The information that defines the logical transformation <math>F\!</math> can be represented in the form of a truth table, as shown in Table&nbsp;60. To cut down on subscripts in this example we continue to use plain letter equivalents for all components of spaces and maps.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:60%"<br />
|+ style="height:30px" | <math>\text{Table 60.} ~~ \text{A Propositional Transformation}\!</math><br />
|- style="height:40px; background:ghostwhite; width:100%"<br />
| style="width:25%" | <math>u\!</math><br />
| style="width:25%" | <math>v\!</math><br />
| style="width:25%; border-left:1px solid black" | <math>f\!</math><br />
| style="width:25%" | <math>g\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="border-top:1px solid black" | <math>u\!</math><br />
| style="border-top:1px solid black" | <math>v\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;61 shows how we might paint a picture of the transformation <math>F\!</math> in the manner of Figure&nbsp;30.<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 61 -- Propositional Transformation.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 61.} ~~ \text{A Propositional Transformation}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;62 extracts the gist of Figure&nbsp;61, exhibiting a style of diagram that is adequate for most purposes.<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 62 -- Propositional Transformation (Short Form).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 62.} ~~ \text{A Propositional Transformation (Short Form)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Local Transformations====<br />
<br />
Figure&nbsp;63 gives a more complete picture of the transformation <math>F,\!</math> showing how the points of <math>U^\bullet\!</math> are transformed into points of <math>X^\bullet.\!</math> The bold lines crossing from one universe to the other trace the action that <math>F\!</math> induces on points, in other words, they show how the transformation acts as a mapping from points to points and chart its effects on the elements that are variously called cells, points, positions, or singular propositions.<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 63 -- Transformation of Positions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 63.} ~~ \text{A Transformation of Positions}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;64 shows how the action of <math>F\!</math> on cells or points can be computed in terms of coordinates.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 64.} ~~ \text{A Transformation of Positions}\!</math><br />
|- style="height:40px; background:ghostwhite; width:100%"<br />
| style="width:8%" | <math>u\!</math><br />
| style="width:8%" | <math>v\!</math><br />
| style="width:12%; border-left:1px solid black" | <math>x\!</math><br />
| style="width:12%" | <math>y\!</math><br />
| style="width:10%; border-left:1px solid black" | <math>x~y\!</math><br />
| style="width:10%" | <math>x \texttt{(} y \texttt{)}\!</math><br />
| style="width:10%" | <math>\texttt{(} x \texttt{)} y\!</math><br />
| style="width:10%" | <math>\texttt{(} x \texttt{)(} y \texttt{)}\!</math><br />
| style="width:20%; border-left:1px solid black" | <math>X^\bullet = [x, y]\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\uparrow<br />
\\[4pt]<br />
F =<br />
\\[4pt]<br />
(f, g)<br />
\\[4pt]<br />
\uparrow<br />
\end{matrix}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="border-top:1px solid black" | <math>u\!</math><br />
| style="border-top:1px solid black" | <math>v\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
| style="border-top:1px solid black" | <math>\texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>u~v\!</math><br />
| style="border-top:1px solid black" | <math>\texttt{(} u \texttt{,} v \texttt{)}\!</math><br />
| style="border-top:1px solid black" | <math>\texttt{(} u \texttt{)(} v \texttt{)}\!</math><br />
| style="border-top:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>U^\bullet = [u, v]\!</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;65 extends this scheme from single cells to arbitrary regions, showing how we might compute the action of a logical transformation on arbitrary propositions in the universe of discourse. The effect of a point-transformation on arbitrary propositions, or any other structures erected on points, is referred to as the ''induced action'' of the transformation on the structures in question.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 65-a.} ~~ \text{An Induced Transformation on Propositions}\!</math><br />
|- style="height:50px; background:ghostwhite"<br />
| style="width:20%" | <math>X^\bullet~\!</math><br />
| style="width:20%; border-right:none" | <math>\longleftarrow\!</math><br />
| style="width:20%; border-left:none; border-right:none" | <math>F = (f, g)\!</math><br />
| style="width:20%; border-left:none" | <math>\longleftarrow\!</math><br />
| style="width:20%" | <math>U^\bullet~\!</math><br />
|- style="background:ghostwhite"<br />
| rowspan="2" | <math>f_i (x, y)\!</math><br />
| align="right" | <math>\begin{matrix}u = \\ v =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~0~0\\1~0~1~0\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= u \\ = v\end{matrix}</math><br />
| rowspan="2" style="width:20%" | <math>f_j (u, v)\!</math><br />
|- style="background:ghostwhite"<br />
| align="right" | <math>\begin{matrix}x = \\ y =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~1~0\\1~0~0~1\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= f(u, v) \\ = g(u, v)\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{2}<br />
\\[2pt]<br />
f_{3}<br />
\\[2pt]<br />
f_{4}<br />
\\[2pt]<br />
f_{5}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{7}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)~ ~}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{~ ~(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{,~} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{~~} y \texttt{)}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~0<br />
\\[2pt]<br />
0~0~0~0<br />
\\[2pt]<br />
0~0~0~1<br />
\\[2pt]<br />
0~0~0~1<br />
\\[2pt]<br />
0~1~1~0<br />
\\[2pt]<br />
0~1~1~0<br />
\\[2pt]<br />
0~1~1~1<br />
\\[2pt]<br />
0~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{,~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{,~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{~~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{~~} v \texttt{)}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[2pt]<br />
f_{0}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{7}<br />
\\[2pt]<br />
f_{7}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{8}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{10}<br />
\\[2pt]<br />
f_{11}<br />
\\[2pt]<br />
f_{12}<br />
\\[2pt]<br />
f_{13}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{15}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[2pt]<br />
\texttt{~ ~ ~ ~} y \texttt{~~}<br />
\\[2pt]<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[2pt]<br />
\texttt{~~} x \texttt{~ ~ ~ ~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
1~0~0~0<br />
\\[2pt]<br />
1~0~0~0<br />
\\[2pt]<br />
1~0~0~1<br />
\\[2pt]<br />
1~0~0~1<br />
\\[2pt]<br />
1~1~1~0<br />
\\[2pt]<br />
1~1~1~0<br />
\\[2pt]<br />
1~1~1~1<br />
\\[2pt]<br />
1~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~~} u \texttt{~~} v \texttt{~~}<br />
\\[2pt]<br />
\texttt{~~} u \texttt{~~} v \texttt{~~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{8}<br />
\\[2pt]<br />
f_{8}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{15}<br />
\\[2pt]<br />
f_{15}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 65-b.} ~~ \text{An Induced Transformation on Propositions}\!</math><br />
|- style="height:50px; background:ghostwhite"<br />
| style="width:20%" | <math>X^\bullet~\!</math><br />
| style="width:20%; border-right:none" | <math>\longleftarrow\!</math><br />
| style="width:20%; border-left:none; border-right:none" | <math>F = (f, g)\!</math><br />
| style="width:20%; border-left:none" | <math>\longleftarrow\!</math><br />
| style="width:20%" | <math>U^\bullet~\!</math><br />
|- style="background:ghostwhite"<br />
| rowspan="2" | <math>f_i (x, y)\!</math><br />
| align="right" | <math>\begin{matrix}u = \\ v =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~0~0\\1~0~1~0\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= u \\ = v\end{matrix}</math><br />
| rowspan="2" style="width:20%" | <math>f_j (u, v)\!</math><br />
|- style="background:ghostwhite"<br />
| align="right" | <math>\begin{matrix}x = \\ y =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~1~0\\1~0~0~1\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= f(u, v) \\ = g(u, v)\end{matrix}</math><br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>f_{0}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[2pt]<br />
f_{2}<br />
\\[2pt]<br />
f_{4}<br />
\\[2pt]<br />
f_{8}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~0<br />
\\[2pt]<br />
0~0~0~1<br />
\\[2pt]<br />
0~1~1~0<br />
\\[2pt]<br />
1~0~0~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{,~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{~} u \texttt{~~} v \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{8}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{3}<br />
\\[2pt]<br />
f_{12}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[2pt]<br />
1~1~1~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} u \texttt{)(} v \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[2pt]<br />
f_{14}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[2pt]<br />
f_{9}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[2pt]<br />
1~0~0~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{~~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{~} u \texttt{~~} v \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[2pt]<br />
f_{8}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{5}<br />
\\[2pt]<br />
f_{10}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[2pt]<br />
1~0~0~1<br />
\end{matrix}~\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} u \texttt{,~} v \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[2pt]<br />
f_{9}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[2pt]<br />
f_{11}<br />
\\[2pt]<br />
f_{13}<br />
\\[2pt]<br />
f_{14}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[2pt]<br />
1~0~0~1<br />
\\[2pt]<br />
1~1~1~0<br />
\\[2pt]<br />
1~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} u \texttt{~~} v \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{15}<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>f_{15}\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Difference Operators and Tangent Functors====<br />
<br />
Given the alphabets <math>\mathcal{U} = \{ u, v \}\!</math> and <math>\mathcal{X} = \{ x, y \},\!</math> along with the corresponding universes of discourse <math>U^\bullet, X^\bullet \cong [\mathbb{B}^2],\!</math> how many logical transformations of the general form <math>G = (G_1, G_2) : U^\bullet \to X^\bullet\!</math> are there? Since <math>G_1\!</math> and <math>G_2\!</math> can be any propositions of the type <math>\mathbb{B}^2 \to \mathbb{B},\!</math> there are <math>2^4 = 16\!</math> choices for each of the maps <math>G_1\!</math> and <math>G_2\!</math> and thus there are <math>2^4 \cdot 2^4 = 2^8 = 256\!</math> different mappings altogether of the form <math>G : U^\bullet \to X^\bullet.\!</math> The set of functions of a given type is denoted by placing its type indicator in parentheses, in the present instance writing <math>(U^\bullet \to X^\bullet) = \{ G : U^\bullet \to X^\bullet \},\!</math> and so the cardinality of the ''function space'' <math>(U^\bullet \to X^\bullet)\!</math> is summed up by writing <math>|(U^\bullet \to X^\bullet)| = |(\mathbb{B}^2 \to \mathbb{B}^2)| = 4^4 = 256.\!</math><br />
<br />
Given a transformation <math>G = (G_1, G_2) : U^\bullet \to X^\bullet\!</math> of this type, we proceed to define a pair of further transformations, related to <math>G,\!</math> that operate between the extended universes, <math>\mathrm{E}U^\bullet\!</math> and <math>\mathrm{E}X^\bullet,\!</math> of its source and target domains.<br />
<br />
First, the ''enlargement map'' (or ''secant transformation'') <math>\mathrm{E}G = (\mathrm{E}G_1, \mathrm{E}G_2) : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet\!</math> is defined by the following set of component equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}G_i<br />
& = & G_i (u + \mathrm{d}u, v + \mathrm{d}v)<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Next, the ''difference map'' (or ''chordal transformation'') <math>\mathrm{D}G = (\mathrm{D}G_1, \mathrm{D}G_2) : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet~\!</math> is defined in component-wise fashion as the boolean sum of the initial proposition <math>G_i\!</math> and the enlarged proposition <math>\mathrm{E}G_i,\!</math> for <math>i = 1, 2,\!</math> according to the following set of equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
\mathrm{D}G_i<br />
& = & G_i (u, v)<br />
& + & \mathrm{E}G_i (u, v, \mathrm{d}u, \mathrm{d}v)<br />
\\[8pt]<br />
& = & G_i (u, v)<br />
& + & G_i (u + \mathrm{d}u, v + \mathrm{d}v)<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Maintaining a strict analogy with ordinary difference calculus would perhaps have us write <math>\mathrm{D}G_i = \mathrm{E}G_i - G_i,\!</math> but the sum and difference operations are the same thing in boolean arithmetic. It is more often natural in the logical context to consider an initial proposition <math>q,\!</math> then to compute the enlargement <math>\mathrm{E}q,\!</math> and finally to determine the difference <math>\mathrm{D}q = q + \mathrm{E}q,\!</math> so we let the variant order of terms reflect this sequence of considerations.<br />
<br />
Viewed in this light the difference operator <math>\mathrm{D}\!</math> is imagined to be a function of very wide scope and polymorphic application, one that is able to realize the association between each transformation <math>G\!</math> and its difference map <math>\mathrm{D}G,\!</math> for example, taking the function space <math>(U^\bullet \to X^\bullet)\!</math> into <math>(\mathrm{E}U^\bullet \to \mathrm{E}X^\bullet).\!</math> When we consider the variety of interpretations permitted to propositions over the contexts in which we put them to use, it should be clear that an operator of this scope is not at all a trivial matter to define in general and that it may take some trouble to work out. For the moment we content ourselves with returning to particular cases.<br />
<br />
Acting on the logical transformation <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;),\!</math> the operators <math>\mathrm{E}\!</math> and <math>\mathrm{D}\!</math> yield the enlarged map <math>\mathrm{E}F = (\mathrm{E}f, \mathrm{E}g)\!</math> and the difference map <math>\mathrm{D}F = (\mathrm{D}f, \mathrm{D}g),\!</math> respectively, whose components are given as follows.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}f<br />
& = & \texttt{((} u + \mathrm{d}u \texttt{)(} v + \mathrm{d}v \texttt{))}<br />
\\[8pt]<br />
\mathrm{E}g<br />
& = & \texttt{((} u + \mathrm{d}u \texttt{,~} v + \mathrm{d}v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
\mathrm{D}f<br />
& = & \texttt{((} u \texttt{)(} v \texttt{))}<br />
& + & \texttt{((} u + \mathrm{d}u \texttt{)(} v + \mathrm{d}v \texttt{))}<br />
\\[8pt]<br />
\mathrm{D}g<br />
& = & \texttt{((} u \texttt{,~} v \texttt{))}<br />
& + & \texttt{((} u + \mathrm{d}u \texttt{,~} v + \mathrm{d}v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
But these initial formulas are purely definitional, and help us little in understanding either the purpose of the operators or the meaning of their results. Working symbolically, let us apply the same method to the separate components <math>f\!</math> and <math>g\!</math> that we earlier used on <math>J.\!</math> This work is recorded in Appendix&nbsp;3 and a summary of the results is presented in Tables&nbsp;66-i and 66-ii.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 66-i.} ~~ \text{Computation Summary for}~ f(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f<br />
& = & u \!\cdot\! v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 1<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}f<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \cdot \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{d}f<br />
& = & u \!\cdot\! v \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}f<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 66-ii.} ~~ \text{Computation Summary for}~ g(u, v) = \texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon g<br />
& = & u \!\cdot\! v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}g<br />
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}g<br />
& = & u \!\cdot\! v \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;67 shows how to compute the analytic series for <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math> in terms of coordinates, and Table&nbsp;68 recaps these results in symbolic terms, agreeing with earlier derivations.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 67.} ~~ \text{Computation of an Analytic Series in Terms of Coordinates}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:6%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}u\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>\mathrm{d}v\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>u'\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>v'\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:4px double black" | <math>f\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>g\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{E}f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{E}g}\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{D}f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{D}g}\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{d}f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{d}g}\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{d}^2\!f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{d}^2\!g}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 68.} ~~ \text{Computation of an Analytic Series in Symbolic Terms}\!</math><br />
|- style="height:40px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black" | <math>f\!</math><br />
| <math>g\!</math><br />
| style="border-left:1px solid black" | <math>{\mathrm{D}f}\!</math><br />
| <math>{\mathrm{D}g}\!</math><br />
| style="border-left:1px solid black" | <math>{\mathrm{d}f}\!</math><br />
| <math>{\mathrm{d}g}\!</math><br />
| style="border-left:1px solid black" | <math>{\mathrm{d}^2\!f}\!</math><br />
| <math>{\mathrm{d}^2\!g}\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[4pt]<br />
\texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
\\[4pt]<br />
\texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
\\[4pt]<br />
\texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;69 gives a graphical picture of the difference map <math>\mathrm{D}F = (\mathrm{D}f, \mathrm{D}g)\!</math> for the transformation <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math> This represents the same information about <math>\mathrm{D}f~\!</math> and <math>\mathrm{D}g~\!</math> that was given in the corresponding rows of Tables&nbsp;66-i and 66-ii, for ease of reference repeated below.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[8pt]<br />
\mathrm{D}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 69 -- Difference Map (Short Form).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 69.} ~~ \text{Difference Map of}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;70-a shows a way of visualizing the tangent functor map <math>\mathrm{d}F = (\mathrm{d}f, \mathrm{d}g)\!</math> for the transformation <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math> This amounts to the same information about <math>\mathrm{d}f~\!</math> and <math>\mathrm{d}g~\!</math> that was given in Tables&nbsp;66-i and 66-ii, the corresponding rows of which are repeated below.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{d}f<br />
& = & u \!\cdot\! v \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[8pt]<br />
\mathrm{d}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 70-a -- Tangent Functor Diagram.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 70-a.} ~~ \text{Tangent Functor Diagram for}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math><br />
|}<br />
<br />
<br><br />
<br />
Figure 70-b shows another way to picture the action of the tangent functor on the logical transformation <math>F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 70-b -- Tangent Functor Diagram.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 70-b.} ~~ \text{Tangent Functor Ferris Wheel for}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math><br />
|}<br />
<br />
<br><br />
<br />
* '''Note.''' The original Figure&nbsp;70-b lost some of its labeling in a succession of platform metamorphoses over the years, so we have included an ASCII version below to indicate where the missing labels go.<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| /////\ /////\ | | /XXXX\ /XXXX\ | | /\\\\\ /\\\\\ |<br />
| ///////o//////\ | | /XXXXXXoXXXXXX\ | | /\\\\\\o\\\\\\\ |<br />
| //////// \//////\ | | /XXXXXX/ \XXXXXX\ | | /\\\\\\/ \\\\\\\\ |<br />
| o/////// \//////o | | oXXXXXX/ \XXXXXXo | | o\\\\\\/ \\\\\\\o |<br />
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |<br />
| |/du//| |//dv/| | | |XXXXX| |XXXXX| | | |\du\\| |\\dv\| |<br />
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |<br />
| o//////\ ///////o | | oXXXXXX\ /XXXXXXo | | o\\\\\\\ /\\\\\\o |<br />
| \//////\ //////// | | \XXXXXX\ /XXXXXX/ | | \\\\\\\\ /\\\\\\/ |<br />
| \//////o/////// | | \XXXXXXoXXXXXX/ | | \\\\\\\o\\\\\\/ |<br />
| \///// \///// | | \XXXX/ \XXXX/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ (u)(v) o-----------------------o dv' @ (u)(v) =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| / \ /////\ | | /\\\\\ /XXXX\ | | /\\\\\ /\\\\\ |<br />
| / o//////\ | | /\\\\\\oXXXXXX\ | | /\\\\\\o\\\\\\\ |<br />
| / //\//////\ | | /\\\\\\//\XXXXXX\ | | /\\\\\\/ \\\\\\\\ |<br />
| o ////\//////o | | o\\\\\\////\XXXXXXo | | o\\\\\\/ \\\\\\\o |<br />
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |<br />
| | du |/////|//dv/| | | |\\\\\|/////|XXXXX| | | |\du\\| |\\dv\| |<br />
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |<br />
| o \//////////o | | o\\\\\\\////XXXXXXo | | o\\\\\\\ /\\\\\\o |<br />
| \ \///////// | | \\\\\\\\//XXXXXX/ | | \\\\\\\\ /\\\\\\/ |<br />
| \ o/////// | | \\\\\\\oXXXXXX/ | | \\\\\\\o\\\\\\/ |<br />
| \ / \///// | | \\\\\/ \XXXX/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ (u) v o-----------------------o dv' @ (u) v =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| /////\ / \ | | /XXXX\ /\\\\\ | | /\\\\\ /\\\\\ |<br />
| ///////o \ | | /XXXXXXo\\\\\\\ | | /\\\\\\o\\\\\\\ |<br />
| /////////\ \ | | /XXXXXX//\\\\\\\\ | | /\\\\\\/ \\\\\\\\ |<br />
| o//////////\ o | | oXXXXXX////\\\\\\\o | | o\\\\\\/ \\\\\\\o |<br />
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |<br />
| |/du//|/////| dv | | | |XXXXX|/////|\\\\\| | | |\du\\| |\\dv\| |<br />
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |<br />
| o//////\//// o | | oXXXXXX\////\\\\\\o | | o\\\\\\\ /\\\\\\o |<br />
| \//////\// / | | \XXXXXX\//\\\\\\/ | | \\\\\\\\ /\\\\\\/ |<br />
| \//////o / | | \XXXXXXo\\\\\\/ | | \\\\\\\o\\\\\\/ |<br />
| \///// \ / | | \XXXX/ \\\\\/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ u (v) o-----------------------o dv' @ u (v) =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| / \ / \ | | /\\\\\ /\\\\\ | | /\\\\\ /\\\\\ |<br />
| / o \ | | /\\\\\\o\\\\\\\ | | /\\\\\\o\\\\\\\ |<br />
| / / \ \ | | /\\\\\\/ \\\\\\\\ | | /\\\\\\/ \\\\\\\\ |<br />
| o / \ o | | o\\\\\\/ \\\\\\\o | | o\\\\\\/ \\\\\\\o |<br />
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |<br />
| | du | | dv | | | |\\\\\| |\\\\\| | | |\du\\| |\\dv\| |<br />
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |<br />
| o \ / o | | o\\\\\\\ /\\\\\\o | | o\\\\\\\ /\\\\\\o |<br />
| \ \ / / | | \\\\\\\\ /\\\\\\/ | | \\\\\\\\ /\\\\\\/ |<br />
| \ o / | | \\\\\\\o\\\\\\/ | | \\\\\\\o\\\\\\/ |<br />
| \ / \ / | | \\\\\/ \\\\\/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ u v o-----------------------o dv' @ u v =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| U | |\U\\\\\\\\\\\\\\\\\\\\\| |\U\\\\\\\\\\\\\\\\\\\\\|<br />
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|<br />
| /////\ /////\ | |\\\\\/////\\/////\\\\\\| |\\\\\/ \\/ \\\\\\|<br />
| ///////o//////\ | |\\\\///////o//////\\\\\| |\\\\/ o \\\\\|<br />
| /////////\//////\ | |\\\////////X\//////\\\\| |\\\/ /\\ \\\\|<br />
| o//////////\//////o | |\\o///////XXX\//////o\\| |\\o /\\\\ o\\|<br />
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|<br />
| |//u//|/////|//v//| | |\\|//u//|XXXXX|//v//|\\| |\\| u |\\\\\| v |\\|<br />
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|<br />
| o//////\//////////o | |\\o//////\XXX///////o\\| |\\o \\\\/ o\\|<br />
| \//////\///////// | |\\\\//////\X////////\\\| |\\\\ \\/ /\\\|<br />
| \//////o/////// | |\\\\\//////o///////\\\\| |\\\\\ o /\\\\|<br />
| \///// \///// | |\\\\\\/////\\/////\\\\\| |\\\\\\ /\\ /\\\\\|<br />
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|<br />
| | |\\\\\\\\\\\\\\\\\\\\\\\| |\\\\\\\\\\\\\\\\\\\\\\\|<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= u' o-----------------------o v' =<br />
= | U' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/u'//|XXXXX|\\v'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
Figure 70-b. Tangent Functor Ferris Wheel for F<u, v> = <((u)(v)), ((u, v))><br />
</pre><br />
|}<br />
<br />
==Epilogue, Enchoiry, Exodus==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
It is time to explain myself . . . . let us stand up.<br />
| width="4%" | &nbsp;<br />
|- <br />
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 79]<br />
|}<br />
<br />
==Appendices==<br />
<br />
===Appendix 1. Propositional Forms and Differential Expansions===<br />
<br />
====Table A1. Propositional Forms on Two Variables====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A1.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="background:ghostwhite"<br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}</math><br />
| width="25%" | <math>\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{0}\\f_{1}\\f_{2}\\f_{3}\\f_{4}\\f_{5}\\f_{6}\\f_{7}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0000}\\f_{0001}\\f_{0010}\\f_{0011}\\f_{0100}\\f_{0101}\\f_{0110}\\f_{0111}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~0\\0~0~0~1\\0~0~1~0\\0~0~1~1\\0~1~0~0\\0~1~0~1\\0~1~1~0\\0~1~1~1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)~ ~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~ ~(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{,~} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{~~} y \texttt{)}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{false}<br />
\\<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\<br />
y ~\text{without}~ x<br />
\\<br />
\text{not}~ x<br />
\\<br />
x ~\text{without}~ y<br />
\\<br />
\text{not}~ y<br />
\\<br />
x ~\text{not equal to}~ y<br />
\\<br />
\text{not both}~ x ~\text{and}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\<br />
\lnot x \land \lnot y<br />
\\<br />
\lnot x \land y<br />
\\<br />
\lnot x<br />
\\<br />
x \land \lnot y<br />
\\<br />
\lnot y<br />
\\<br />
x \ne y<br />
\\<br />
\lnot x \lor \lnot y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{8}\\f_{9}\\f_{10}\\f_{11}\\f_{12}\\f_{13}\\f_{14}\\f_{15}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{1000}\\f_{1001}\\f_{1010}\\f_{1011}\\f_{1100}\\f_{1101}\\f_{1110}\\f_{1111}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1~0~0~0\\1~0~0~1\\1~0~1~0\\1~0~1~1\\1~1~0~0\\1~1~0~1\\1~1~1~0\\1~1~1~1<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~ ~ ~ ~} y \texttt{~~}<br />
\\<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{~~} x \texttt{~ ~ ~ ~}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{((~))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{and}~ y<br />
\\<br />
x ~\text{equal to}~ y<br />
\\<br />
y<br />
\\<br />
\text{not}~ x ~\text{without}~ y<br />
\\<br />
x<br />
\\<br />
\text{not}~ y ~\text{without}~ x<br />
\\<br />
x ~\text{or}~ y<br />
\\<br />
\text{true}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x \land y<br />
\\<br />
x = y<br />
\\<br />
y<br />
\\<br />
x \Rightarrow y<br />
\\<br />
x<br />
\\<br />
x \Leftarrow y<br />
\\<br />
x \lor y<br />
\\<br />
1<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A2. Propositional Forms on Two Variables====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A2.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="background:ghostwhite"<br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}</math><br />
| width="25%" | <math>\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>f_{0000}\!</math><br />
| <math>0~0~0~0</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0001}\\f_{0010}\\f_{0100}\\f_{1000}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~1\\0~0~1~0\\0~1~0~0\\1~0~0~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\<br />
y ~\text{without}~ x<br />
\\<br />
x ~\text{without}~ y<br />
\\<br />
x ~\text{and}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot x \land \lnot y<br />
\\<br />
\lnot x \land y<br />
\\<br />
x \land \lnot y<br />
\\<br />
x \land y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{3}\\f_{12}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0011}\\f_{1100}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~1~1\\1~1~0~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ x<br />
\\<br />
x<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot x<br />
\\<br />
x<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{6}\\f_{9}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0110}\\f_{1001}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~0\\1~0~0~1<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{not equal to}~ y<br />
\\<br />
x ~\text{equal to}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x \ne y<br />
\\<br />
x = y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{5}\\f_{10}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0101}\\f_{1010}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~0~1\\1~0~1~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ y<br />
\\<br />
y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot y<br />
\\<br />
y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{7}\\f_{11}\\f_{13}\\f_{14}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0111}\\f_{1011}\\f_{1101}\\f_{1110}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~1\\1~0~1~1\\1~1~0~1\\1~1~1~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not both}~ x ~\text{and}~ y<br />
\\<br />
\text{not}~ x ~\text{without}~ y<br />
\\<br />
\text{not}~ y ~\text{without}~ x<br />
\\<br />
x ~\text{or}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot x \lor \lnot y<br />
\\<br />
x \Rightarrow y<br />
\\<br />
x \Leftarrow y<br />
\\<br />
x \lor y<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>f_{1111}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A3. E''f'' Expanded Over Differential Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A3.} ~~ \mathrm{E}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{T}_{11}f\\\mathrm{E}f|_{\mathrm{d}x ~ \mathrm{d}y}\end{matrix}</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{T}_{10}f\\\mathrm{E}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}\end{matrix}</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{T}_{01}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}\end{matrix}</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{T}_{00}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{3}\\f_{12}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{6}\\f_{9}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{5}\\f_{10}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{7}\\f_{11}\\f_{13}\\f_{14}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
|- style="background:ghostwhite"<br />
| style="border-top:1px solid black" colspan="2" | <math>\text{Fixed Point Total}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>4\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>4\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>4\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>16\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A4. D''f'' Expanded Over Differential Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A4.} ~~ \mathrm{D}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\mathrm{D}f|_{\mathrm{d}x ~ \mathrm{d}y}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}~\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
y<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
y<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
x<br />
\\<br />
x<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
x<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
y<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
y<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <br />
<math>\begin{matrix}<br />
x<br />
\\<br />
x<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A5. E''f'' Expanded Over Ordinary Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A5.} ~~ \mathrm{E}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\mathrm{E}f|_{xy}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{E}f|_{x \texttt{(} y \texttt{)}}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{E}f|_{\texttt{(} x \texttt{)} y}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{E}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{3}\\f_{12}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{6}\\f_{9}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{5}\\f_{10}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{7}\\f_{11}\\f_{13}\\f_{14}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A6. D''f'' Expanded Over Ordinary Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A6.} ~~ \mathrm{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\mathrm{D}f|_{xy}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{x \texttt{(} y \texttt{)}}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} x \texttt{)} y}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\texttt{(} x \texttt{)}\\\texttt{~} x \texttt{~}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Appendix 2. Differential Forms===<br />
<br />
The actions of the difference operator <math>\mathrm{D}\!</math> and the tangent operator <math>\mathrm{d}\!</math> on the 16 bivariate propositions are shown in Tables&nbsp;A7 and A8.<br />
<br />
Table A7 expands the differential forms that result over a ''logical basis'':<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
|<br />
<math>\{~ \texttt{(}\mathrm{d}x\texttt{)(}\mathrm{d}y\texttt{)}, ~\mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}, ~\texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.\!</math><br />
|}<br />
<br />
This set consists of the singular propositions in the first order differential variables, indicating mutually exclusive and exhaustive ''cells'' of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the cell-basis, point-basis, or singular differential basis. In this setting it is frequently convenient to use the following abbreviations:<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
|<br />
<math>\partial x ~=~ \mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}\!</math> &nbsp; &nbsp; and &nbsp; &nbsp; <math>\partial y ~=~ \texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y.\!</math><br />
|}<br />
<br />
Table A8 expands the differential forms that result over an ''algebraic basis'':<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
| <math>\{~ 1, ~\mathrm{d}x, ~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.\!</math><br />
|}<br />
<br />
This set consists of the ''positive propositions'' in the first order differential variables, indicating overlapping positive regions of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the ''positive differential basis''.<br />
<br />
====Table A7. Differential Forms Expanded on a Logical Basis====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A7.} ~~ \text{Differential Forms Expanded on a Logical Basis}\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| &nbsp;<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" | <math>\mathrm{D}f~\!</math><br />
| <math>\mathrm{d}f~\!</math><br />
|-<br />
| <math>f_{0}\!</math><br />
| style="border-right:none" | <math>\texttt{(~)}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x) ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\\<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\\<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\\<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\partial x<br />
\\<br />
\partial x<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
\\<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\partial x & + & \partial y<br />
\\<br />
\partial x & + & \partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\partial y<br />
\\<br />
\partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x) ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\\<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\\<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\\<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| style="border-right:none" | <math>\texttt{((~))}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A8. Differential Forms Expanded on an Algebraic Basis====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A8.} ~~ \text{Differential Forms Expanded on an Algebraic Basis}\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| &nbsp;<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" | <math>\mathrm{D}f~\!</math><br />
| <math>\mathrm{d}f~\!</math><br />
|-<br />
| <math>f_{0}\!</math><br />
| style="border-right:none" | <math>\texttt{(~)}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x<br />
\\<br />
\mathrm{d}x<br />
\end{matrix}\!</math><br />
| <math>\begin{matrix}<br />
\mathrm{d}x<br />
\\<br />
\mathrm{d}x<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\\<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\\<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}y<br />
\\<br />
\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y<br />
\\<br />
\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| style="border-right:none" | <math>\texttt{((~))}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A9. Tangent Proposition as Pointwise Linear Approximation====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A9.} ~~ \text{Tangent Proposition}~ \mathrm{d}f = \text{Pointwise Linear Approximation to the Difference Map}~ \mathrm{D}f\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}f =<br />
\\[2pt]<br />
\partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}^2\!f =<br />
\\[2pt]<br />
\partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\mathrm{d}f|_{x \, y}</math><br />
| <math>\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math><br />
|-<br />
| style="border-right:none" | <math>f_0\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{5}\\f_{10}\end{matrix}\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\end{matrix}\!</math><br />
| <math>\begin{matrix}<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>f_{15}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A10. Taylor Series Expansion Df = d''f'' + d<sup>2</sup>''f''====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" |<br />
<math>\text{Table A10.} ~~ \text{Taylor Series Expansion}~ {\mathrm{D}f = \mathrm{d}f + \mathrm{d}^2\!f}\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{D}f<br />
\\<br />
= & \mathrm{d}f & + & \mathrm{d}^2\!f<br />
\\<br />
= & \partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y & + & \partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\mathrm{d}f|_{x \, y}</math><br />
| <math>\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math><br />
|-<br />
| style="border-right:none" | <math>f_0\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>f_{15}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A11. Partial Differentials and Relative Differentials====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A11.} ~~ \text{Partial Differentials and Relative Differentials}\!</math><br />
|- style="background:ghostwhite; height:50px"<br />
| &nbsp;<br />
| <math>f\!</math><br />
| <math>\frac{\partial f}{\partial x}\!</math><br />
| <math>\frac{\partial f}{\partial y}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}f =<br />
\\[2pt]<br />
\partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\left. \frac{\partial x}{\partial y} \right| f\!</math><br />
| <math>\left. \frac{\partial y}{\partial x} \right| f\!</math><br />
|-<br />
| <math>f_0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A12. Detail of Calculation for the Difference Map====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:4px double black; border-left:4px double black; border-right:4px double black; border-top:4px double black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A12.} ~~ \text{Detail of Calculation for}~ {\mathrm{E}f + f = \mathrm{D}f}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:6%" | &nbsp;<br />
| style="width:14%; border-left:1px solid black" | <math>f\!</math><br />
| style="width:20%; border-left:4px double black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\mathrm{d}x ~ \mathrm{d}y}<br />
\\[4pt]<br />
+ & f|_{\mathrm{d}x ~ \mathrm{d}y}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\mathrm{d}x ~ \mathrm{d}y}<br />
\end{array}</math><br />
| style="width:20%; border-left:1px solid black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}<br />
\\[4pt]<br />
+ & f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}<br />
\end{array}</math><br />
| style="width:20%; border-left:1px solid black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
+ & f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}<br />
\end{array}</math><br />
| style="width:20%; border-left:1px solid black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
+ & f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}<br />
\end{array}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{0}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:4px double black; border-left:4px double black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{1}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{)(} y \texttt{)~}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{2}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{)~} y \texttt{~~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{4}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~~} x \texttt{~(} y \texttt{)~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{8}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~~} x \texttt{~~} y \texttt{~~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{3}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{(} x \texttt{)}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{12}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>x\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{6}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{,~} y \texttt{)~}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{9}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} x \texttt{,~} y \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{5}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{(} y \texttt{)}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{10}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>y\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{7}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{~~} y \texttt{)~}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{11}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{~(} y \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{13}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} x \texttt{)~} y \texttt{)~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{14}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} x \texttt{)(} y \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{15}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:4px double black; border-left:4px double black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Appendix 3. Computational Details===<br />
<br />
====Operator Maps for the Logical Conjunction ''f''<sub>8</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{8}~\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{8}<br />
& = && f_{8}(u, v)<br />
\\[4pt]<br />
& = && uv<br />
\\[4pt]<br />
& = && uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & uv \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & uv \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{8}<br />
& = && uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & uv \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & uv \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & uv \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.2-i} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{E}f_{8}<br />
& = & f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{8}(\mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{8}(\mathrm{d}u, \mathrm{d}v)<br />
\\[4pt]<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[20pt]<br />
\mathrm{E}f_{8}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
\\[4pt]<br />
&&&&& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&&&&&& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.2-ii} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{c}}<br />
\mathrm{E}f_{8}<br />
& = & (u + \mathrm{d}u) \cdot (v + \mathrm{d}v)<br />
\\[6pt]<br />
& = & u \cdot v<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}f_{8}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{8}<br />
& = && \mathrm{E}f_{8}<br />
& + & \boldsymbol\varepsilon f_{8}<br />
\\[4pt]<br />
& = && f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{8}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & uv<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{~}<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}f_{8}<br />
& = & \boldsymbol\varepsilon f_{8}<br />
& + & \mathrm{E}f_{8}<br />
\\[6pt]<br />
& = & f_{8}(u, v)<br />
& + & f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[6pt]<br />
& = & uv<br />
& + & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
& = & 0<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}f_{8}<br />
& = & 0<br />
& + & u \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.3-iii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 3)}\!</math><br />
|<br />
<math>\begin{array}{c*{9}{l}}<br />
\mathrm{D}f_{8}<br />
& = & \boldsymbol\varepsilon f_{8} ~+~ \mathrm{E}f_{8}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{8}<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ u \,\cdot\, v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~ u \;\cdot\; v \;\cdot\; \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}f_{8}<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u ~ \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} ~ v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot\, \mathrm{d}u ~ \mathrm{d}v<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = & ~ ~ 0 ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~ ~ u ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ ~ ~ v ~~ \cdot ~ \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
=====Computation of d''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.4} ~~ \text{Computation of}~ \mathrm{d}f_{8}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{8}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.5} ~~ \text{Computation of}~ \mathrm{r}f_{8}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{8} & = & \mathrm{D}f_{8} ~+~ \mathrm{d}f_{8}<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[20pt]<br />
\mathrm{r}f_{8}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Conjunction=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.6} ~~ \text{Computation Summary for}~ f_{8}(u, v) = uv\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{8}<br />
& = & uv \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}f_{8}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{r}f_{8}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Operator Maps for the Logical Equality ''f''<sub>9</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{9}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{9}<br />
& = && f_{9}(u, v)<br />
\\[4pt]<br />
& = && \texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{9}(1, 1)<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{9}(1, 0)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{9}(0, 1)<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{9}(0, 0)<br />
\\[4pt]<br />
& = && u v & + & 0 & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{9}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.2} ~~ \text{Computation of}~ \mathrm{E}f_{9}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{E}f_{9}<br />
& = && f_{9}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! f_{9}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! f_{9}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{ } \mathrm{d}v \texttt{ })<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! f_{9}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! f_{9}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{ } \mathrm{d}v \texttt{ })<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{ (} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{) }<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{ (} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{) }<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{E}f_{9}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{9}<br />
& = && \mathrm{E}f_{9}<br />
& + & \boldsymbol\varepsilon f_{9}<br />
\\[4pt]<br />
& = && f_{9}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{9}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{9}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[20pt]<br />
\mathrm{D}f_{9}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}f_{9}<br />
& = & 0 \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & 1 \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & 1 \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & 0 \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of d''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.4} ~~ \text{Computation of}~ \mathrm{d}f_{9}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.5} ~~ \text{Computation of}~ \mathrm{r}f_{9}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{9} & = & \mathrm{D}f_{9} ~+~ \mathrm{d}f_{9}<br />
\\[20pt]<br />
\mathrm{D}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[20pt]<br />
\mathrm{r}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Equality=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.6} ~~ \text{Computation Summary for}~ f_{9}(u, v) = \texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{9}<br />
& = & uv \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}f_{9}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f_{9}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{9}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}f_{9}<br />
& = & uv \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Operator Maps for the Logical Implication ''f''<sub>11</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{11}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{11}<br />
& = && f_{11}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(} u \texttt{(} v \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{11}(1, 1)<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{11}(1, 0)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{11}(0, 1)<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{11}(0, 0)<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ }<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ }<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{11}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.2} ~~ \text{Computation of}~ \mathrm{E}f_{11}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{E}f_{11}<br />
& = && f_{11}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = &&<br />
\texttt{(}<br />
\\<br />
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\\<br />
&&& \texttt{(}<br />
\\<br />
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\<br />
&&& \texttt{))}<br />
\\[4pt]<br />
& = &&<br />
u v<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)((} \mathrm{d}v \texttt{)))}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u \texttt{((} \mathrm{d}v \texttt{)))}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[4pt]<br />
& = &&<br />
u v<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{E}f_{11}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{11}<br />
& = && \mathrm{E}f_{11}<br />
& + & \boldsymbol\varepsilon f_{11}<br />
\\[4pt]<br />
& = && f_{11}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{11}(u, v)<br />
\\[4pt]<br />
& = &&<br />
\texttt{(} \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\texttt{(} \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{(} v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & u v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & 0<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & 0<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = && u v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{c*{9}{l}}<br />
\mathrm{D}f_{11}<br />
& = & \boldsymbol\varepsilon f_{11} ~+~ \mathrm{E}f_{11}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{11}<br />
& = & u v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}f_{11}<br />
& = &<br />
u v<br />
\cdot<br />
\texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\cdot<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\cdot<br />
\texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\cdot<br />
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = &<br />
u v<br />
\cdot<br />
\texttt{~(} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{~}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\cdot<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\cdot<br />
\texttt{~} \mathrm{d}u ~ \mathrm{d}v \texttt{~}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\cdot<br />
\texttt{~} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of d''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.4} ~~ \text{Computation of}~ \mathrm{d}f_{11}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{11}<br />
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{11}<br />
& = & u v \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.5} ~~ \text{Computation of}~ \mathrm{r}f_{11}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{11} & = & \mathrm{D}f_{11} ~+~ \mathrm{d}f_{11}<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{11}<br />
& = & u v \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u<br />
\\[20pt]<br />
\mathrm{r}f_{11}<br />
& = & u v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Implication=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.6} ~~ \text{Computation Summary for}~ f_{11}(u, v) = \texttt{(} u \texttt{(} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{11}<br />
& = & u v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}f_{11}<br />
& = & u v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f_{11}<br />
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{11}<br />
& = & u v \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u<br />
\\[6pt]<br />
\mathrm{r}f_{11}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Operator Maps for the Logical Disjunction ''f''<sub>14</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{14}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{14}<br />
& = && f_{14}(u, v)<br />
\\[4pt]<br />
& = && \texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{14}(1, 1)<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{14}(1, 0)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{14}(0, 1)<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{14}(0, 0)<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ }<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ }<br />
& + & 0<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{14}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.2} ~~ \text{Computation of}~ \mathrm{E}f_{14}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{E}f_{14}<br />
& = && f_{14}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = &&<br />
\texttt{((}<br />
\\<br />
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\\<br />
&&& \texttt{)(}<br />
\\<br />
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\<br />
&&& \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! f_{14}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! f_{14}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{ } \mathrm{d}v \texttt{ })<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! f_{14}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! f_{14}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{ } \mathrm{d}v \texttt{ })<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{E}f_{14}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{14}<br />
& = && \mathrm{E}f_{14}<br />
& + & \boldsymbol\varepsilon f_{14}<br />
\\[4pt]<br />
& = && f_{14}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{14}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{))((} v \texttt{,} \mathrm{d}v \texttt{)))}<br />
& + & \texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{14}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
\\[4pt]<br />
&& + & 0<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
\\[4pt]<br />
&& + & uv \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
\\[20pt]<br />
\mathrm{D}f_{14}<br />
& = && uv \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}f_{14}<br />
& = & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & 0 \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of d''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.4} ~~ \text{Computation of}~ \mathrm{d}f_{14}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.5} ~~ \text{Computation of}~ \mathrm{r}f_{14}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{14} & = & \mathrm{D}f_{14} ~+~ \mathrm{d}f_{14}<br />
\\[20pt]<br />
\mathrm{D}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{d}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[20pt]<br />
\mathrm{r}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Disjunction=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.6} ~~ \text{Computation Summary for}~ f_{14}(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{14}<br />
& = & uv \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 1<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}f_{14}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f_{14}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{d}f_{14}<br />
& = & uv \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}f_{14}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
===Appendix 4. Source Materials===<br />
<br />
===Appendix 5. Various Definitions of the Tangent Vector===<br />
<br />
==References==<br />
<br />
===Works Cited===<br />
<br />
{| cellpadding=3<br />
| valign=top | [AuM]<br />
| Auslander, L., and MacKenzie, R.E., ''Introduction to Differentiable Manifolds'', McGraw-Hill, 1963. Reprinted, Dover, New York, NY, 1977.<br />
|-<br />
| valign=top | [BiG]<br />
| Bishop, R.L., and Goldberg, S.I., ''Tensor Analysis on Manifolds'', Macmillan, 1968. Reprinted, Dover, New York, NY, 1980.<br />
|-<br />
| valign=top | [Boo]<br />
| Boole, G., ''An Investigation of The Laws of Thought'', Macmillan, 1854. Reprinted, Dover, New York, NY, 1958.<br />
|-<br />
| valign=top | [BoT]<br />
| Bott, R., and Tu, L.W., ''Differential Forms in Algebraic Topology'', Springer-Verlag, New York, NY, 1982.<br />
|-<br />
| valign=top | [dCa]<br />
| do Carmo, M.P., ''Riemannian Geometry''. Originally published in Portuguese, 1st editiom 1979, 2nd edition 1988. Translated by F. Flaherty, Birkhäuser, Boston, MA, 1992.<br />
|-<br />
| valign=top | [Che46]<br />
| Chevalley, C., ''Theory of Lie Groups'', Princeton University Press, Princeton, NJ, 1946.<br />
|-<br />
| valign=top | [Che56]<br />
| Chevalley, C., ''Fundamental Concepts of Algebra'', Academic Press, 1956.<br />
|-<br />
| valign=top | [Cho86]<br />
| Chomsky, N., ''Knowledge of Language : Its Nature, Origin, and Use'', Praeger, New York, NY, 1986.<br />
|-<br />
| valign=top | [Cho93]<br />
| Chomsky, N., ''Language and Thought'', Moyer Bell, Wakefield, RI, 1993.<br />
|-<br />
| valign=top | [DoM]<br />
| Doolin, B.F., and Martin, C.F., ''Introduction to Differential Geometry for Engineers'', Marcel Dekker, New York, NY, 1990.<br />
|-<br />
| valign=top | [Fuji]<br />
| Fujiwara, H., ''Logic Testing and Design for Testability'', MIT Press, Cambridge, MA, 1985.<br />
|-<br />
| valign=top | [Hic]<br />
| Hicks, N.J., ''Notes on Differential Geometry'', Van Nostrand, Princeton, NJ, 1965.<br />
|-<br />
| valign=top | [Hir]<br />
| Hirsch, M.W., ''Differential Topology'', Springer-Verlag, New York, NY, 1976.<br />
|-<br />
| valign=top | [How]<br />
| Howard, W.A., "The Formulae-as-Types Notion of Construction", Notes circulated from 1969. Reprinted in [SeH, 479-490].<br />
|-<br />
| valign=top | [JGH]<br />
| Jones, A., Gray, A., and Hutton, R., ''Manifolds and Mechanics'', Cambridge University Press, Cambridge, UK, 1987.<br />
|-<br />
| valign=top | [KoA]<br />
| Kosinski, A.A., ''Differential Manifolds'', Academic Press, San Diego, CA, 1993.<br />
|-<br />
| valign=top | [Koh]<br />
| Kohavi, Z., ''Switching and Finite Automata Theory'', 2nd edition, McGraw-Hill, New York, NY, 1978.<br />
|-<br />
| valign=top | [LaS]<br />
| Lambek, J., and Scott, P.J., ''Introduction to Higher Order Categorical Logic'', Cambridge University Press, Cambridge, UK, 1986.<br />
|-<br />
| valign=top | [La83]<br />
| Lang, S., ''Real Analysis'', 2nd edition, Addison-Wesley, Reading, MA, 1983.<br />
|-<br />
| valign=top | [La84]<br />
| Lang, S., ''Algebra'', 2nd edition, Addison-Wesley, Menlo Park, CA, 1984.<br />
|-<br />
| valign=top | [La85]<br />
| Lang, S., ''Differential Manifolds'', Springer-Verlag, New York, NY, 1985.<br />
|-<br />
| valign=top | [La93]<br />
| Lang, S., ''Real and Functional Analysis'', 3rd edition, Springer-Verlag, New York, NY, 1993.<br />
|-<br />
| valign=top | [Lie80]<br />
| Lie, S., "Sophus Lie's 1880 Transformation Group Paper", in ''Lie Groups : History, Frontiers, and Applications, Volume 1'', translated by M. Ackerman, comments by R. Hermann, Math Sci Press, Brookline, MA, 1975. Original paper 1880.<br />
|-<br />
| valign=top | [Lie84]<br />
| Lie, S., "Sophus Lie's 1884 Differential Invariant Paper", in ''Lie Groups : History, Frontiers, and Applications, Volume 3'', translated by M. Ackerman, comments by R. Hermann, Math Sci Press, Brookline, MA, 1976. Original paper 1884.<br />
|-<br />
| valign=top | [LoS]<br />
| Loomis, L.H., and Sternberg, S., ''Advanced Calculus'', Addison-Wesley, Reading, MA, 1968.<br />
|-<br />
| valign=top | [Mel]<br />
| Melzak, Z.A., ''Companion to Concrete Mathematics, Volume 2 : Mathematical Ideas, Modeling, and Applications'', John Wiley amd Sons, New York, NY, 1976.<br />
|-<br />
| valign=top | [Men]<br />
| Menabrea, L.F., "Sketch of the Analytical Engine Invented by Charles Babbage" with Notes by the Translator, Ada Augusta (Byron), Countess of Lovelace'', in [M&M, 225–297]. Originally published 1842.<br />
|-<br />
| valign=top | [M&M]<br />
| Morrison, P., and Morrison, E. (eds.), ''Charles Babbage on the Principles and Development of the Calculator, and Other Seminal Writings by Charles Babbage and Others, With an Introduction by the Editors'', Dover, Mineola, NY, 1961.<br />
|-<br />
| valign=top | [P1]<br />
| Peirce, C.S., ''Collected Papers of Charles Sanders Peirce'', vols. 1–8, C. Hartshorne, P. Weiss, and A.W. Burks (eds.), Harvard University Press, Cambridge, MA, 1931–1960. Cited as CP [volume].[paragraph].<br />
|-<br />
| valign=top | [P2]<br />
| Peirce, C.S., "Qualitative Logic", in ''The New Elements of Mathematics, Volume 4'', C. Eisele (ed.), Mouton, The Hague, 1976. Cited as NE [volume], [page].<br />
|-<br />
| valign=top | [Rob]<br />
| Roberts, D.D., ''The Existential Graphs of Charles S. Peirce'', Mouton, The Hague, 1973.<br />
|-<br />
| valign=top | [SeH]<br />
| Seldin, J.P., and Hindley, J.R. (eds.), ''To H.B. Curry : Essays on Combinatory Logic, Lambda Calculus, and Formalism'', Academic Press, London, UK, 1980.<br />
|-<br />
| valign=top | [SpB]<br />
| Spencer-Brown, G., ''Laws of Form'', George Allen and Unwin, London, UK, 1969.<br />
|-<br />
| valign=top | [Sp65]<br />
| Spivak, M., ''Calculus on Manifolds : A Modern Approach to Classical Theorems of Advanced Calculus'', W.A. Benjamin, New York, NY, 1965.<br />
|-<br />
| valign=top | [Sp79]<br />
| Spivak, M., ''A Comprehensive Introduction to Differential Geometry'', vols. 1–2. 1st edition 1970. 2nd edition, Publish or Perish Inc., Houston, TX, 1979.<br />
|-<br />
| valign=top | [Sty]<br />
| Styazhkin, N.I., ''History of Mathematical Logic from Leibniz to Peano'', 1st published in Russian, Nauka, Moscow, 1964. MIT Press, Cambridge, MA, 1969.<br />
|-<br />
| valign=top | [Wie]<br />
| Wiener, N., ''Cybernetics : or Control and Communication in the Animal and the Machine'', 1st edition 1948. 2nd edition, MIT Press, Cambridge, MA, 1961.<br />
|}<br />
<br />
===Works Consulted===<br />
<br />
{| cellpadding=3<br />
| valign=top | [Ami]<br />
| Amit, D.J., ''Modeling Brain Function : The World of Attractor Neural Networks'', Cambridge University Press, Cambridge, UK, 1989.<br />
|-<br />
| valign=top | [Ed87]<br />
| Edelman, G.M., ''Neural Darwinism : The Theory of Neuronal Group Selection'', Basic Books, New York, NY, 1987.<br />
|-<br />
| valign=top | [Ed88]<br />
| Edelman, G.M., ''Topobiology : An Introduction to Molecular Embryology'', Basic Books, New York, NY, 1988.<br />
|-<br />
| valign=top | [Fla]<br />
| Flanders, H., ''Differential Forms with Applications to the Physical Sciences'', Academic Press, 1963. Reprinted, Dover, Mineola, NY, 1989. <br />
|-<br />
| valign=top | [Has]<br />
| Hassoun, M.H. (ed.), ''Associative Neural Memories : Theory and Implementation'', Oxford University Press, New York, NY, 1993.<br />
|-<br />
| valign=top | [KoB]<br />
| Kosko, B., ''Neural Networks and Fuzzy Systems : A Dynamical Systems Approach to Machine Intelligence'', Prentice-Hall, Englewood Cliffs, NJ, 1992.<br />
|-<br />
| valign=top | [MaB]<br />
| Mac Lane, S., and Birkhoff, G., ''Algebra'', 3rd edition, Chelsea, New York, NY, 1993.<br />
|-<br />
| valign=top | [Mac]<br />
| Mac Lane, S., ''Categories for the Working Mathematician'', Springer-Verlag, New York, NY, 1971.<br />
|-<br />
| valign=top | [McC]<br />
| McCulloch, W.S., ''Embodiments of Mind'', MIT Press, Cambridge, MA, 1965.<br />
|-<br />
| valign=top | [Mc1]<br />
| McCulloch, W.S., "A Heterarchy of Values Determined by the Topology of Nervous Nets", Bulletin of Mathematical Biophysics, vol. 7 (1945), pp. 89–93. Reprinted in [McC].<br />
|-<br />
| valign=top | [MiP]<br />
| Minsky, M.L., and Papert, S.A., ''Perceptrons : An Introduction to Computational Geometry'', MIT Press, Cambridge, MA, 1969. 2nd printing 1972. Expanded edition 1988.<br />
|-<br />
| valign=top | [Rum]<br />
| Rumelhart, D.E., Hinton, G.E., and McClelland, J.L., "A General Framework for Parallel Distributed Processing" = Chapter 2 in Rumelhart, McClelland, and the PDP Research Group, ''Parallel Distributed Processing, Explorations in the Microstructure of Cognition, Volume 1 : Foundations'', MIT Press, Cambridge, MA, 1986.<br />
|}<br />
<br />
===Incidental Works===<br />
<br />
{| cellpadding=3<br />
| valign=top | [Dew]<br />
| Dewey, John, ''How We Think'', D.C. Heath, Lexington, MA, 1910. Reprinted, Prometheus Books, Buffalo, NY, 1991.<br />
|-<br />
| valign=top | [Fou]<br />
| Foucault, Michel, ''The Archaeology of Knowledge and The Discourse on Language'', A.M. Sheridan-Smith and Rupert Swyer (trans.), Pantheon, New York, NY, 1972. Originally published as ''L´Archéologie du Savoir et L´ordre du discours'', Editions Gallimard, 1969 & 1971.<br />
|-<br />
| valign=top | [Hom]<br />
| Homer, ''The Odyssey'', with an English translation by A.T. Murray, Loeb Classical Library, Harvard University Press, Cambridge, MA, 1980. First printed 1919.<br />
|-<br />
| valign=top | [Jam]<br />
| James, William, ''Pragmatism : A New Name for Some Old Ways of Thinking'', Longmans, Green, and Company, New York, NY, 1907.<br />
|-<br />
| valign=top | [Ler]<br />
| Leroux, Gaston, ''The Phantom of the Opera'', foreword by P. Haining, Dorset Press, New York, NY, 1988. Originally published in French, 1911.<br />
|-<br />
| valign=top | [Mus]<br />
| Musil, Robert, ''The Man Without Qualities'', 3 volumes, translated with a foreword by Eithne Wilkins and Ernst Kaiser, Pan Books, London, UK, 1979. English edition first published by Secker and Warburg, 1954. Originally published in German, ''Der Mann ohne Eigenschaften'', 1930 & 1932.<br />
|-<br />
| valign=top | [PlaR]<br />
| Plato, ''The Republic'', with an English translation by Paul Shorey, Loeb Classical Library, Harvard University Press, Cambridge, MA, 1980. First printed 1930 & 1935.<br />
|-<br />
| valign=top | [PlaS]<br />
| Plato, ''The Sophist'', Loeb Classical Library, William Heinemann, London, 1921, 1987.<br />
|-<br />
| valign=top | [Qui]<br />
| Quine, W.V., ''Mathematical Logic'', 1st edition, 1940. Revised edition, 1951. Harvard University Press, Cambridge, MA, 1981.<br />
|-<br />
| valign=top | [SaD]<br />
| de Santillana, Giorgio, and von Dechend, Hertha, ''Hamlet's Mill : An Essay on Myth and the Frame of Time'', David R. Godine, Publisher, Boston, MA, 1977. 1st published 1969.<br />
|-<br />
| valign=top | [Sha]<br />
| Shakespeare, William, '' William Shakespeare : The Complete Works'', Compact Edition, S. Wells and G. Taylor (eds.), Oxford University Press, Oxford, UK, 1988.<br />
|-<br />
| valign=top | [Sh1]<br />
| Shakespeare, William, ''A Midsummer Night's Dream'', Washington Square Press, New York, NY, 1958.<br />
|-<br />
| valign=top | [Sh2]<br />
| Shakespeare, William, ''The Tragedy of Hamlet, Prince of Denmark'', In [Sha], pp. 654&ndash;690.<br />
|-<br />
| valign=top | [Sh3]<br />
| Shakespeare, William, ''Measure for Measure'', Washington Square Press, New York, NY, 1965.<br />
|-<br />
| valign=top | [Web]<br />
| ''Webster's Ninth New Collegiate Dictionary'', Merriam-Webster, Springfield, MA, 1983.<br />
|-<br />
| valign=top | [Whi]<br />
| Whitman, Walt, ''Leaves of Grass'', Vintage Books / The Library of America, New York, NY, 1992. Originally published in numerous editions, 1855&ndash;1892.<br />
|-<br />
| valign=top | [Wil]<br />
| Wilhelm, R., and Baynes, C.F. (trans.), ''The I Ching, or Book of Changes'', foreword by C.G. Jung, preface by H. Wilhelm, 3rd edition, Bollingen Series XIX, Princeton University Press, Princeton, NJ, 1967.<br />
|}<br />
<br />
==Document History==<br />
<br />
<pre><br />
Author: Jon Awbrey<br />
Created: 16 Dec 1993<br />
Relayed: 31 Oct 1994<br />
Revised: 03 Jun 2003<br />
Recoded: 03 Jun 2007<br />
</pre><br />
<br />
[[Category:Adaptive Systems]]<br />
[[Category:Artificial Intelligence]]<br />
[[Category:Boolean Algebra]]<br />
[[Category:Boolean Functions]]<br />
[[Category:Charles Sanders Peirce]]<br />
[[Category:Combinatorics]]<br />
[[Category:Computer Science]]<br />
[[Category:Cybernetics]]<br />
[[Category:Differential Logic]]<br />
[[Category:Discrete Systems]]<br />
[[Category:Dynamical Systems]]<br />
[[Category:Formal Languages]]<br />
[[Category:Formal Sciences]]<br />
[[Category:Formal Systems]]<br />
[[Category:Functional Logic]]<br />
[[Category:Graph Theory]]<br />
[[Category:Group Theory]]<br />
[[Category:Inquiry]]<br />
[[Category:Knowledge Representation]]<br />
[[Category:Linguistics]]<br />
[[Category:Logic]]<br />
[[Category:Logical Graphs]]<br />
[[Category:Mathematics]]<br />
[[Category:Mathematical Systems Theory]]<br />
[[Category:Philosophy]]<br />
[[Category:Science]]<br />
[[Category:Semiotics]]<br />
[[Category:Systems Science]]<br />
[[Category:Visualization]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Directory:Jon_Awbrey/Differential_Logic_and_Dynamic_Systems_2.0&diff=469879Directory:Jon Awbrey/Differential Logic and Dynamic Systems 2.02021-01-13T18:31:31Z<p>Jon Awbrey: parse test</p>
<hr />
<div>{{DISPLAYTITLE:Differential Logic and Dynamic Systems 2.0}}<br />
'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''<br />
<br />
{| align="center" cellpadding="10"<br />
| [[Image:Tangent_Functor_Ferris_Wheel.gif]]<br />
|}<br />
<br />
{| style="height:36px; width:100%"<br />
| align="left" | ''Stand and unfold yourself.''<br />
| align="right" | Hamlet: Francsico&mdash;1.1.2<br />
|}<br />
<br />
This article develops a differential extension of propositional calculus and applies it to a context of problems arising in dynamic systems. The work pursued here is coordinated with a parallel application that focuses on neural network systems, but the dependencies are arranged to make the present article the main and the more self-contained work, to serve as a conceptual frame and a technical background for the network project.<br />
<br />
==Review and Transition==<br />
<br />
This note continues a previous discussion on the problem of dealing with change and diversity in logic-based intelligent systems. It is useful to begin by summarizing essential material from previous reports.<br />
<br />
Table 1 outlines a notation for propositional calculus based on two types of logical connectives, both of variable <math>k\!</math>-ary scope.<br />
<br />
* A bracketed list of propositional expressions in the form <math>\texttt{(} e_1, e_2, \ldots, e_{k-1}, e_k \texttt{)}\!</math> indicates that exactly one of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> is false.<br />
<br />
* A concatenation of propositional expressions in the form <math>e_1 ~ e_2 ~ \ldots ~ e_{k-1} ~ e_k</math> indicates that all of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> are true, in other words, that their [[logical conjunction]] is true.<br />
<br />
All other propositional connectives can be obtained in a very efficient style of representation through combinations of these two forms. Strictly speaking, the concatenation form is dispensable in light of the bracketed form, but it is convenient to maintain it as an abbreviation of more complicated bracket expressions.<br />
<br />
This treatment of propositional logic is derived from the work of C.S. Peirce [P1, P2], who gave this approach an extensive development in his graphical systems of predicate, relational, and modal logic [Rob]. More recently, these ideas were revived and supplemented in an alternative interpretation by George Spencer-Brown [SpB]. Both of these authors used other forms of enclosure where I use parentheses, but the structural topologies of expression and the functional varieties of interpretation are fundamentally the same.<br />
<br />
While working with expressions solely in propositional calculus, it is easiest to use plain parentheses for logical connectives. In contexts where parentheses are needed for other purposes &ldquo;teletype&rdquo; parentheses <math>\texttt{(} \ldots \texttt{)}\!</math> or barred parentheses <math>(\!| \ldots |\!)</math> may be used for logical operators.<br />
<br />
The briefest expression for logical truth is the empty word, usually denoted by <math>{}^{\backprime\backprime} \boldsymbol\varepsilon {}^{\prime\prime}\!</math> or <math>{}^{\backprime\backprime} \boldsymbol\lambda {}^{\prime\prime}\!</math> in formal languages, where it forms the identity element for concatenation. To make it visible in this text, it may be denoted by the equivalent expression <math>{}^{\backprime\backprime} \texttt{((~))} {}^{\prime\prime},\!</math> or, especially if operating in an algebraic context, by a simple <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}.\!</math> Also when working in an algebraic mode, the plus sign <math>{}^{\backprime\backprime} + {}^{\prime\prime}\!</math> may be used for [[exclusive disjunction]]. For example, we have the following paraphrases of algebraic expressions by bracket expressions:<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
|<br />
<math>\begin{matrix}<br />
x + y ~=~ \texttt{(} x, y \texttt{)}<br />
\\[6pt]<br />
x + y + z ~=~ \texttt{((} x, y \texttt{)}, z \texttt{)} ~=~ \texttt{(} x, \texttt{(} y, z \texttt{))}<br />
\end{matrix}</math><br />
|}<br />
<br />
It is important to note that the last expressions are not equivalent to the triple bracket <math>\texttt{(} x, y, z \texttt{)}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 1.} ~~ \text{Syntax and Semantics of a Calculus for Propositional Logic}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Expression}~\!</math><br />
| <math>\text{Interpretation}\!</math><br />
| <math>\text{Other Notations}\!</math><br />
|-<br />
| &nbsp;<br />
| <math>\text{True}\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{False}\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>x\!</math><br />
| <math>x\!</math><br />
| <math>x\!</math><br />
|-<br />
| <math>\texttt{(} x \texttt{)}\!</math><br />
| <math>\text{Not}~ x\!</math><br />
|<br />
<math>\begin{matrix}<br />
x'<br />
\\<br />
\tilde{x}<br />
\\<br />
\lnot x<br />
\end{matrix}\!</math><br />
|-<br />
| <math>x~y~z\!</math><br />
| <math>x ~\text{and}~ y ~\text{and}~ z\!</math><br />
| <math>x \land y \land z\!</math><br />
|-<br />
| <math>\texttt{((} x \texttt{)(} y \texttt{)(} z \texttt{))}\!</math><br />
| <math>x ~\text{or}~ y ~\text{or}~ z\!</math><br />
| <math>x \lor y \lor z\!</math><br />
|-<br />
| <math>\texttt{(} x ~ \texttt{(} y \texttt{))}\!</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{implies}~ y<br />
\\<br />
\mathrm{If}~ x ~\text{then}~ y<br />
\end{matrix}</math><br />
| <math>x \Rightarrow y\!</math><br />
|-<br />
| <math>\texttt{(} x \texttt{,} y \texttt{)}\!</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{not equal to}~ y<br />
\\<br />
x ~\text{exclusive or}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x \ne y<br />
\\<br />
x + y<br />
\end{matrix}</math><br />
|-<br />
| <math>\texttt{((} x \texttt{,} y \texttt{))}\!</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{is equal to}~ y<br />
\\<br />
x ~\text{if and only if}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x = y<br />
\\<br />
x \Leftrightarrow y<br />
\end{matrix}</math><br />
|-<br />
| <math>\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Just one of}<br />
\\<br />
x, y, z<br />
\\<br />
\text{is false}.<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x'y~z~ & \lor<br />
\\<br />
x~y'z~ & \lor<br />
\\<br />
x~y~z' &<br />
\end{matrix}</math><br />
|-<br />
| <math>\texttt{((} x \texttt{),(} y \texttt{),(} z \texttt{))}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Just one of}<br />
\\<br />
x, y, z<br />
\\<br />
\text{is true}.<br />
\\<br />
&<br />
\\<br />
\text{Partition all}<br />
\\<br />
\text{into}~ x, y, z.<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x~y'z' & \lor<br />
\\<br />
x'y~z' & \lor<br />
\\<br />
x'y'z~ &<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,} y \texttt{),} z \texttt{)}<br />
\\<br />
&<br />
\\<br />
\texttt{(} x \texttt{,(} y \texttt{,} z \texttt{))}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Oddly many of}<br />
\\<br />
x, y, z<br />
\\<br />
\text{are true}.<br />
\end{matrix}\!</math><br />
|<br />
<p><math>x + y + z\!</math></p><br />
<br><br />
<p><math>\begin{matrix}<br />
x~y~z~ & \lor<br />
\\<br />
x~y'z' & \lor<br />
\\<br />
x'y~z' & \lor<br />
\\<br />
x'y'z~ &<br />
\end{matrix}\!</math></p><br />
|-<br />
| <math>\texttt{(} w \texttt{,(} x \texttt{),(} y \texttt{),(} z \texttt{))}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Partition}~ w<br />
\\<br />
\text{into}~ x, y, z.<br />
\\<br />
&<br />
\\<br />
\text{Genus}~ w ~\text{comprises}<br />
\\<br />
\text{species}~ x, y, z.<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
w'x'y'z' & \lor<br />
\\<br />
w~x~y'z' & \lor<br />
\\<br />
w~x'y~z' & \lor<br />
\\<br />
w~x'y'z~ &<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
'''Note.''' The usage that one often sees, of a plus sign "<math>+\!</math>" to represent inclusive disjunction, and the reference to this operation as ''boolean addition'', is a misnomer on at least two counts. Boole used the plus sign to represent exclusive disjunction (at any rate, an operation of aggregation restricted in its logical interpretation to cases where the represented sets are disjoint (Boole, 32)), as any mathematician with a sensitivity to the ring and field properties of algebra would do:<br />
<br />
<blockquote><br />
The expression <math>x + y\!</math> seems indeed uninterpretable, unless it be assumed that the things represented by <math>x\!</math> and the things represented by <math>y\!</math> are entirely separate; that they embrace no individuals in common. (Boole, 66).<br />
</blockquote><br />
<br />
It was only later that Peirce and Jevons treated inclusive disjunction as a fundamental operation, but these authors, with a respect for the algebraic properties that were already associated with the plus sign, used a variety of other symbols for inclusive disjunction (Sty, 177, 189). It seems to have been Schröder who later reassigned the plus sign to inclusive disjunction (Sty, 208). Additional information, discussion, and references can be found in (Boole) and (Sty, 177&ndash;263). Aside from these historical points, which never really count against a current practice that has gained a life of its own, this usage does have a further disadvantage of cutting or confounding the lines of communication between algebra and logic. For this reason, it will be avoided here.<br />
<br />
==A Functional Conception of Propositional Calculus==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Out of the dimness opposite equals advance . . . .<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Always substance and increase,<br><br />
Always a knit of identity . . . . always distinction . . . .<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;always a breed of life.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]<br />
|}<br />
<br />
In the general case, we start with a set of logical features <math>\{a_1, \ldots, a_n\}</math> that represent properties of objects or propositions about the world. In concrete examples the features <math>\{a_i\!\}</math> commonly appear as capital letters from an ''alphabet'' like <math>\{A, B, C, \ldots\}</math> or as meaningful words from a linguistic ''vocabulary'' of codes. This language can be drawn from any sources, whether natural, technical, or artificial in character and interpretation. In the application to dynamic systems we tend to use the letters <math>\{x_1, \ldots, x_n\}</math> as our coordinate propositions, and to interpret them as denoting properties of a system's ''state'', that is, as propositions about its location in configuration space. Because I have to consider non-deterministic systems from the outset, I often use the word ''state'' in a loose sense, to denote the position or configuration component of a contemplated state vector, whether or not it ever gets a deterministic completion.<br />
<br />
The set of logical features <math>\{a_1, \ldots, a_n\}</math> provides a basis for generating an <math>n\!</math>-dimensional ''[[universe of discourse]]'' that I denote as <math>[a_1, \ldots, a_n].</math> It is useful to consider each universe of discourse as a unified categorical object that incorporates both the set of points <math>\langle a_1, \ldots, a_n \rangle</math> and the set of propositions <math>f : \langle a_1, \ldots, a_n \rangle \to \mathbb{B}</math> that are implicit with the ordinary picture of a venn diagram on <math>n\!</math> features. Thus, we may regard the universe of discourse <math>[a_1, \ldots, a_n]</math> as an ordered pair having the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}),</math> and we may abbreviate this last type designation as <math>\mathbb{B}^n\ +\!\to \mathbb{B},</math> or even more succinctly as <math>[\mathbb{B}^n].</math> (Used this way, the angle brackets <math>\langle\ldots\rangle</math> are referred to as ''generator brackets''.)<br />
<br />
Table 2 exhibits the scheme of notation I use to formalize the domain of propositional calculus, corresponding to the logical content of truth tables and venn diagrams. Although it overworks the square brackets a bit, I also use either one of the equivalent notations <math>[n]\!</math> or <math>\mathbf{n}</math> to denote the data type of a finite set on <math>n\!</math> elements.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 2.} ~~ \text{Propositional Calculus : Basic Notation}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Symbol}\!</math><br />
| <math>\text{Notation}\!</math><br />
| <math>\text{Description}\!</math><br />
| <math>\text{Type}\!</math><br />
|-<br />
| <math>\mathfrak{A}\!</math><br />
| <math>\{ {}^{\backprime\backprime} a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} a_n {}^{\prime\prime} \}\!</math><br />
| <math>\text{Alphabet}\!</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>\mathcal{A}\!</math><br />
| <math>\{ a_1, \ldots, a_n \}\!</math><br />
| <math>\text{Basis}\!</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>A_i\!</math><br />
| <math>\{ \texttt{(} a_i \texttt{)}, a_i \}\!</math><br />
| <math>\text{Dimension}~ i\!</math><br />
| <math>\mathbb{B}\!</math><br />
|-<br />
| <math>A\!</math><br />
|<br />
<math>\begin{matrix}<br />
\langle \mathcal{A} \rangle<br />
\\[2pt]<br />
\langle a_1, \ldots, a_n \rangle<br />
\\[2pt]<br />
\{ (a_1, \ldots, a_n) \}<br />
\\[2pt]<br />
A_1 \times \ldots \times A_n<br />
\\[2pt]<br />
\textstyle \prod_{i=1}^n A_i<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Set of cells},<br />
\\[2pt]<br />
\text{coordinate tuples},<br />
\\[2pt]<br />
\text{points, or vectors}<br />
\\[2pt]<br />
\text{in the universe}<br />
\\[2pt]<br />
\text{of discourse}<br />
\end{matrix}</math><br />
| <math>\mathbb{B}^n\!</math><br />
|-<br />
| <math>A^*\!</math><br />
| <math>(\mathrm{hom} : A \to \mathbb{B})\!</math><br />
| <math>\text{Linear functions}\!</math><br />
| <math>(\mathbb{B}^n)^* \cong \mathbb{B}^n\!</math><br />
|-<br />
| <math>A^\uparrow\!</math><br />
| <math>(A \to \mathbb{B})\!</math><br />
| <math>\text{Boolean functions}\!</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}\!</math><br />
|-<br />
| <math>A^\bullet\!</math><br />
|<br />
<math>\begin{matrix}<br />
[\mathcal{A}]<br />
\\[2pt]<br />
(A, A^\uparrow)<br />
\\[2pt]<br />
(A ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
(A, (A \to \mathbb{B}))<br />
\\[2pt]<br />
[a_1, \ldots, a_n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Universe of discourse}<br />
\\[2pt]<br />
\text{based on the features}<br />
\\[2pt]<br />
\{ a_1, \ldots, a_n \}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))<br />
\\[2pt]<br />
(\mathbb{B}^n ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
[\mathbb{B}^n]<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
===Qualitative Logic and Quantitative Analogy===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
''Logical'', however, is used in a third sense, which is at once more vital and more practical; to denote, namely, the systematic care, negative and positive, taken to safeguard reflection so that it may yield the best results under the given conditions.<br />
| width="4%" | &nbsp;<br />
|- <br />
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 56]<br />
|}<br />
<br />
These concepts and notations may now be explained in greater detail. In order to begin as simply as possible, let us distinguish two levels of analysis and set out initially on the easier path. On the first level of analysis we take spaces like <math>\mathbb{B},</math> <math>\mathbb{B}^n,</math> and <math>(\mathbb{B}^n \to \mathbb{B})</math> at face value and treat them as the primary objects of interest. On the second level of analysis we use these spaces as coordinate charts for talking about points and functions in more fundamental spaces.<br />
<br />
A pair of spaces, of types <math>\mathbb{B}^n</math> and <math>(\mathbb{B}^n \to \mathbb{B}),</math> give typical expression to everything that we commonly associate with the ordinary picture of a venn diagram. The dimension, <math>n,\!</math> counts the number of "circles" or simple closed curves that are inscribed in the universe of discourse, corresponding to its relevant logical features or basic propositions. Elements of type <math>\mathbb{B}^n</math> correspond to what are often called propositional ''interpretations'' in logic, that is, the different assignments of truth values to sentence letters. Relative to a given universe of discourse, these interpretations are visualized as its ''cells'', in other words, the smallest enclosed areas or undivided regions of the venn diagram. The functions <math>f : \mathbb{B}^n \to \mathbb{B}</math> correspond to the different ways of shading the venn diagram to indicate arbitrary propositions, regions, or sets. Regions included under a shading indicate the ''models'', and regions excluded represent the ''non-models'' of a proposition. To recognize and formalize the natural cohesion of these two layers of concepts into a single universe of discourse, we introduce the type notations <math>[\mathbb{B}^n] = \mathbb{B}^n\ +\!\to \mathbb{B}</math> to stand for the pair of types <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})).</math> The resulting "stereotype" serves to frame the universe of discourse as a unified categorical object, and makes it subject to prescribed sets of evaluations and transformations (categorical morphisms or ''arrows'') that affect the universe of discourse as an integrated whole.<br />
<br />
Most of the time we can serve the algebraic, geometric, and logical interests of our study without worrying about their occasional conflicts and incidental divergences. The conventions and definitions already set down will continue to cover most of the algebraic and functional aspects of our discussion, but to handle the logical and qualitative aspects we will need to add a few more. In general, abstract sets may be denoted by gothic, greek, or script capital variants of <math>A, B, C,\!</math> and so on, with their elements being denoted by a corresponding set of subscripted letters in plain lower case, for example, <math>\mathcal{A} = \{a_i\}.</math> Most of the time, a set such as <math>\mathcal{A} = \{a_i\}\!</math> will be employed as the ''alphabet'' of a [[formal language]]. These alphabet letters serve to name the logical features (properties or propositions) that generate a particular universe of discourse. When we want to discuss the particular features of a universe of discourse, beyond the abstract designation of a type like <math>(\mathbb{B}^n\ +\!\to \mathbb{B}),</math> then we may use the following notations. If <math>\mathcal{A} = \{a_1, \ldots, a_n\}</math> is an alphabet of logical features, then <math>A = \langle \mathcal{A} \rangle = \langle a_1, \ldots, a_n \rangle</math> is the set of interpretations, <math>A^\uparrow = (A \to \mathbb{B})</math> is the set of propositions, and <math>A^\bullet = [\mathcal{A}] = [a_1, \ldots, a_n]</math> is the combination of these interpretations and propositions into the universe of discourse that is based on the features <math>\{a_1, \ldots, a_n\}.</math><br />
<br />
As always, especially in concrete examples, these rules may be dropped whenever necessary, reverting to a free assortment of feature labels. However, when we need to talk about the logical aspects of a space that is already named as a vector space, it will be necessary to make special provisions. At any rate, these elaborations can be deferred until actually needed.<br />
<br />
===Philosophy of Notation : Formal Terms and Flexible Types===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Where number is irrelevant, regimented mathematical technique has hitherto tended to be lacking. Thus it is that the progress of natural science has depended so largely upon the discernment of measurable quantity of one sort or another.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]<br />
|}<br />
<br />
For much of our discussion propositions and boolean functions are treated as the same formal objects, or as different interpretations of the same formal calculus. This rule of interpretation has exceptions, though. There is a distinctively logical interest in the use of propositional calculus that is not exhausted by its functional interpretation. It is part of our task in this study to deal with these uniquely logical characteristics as they present themselves both in our subject matter and in our formal calculus. Just to provide a hint of what's at stake: In logic, as opposed to the more imaginative realms of mathematics, we consider it a good thing to always know what we are talking about. Where mathematics encourages tolerance for uninterpreted symbols as intermediate terms, logic exerts a keener effort to interpret directly each oblique carrier of meaning, no matter how slight, and to unfold the complicities of every indirection in the flow of information. Translated into functional terms, this means that we want to maintain a continual, immediate, and persistent sense of both the converse relation <math>f^{-1} \subseteq \mathbb{B} \times \mathbb{B}^n,</math> or what is the same thing, <math>f^{-1} : \mathbb{B} \to \mathcal{P}(\mathbb{B}^n),</math> and the ''fibers'' or inverse images <math>f^{-1}(0)\!</math> and <math>f^{-1}(1),\!</math> associated with each boolean function <math>f : \mathbb{B}^n \to \mathbb{B}</math> that we use. In practical terms, the desired implementation of a propositional interpreter should incorporate our intuitive recognition that the induced partition of the functional domain into level sets <math>f^{-1}(b),\!</math> for <math>b \in \mathbb{B},</math> is part and parcel of understanding the denotative uses of each propositional function <math>f.\!</math><br />
<br />
===Special Classes of Propositions===<br />
<br />
It is important to remember that the coordinate propositions <math>\{a_i\},\!</math> besides being projection maps <math>a_i : \mathbb{B}^n \to \mathbb{B},</math> are propositions on an equal footing with all others, even though employed as a basis in a particular moment. This set of <math>n\!</math> propositions may sometimes be referred to as the ''basic propositions'', the ''coordinate propositions'', or the ''simple propositions'' that found a universe of discourse. Either one of the equivalent notations, <math>\{a_i : \mathbb{B}^n \to \mathbb{B}\}</math> or <math>(\mathbb{B}^n \xrightarrow{i} \mathbb{B}),</math> may be used to indicate the adoption of the propositions <math>a_i\!</math> as a basis for describing a universe of discourse.<br />
<br />
Among the <math>2^{2^n}</math> propositions in <math>[a_1, \ldots, a_n]</math> are several families of <math>2^n\!</math> propositions each that take on special forms with respect to the basis <math>\{ a_1, \ldots, a_n \}.</math> Three of these families are especially prominent in the present context, the ''linear'', the ''positive'', and the ''singular'' propositions. Each family is naturally parameterized by the coordinate <math>n\!</math>-tuples in <math>\mathbb{B}^n</math> and falls into <math>n + 1\!</math> ranks, with a binomial coefficient <math>\tbinom{n}{k}</math> giving the number of propositions that have rank or weight <math>k.\!</math><br />
<br />
<ul><br />
<br />
<li><br />
<p>The ''linear propositions'', <math>\{ \ell : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{\ell} \mathbb{B}),\!</math> may be written as sums:</p><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\sum_{i=1}^n e_i ~=~ e_1 + \ldots + e_n<br />
~\text{where}~<br />
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 0 \end{matrix}\right\}<br />
~\text{for}~ i = 1 ~\text{to}~ n.\!</math><br />
|}<br />
</li><br />
<br />
<li><br />
<p>The ''positive propositions'', <math>\{ p : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{p} \mathbb{B}),\!</math> may be written as products:</p><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\prod_{i=1}^n e_i ~=~ e_1 \cdot \ldots \cdot e_n<br />
~\text{where}~<br />
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 1 \end{matrix}\right\}<br />
~\text{for}~ i = 1 ~\text{to}~ n.\!</math><br />
|}<br />
</li><br />
<br />
<li><br />
<p>The ''singular propositions'', <math>\{ \mathbf{x} : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{s} \mathbb{B}),\!</math> may be written as products:</p><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\prod_{i=1}^n e_i ~=~ e_1 \cdot \ldots \cdot e_n<br />
~\text{where}~<br />
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = \texttt{(} a_i \texttt{)} \end{matrix}\right\}<br />
~\text{for}~ i = 1 ~\text{to}~ n.\!</math><br />
|}<br />
</li><br />
<br />
</ul><br />
<br />
In each case the rank <math>k\!</math> ranges from <math>0\!</math> to <math>n\!</math> and counts the number of positive appearances of the coordinate propositions <math>a_1, \ldots, a_n\!</math> in the resulting expression. For example, for <math>{n = 3},\!</math> the linear proposition of rank <math>0\!</math> is <math>0,\!</math> the positive proposition of rank <math>0\!</math> is <math>1,\!</math> and the singular proposition of rank <math>0\!</math> is <math>\texttt{(} a_1 \texttt{)(} a_2 \texttt{)(} a_3\texttt{)}.\!</math><br />
<br />
The basic propositions <math>a_i : \mathbb{B}^n \to \mathbb{B}</math> are both linear and positive. So these two kinds of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions.<br />
<br />
Linear propositions and positive propositions are generated by taking boolean sums and products, respectively, over selected subsets of basic propositions, so both families of propositions are parameterized by the powerset <math>\mathcal{P}(\mathcal{I}),</math> that is, the set of all subsets <math>J\!</math> of the basic index set <math>\mathcal{I} = \{1, \ldots, n\}.\!</math><br />
<br />
Let us define <math>\mathcal{A}_J</math> as the subset of <math>\mathcal{A}</math> that is given by <math>\{a_i : i \in J\}.\!</math> Then we may comprehend the action of the linear and the positive propositions in the following terms:<br />
<br />
* The linear proposition <math>\ell_J : \mathbb{B}^n \to \mathbb{B}</math> evaluates each cell <math>\mathbf{x}</math> of <math>\mathbb{B}^n</math> by looking at the coefficients of <math>\mathbf{x}</math> with respect to the features that <math>\ell_J</math> "likes", namely those in <math>\mathcal{A}_J,</math> and then adds them up in <math>\mathbb{B}.</math> Thus, <math>\ell_J(\mathbf{x})</math> computes the parity of the number of features that <math>\mathbf{x}</math> has in <math>\mathcal{A}_J,</math> yielding one for odd and zero for even. Expressed in this idiom, <math>\ell_J(\mathbf{x}) = 1</math> says that <math>\mathbf{x}</math> seems ''odd'' (or ''oddly true'') to <math>\mathcal{A}_J,</math> whereas <math>\ell_J(\mathbf{x}) = 0</math> says that <math>\mathbf{x}</math> seems ''even'' (or ''evenly true'') to <math>\mathcal{A}_J,</math> so long as we recall that ''zero times'' is evenly often, too.<br />
<br />
* The positive proposition <math>p_J : \mathbb{B}^n \to \mathbb{B}</math> evaluates each cell <math>\mathbf{x}</math> of <math>\mathbb{B}^n</math> by looking at the coefficients of <math>\mathbf{x}</math> with regard to the features that <math>p_J\!</math> "likes", namely those in <math>\mathcal{A}_J,</math> and then takes their product in <math>\mathbb{B}.</math> Thus, <math>p_J(\mathbf{x})</math> assesses the unanimity of the multitude of features that <math>\mathbf{x}</math> has in <math>\mathcal{A}_J,</math> yielding one for all and aught for else. In these consensual or contractual terms, <math>p_J(\mathbf{x}) = 1</math> means that <math>\mathbf{x}</math> is ''AOK'' or congruent with all of the conditions of <math>\mathcal{A}_J,</math> while <math>p_J(\mathbf{x}) = 0</math> means that <math>\mathbf{x}</math> defaults or dissents from some condition of <math>\mathcal{A}_J.</math><br />
<br />
===Basis Relativity and Type Ambiguity===<br />
<br />
Finally, two things are important to keep in mind with regard to the simplicity, linearity, positivity, and singularity of propositions.<br />
<br />
First, all of these properties are relative to a particular basis. For example, a singular proposition with respect to a basis <math>\mathcal{A}</math> will not remain singular if <math>\mathcal{A}</math> is extended by a number of new and independent features. Even if we stick to the original set of pairwise options <math>\{a_i\} \cup \{ \texttt{(} a_i \texttt{)} \}\!</math> to select a new basis, the sets of linear and positive propositions are determined by the choice of simple propositions, and this determination is tantamount to the conventional choice of a cell as origin.<br />
<br />
Second, the singular propositions <math>\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B},</math> picking out as they do a single cell or a coordinate tuple <math>\mathbf{x}</math> of <math>\mathbb{B}^n,</math> become the carriers or the vehicles of a certain type-ambiguity that vacillates between the dual forms <math>\mathbb{B}^n</math> and <math>(\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B})</math> and infects the whole hierarchy of types built on them. In other words, the terms that signify the interpretations <math>\mathbf{x} : \mathbb{B}^n</math> and the singular propositions <math>\mathbf{x} : \mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B}</math> are fully equivalent in information, and this means that every token of the type <math>\mathbb{B}^n</math> can be reinterpreted as an appearance of the subtype <math>\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B}.</math> And vice versa, the two types can be exchanged with each other everywhere that they turn up. In practical terms, this allows the use of singular propositions as a way of denoting points, forming an alternative to coordinate tuples.<br />
<br />
For example, relative to the universe of discourse <math>[a_1, a_2, a_3]\!</math> the singular proposition <math>a_1 a_2 a_3 : \mathbb{B}^3 \xrightarrow{s} \mathbb{B}</math> could be explicitly retyped as <math>a_1 a_2 a_3 : \mathbb{B}^3</math> to indicate the point <math>(1, 1, 1)\!</math> but in most cases the proper interpretation could be gathered from context. Both notations remain dependent on a particular basis, but the code that is generated under the singular option has the advantage in its self-commenting features, in other words, it constantly reminds us of its basis in the process of denoting points. When the time comes to put a multiplicity of different bases into play, and to search for objects and properties that remain invariant under the transformations between them, this infinitesimal potential advantage may well evolve into an overwhelming practical necessity.<br />
<br />
===The Analogy Between Real and Boolean Types===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Measurement consists in correlating our subject matter with the series of real numbers; and such correlations are desirable because, once they are set up, all the well-worked theory of numerical mathematics lies ready at hand as a tool for our further reasoning.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]<br />
|}<br />
<br />
There are two further reasons why it useful to spend time on a careful treatment of types, and they both have to do with our being able to take full computational advantage of certain dimensions of flexibility in the types that apply to terms. First, the domains of differential geometry and logic programming are connected by analogies between real and boolean types of the same pattern. Second, the types involved in these patterns have important isomorphisms connecting them that apply on both the real and the boolean sides of the picture.<br />
<br />
Amazingly enough, these isomorphisms are themselves schematized by the axioms and theorems of propositional logic. This fact is known as the ''propositions as types'' analogy or the Curry&ndash;Howard isomorphism [How]. In another formulation it says that terms are to types as proofs are to propositions. See [LaS, 42&ndash;46] and [SeH] for a good discussion and further references. To anticipate the bearing of these issues on our immediate topic, Table&nbsp;3 sketches a partial overview of the Real to Boolean analogy that may serve to illustrate the paradigm in question.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 3.} ~~ \text{Analogy Between Real and Boolean Types}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Real Domain} ~ \mathbb{R}\!</math><br />
| <math>\longleftrightarrow\!</math><br />
| <math>\text{Boolean Domain} ~ \mathbb{B}\!</math><br />
|-<br />
| <math>\mathbb{R}^n\!</math><br />
| <math>\text{Basic Space}\!</math><br />
| <math>\mathbb{B}^n\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}\!</math><br />
| <math>\text{Function Space}\!</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R}\!</math><br />
| <math>\text{Tangent Vector}\!</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to ((\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R})\!</math><br />
| <math>\text{Vector Field}\!</math><br />
| <math>\mathbb{B}^n \to ((\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B})\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \times (\mathbb{R}^n \to \mathbb{R})) \to \mathbb{R}\!</math><br />
| '''<font size="4">"</font>'''<br />
| <math>(\mathbb{B}^n \times (\mathbb{B}^n \to \mathbb{B})) \to \mathbb{B}\!</math><br />
|-<br />
| <math>((\mathbb{R}^n \to \mathbb{R}) \times \mathbb{R}^n) \to \mathbb{R}\!</math><br />
| '''<font size="4">"</font>'''<br />
| <math>((\mathbb{B}^n \to \mathbb{B}) \times \mathbb{B}^n) \to \mathbb{B}\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^n \to \mathbb{R})~\!</math><br />
| <math>\text{Derivation}\!</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^n \to \mathbb{B})\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}^m\!</math><br />
| <math>\begin{matrix}\text{Basic}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}^m\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^m \to \mathbb{R})\!</math><br />
| <math>\begin{matrix}\text{Function}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^m \to \mathbb{B})\!</math><br />
|}<br />
<br />
<br><br />
<br />
The Table exhibits a sample of likely parallels between the real and boolean domains. The central column gives a selection of terminology that is borrowed from differential geometry and extended in its meaning to the logical side of the Table. These are the varieties of spaces that come up when we turn to analyzing the dynamics of processes that pursue their courses through the states of an arbitrary space <math>X.\!</math> Moreover, when it becomes necessary to approach situations of overwhelming dynamic complexity in a succession of qualitative reaches, then the methods of logic that are afforded by the boolean domains, with their declarative means of synthesis and deductive modes of analysis, supply a natural battery of tools for the task.<br />
<br />
It is usually expedient to take these spaces two at a time, in dual pairs of the form <math>X\!</math> and <math>(X \to \mathbb{K}).</math> In general, one creates pairs of type schemas by replacing any space <math>X\!</math> with its dual <math>(X \to \mathbb{K}),</math> for example, pairing the type <math>X \to Y</math> with the type <math>(X \to \mathbb{K}) \to (Y \to \mathbb{K}),</math> and <math>X \times Y</math> with <math>(X \to \mathbb{K}) \times (Y \to \mathbb{K}).</math> The word ''dual'' is used here in its broader sense to mean all of the functionals, not just the linear ones. Given any function <math>f : X \to \mathbb{K},</math> the ''converse'' or inverse relation corresponding to <math>f\!</math> is denoted <math>f^{-1},\!</math> and the subsets of <math>X\!</math> that are defined by <math>f^{-1}(k),\!</math> taken over <math>k\!</math> in <math>\mathbb{K},</math> are called the ''fibers'' or the ''level sets'' of the function <math>f.\!</math><br />
<br />
===Theory of Control and Control of Theory===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
You will hardly know who I am or what I mean,<br><br />
But I shall be good health to you nevertheless,<br><br />
And filter and fibre your blood.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]<br />
|}<br />
<br />
In the boolean context a function <math>f : X \to \mathbb{B}</math> is tantamount to a ''proposition'' about elements of <math>X,\!</math> and the elements of <math>X\!</math> constitute the ''interpretations'' of that proposition. The fiber <math>f^{-1}(1)\!</math> comprises the set of ''models'' of <math>f,\!</math> or examples of elements in <math>X\!</math> satisfying the proposition <math>f.\!</math> The fiber <math>f^{-1}(0)\!</math> collects the complementary set of ''anti-models'', or the exceptions to the proposition <math>f\!</math> that exist in <math>X.\!</math> Of course, the space of functions <math>(X \to \mathbb{B})\!</math> is isomorphic to the set of all subsets of <math>X,\!</math> called the ''power set'' of <math>X,\!</math> and often denoted <math>\mathcal{P}(X)</math> or <math>2^X.\!</math><br />
<br />
The operation of replacing <math>X\!</math> by <math>(X \to \mathbb{B})\!</math> in a type schema corresponds to a certain shift of attitude towards the space <math>X,\!</math> in which one passes from a focus on the ostensibly individual elements of <math>X\!</math> to a concern with the states of information and uncertainty that one possesses about objects and situations in <math>X.\!</math> The conceptual obstacles in the path of this transition can be smoothed over by using singular functions <math>(\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B})</math> as stepping stones. First of all, it's an easy step from an element <math>\mathbf{x}</math> of type <math>\mathbb{B}^n</math> to the equivalent information of a singular proposition <math>\mathbf{x} : X \xrightarrow{s} \mathbb{B}, </math> and then only a small jump of generalization remains to reach the type of an arbitrary proposition <math>f : X \to \mathbb{B},</math> perhaps understood to indicate a relaxed constraint on the singularity of points or a neighborhood circumscribing the original <math>\mathbf{x}.</math> This is frequently a useful transformation, communicating between the ''objective'' and the ''intentional'' perspectives, in spite perhaps of the open objection that this distinction is transient in the mean time and ultimately superficial.<br />
<br />
It is hoped that this measure of flexibility, allowing us to stretch a point into a proposition, can be useful in the examination of inquiry driven systems, where the differences between empirical, intentional, and theoretical propositions constitute the discrepancies and the distributions that drive experimental activity. I can give this model of inquiry a cybernetic cast by realizing that theory change and theory evolution, as well as the choice and the evaluation of experiments, are actions that are taken by a system or its agent in response to the differences that are detected between observational contents and theoretical coverage.<br />
<br />
All of the above notwithstanding, there are several points that distinguish these two tasks, namely, the ''theory of control'' and the ''control of theory'', features that are often obscured by too much precipitation in the quickness with which we understand their similarities. In the control of uncertainty through inquiry, some of the actuators that we need to be concerned with are axiom changers and theory modifiers, operators with the power to compile and to revise the theories that generate expectations and predictions, effectors that form and edit our grammars for the languages of observational data, and agencies that rework the proposed model to fit the actual sequences of events and the realized relationships of values that are observed in the environment. Moreover, when steps must be taken to carry out an experimental action, there must be something about the particular shape of our uncertainty that guides us in choosing what directions to explore, and this impression is more than likely influenced by previous accumulations of experience. Thus it must be anticipated that much of what goes into scientific progress, or any sustainable effort toward a goal of knowledge, is necessarily predicated on long term observation and modal expectations, not only on the more local or short term prediction and correction.<br />
<br />
===Propositions as Types and Higher Order Types===<br />
<br />
The types collected in Table&nbsp;3 (repeated below) serve to illustrate the themes of ''higher order propositional expressions'' and the ''propositions as types'' (PAT) analogy.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 3.} ~~ \text{Analogy Between Real and Boolean Types}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Real Domain} ~ \mathbb{R}\!</math><br />
| <math>\longleftrightarrow\!</math><br />
| <math>\text{Boolean Domain} ~ \mathbb{B}\!</math><br />
|-<br />
| <math>\mathbb{R}^n\!</math><br />
| <math>\text{Basic Space}\!</math><br />
| <math>\mathbb{B}^n\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}\!</math><br />
| <math>\text{Function Space}\!</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R}\!</math><br />
| <math>\text{Tangent Vector}\!</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to ((\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R})\!</math><br />
| <math>\text{Vector Field}\!</math><br />
| <math>\mathbb{B}^n \to ((\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B})\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \times (\mathbb{R}^n \to \mathbb{R})) \to \mathbb{R}\!</math><br />
| '''<font size="4">"</font>'''<br />
| <math>(\mathbb{B}^n \times (\mathbb{B}^n \to \mathbb{B})) \to \mathbb{B}\!</math><br />
|-<br />
| <math>((\mathbb{R}^n \to \mathbb{R}) \times \mathbb{R}^n) \to \mathbb{R}\!</math><br />
| '''<font size="4">"</font>'''<br />
| <math>((\mathbb{B}^n \to \mathbb{B}) \times \mathbb{B}^n) \to \mathbb{B}\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^n \to \mathbb{R})~\!</math><br />
| <math>\text{Derivation}\!</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^n \to \mathbb{B})\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}^m\!</math><br />
| <math>\begin{matrix}\text{Basic}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}^m\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^m \to \mathbb{R})\!</math><br />
| <math>\begin{matrix}\text{Function}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^m \to \mathbb{B})\!</math><br />
|}<br />
<br />
<br><br />
<br />
First, observe that the type of a ''tangent vector at a point'', also known as a ''directional derivative'' at that point, has the form <math>(\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K},</math> where <math>\mathbb{K}</math> is the chosen ground field, in the present case either <math>\mathbb{R}</math> or <math>\mathbb{B}.</math> At a point in a space of type <math>\mathbb{K}^n,</math> a directional derivative operator <math>\vartheta\!</math> takes a function on that space, an <math>f\!</math> of type <math>(\mathbb{K}^n \to \mathbb{K}),</math> and maps it to a ground field value of type <math>\mathbb{K}.</math> This value is known as the ''derivative'' of <math>f\!</math> in the direction <math>\vartheta\!</math> [Che46, 76&ndash;77]. In the boolean case <math>\vartheta : (\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math> has the form of a proposition about propositions, in other words, a proposition of the next higher type.<br />
<br />
Next, by way of illustrating the propositions as types idea, consider a proposition of the form <math>X \Rightarrow (Y \Rightarrow Z).</math> One knows from propositional calculus that this is logically equivalent to a proposition of the form <math>(X \land Y) \Rightarrow Z.</math> But this equivalence should remind us of the functional isomorphism that exists between a construction of the type <math>X \to (Y \to Z)</math> and a construction of the type <math>(X \times Y) \to Z.</math> The propositions as types analogy permits us to take a functional type like this and, under the right conditions, replace the functional arrows &ldquo;<math>\to~\!</math>&rdquo; and products &ldquo;<math>\times\!</math>&rdquo; with the respective logical arrows &ldquo;<math>\Rightarrow\!</math>&rdquo; and products &ldquo;<math>\land\!</math>&rdquo;. Accordingly, viewing the result as a proposition, we can employ axioms and theorems of propositional calculus to suggest appropriate isomorphisms among the categorical and functional constructions.<br />
<br />
Finally, examine the middle four rows of Table&nbsp;3. These display a series of isomorphic types that stretch from the categories that are labeled ''Vector Field'' to those that are labeled ''Derivation''. A ''vector field'', also known as an ''infinitesimal transformation'', associates a tangent vector at a point with each point of a space. In symbols, a vector field is a function of the form <math>\textstyle \xi : X \to \bigcup_{x \in X} \xi_x\!</math> that assigns to each point <math>x\!</math> of the space <math>X\!</math> a tangent vector to <math>X\!</math> at that point, namely, the tangent vector <math>\xi_x\!</math> [Che46, 82&ndash;83]. If <math>X\!</math> is of the type <math>\mathbb{K}^n,</math> then <math>\xi\!</math> is of the type <math>\mathbb{K}^n \to ((\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K}).</math> This has the pattern <math>X \to (Y \to Z),</math> with <math>X = \mathbb{K}^n,</math> <math>Y = (\mathbb{K}^n \to \mathbb{K}),</math> and <math>Z = \mathbb{K}.</math><br />
<br />
Applying the propositions as types analogy, one can follow this pattern through a series of metamorphoses from the type of a vector field to the type of a derivation, as traced out in Table&nbsp;4. Observe how the function <math>f : X \to \mathbb{K},</math> associated with the place of <math>Y\!</math> in the pattern, moves through its paces from the second to the first position. In this way, the vector field <math>\xi,\!</math> initially viewed as attaching each tangent vector <math>\xi_x\!</math> to the site <math>x\!</math> where it acts in <math>X,\!</math> now comes to be seen as acting on each scalar potential <math>f : X \to \mathbb{K}</math> like a generalized species of differentiation, producing another function <math>\xi f : X \to \mathbb{K}</math> of the same type.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 4.} ~~ \text{An Equivalence Based on the Propositions as Types Analogy}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Pattern}\!</math><br />
| <math>\text{Construct}\!</math><br />
| <math>\text{Instance}\!</math><br />
|-<br />
| <math>X \to (Y \to Z)\!</math><br />
| <math>\text{Vector Field}\!</math><br />
| <math>\mathbb{K}^n \to ((\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K})\!</math><br />
|-<br />
| <math>(X \times Y) \to Z\!</math><br />
| <math>\Uparrow\!</math><br />
| <math>(\mathbb{K}^n \times (\mathbb{K}^n \to \mathbb{K})) \to \mathbb{K}\!</math><br />
|-<br />
| <math>(Y \times X) \to Z\!</math><br />
| <math>\Downarrow\!</math><br />
| <math>((\mathbb{K}^n \to \mathbb{K}) \times \mathbb{K}^n) \to \mathbb{K}\!</math><br />
|-<br />
| <math>Y \to (X \to Z)\!</math><br />
| <math>\text{Derivation}\!</math><br />
| <math>(\mathbb{K}^n \to \mathbb{K}) \to (\mathbb{K}^n \to \mathbb{K})\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Reality at the Threshold of Logic===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
But no science can rest entirely on measurement, and many scientific investigations are quite out of reach of that device. To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]<br />
|}<br />
<br />
Table 5 accumulates an array of notation that I hope will not be too distracting. Some of it is rarely needed, but has been filled in for the sake of completeness. Its purpose is simple, to give literal expression to the visual intuitions that come with venn diagrams, and to help build a bridge between our qualitative and quantitative outlooks on dynamic systems.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 5.} ~~ \text{A Bridge Over Troubled Waters}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| align="center" | <math>\text{Linear Space}\!</math><br />
| align="center" | <math>\text{Liminal Space}\!</math><br />
| align="center" | <math>\text{Logical Space}\!</math><br />
|-<br />
| <math>\begin{matrix}\mathcal{X} & = & \{ x_1, \ldots, x_n \}\end{matrix}</math><br />
| <math>\begin{matrix}\underline{\mathcal{X}} & = & \{ \underline{x}_1, \ldots, \underline{x}_n \}\end{matrix}</math><br />
| <math>\begin{matrix}\mathcal{A} & = & \{ a_1, \ldots, a_n \}\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X_i & = & \langle x_i \rangle<br />
\\<br />
& \cong & \mathbb{K}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}_i & = & \{ \texttt{(} \underline{x}_i \texttt{)}, \underline{x}_i \}<br />
\\<br />
& \cong & \mathbb{B}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A_i & = & \{ \texttt{(} a_i \texttt{)}, a_i \}<br />
\\<br />
& \cong & \mathbb{B}<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X<br />
\\<br />
= & \langle \mathcal{X} \rangle<br />
\\<br />
= & \langle x_1, \ldots, x_n \rangle<br />
\\<br />
= & X_1 \times \ldots \times X_n<br />
\\<br />
= & \prod_{i=1}^n X_i<br />
\\<br />
\cong & \mathbb{K}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}<br />
\\<br />
= & \langle \underline{\mathcal{X}} \rangle<br />
\\<br />
= & \langle \underline{x}_1, \ldots, \underline{x}_n \rangle<br />
\\<br />
= & \underline{X}_1 \times \ldots \times \underline{X}_n<br />
\\<br />
= & \prod_{i=1}^n \underline{X}_i<br />
\\<br />
\cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A<br />
\\<br />
= & \langle \mathcal{A} \rangle<br />
\\<br />
= & \langle a_1, \ldots, a_n \rangle<br />
\\<br />
= & A_1 \times \ldots \times A_n<br />
\\<br />
= & \prod_{i=1}^n A_i<br />
\\<br />
\cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X^* & = & (\ell : X \to \mathbb{K})<br />
\\<br />
& \cong & \mathbb{K}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}^* & = & (\ell : \underline{X} \to \mathbb{B})<br />
\\<br />
& \cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A^* & = & (\ell : A \to \mathbb{B})<br />
\\<br />
& \cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X^\uparrow & = & (X \to \mathbb{K})<br />
\\<br />
& \cong & (\mathbb{K}^n \to \mathbb{K})<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}^\uparrow & = & (\underline{X} \to \mathbb{B})<br />
\\<br />
& \cong & (\mathbb{B}^n \to \mathbb{B})<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A^\uparrow & = & (A \to \mathbb{B})<br />
\\<br />
& \cong & (\mathbb{B}^n \to \mathbb{B})<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X^\bullet<br />
\\<br />
= & [\mathcal{X}]<br />
\\<br />
= & [x_1, \ldots, x_n]<br />
\\<br />
= & (X, X^\uparrow)<br />
\\<br />
= & (X ~+\!\to \mathbb{K})<br />
\\<br />
= & (X, (X \to \mathbb{K}))<br />
\\<br />
\cong & (\mathbb{K}^n, (\mathbb{K}^n \to \mathbb{K}))<br />
\\<br />
= & (\mathbb{K}^n ~+\!\to \mathbb{K})<br />
\\<br />
= & [\mathbb{K}^n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}^\bullet<br />
\\<br />
= & [\underline{\mathcal{X}}]<br />
\\<br />
= & [\underline{x}_1, \ldots, \underline{x}_n]<br />
\\<br />
= & (\underline{X}, \underline{X}^\uparrow)<br />
\\<br />
= & (\underline{X} ~+\!\to \mathbb{B})<br />
\\<br />
= & (\underline{X}, (\underline{X} \to \mathbb{B}))<br />
\\<br />
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))<br />
\\<br />
= & (\mathbb{B}^n ~+\!\to \mathbb{B})<br />
\\<br />
= & [\mathbb{B}^n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A^\bullet<br />
\\<br />
= & [\mathcal{A}]<br />
\\<br />
= & [a_1, \ldots, a_n]<br />
\\<br />
= & (A, A^\uparrow)<br />
\\<br />
= & (A ~+\!\to \mathbb{B})<br />
\\<br />
= & (A, (A \to \mathbb{B}))<br />
\\<br />
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))<br />
\\<br />
= & (\mathbb{B}^n ~+\!\to \mathbb{B})<br />
\\<br />
= & [\mathbb{B}^n]<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
The left side of the Table collects mostly standard notation for an <math>n\!</math>-dimensional vector space over a field <math>\mathbb{K}.</math> The right side of the table repeats the first elements of a notation that I sketched above, to be used in further developments of propositional calculus. (I plan to use this notation in the logical analysis of neural network systems.) The middle column of the table is designed as a transitional step from the case of an arbitrary field <math>\mathbb{K},</math> with a special interest in the continuous line <math>\mathbb{R},</math> to the qualitative and discrete situations that are instanced and typified by <math>\mathbb{B}.</math><br />
<br />
I now proceed to explain these concepts in more detail. The most important ideas developed in Table&nbsp;5 are these:<br />
<br />
* The idea of a universe of discourse, which includes both a space of ''points'' and a space of ''maps'' on those points.<br />
<br />
* The idea of passing from a more complex universe to a simpler universe by a process of ''thresholding'' each dimension of variation down to a single bit of information.<br />
<br />
For the sake of concreteness, let us suppose that we start with a continuous <math>n\!</math>-dimensional vector space like <math>X = \langle x_1, \ldots, x_n \rangle \cong \mathbb{R}^n.</math> The coordinate system <math>\mathcal{X} = \{x_i\}</math> is a set of maps <math>x_i : \mathbb{R}^n \to \mathbb{R},</math> also known as the ''coordinate projections''. Given a "dataset" of points <math>\mathbf{x}</math> in <math>\mathbb{R}^n,</math> there are numerous ways of sensibly reducing the data down to one bit for each dimension. One strategy that is general enough for our present purposes is as follows. For each <math>i\!</math> we choose an <math>n\!</math>-ary relation <math>L_i\!</math> on <math>\mathbb{R}^n,</math> that is, a subset of <math>\mathbb{R}^n,</math> and then we define the <math>i^\mathrm{th}\!</math> threshold map, or ''limen'' <math>\underline{x}_i</math> as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\underline{x}_i : \mathbb{R}^n \to \mathbb{B}\ \text{such that:}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
\underline{x}_i(\mathbf{x}) = 1 & \text{if} & \mathbf{x} \in L_i,<br />
\\[4pt]<br />
\underline{x}_i(\mathbf{x}) = 0 & \text{if} & \mathbf{x} \not\in L_i.<br />
\end{matrix}</math><br />
|}<br />
<br />
In other notations that are sometimes used, the operator <math>\chi (\ldots)</math> or the corner brackets <math>\lceil\ldots\rceil</math> can be used to denote a ''characteristic function'', that is, a mapping from statements to their truth values in <math>\mathbb{B}.</math> Finally, it is not uncommon to use the name of the relation itself as a predicate that maps <math>n\!</math>-tuples into truth values. Thus we have the following notational variants of the above definition:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
\underline{x}_i (\mathbf{x}) & = & \chi (\mathbf{x} \in L_i) & = & \lceil \mathbf{x} \in L_i \rceil & = & L_i (\mathbf{x}).<br />
\end{matrix}</math><br />
|}<br />
<br />
Notice that, as defined here, there need be no actual relation between the <math>n\!</math>-dimensional subsets <math>\{L_i\}\!</math> and the coordinate axes corresponding to <math>\{x_i\},\!</math> aside from the circumstance that the two sets have the same cardinality. In concrete cases, though, one usually has some reason for associating these "volumes" with these "lines", for instance, <math>L_i\!</math> is bounded by some hyperplane that intersects the <math>i^\text{th}\!</math> axis at a unique threshold value <math>r_i \in \mathbb{R}.\!</math> Often, the hyperplane is chosen normal to the axis. In recognition of this motive, let us make the following convention. When the set <math>L_i\!</math> has points on the <math>i^\text{th}\!</math> axis, that is, points of the form <math>(0, \ldots, 0, r_i, 0, \ldots, 0)</math> where only the <math>x_i\!</math> coordinate is possibly non-zero, we may pick any one of these coordinate values as a parametric index of the relation. In this case we say that the indexing is ''real'', otherwise the indexing is ''imaginary''. For a knowledge based system <math>X,\!</math> this should serve once again to mark the distinction between ''acquaintance'' and ''opinion''.<br />
<br />
States of knowledge about the location of a system or about the distribution of a population of systems in a state space <math>X = \mathbb{R}^n</math> can now be expressed by taking the set <math>\underline{\mathcal{X}} = \{\underline{x}_i\}</math> as a basis of logical features. In picturesque terms, one may think of the underscore and the subscript as combining to form a subtextual spelling for the <math>i^\text{th}\!</math> threshold map. This can help to remind us that the ''threshold operator'' <math>(\underline{~})_i</math> acts on <math>\mathbf{x}</math> by setting up a kind of a &ldquo;hurdle&rdquo; for it. In this interpretation the coordinate proposition <math>\underline{x}_i</math> asserts that the representative point <math>\mathbf{x}</math> resides ''above'' the <math>i^\mathrm{th}\!</math> threshold.<br />
<br />
Primitive assertions of the form <math>\underline{x}_i (\mathbf{x})</math> may then be negated and joined by means of propositional connectives in the usual ways to provide information about the state <math>\mathbf{x}</math> of a contemplated system or a statistical ensemble of systems. Parentheses <math>\texttt{(} \ldots \texttt{)}\!</math> may be used to indicate logical negation. Eventually one discovers the usefulness of the <math>k\!</math>-ary ''just one false'' operators of the form <math>\texttt{(} a_1 \texttt{,} \ldots \texttt{,} a_k \texttt{)},\!</math> as treated in earlier reports. This much tackle generates a space of points (cells, interpretations), <math>\underline{X} \cong \mathbb{B}^n,</math> and a space of functions (regions, propositions), <math>\underline{X}^\uparrow \cong (\mathbb{B}^n \to \mathbb{B}).</math> Together these form a new universe of discourse <math>\underline{X}^\bullet</math> of the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})),</math> which we may abbreviate as <math>\mathbb{B}^n\ +\!\to \mathbb{B}</math> or most succinctly as <math>[\mathbb{B}^n].</math><br />
<br />
The square brackets have been chosen to recall the rectangular frame of a venn diagram. In thinking about a universe of discourse it is a good idea to keep this picture in mind, graphically illustrating the links among the elementary cells <math>\underline{\mathbf{x}},</math> the defining features <math>\underline{x}_i,</math> and the potential shadings <math>f : \underline{X} \to \mathbb{B}</math> all at the same time, not to mention the arbitrariness of the way we choose to inscribe our distinctions in the medium of a continuous space.<br />
<br />
Finally, let <math>X^*\!</math> denote the space of linear functions, <math>(\ell : X \to \mathbb{K}),</math> which has in the finite case the same dimensionality as <math>X,\!</math> and let the same notation be extended across the Table.<br />
<br />
We have just gone through a lot of work, apparently doing nothing more substantial than spinning a complex spell of notational devices through a labyrinth of baffled spaces and baffling maps. The reason for doing this was to bind together and to constitute the intuitive concept of a universe of discourse into a coherent categorical object, the kind of thing, once grasped, that can be turned over in the mind and considered in all its manifold changes and facets. The effort invested in these preliminary measures is intended to pay off later, when we need to consider the state transformations and the time evolution of neural network systems.<br />
<br />
===Tables of Propositional Forms===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope. It provides explicit techniques for manipulating the most basic ingredients of discourse.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7&ndash;8]<br />
|}<br />
<br />
To prepare for the next phase of discussion, Tables&nbsp;6 and 7 collect and summarize all of the propositional forms on one and two variables. These propositional forms are represented over bases of boolean variables as complete sets of boolean-valued functions. Adjacent to their names and specifications are listed what are roughly the simplest expressions in the ''[[cactus language]]'', the particular syntax for propositional calculus that I use in formal and computational contexts. For the sake of orientation, the English paraphrases and the more common notations are listed in the last two columns. As simple and circumscribed as these low-dimensional universes may appear to be, a careful exploration of their differential extensions will involve us in complexities sufficient to demand our attention for some time to come.<br />
<br />
Propositional forms on one variable correspond to boolean functions <math>f : \mathbb{B}^1 \to \mathbb{B}.</math> In Table&nbsp;6 these functions are listed in a variant form of [[truth table]], one in which the axes of the usual arrangement are rotated through a right angle. Each function <math>f_i\!</math> is indexed by the string of values that it takes on the points of the universe <math>X^\bullet = [x] \cong \mathbb{B}^1.</math> The binary index generated in this way is converted to its decimal equivalent and these are used as conventional names for the <math>f_i,\!</math> as shown in the first column of the Table. In their own right the <math>2^1\!</math> points of the universe <math>X^\bullet</math> are coordinated as a space of type <math>\mathbb{B}^1,</math> this in light of the universe <math>X^\bullet</math> being a functional domain where the coordinate projection <math>x\!</math> takes on its values in <math>\mathbb{B}.</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 6.} ~~ \text{Propositional Forms on One Variable}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_0\!</math><br />
| <math>f_{00}\!</math><br />
| <math>0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>f_1\!</math><br />
| <math>f_{01}\!</math><br />
| <math>0~1\!</math><br />
| <math>\texttt{(} x \texttt{)}\!</math><br />
| <math>\text{not}~ x\!</math><br />
| <math>\lnot x\!</math><br />
|-<br />
| <math>f_2\!</math><br />
| <math>f_{10}\!</math><br />
| <math>1~0\!</math><br />
| <math>x\!</math><br />
| <math>x\!</math><br />
| <math>x\!</math><br />
|-<br />
| <math>f_3\!</math><br />
| <math>f_{11}\!</math><br />
| <math>1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
Propositional forms on two variables correspond to boolean functions <math>f : \mathbb{B}^2 \to \mathbb{B}.</math> In Table&nbsp;7 each function <math>f_i\!</math> is indexed by the values that it takes on the points of the universe <math>X^\bullet = [x, y] \cong \mathbb{B}^2.</math> Converting the binary index thus generated to a decimal equivalent, we obtain the functional nicknames that are listed in the first column. The <math>2^2\!</math> points of the universe <math>X^\bullet</math> are coordinated as a space of type <math>\mathbb{B}^2,</math> as indicated under the heading of the Table, where the coordinate projections <math>x\!</math> and <math>y\!</math> run through the various combinations of their values in <math>\mathbb{B}.</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 7-a.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}\!</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}\!</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}\!</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}\!</math><br />
| style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}\!</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}\!</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[4pt]<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\\[4pt]<br />
f_{3}<br />
\\[4pt]<br />
f_{4}<br />
\\[4pt]<br />
f_{5}<br />
\\[4pt]<br />
f_{6}<br />
\\[4pt]<br />
f_{7}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0000}<br />
\\[4pt]<br />
f_{0001}<br />
\\[4pt]<br />
f_{0010}<br />
\\[4pt]<br />
f_{0011}<br />
\\[4pt]<br />
f_{0100}<br />
\\[4pt]<br />
f_{0101}<br />
\\[4pt]<br />
f_{0110}<br />
\\[4pt]<br />
f_{0111}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~0<br />
\\[4pt]<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~0~1~1<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
0~1~0~1<br />
\\[4pt]<br />
0~1~1~0<br />
\\[4pt]<br />
0~1~1~1<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{)} ~ y ~<br />
\\[4pt]<br />
\texttt{(} x \texttt{)}<br />
\\[4pt]<br />
~ x ~ \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{,} ~ y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x ~ y \texttt{)}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{false}<br />
\\[4pt]<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\[4pt]<br />
y ~\text{without}~ x<br />
\\[4pt]<br />
\text{not}~ x<br />
\\[4pt]<br />
x ~\text{without}~ y<br />
\\[4pt]<br />
\text{not}~ y<br />
\\[4pt]<br />
x ~\text{not equal to}~ y<br />
\\[4pt]<br />
\text{not both}~ x ~\text{and}~ y<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
\lnot x \land \lnot y<br />
\\[4pt]<br />
\lnot x \land y<br />
\\[4pt]<br />
\lnot x<br />
\\[4pt]<br />
x \land \lnot y<br />
\\[4pt]<br />
\lnot y<br />
\\[4pt]<br />
x \ne y<br />
\\[4pt]<br />
\lnot x \lor \lnot y<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{8}<br />
\\[4pt]<br />
f_{9}<br />
\\[4pt]<br />
f_{10}<br />
\\[4pt]<br />
f_{11}<br />
\\[4pt]<br />
f_{12}<br />
\\[4pt]<br />
f_{13}<br />
\\[4pt]<br />
f_{14}<br />
\\[4pt]<br />
f_{15}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1000}<br />
\\[4pt]<br />
f_{1001}<br />
\\[4pt]<br />
f_{1010}<br />
\\[4pt]<br />
f_{1011}<br />
\\[4pt]<br />
f_{1100}<br />
\\[4pt]<br />
f_{1101}<br />
\\[4pt]<br />
f_{1110}<br />
\\[4pt]<br />
f_{1111}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
1~0~0~0<br />
\\[4pt]<br />
1~0~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\\[4pt]<br />
1~1~1~1<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x ~ y<br />
\\[4pt]<br />
\texttt{((} x \texttt{,} ~ y \texttt{))}<br />
\\[4pt]<br />
y<br />
\\[4pt]<br />
\texttt{(} x ~ \texttt{(} y \texttt{))}<br />
\\[4pt]<br />
x<br />
\\[4pt]<br />
\texttt{((} x \texttt{)} ~ y \texttt{)}<br />
\\[4pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
\texttt{((~))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x ~\text{and}~ y<br />
\\[4pt]<br />
x ~\text{equal to}~ y<br />
\\[4pt]<br />
y<br />
\\[4pt]<br />
\text{not}~ x ~\text{without}~ y<br />
\\[4pt]<br />
x<br />
\\[4pt]<br />
\text{not}~ y ~\text{without}~ x<br />
\\[4pt]<br />
x ~\text{or}~ y<br />
\\[4pt]<br />
\text{true}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x \land y<br />
\\[4pt]<br />
x = y<br />
\\[4pt]<br />
y<br />
\\[4pt]<br />
x \Rightarrow y<br />
\\[4pt]<br />
x<br />
\\[4pt]<br />
x \Leftarrow y<br />
\\[4pt]<br />
x \lor y<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 7-b.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}</math><br />
| style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>f_{0000}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\\[4pt]<br />
f_{4}<br />
\\[4pt]<br />
f_{8}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0001}<br />
\\[4pt]<br />
f_{0010}<br />
\\[4pt]<br />
f_{0100}<br />
\\[4pt]<br />
f_{1000}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
1~0~0~0<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{)} ~ y ~<br />
\\[4pt]<br />
~ x ~ \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
~ x ~~ y ~<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\[4pt]<br />
y ~\text{without}~ x<br />
\\[4pt]<br />
x ~\text{without}~ y<br />
\\[4pt]<br />
x ~\text{and}~ y<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot x \land \lnot y<br />
\\[4pt]<br />
\lnot x \land y<br />
\\[4pt]<br />
x \land \lnot y<br />
\\[4pt]<br />
x \land y<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{3}<br />
\\[4pt]<br />
f_{12}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0011}<br />
\\[4pt]<br />
f_{1100}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\[4pt]<br />
x<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{not}~ x<br />
\\[4pt]<br />
x<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot x<br />
\\[4pt]<br />
x<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[4pt]<br />
f_{9}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0110}<br />
\\[4pt]<br />
f_{1001}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[4pt]<br />
1~0~0~1<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{,} y \texttt{)}<br />
\\[4pt]<br />
\texttt{((} x \texttt{,} y \texttt{))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x ~\text{not equal to}~ y<br />
\\[4pt]<br />
x ~\text{equal to}~ y<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x \ne y<br />
\\[4pt]<br />
x = y<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{5}<br />
\\[4pt]<br />
f_{10}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0101}<br />
\\[4pt]<br />
f_{1010}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\[4pt]<br />
y<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{not}~ y<br />
\\[4pt]<br />
y<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot y<br />
\\[4pt]<br />
y<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[4pt]<br />
f_{11}<br />
\\[4pt]<br />
f_{13}<br />
\\[4pt]<br />
f_{14}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0111}<br />
\\[4pt]<br />
f_{1011}<br />
\\[4pt]<br />
f_{1101}<br />
\\[4pt]<br />
f_{1110}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} ~ x ~~ y ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{(} ~ x ~ \texttt{(} y \texttt{))}<br />
\\[4pt]<br />
\texttt{((} x \texttt{)} ~ y ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{not both}~ x ~\text{and}~ y<br />
\\[4pt]<br />
\text{not}~ x ~\text{without}~ y<br />
\\[4pt]<br />
\text{not}~ y ~\text{without}~ x<br />
\\[4pt]<br />
x ~\text{or}~ y<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot x \lor \lnot y<br />
\\[4pt]<br />
x \Rightarrow y<br />
\\[4pt]<br />
x \Leftarrow y<br />
\\[4pt]<br />
x \lor y<br />
\end{matrix}\!</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>f_{1111}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
==A Differential Extension of Propositional Calculus==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Fire over water:<br><br />
The image of the condition before transition.<br><br />
Thus the superior man is careful<br><br />
In the differentiation of things,<br><br />
So that each finds its place.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; ''I Ching'', Hexagram 64, [Wil, 249]<br />
|}<br />
<br />
This much preparation is enough to begin introducing my subject, if I excuse myself from giving full arguments for my definitional choices until some later stage. I am trying to develop a ''differential theory of qualitative equations'' that parallels the application of differential geometry to dynamic systems. The idea of a tangent vector is key to this work and a major goal is to find the right logical analogues of tangent spaces, bundles, and functors. The strategy is taken of looking for the simplest versions of these constructions that can be discovered within the realm of propositional calculus, so long as they serve to fill out the general theme.<br />
<br />
===Differential Propositions : Qualitative Analogues of Differential Equations===<br />
<br />
In order to define the differential extension of a universe of discourse <math>[\mathcal{A}],</math> the initial alphabet <math>\mathcal{A}</math> must be extended to include a collection of symbols for ''differential features'', or basic ''changes'' that are capable of occurring in <math>[\mathcal{A}].</math> Intuitively, these symbols may be construed as denoting primitive features of change, qualitative attributes of motion, or propositions about how things or points in the universe may be changing or moving with respect to the features that are noted in the initial alphabet.<br />
<br />
Therefore, let us define the corresponding ''differential alphabet'' or ''tangent alphabet'' as <math>\mathrm{d}\mathcal{A}\!</math> <math>=\!</math> <math>\{\mathrm{d}a_1, \ldots, \mathrm{d}a_n\},</math> in principle just an arbitrary alphabet of symbols, disjoint from the initial alphabet <math>\mathcal{A}\!</math> <math>=\!</math> <math>\{ a_1, \ldots, a_n \},\!</math> that is intended to be interpreted in the way just indicated. It only remains to be understood that the precise interpretation of the symbols in <math>\mathrm{d}\mathcal{A}\!</math> is often conceived to be changeable from point to point of the underlying space <math>A.\!</math> Indeed, for all we know, the state space <math>A\!</math> might well be the state space of a language interpreter, one that is concerned, among other things, with the idiomatic meanings of the dialect generated by <math>\mathcal{A}</math> and <math>\mathrm{d}\mathcal{A}.\!</math><br />
<br />
The ''tangent space'' to <math>A\!</math> at one of its points <math>x,\!</math> sometimes written <math>\mathrm{T}_x(A),</math> takes the form <math>\mathrm{d}A</math> <math>=\!</math> <math>\langle \mathrm{d}\mathcal{A} \rangle</math> <math>=\!</math> <math>\langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle.\!</math> Strictly speaking, the name ''cotangent space'' is probably more correct for this construction, but the fact that we take up spaces and their duals in pairs to form our universes of discourse allows our language to be pliable here.<br />
<br />
Proceeding as we did with the base space <math>A,\!</math> the tangent space <math>\mathrm{d}A</math> at a point of <math>A\!</math> can be analyzed as a product of distinct and independent factors:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\mathrm{d}A ~=~ \prod_{i=1}^n \mathrm{d}A_i ~=~ \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n.\!</math><br />
|}<br />
<br />
Here, <math>\mathrm{d}A_i\!</math> is a set of two differential propositions, <math>\mathrm{d}A_i = \{ (\mathrm{d}a_i), \mathrm{d}a_i \},\!</math> where <math>\texttt{(} \mathrm{d}a_i \texttt{)}\!</math> is a proposition with the logical value of <math>\text{not} ~ \mathrm{d}a_i.\!</math> Each component <math>\mathrm{d}A_i\!</math> has the type <math>\mathbb{B},\!</math> operating under the ordered correspondence <math>\{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \} \cong \{ 0, 1 \}.\!</math> However, clarity is often served by acknowledging this differential usage with a superficially distinct type <math>\mathbb{D},\!</math> whose intension may be indicated as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\mathbb{D} = \{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \} = \{ \text{same}, \text{different} \} = \{ \text{stay}, \text{change} \} = \{ \text{stop}, \text{step} \}.\!</math><br />
|}<br />
<br />
Viewed within a coordinate representation, spaces of type <math>\mathbb{B}^n\!</math> and <math>\mathbb{D}^n\!</math> may appear to be identical sets of binary vectors, but taking a view at this level of abstraction would be like ignoring the qualitative units and the diverse dimensions that distinguish position and momentum, or the different roles of quantity and impulse.<br />
<br />
===An Interlude on the Path===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
There would have been no beginnings: instead, speech would proceed from me, while I stood in its path &ndash; a slender gap &ndash; the point of its possible disappearance.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
A sense of the relation between <math>\mathbb{B}</math> and <math>\mathbb{D}</math> may be obtained by considering the ''path classifier'' (or the ''equivalence class of curves'') approach to tangent vectors. Consider a universe <math>[\mathcal{X}].\!</math> Given the boolean value system, a path in the space <math>X = \langle \mathcal{X} \rangle</math> is a map <math>q : \mathbb{B} \to X.</math> In this context the set of paths <math>(\mathbb{B} \to X)</math> is isomorphic to the cartesian square <math>X^2 = X \times X,</math> or the set of ordered pairs chosen from <math>X.\!</math><br />
<br />
We may analyze <math>X^2 = \{ (u, v) : u, v \in X \}</math> into two parts, specifically, the ordered pairs <math>(u, v)\!</math> that lie on and off the diagonal:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}X^2 & = & \{ (u, v) : u = v \} & \cup & \{ (u, v) : u \ne v \}.\end{matrix}</math><br />
|}<br />
<br />
This partition may also be expressed in the following symbolic form:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}X^2 & \cong & \operatorname{diag} (X) & + & 2 \binom{X}{2}.\end{matrix}</math><br />
|}<br />
<br />
The separate terms of this formula are defined as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}\operatorname{diag} (X) & = & \{ (x, x) : x \in X \}.\end{matrix}\!</math><br />
|}<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}\binom{X}{k} & = & X ~\text{choose}~ k & = & \{ k\text{-sets from}~ X \}.\end{matrix}\!</math><br />
|}<br />
<br />
Thus we have:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}\binom{X}{2} & = & \{ \{ u, v \} : u, v \in X \}.\end{matrix}</math><br />
|}<br />
<br />
We may now use the features in <math>\mathrm{d}\mathcal{X} = \{ \mathrm{d}x_i \} = \{ \mathrm{d}x_1, \ldots, \mathrm{d}x_n \}</math> to classify the paths of <math>(\mathbb{B} \to X)</math> by way of the pairs in <math>X^2.\!</math> If <math>X \cong \mathbb{B}^n,</math> then a path <math>q\!</math> in <math>X\!</math> has the following form:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
q : (\mathbb{B} \to \mathbb{B}^n) & \cong & \mathbb{B}^n \times \mathbb{B}^n & \cong & \mathbb{B}^{2n} & \cong & (\mathbb{B}^2)^n.<br />
\end{matrix}</math><br />
|}<br />
<br />
Intuitively, we want to map this <math>(\mathbb{B}^2)^n</math> onto <math>\mathbb{D}^n</math> by mapping each component <math>\mathbb{B}^2</math> onto a copy of <math>\mathbb{D}.</math> But in the presenting context <math>{}^{\backprime\backprime} \mathbb{D} {}^{\prime\prime}</math> is just a name associated with, or an incidental quality attributed to, coefficient values in <math>\mathbb{B}</math> when they are attached to features in <math>\mathrm{d}\mathcal{X}.</math><br />
<br />
Taking these intentions into account, define <math>\mathrm{d}x_i : X^2 \to \mathbb{B}</math> in the following manner:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcrcl}<br />
\mathrm{d}x_i(u, v)<br />
& = & \texttt{(} ~ x_i(u) & \texttt{,} & x_i(v) ~ \texttt{)}<br />
\\<br />
& = & x_i(u) & + & x_i(v)<br />
\\<br />
& = & x_i(v) & - & x_i(u).<br />
\end{array}</math><br />
|}<br />
<br />
In the above transcription, the operator bracket of the form <math>\texttt{(} \ldots \texttt{,} \ldots \texttt{)}\!</math> is a ''cactus lobe'', in general signifying that just one of the arguments listed is false. In the case of two arguments this is the same thing as saying that the arguments are not equal. The plus sign signifies boolean addition, in the sense of addition in <math>\mathrm{GF}(2),</math> and thus means the same thing in this context as the minus sign, in the sense of adding the additive inverse.<br />
<br />
The above definition of <math>\mathrm{d}x_i : X^2 \to \mathbb{B}</math> is equivalent to defining <math>\mathrm{d}x_i : (\mathbb{B} \to X) \to \mathbb{B}\!</math> in the following way:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcrcl}<br />
\mathrm{d}x_i (q)<br />
& = & \texttt{(} ~ x_i(q_0) & \texttt{,} & x_i(q_1) ~ \texttt{)}<br />
\\<br />
& = & x_i(q_0) & + & x_i(q_1)<br />
\\<br />
& = & x_i(q_1) & - & x_i(q_0).<br />
\end{array}</math><br />
|}<br />
<br />
In this definition <math>q_b = q(b),\!</math> for each <math>b\!</math> in <math>\mathbb{B}.</math> Thus, the proposition <math>\mathrm{d}x_i</math> is true of the path <math>q = (u, v)\!</math> exactly if the terms of <math>q,\!</math> the endpoints <math>u\!</math> and <math>v,\!</math> lie on different sides of the question <math>x_i.\!</math><br />
<br />
The language of features in <math>\langle \mathrm{d}\mathcal{X} \rangle,</math> indeed the whole calculus of propositions in <math>[\mathrm{d}\mathcal{X}],</math> may now be used to classify paths and sets of paths. In other words, the paths can be taken as models of the propositions <math>g : \mathrm{d}X \to \mathbb{B}.</math> For example, the paths corresponding to <math>\mathrm{diag}(X)</math> fall under the description <math>\texttt{(} \mathrm{d}x_1 \texttt{)} \cdots \texttt{(} \mathrm{d}x_n \texttt{)},\!</math> which says that nothing changes against the backdrop of the coordinate frame <math>\{ x_1, \ldots, x_n \}.\!</math><br />
<br />
Finally, a few words of explanation may be in order. If this concept of a path appears to be described in a roundabout fashion, it is because I am trying to avoid using any assumption of vector space properties for the space <math>X\!</math> that contains its range. In many ways the treatment is still unsatisfactory, but improvements will have to wait for the introduction of substitution operators acting on singular propositions.<br />
<br />
===The Extended Universe of Discourse===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
At the moment of speaking, I would like to have perceived a nameless voice, long preceding me, leaving me merely to enmesh myself in it, taking up its cadence, and to lodge myself, when no one was looking, in its interstices as if it had paused an instant, in suspense, to beckon to me.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
Next we define the ''extended alphabet'' or ''bundled alphabet'' <math>\mathrm{E}\mathcal{A}</math> as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lclcl}<br />
\mathrm{E}\mathcal{A}<br />
& = & \mathcal{A} \cup \mathrm{d}\mathcal{A}<br />
& = & \{ a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}.<br />
\end{array}</math><br />
|}<br />
<br />
This supplies enough material to construct the ''differential extension'' <math>\mathrm{E}A,</math> or the ''tangent bundle'' over the initial space <math>A,\!</math> in the following fashion:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcl}<br />
\mathrm{E}A<br />
& = & \langle \mathrm{E}\mathcal{A} \rangle<br />
\\[4pt]<br />
& = & \langle \mathcal{A} \cup \mathrm{d}\mathcal{A} \rangle<br />
\\[4pt]<br />
& = & \langle a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle,<br />
\end{array}</math><br />
|}<br />
<br />
and also:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcl}<br />
\mathrm{E}A<br />
& = & A \times \mathrm{d}A<br />
\\[4pt]<br />
& = & A_1 \times \ldots \times A_n \times \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n.<br />
\end{array}</math><br />
|}<br />
<br />
This gives <math>\mathrm{E}A</math> the type <math>\mathbb{B}^n \times \mathbb{D}^n.</math><br />
<br />
Finally, the tangent universe <math>\mathrm{E}A^\bullet = [\mathrm{E}\mathcal{A}]\!</math> is constituted from the totality of points and maps, or interpretations and propositions, that are based on the extended set of features <math>\mathrm{E}\mathcal{A},</math> and this fact is summed up in the following notation:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lclcl}<br />
\mathrm{E}A^\bullet<br />
& = & [\mathrm{E}\mathcal{A}]<br />
& = & [a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n].<br />
\end{array}</math><br />
|}<br />
<br />
This gives the tangent universe <math>\mathrm{E}A^\bullet\!</math> the type:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcl}<br />
(\mathbb{B}^n \times \mathbb{D}^n\ +\!\to \mathbb{B})<br />
& = & (\mathbb{B}^n \times \mathbb{D}^n, (\mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B})).<br />
\end{array}</math><br />
|}<br />
<br />
A proposition in the tangent universe <math>[\mathrm{E}\mathcal{A}]</math> is called a ''differential proposition'' and forms the analogue of a system of differential equations, constraints, or relations in ordinary calculus.<br />
<br />
With these constructions, the differential extension <math>\mathrm{E}A</math> and the space of differential propositions <math>(\mathrm{E}A \to \mathbb{B}),\!</math> we have arrived, in main outline, at one of the major subgoals of this study. Table&nbsp;8 summarizes the concepts that have been introduced for working with differentially extended universes of discourse.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 8.} ~~ \text{Differential Extension : Basic Notation}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Symbol}\!</math><br />
| <math>\text{Notation}\!</math><br />
| <math>\text{Description}\!</math><br />
| <math>\text{Type}\!</math><br />
|-<br />
| <math>\mathrm{d}\mathfrak{A}\!</math><br />
| <math>\{ {}^{\backprime\backprime} \mathrm{d}a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} \mathrm{d}a_n {}^{\prime\prime} \}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Alphabet of}<br />
\\[2pt]<br />
\text{differential symbols}<br />
\end{matrix}</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>\mathrm{d}\mathcal{A}\!</math><br />
| <math>\{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Basis of}<br />
\\[2pt]<br />
\text{differential features}<br />
\end{matrix}</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>\mathrm{d}A_i\!</math><br />
| <math>\{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \}\!</math><br />
| <math>\text{Differential dimension}~ i\!</math><br />
| <math>\mathbb{D}\!</math><br />
|-<br />
| <math>\mathrm{d}A\!</math><br />
|<br />
<math>\begin{matrix}<br />
\langle \mathrm{d}\mathcal{A} \rangle<br />
\\[2pt]<br />
\langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle<br />
\\[2pt]<br />
\{ (\mathrm{d}a_1, \ldots, \mathrm{d}a_n) \}<br />
\\[2pt]<br />
\mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n<br />
\\[2pt]<br />
\textstyle \prod_i \mathrm{d}A_i<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Tangent space at a point:}<br />
\\[2pt]<br />
\text{Set of changes, motions,}<br />
\\[2pt]<br />
\text{steps, tangent vectors}<br />
\\[2pt]<br />
\text{at a point}<br />
\end{matrix}</math><br />
| <math>\mathbb{D}^n\!</math><br />
|-<br />
| <math>\mathrm{d}A^*\!</math><br />
| <math>(\mathrm{hom} : \mathrm{d}A \to \mathbb{B})\!</math><br />
| <math>\text{Linear functions on}~ \mathrm{d}A\!</math><br />
| <math>(\mathbb{D}^n)^* \cong \mathbb{D}^n\!</math><br />
|-<br />
| <math>\mathrm{d}A^\uparrow\!</math><br />
| <math>(\mathrm{d}A \to \mathbb{B})\!</math><br />
| <math>\text{Boolean functions on}~ \mathrm{d}A\!</math><br />
| <math>\mathbb{D}^n \to \mathbb{B}\!</math><br />
|-<br />
| <math>\mathrm{d}A^\bullet\!</math><br />
|<br />
<math>\begin{matrix}<br />
[\mathrm{d}\mathcal{A}]<br />
\\[2pt]<br />
(\mathrm{d}A, \mathrm{d}A^\uparrow)<br />
\\[2pt]<br />
(\mathrm{d}A ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
(\mathrm{d}A, (\mathrm{d}A \to \mathbb{B}))<br />
\\[2pt]<br />
[\mathrm{d}a_1, \ldots, \mathrm{d}a_n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Tangent universe at a point of}~ A^\bullet,<br />
\\[2pt]<br />
\text{based on the tangent features}<br />
\\[2pt]<br />
\{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
(\mathbb{D}^n, (\mathbb{D}^n \to \mathbb{B}))<br />
\\[2pt]<br />
(\mathbb{D}^n ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
[\mathbb{D}^n]<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
The adjectives ''differential'' or ''tangent'' are systematically attached to every construct based on the differential alphabet <math>\mathrm{d}\mathfrak{A},</math> taken by itself. Strictly speaking, we probably ought to call <math>\mathrm{d}\mathcal{A}</math> the set of ''cotangent'' features derived from <math>\mathcal{A},</math> but the only time this distinction really seems to matter is when we need to distinguish the tangent vectors as maps of type <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math> from cotangent vectors as elements of type <math>\mathbb{D}^n.</math> In like fashion, having defined <math>\mathrm{E}\mathcal{A} = \mathcal{A} \cup \mathrm{d}\mathcal{A},</math> we can systematically attach the adjective ''extended'' or the substantive ''bundle'' to all of the constructs associated with this full complement of <math>{2n}\!</math> features.<br />
<br />
It eventually becomes necessary to extend the initial alphabet even further, to allow for the discussion of higher order differential expressions. Table&nbsp;9 provides a suggestion of how these further extensions can be carried out.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 9.} ~~ \text{Higher Order Differential Features}\!</math><br />
|<br />
<p><math>\begin{array}{lllll}<br />
\mathrm{d}^0 \mathcal{A} & = & \{ a_1, \ldots, a_n \} & = & \mathcal{A}<br />
\\<br />
\mathrm{d}^1 \mathcal{A} & = & \{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \} & = & \mathrm{d}\mathcal{A}<br />
\end{array}</math></p><br />
<br />
<p><math>\begin{array}{lll}<br />
\mathrm{d}^k \mathcal{A} & = & \{ \mathrm{d}^k a_1, \ldots, \mathrm{d}^k a_n \}<br />
\\<br />
\mathrm{d}^* \mathcal{A} & = & \{ \mathrm{d}^0 \mathcal{A}, \ldots, \mathrm{d}^k \mathcal{A}, \ldots \}<br />
\end{array}</math></p><br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}^0 \mathcal{A} & = & \mathrm{d}^0 \mathcal{A}<br />
\\<br />
\mathrm{E}^1 \mathcal{A} & = & \mathrm{d}^0 \mathcal{A} ~\cup~ \mathrm{d}^1 \mathcal{A}<br />
\\<br />
\mathrm{E}^k \mathcal{A} & = & \mathrm{d}^0 \mathcal{A} ~\cup~ \ldots ~\cup~ \mathrm{d}^k \mathcal{A}<br />
\\<br />
\mathrm{E}^\infty \mathcal{A} & = & \bigcup~ \mathrm{d}^* \mathcal{A}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
===Intentional Propositions===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Do you guess I have some intricate purpose?<br><br />
Well I have . . . . for the April rain has, and the mica on<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the side of a rock has.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 45]<br />
|}<br />
<br />
In order to analyze the behavior of a system at successive moments in time, while staying within the limitations of propositional logic, it is necessary to create independent alphabets of logical features for each moment of time that we contemplate using in our discussion. These moments have reference to typical instances and relative intervals, not actual or absolute times. For example, to discuss ''velocities'' (first order rates of change) we need to consider points of time in pairs. There are a number of natural ways of doing this. Given an initial alphabet, we could use its symbols as a lexical basis to generate successive alphabets of compound symbols, say, with temporal markers appended as suffixes.<br />
<br />
As a standard way of dealing with these situations, the following scheme of notation suggests a way of extending any alphabet of logical features through as many temporal moments as a particular order of analysis may demand. The lexical operators <math>\mathrm{p}^k</math> and <math>\mathrm{Q}^k</math> are convenient in many contexts where the accumulation of prime symbols and union symbols would otherwise be cumbersome.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 10.} ~~ \text{A Realm of Intentional Features}\!</math><br />
|<br />
<p><math>\begin{array}{lllll}<br />
\mathrm{p}^0 \mathcal{A} & = & \{ a_1, \ldots, a_n \} & = & \mathcal{A}<br />
\\<br />
\mathrm{p}^1 \mathcal{A} & = & \{ a_1^\prime, \ldots, a_n^\prime \} & = & \mathcal{A}^\prime<br />
\\<br />
\mathrm{p}^2 \mathcal{A} & = & \{ a_1^{\prime\prime}, \ldots, a_n^{\prime\prime} \} & = & \mathcal{A}^{\prime\prime}<br />
\\<br />
\cdots & & \cdots &<br />
\end{array}</math></p><br />
<br />
<p><math>\begin{array}{lll}<br />
\mathrm{p}^k \mathcal{A} & = & \{ \mathrm{p}^k a_1, \ldots, \mathrm{p}^k a_n \}<br />
\end{array}</math></p><br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{Q}^0 \mathcal{A} & = & \mathcal{A}<br />
\\<br />
\mathrm{Q}^1 \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}'<br />
\\<br />
\mathrm{Q}^2 \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}' \cup \mathcal{A}''<br />
\\<br />
\cdots & & \cdots<br />
\\<br />
\mathrm{Q}^k \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}' \cup \ldots \cup \mathrm{p}^k \mathcal{A}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
The resulting augmentations of our logical basis determine a series of discursive universes that may be called the ''intentional extension'' of propositional calculus. This extension follows a pattern analogous to the differential extension, which was developed in terms of the operators <math>\mathrm{d}^k</math> and <math>\mathrm{E}^k,</math> and there is a natural relation between these two extensions that bears further examination. In contexts displaying this pattern, where a sequence of domains stretches from an anchoring domain <math>X\!</math> through an indefinite number of higher reaches, a particular collection of domains based on <math>X\!</math> will be referred to as a ''realm'' of <math>X,\!</math> and when the succession exhibits a temporal aspect, as a ''reign'' of <math>X.\!</math><br />
<br />
For the purposes of this discussion, an ''intentional proposition'' is defined as a proposition in the universe of discourse <math>\mathrm{Q}X^\bullet = [\mathrm{Q}\mathcal{X}],</math> in other words, a map <math>q : \mathrm{Q}X \to \mathbb{B}.</math> The sense of this definition may be seen if we consider the following facts. First, the equivalence <math>\mathrm{Q}X = X \times X'</math> motivates the following chain of isomorphisms between spaces:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lllcl}<br />
(\mathrm{Q}X \to \mathbb{B})<br />
& \cong & (X & \times & ~X' \to \mathbb{B})<br />
\\[4pt]<br />
& \cong & (X & \to & (X' \to \mathbb{B}))<br />
\\[4pt]<br />
& \cong & (X' & \to & (X~ \to \mathbb{B})).<br />
\end{array}</math><br />
|}<br />
<br />
Viewed in this light, an intentional proposition <math>q\!</math> may be rephrased as a map <math>q : X \times X' \to \mathbb{B},</math> which judges the juxtaposition of states in <math>X\!</math> from one moment to the next. Alternatively, <math>q\!</math> may be parsed in two stages in two different ways, as <math>q : X \to (X' \to \mathbb{B})</math> and as <math>q : X' \to (X \to \mathbb{B}),</math> which associate to each point of <math>X\!</math> or <math>X'\!</math> a proposition about states in <math>X'\!</math> or <math>X,\!</math> respectively. In this way, an intentional proposition embodies a type of value system, in effect, a proposal that places a value on a collection of ends-in-view, or a project that evaluates a set of goals as regarded from each point of view in the state space of a system.<br />
<br />
In sum, the intentional proposition <math>q\!</math> indicates a method for the systematic selection of local goals. As a general form of description, a map of the type <math>q : \mathrm{Q}^i X \to \mathbb{B}\!</math> may be referred to as an "<math>i^\text{th}</math> order intentional proposition". Naturally, when we speak of intentional propositions without qualification, we usually mean first order intentions.<br />
<br />
Many different realms of discourse have the same structure as the extensions that have been indicated here. From a strictly logical point of view, each new layer of terms is composed of independent logical variables that are no different in kind from those that go before, and each further course of logical atoms is treated like so many individual, but otherwise indifferent bricks by the prototype computer program that I use as a propositional interpreter. Thus, the names that I use to single out the differential and the intentional extensions, and the lexical paradigms that I follow to construct them, are meant to suggest the interpretations that I have in mind, but they can only hint at the extra meanings that human communicators may pack into their terms and inflections.<br />
<br />
As applied here, the word ''intentional'' is drawn from common use and may have little bearing on its technical use in other, more properly philosophical, contexts. I am merely using the complex of intentional concepts &mdash; aims, ends, goals, objectives, purposes, and so on &mdash; metaphorically to flesh out and vividly to represent any situation where one needs to contemplate a system in multiple aspects of state and destination, that is, its being in certain states and at the same time acting as if headed through certain states. If confusion arises, more neutral words like ''conative'', ''contingent'', ''discretionary'', ''experimental'', ''kinetic'', ''progressive'', ''tentative'', or ''trial'' would probably serve as well.<br />
<br />
===Life on Easy Street===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Failing to fetch me at first keep encouraged,<br><br />
Missing me one place search another,<br><br />
I stop some where waiting for you<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]<br />
|}<br />
<br />
The finite character of the extended universe <math>[\mathrm{E}\mathcal{A}]</math> makes the problem of solving differential propositions relatively straightforward, at least, in principle. The solution set of the differential proposition <math>q : \mathrm{E}A \to \mathbb{B}</math> is the set of models <math>q^{-1}(1)\!</math> in <math>\mathrm{E}A.</math> Finding all the models of <math>q,\!</math> the extended interpretations in <math>\mathrm{E}A</math> that satisfy <math>q,\!</math> can be carried out by a finite search. Being in possession of complete algorithms for propositional calculus modeling, theorem checking, or theorem proving makes the analytic task fairly simple in principle, though the question of efficiency in the face of arbitrary complexity may always remain another matter entirely. While the fact that propositional satisfiability is NP-complete may be discouraging for the prospects of a single efficient algorithm that covers the whole space <math>[\mathrm{E}\mathcal{A}]</math> with equal facility, there appears to be much room for improvement in classifying special forms and in developing algorithms that are tailored to their practical processing.<br />
<br />
In view of these constraints and contingencies, our focus shifts to the tasks of approximation and interpretation that support intuition, especially in dealing with the natural kinds of differential propositions that arise in applications, and in the effort to understand, in succinct and adaptive forms, their dynamic implications. In the absence of direct insights, these tasks are partially carried out by forging analogies with the familiar situations and customary routines of ordinary calculus. But the indirect approach, going by way of specious analogy and intuitive habit, forces us to remain on guard against the circumstance that occurs when the word ''forging'' takes on its shadier nuance, indicting the constant risk of a counterfeit in the proportion.<br />
<br />
==Back to the Beginning : Exemplary Universes==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
I would have preferred to be enveloped in words, borne way beyond all possible beginnings.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
To anchor our understanding of differential logic, let us look at how the various concepts apply in the simplest possible concrete cases, where the initial dimension is only 1 or 2. In spite of the obvious simplicity of these cases, it is possible to observe how central difficulties of the subject begin to arise already at this stage.<br />
<br />
===A One-Dimensional Universe===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
There was never any more inception than there is now,<br><br />
Nor any more youth or age than there is now;<br><br />
And will never be any more perfection than there is now,<br><br />
Nor any more heaven or hell than there is now.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]<br />
|}<br />
<br />
Let <math>\mathcal{X} = \{ x_1 \} = \{ A \}</math> be an alphabet that represents one boolean variable or a single logical feature. In this example the capital letter <math>{}^{\backprime\backprime} A {}^{\prime\prime}\!</math> is used usual informally, to name a feature and not a space, in departure from our formerly stated formal conventions. At any rate, the basis element <math>A = x_1\!</math> may be interpreted as a simple proposition or a coordinate projection <math>A = x_1 : \mathbb{B} \xrightarrow{i} \mathbb{B}.</math> The space <math>X = \langle A \rangle = \{ \texttt{(} A \texttt{)}, A \}</math> of points (cells, vectors, interpretations) has cardinality <math>2^n = 2^1 = 2\!</math> and is isomorphic to <math>\mathbb{B} = \{ 0, 1 \}.</math> Moreover, <math>X\!</math> may be identified with the set of singular propositions <math>\{ x : \mathbb{B} \xrightarrow{s} \mathbb{B} \}.</math> The space of linear propositions <math>X^* = \{ \mathrm{hom} : \mathbb{B} \xrightarrow{\ell} \mathbb{B} \} = \{ 0, A \}</math> is algebraically dual to <math>X\!</math> and also has cardinality <math>2.\!</math> Here, <math>{}^{\backprime\backprime} 0 {}^{\prime\prime}\!</math> is interpreted as denoting the constant function <math>0 : \mathbb{B} \to \mathbb{B},</math> amounting to the linear proposition of rank <math>0,\!</math> while <math>A\!</math> is the linear proposition of rank <math>1.\!</math> Last but not least we have the positive propositions <math>\{ \mathrm{pos} : \mathbb{B} \xrightarrow{p} \mathbb{B} \} = \{ A, 1 \},\!</math> of rank <math>1\!</math> and <math>0,\!</math> respectively, where <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}\!</math> is understood as denoting the constant function <math>1 : \mathbb{B} \to \mathbb{B}.</math> In sum, there are <math>2^{2^n} = 2^{2^1} = 4</math> propositions altogether in the universe of discourse, comprising the set <math>X^\uparrow = \{ f : X \to \mathbb{B} \} = \{ 0, \texttt{(} A \texttt{)}, A, 1 \} \cong (\mathbb{B} \to \mathbb{B}).</math><br />
<br />
The first order differential extension of <math>\mathcal{X}</math> is <math>\mathrm{E}\mathcal{X} = \{ x_1, \mathrm{d}x_1 \} = \{ A, \mathrm{d}A \}.</math> If the feature <math>A\!</math> is understood as applying to some object or state, then the feature <math>\mathrm{d}A</math> may be interpreted as an attribute of the same object or state that says that it is changing ''significantly'' with respect to the property <math>A,\!</math> or that it has an ''escape velocity'' with respect to the state <math>A.\!</math> In practice, differential features acquire their logical meaning through a class of ''temporal inference rules''.<br />
<br />
For example, relative to a frame of observation that is left implicit for now, one is permitted to make the following sorts of inference: From the fact that <math>A\!</math> and <math>\mathrm{d}A</math> are true at a given moment one may infer that <math>\texttt{(} A \texttt{)}\!</math> will be true in the next moment of observation. Altogether in the present instance, there is the fourfold scheme of inference that is shown below:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:50%"<br />
|<br />
<math>\begin{matrix}<br />
\text{From} & \texttt{(} A \texttt{)}<br />
& \text{and} & \texttt{(} \mathrm{d}A \texttt{)}<br />
& \text{infer} & \texttt{(} A \texttt{)} & \text{next.}<br />
\\[8pt]<br />
\text{From} & \texttt{(} A \texttt{)}<br />
& \text{and} & \mathrm{d}A<br />
& \text{infer} & A & \text{next.}<br />
\\[8pt]<br />
\text{From} & A<br />
& \text{and} & \texttt{(} \mathrm{d}A \texttt{)}<br />
& \text{infer} & A & \text{next.}<br />
\\[8pt]<br />
\text{From} & A<br />
& \text{and} & \mathrm{d}A<br />
& \text{infer} & \texttt{(} A \texttt{)} & \text{next.}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
It might be thought that an independent time variable needs to be brought in at this point, but it is an insight of fundamental importance that the idea of process is logically prior to the notion of time. A time variable is a reference to a ''clock'' &mdash; a canonical, conventional process that is accepted or established as a standard of measurement, but in essence no different than any other process. This raises the question of how different subsystems in a more global process can be brought into comparison, and what it means for one process to serve the function of a local standard for others. But these inquiries only wrap up puzzles in further riddles, and are obviously too involved to be handled at our current level of approximation.<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
The clock indicates the moment . . . . but what does<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;eternity indicate?<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 79]<br />
|}<br />
<br />
Observe that the secular inference rules, used by themselves, involve a loss of information, since nothing in them can tell us whether the momenta <math>\{ \texttt{(} \mathrm{d}A \texttt{)}, \mathrm{d}A \}\!</math> are changed or unchanged in the next instance. In order to know this, one would have to determine <math>\mathrm{d}^2 A,\!</math> and so on, pursuing an infinite regress. Ultimately, in order to rest with a finitely determinate system, it is necessary to make an infinite assumption, for example, that <math>\mathrm{d}^k A = 0\!</math> for all <math>k\!</math> greater than some fixed value <math>M.\!</math> Another way to escape the regress is through the provision of a dynamic law, in typical form making higher order differentials dependent on lower degrees and estates.<br />
<br />
===Example 1. A Square Rigging===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Urge and urge and urge,<br><br />
Always the procreant urge of the world.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]<br />
|}<br />
<br />
By way of example, suppose that we are given the initial condition <math>A = \mathrm{d}A\!</math> and the law <math>\mathrm{d}^2 A = \texttt{(} A \texttt{)}.\!</math> Since the equation <math>A = \mathrm{d}A\!</math> is logically equivalent to the disjunction <math>A ~ \mathrm{d}A ~\text{or}~ \texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)},\!</math> we may infer two possible trajectories, as displayed in Table&nbsp;11. In either case the state <math>A ~ \texttt{(} \mathrm{d}A \texttt{)(} \mathrm{d}^2 A \texttt{)}\!</math> is a stable attractor or a terminal condition for both starting points.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 11.} ~~ \text{A Pair of Commodious Trajectories}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Time}\!</math><br />
| <math>\text{Trajectory 1}\!</math><br />
| <math>\text{Trajectory 2}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
2<br />
\\[4pt]<br />
3<br />
\\[4pt]<br />
4<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A & \mathrm{d}A & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
\texttt{(} A \texttt{)} & \mathrm{d}A & \mathrm{d}^2 A<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
{}^{\shortparallel} & {}^{\shortparallel} & {}^{\shortparallel}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{)} & \texttt{(} \mathrm{d}A \texttt{)} & \mathrm{d}^2 A<br />
\\[4pt]<br />
\texttt{(} A \texttt{)} & \mathrm{d}A & \mathrm{d}^2 A<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
{}^{\shortparallel} & {}^{\shortparallel} & {}^{\shortparallel}<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Because the initial space <math>X = \langle A \rangle\!</math> is one-dimensional, we can easily fit the second order extension <math>\mathrm{E}^2 X = \langle A, \mathrm{d}A, \mathrm{d}^2 A \rangle\!</math> within the compass of a single venn diagram, charting the couple of converging trajectories as shown in Figure&nbsp;12.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 12 -- The Anchor.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 12.} ~~ \text{The Anchor}\!</math><br />
|}<br />
<br />
If we eliminate from view the regions of <math>\mathrm{E}^2 X\!</math> that are ruled out by the dynamic law <math>\mathrm{d}^2 A = \texttt{(} A \texttt{)},\!</math> then what remains is the quotient structure that is shown in Figure&nbsp;13. This picture makes it easy to see that the dynamically allowable portion of the universe is partitioned between the properties <math>A\!</math> and <math>\mathrm{d}^2 A\!.</math> As it happens, this fact might have been expressed &ldquo;right off the bat&rdquo; by an equivalent formulation of the differential law, one that uses the exclusive disjunction to state the law as <math>\texttt{(} A \texttt{,} \mathrm{d}^2 A \texttt{)}\!.</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 13 -- The Tiller.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 13.} ~~ \text{The Tiller}\!</math><br />
|}<br />
<br />
What we have achieved in this example is to give a differential description of a simple dynamic process. In effect, we did this by embedding a directed graph, which can be taken to represent the state transitions of a finite automaton, in a dynamically allotted quotient structure that is created from a boolean lattice or an <math>n\!</math>-cube by nullifying all of the regions that the dynamics outlaws. With growth in the dimensions of our contemplated universes, it becomes essential, both for human comprehension and for computer implementation, that the dynamic structures of interest to us be represented not actually, by acquaintance, but virtually, by description. In our present study, we are using the language of propositional calculus to express the relevant descriptions, and to comprehend the structure that is implicit in the subsets of a <math>n\!</math>-cube without necessarily being forced to actualize all of its points.<br />
<br />
One of the reasons for engaging in this kind of extremely reduced, but explicitly controlled case study is to throw light on the general study of languages, formal and natural, in their full array of syntactic, semantic, and pragmatic aspects. Propositional calculus is one of the last points of departure where we can view these three aspects interacting in a non-trivial way without being immediately and totally overwhelmed by the complexity they generate. Often this complexity causes investigators of formal and natural languages to adopt the strategy of focusing on a single aspect and to abandon all hope of understanding the whole, whether it's the still living natural language or the dynamics of inquiry that lies crystallized in formal logic.<br />
<br />
From the perspective that I find most useful here, a language is a syntactic system that is designed or evolved in part to express a set of descriptions. When the explicit symbols of a language have extensions in its object world that are actually infinite, or when the implicit categories and generative devices of a linguistic theory have extensions in its subject matter that are potentially infinite, then the finite characters of terms, statements, arguments, grammars, logics, and rhetorics force an excess of intension to reside in all these symbols and functions, across the spectrum from the object language to the metalinguistic uses. In the aphorism from W. von Humboldt that Chomsky often cites, for example, in [Cho86, 30] and [Cho93, 49], language requires &ldquo;the infinite use of finite means&rdquo;. This is necessarily true when the extensions are infinite, when the referential symbols and grammatical categories of a language possess infinite sets of models and instances. But it also voices a practical truth when the extensions, though finite at every stage, tend to grow at exponential rates.<br />
<br />
This consequence of dealing with extensions that are &ldquo;practically infinite&rdquo; becomes crucial when one tries to build neural network systems that learn, since the learning competence of any intelligent system is limited to the objects and domains that it is able to represent. If we want to design systems that operate intelligently with the full deck of propositions dealt by intact universes of discourse, then we must supply them with succinct representations and efficient transformations in this domain. Furthermore, in the project of constructing inquiry driven systems, we find ourselves forced to contemplate the level of generality that is embodied in propositions, because the dynamic evolution of these systems is driven by the measurable discrepancies that occur among their expectations, intentions, and observations, and because each of these subsystems or components of knowledge constitutes a propositional modality that can take on the fully generic character of an empirical summary or an axiomatic theory.<br />
<br />
A compression scheme by any other name is a symbolic representation, and this is what the differential extension of propositional calculus, through all of its many universes of discourse, is intended to supply. Why is this particular program of mental calisthenics worth carrying out in general? By providing a uniform logical medium for describing dynamic systems we can make the task of understanding complex systems much easier, both in looking for invariant representations of individual cases and in finding points of comparison among diverse structures that would otherwise appear as isolated systems. All of this goes to facilitate the search for compact knowledge and to adapt what is learned from individual cases to the general realm.<br />
<br />
===Back to the Feature===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
I guess it must be the flag of my disposition, out of hopeful<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;green stuff woven.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 31]<br />
|}<br />
<br />
Let us assume that the sense intended for differential features is well enough established in the intuition, for now, that we may continue with outlining the structure of the differential extension <math>[\mathrm{E}\mathcal{X}] = [A, \mathrm{d}A].\!</math> Over the extended alphabet <math>\mathrm{E}\mathcal{X} = \{ x_1, \mathrm{d}x_1 \} = \{ A, \mathrm{d}A \}\!</math> of cardinality <math>2^n = 2\!</math> we generate the set of points <math>\mathrm{E}X\!</math> of cardinality <math>2^{2n} = 4\!</math> that bears the following chain of equivalent descriptions:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}X & = & \langle A, \mathrm{d}A \rangle<br />
\\[4pt]<br />
& = & \{ \texttt{(} A \texttt{)}, A \} ~\times~ \{ \texttt{(} \mathrm{d}A \texttt{)}, \mathrm{d}A \}<br />
\\[4pt]<br />
& = &<br />
\{<br />
\texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)},~<br />
\texttt{(} A \texttt{)} \mathrm{d}A,~<br />
A \texttt{(} \mathrm{d}A \texttt{)},~<br />
A ~ \mathrm{d}A<br />
\}.<br />
\end{array}</math><br />
|}<br />
<br />
The space <math>\mathrm{E}X\!</math> may be assigned the mnemonic type <math>\mathbb{B} \times \mathbb{D},\!</math> which is really no different than <math>\mathbb{B} \times \mathbb{B} = \mathbb{B}^2.\!</math> An individual element of <math>\mathrm{E}X\!</math> may be regarded as a ''disposition at a point'' or a ''situated direction'', in effect, a singular mode of change occurring at a single point in the universe of discourse. In applications, the modality of this change can be interpreted in various ways, for example, as an expectation, an intention, or an observation with respect to the behavior of a system.<br />
<br />
To complete the construction of the extended universe of discourse <math>\mathrm{E}X^\bullet = [x_1, \mathrm{d}x_1] = [A, \mathrm{d}A]\!</math> one must add the set of differential propositions <math>\mathrm{E}X^\uparrow = \{ g : \mathrm{E}X \to \mathbb{B} \} \cong (\mathbb{B} \times \mathbb{D} \to \mathbb{B})\!</math> to the set of dispositions in <math>\mathrm{E}X.\!</math> There are <math>2^{2^{2n}} = 16\!</math> propositions in <math>\mathrm{E}X^\uparrow,\!</math> as detailed in Table&nbsp;14.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 14.} ~~ \text{Differential Propositions}\!</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>A\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>\mathrm{d}A\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>g_{0}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{1}<br />
\\[4pt]<br />
g_{2}<br />
\\[4pt]<br />
g_{4}<br />
\\[4pt]<br />
g_{8}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
1~0~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
\texttt{(} A \texttt{)} ~ \mathrm{d}A ~<br />
\\[4pt]<br />
~ A ~ \texttt{(} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
~ A ~~ \mathrm{d}A ~<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{neither}~ A ~\text{nor}~ \mathrm{d}A<br />
\\[4pt]<br />
\mathrm{d}A ~\text{and not}~ A<br />
\\[4pt]<br />
A ~\text{and not}~ \mathrm{d}A<br />
\\[4pt]<br />
A ~\text{and}~ \mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot A \land \lnot \mathrm{d}A<br />
\\[4pt]<br />
\lnot A \land \mathrm{d}A<br />
\\[4pt]<br />
A \land \lnot \mathrm{d}A<br />
\\[4pt]<br />
A \land \mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
g_{3}<br />
\\[4pt]<br />
g_{12}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{)}<br />
\\[4pt]<br />
A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ A<br />
\\[4pt]<br />
A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot A<br />
\\[4pt]<br />
A<br />
\end{matrix}\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{6}<br />
\\[4pt]<br />
g_{9}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[4pt]<br />
1~0~0~1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{,} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
\texttt{((} A \texttt{,} \mathrm{d}A \texttt{))}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
A ~\text{not equal to}~ \mathrm{d}A<br />
\\[4pt]<br />
A ~\text{equal to}~ \mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
A \ne \mathrm{d}A<br />
\\[4pt]<br />
A = \mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{5}<br />
\\[4pt]<br />
g_{10}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
\mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ \mathrm{d}A<br />
\\[4pt]<br />
\mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot \mathrm{d}A<br />
\\[4pt]<br />
\mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{7}<br />
\\[4pt]<br />
g_{11}<br />
\\[4pt]<br />
g_{13}<br />
\\[4pt]<br />
g_{14}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} ~ A ~~ \mathrm{d}A ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{(} ~ A ~ \texttt{(} \mathrm{d}A \texttt{))}<br />
\\[4pt]<br />
\texttt{((} A \texttt{)} ~ \mathrm{d}A ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{((} A \texttt{)(} \mathrm{d}A \texttt{))}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not both}~ A ~\text{and}~ \mathrm{d}A<br />
\\[4pt]<br />
\text{not}~ A ~\text{without}~ \mathrm{d}A<br />
\\[4pt]<br />
\text{not}~ \mathrm{d}A ~\text{without}~ A<br />
\\[4pt]<br />
A ~\text{or}~ \mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot A \lor \lnot \mathrm{d}A<br />
\\[4pt]<br />
A \Rightarrow \mathrm{d}A<br />
\\[4pt]<br />
A \Leftarrow \mathrm{d}A<br />
\\[4pt]<br />
A \lor \mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
| <math>f_{3}\!</math><br />
| <math>g_{15}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
Aside from changing the names of variables and shuffling the order of rows, this Table follows the format that was used previously for boolean functions of two variables. The rows are grouped to reflect natural similarity classes among the propositions. In a future discussion, these classes will be given additional explanation and motivation as the orbits of a certain transformation group acting on the set of 16 propositions. Notice that four of the propositions, in their logical expressions, resemble those given in the table for <math>X^\uparrow.\!</math> Thus the first set of propositions <math>\{ f_i \}\!</math> is automatically embedded in the present set <math>\{ g_j \}\!</math> and the corresponding inclusions are indicated at the far left margin of the Table.<br />
<br />
===Tacit Extensions===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
I would really like to have slipped imperceptibly into this lecture, as into all the others I shall be delivering, perhaps over the years ahead.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
Strictly speaking, however, there is a subtle distinction in type between the function <math>f_i : X \to \mathbb{B}</math> and the corresponding function <math>g_j : \mathrm{E}X \to \mathbb{B},</math> even though they share the same logical expression. Naturally, we want to maintain the logical equivalence of expressions that represent the same proposition while appreciating the full diversity of that proposition's functional and typical representatives. Both perspectives, and all the levels of abstraction extending through them, have their reasons, as will develop in time.<br />
<br />
Because this special circumstance points up an important general theme, it is a good idea to discuss it more carefully. Whenever there arises a situation like this, where one alphabet <math>\mathcal{X}</math> is a subset of another alphabet <math>\mathcal{Y},</math> then we say that any proposition <math>f : \langle \mathcal{X} \rangle \to \mathbb{B}</math> has a ''tacit extension'' to a proposition <math>\boldsymbol\varepsilon f : \langle \mathcal{Y} \rangle \to \mathbb{B},\!</math> and that the space <math>(\langle \mathcal{X} \rangle \to \mathbb{B})</math> has an ''automatic embedding'' within the space <math>(\langle \mathcal{Y} \rangle \to \mathbb{B}).</math> The extension is defined in such a way that <math>\boldsymbol\varepsilon f\!</math> puts the same constraint on the variables of <math>\mathcal{X}</math> that are contained in <math>\mathcal{Y}</math> as the proposition <math>f\!</math> initially did, while it puts no constraint on the variables of <math>\mathcal{Y}</math> outside of <math>\mathcal{X},</math> in effect, conjoining the two constraints.<br />
<br />
If the variables in question are indexed as <math>\mathcal{X} = \{ x_1, \ldots, x_n \}</math> and <math>\mathcal{Y} = \{ x_1, \ldots, x_n, \ldots, x_{n+k} \},</math> then the definition of the tacit extension from <math>\mathcal{X}</math> to <math>\mathcal{Y}</math> may be expressed in the form of an equation:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\boldsymbol\varepsilon f(x_1, \ldots, x_n, \ldots, x_{n+k}) ~=~ f(x_1, \ldots, x_n).\!</math><br />
|}<br />
<br />
On formal occasions, such as the present context of definition, the tacit extension from <math>\mathcal{X}</math> to <math>\mathcal{Y}</math> is explicitly symbolized by the operator <math>\boldsymbol\varepsilon : (\langle \mathcal{X} \rangle \to \mathbb{B}) \to (\langle \mathcal{Y} \rangle \to \mathbb{B}),</math> where the appropriate alphabets <math>\mathcal{X}</math> and <math>\mathcal{Y}</math> are understood from context, but normally one may leave the "<math>\boldsymbol\varepsilon\!</math>" silent.<br />
<br />
Let's explore what this means for the present Example. Here, <math>\mathcal{X} = \{ A \}</math> and <math>\mathcal{Y} = \mathrm{E}\mathcal{X} = \{ A, \mathrm{d}A \}.</math> For each of the propositions <math>f_i\!</math> over <math>X\!,</math> specifically, those whose expression <math>e_i\!</math> lies in the collection <math>\{ 0, \texttt{(} A \texttt{)}, A, 1 \},\!</math> the tacit extension <math>\boldsymbol\varepsilon f\!</math> of <math>f\!</math> to <math>\mathrm{E}X</math> can be phrased as a logical conjunction of two factors, <math>f_i = e_i \cdot \tau ~ ,\!</math> where <math>\tau\!</math> is a logical tautology that uses all the variables of <math>\mathcal{Y} - \mathcal{X}.</math> Working in these terms, the tacit extensions <math>\boldsymbol\varepsilon f\!</math> of <math>f\!</math> to <math>\mathrm{E}X</math> may be explicated as shown in Table&nbsp;15.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:60%"<br />
|+ style="height:30px" | <math>\text{Table 15.} ~~ \text{Tacit Extension of}~ [A] ~\text{to}~ [A, \mathrm{d}A]\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0<br />
& = & 0 & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))} & = & & 0<br />
\\[8pt]<br />
\texttt{(} A \texttt{)}<br />
& = & \texttt{(} A \texttt{)} & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))}<br />
& = & \texttt{(} A \texttt{)} \, \mathrm{d}A ~ & + & \texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)}<br />
\\[8pt]<br />
A<br />
& = & ~A~ & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))}<br />
& = & ~A~ ~\mathrm{d}A~ & + & ~A~ \texttt{(} \mathrm{d}A \texttt{)}<br />
\\[8pt]<br />
1<br />
& = & 1 & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))} & = & & 1<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
In its effect on the singular propositions over <math>X,\!</math> this analysis has an interesting interpretation. The tacit extension takes us from thinking about a particular state, like <math>A\!</math> or <math>\texttt{(} A \texttt{)},\!</math> to considering the collection of outcomes, the outgoing changes or the singular dispositions, that spring from that state.<br />
<br />
===Example 2. Drives and Their Vicissitudes===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
I open my scuttle at night and see the far-sprinkled systems,<br><br />
And all I see, multiplied as high as I can cipher, edge but<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the rim of the farther systems.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 81]<br />
|}<br />
<br />
Before we leave the one-feature case let's look at a more substantial example, one that illustrates a general class of curves that can be charted through the extended feature spaces and that provides an opportunity to discuss a number of important themes concerning their structure and dynamics.<br />
<br />
Again, let <math>\mathcal{X} = \{ x_1 \} = \{ A \}.\!</math> In the discussion that follows we will consider a class of trajectories having the property that <math>\mathrm{d}^k A = 0\!</math> for all <math>k\!</math> greater than some fixed <math>m\!</math> and we may indulge in the use of some picturesque terms that describe salient classes of such curves. Given the finite order condition, there is a highest order non-zero difference <math>\mathrm{d}^m A\!</math> exhibited at each point in the course of any determinate trajectory that one may wish to consider. With respect to any point of the corresponding orbit or curve let us call this highest order differential feature <math>\mathrm{d}^m A\!</math> the ''drive'' at that point. Curves of constant drive <math>\mathrm{d}^m A\!</math> are then referred to as ''<math>m^\text{th}\!</math>-gear curves''.<br />
<br />
* '''Scholium.''' The fact that a difference calculus can be developed for boolean functions is well known [Fuji], [Koh, &sect; 8-4] and was probably familiar to Boole, who was an expert in difference equations before he turned to logic. And of course there is the strange but true story of how the Turin machines of the 1840s prefigured the Turing machines of the 1940s [Men, 225-297]. At the very outset of general purpose, mechanized computing we find that the motive power driving the Analytical Engine of Babbage, the kernel of an idea behind all of his wheels, was exactly his notion that difference operations, suitably trained, can serve as universal joints for any conceivable computation [M&M], [Mel, ch. 4].<br />
<br />
Given this language, the Example we take up here can be described as the family of <math>4^\text{th}\!</math>-gear curves through <math>\mathrm{E}^4 X\!</math> <math>=\!</math> <math>\langle A, ~\mathrm{d}A, ~\mathrm{d}^2\!A, ~\mathrm{d}^3\!A, ~\mathrm{d}^4\!A \rangle.</math> These are the trajectories generated subject to the dynamic law <math>\mathrm{d}^4 A = 1,\!</math> where it is understood in such a statement that all higher order differences are equal to <math>0.\!</math> Since <math>\mathrm{d}^4 A\!</math> and all higher <math>\mathrm{d}^k A\!</math> are fixed, the temporal or transitional conditions (initial, mediate, terminal &mdash; transient or stable states) vary only with respect to their projections as points of <math>\mathrm{E}^3 X = \langle A, ~\mathrm{d}A, ~\mathrm{d}^2\!A, ~\mathrm{d}^3\!A \rangle.</math> Thus, there is just enough space in a planar venn diagram to plot all of these orbits and to show how they partition the points of <math>\mathrm{E}^3 X.\!</math> It turns out that there are exactly two possible orbits, of eight points each, as illustrated in Figure&nbsp;16.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 16 -- A Couple of Fourth Gear Orbits.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 16.} ~~ \text{A Couple of Fourth Gear Orbits}\!</math><br />
|}<br />
<br />
With a little thought it is possible to devise an indexing scheme for the general run of dynamic states that allows for comparing universes of discourse that weigh in on different scales of observation. With this end in sight, let us index the states <math>q \in \mathrm{E}^m X\!</math> with the dyadic rationals (or the binary fractions) in the half-open interval <math>[0, 2).\!</math> Formally and canonically, a state <math>q_r\!</math> is indexed by a fraction <math>r = \tfrac{s}{t}\!</math> whose denominator is the power of two <math>t = 2^m\!</math> and whose numerator is a binary numeral formed from the coefficients of state in a manner to be described next. The ''differential coefficients'' of the state <math>q\!</math> are just the values <math>\mathrm{d}^k\!A(q)</math> for <math>k = 0 ~\text{to}~ m,\!</math> where <math>\mathrm{d}^0\!A</math> is defined as being identical to <math>A.\!</math> To form the binary index <math>d_0.d_1 \ldots d_m\!</math> of the state <math>q\!</math> the coefficient <math>\mathrm{d}^k\!A(q)</math> is read off as the binary digit <math>d_k\!</math> associated with the place value <math>2^{-k}.\!</math> Expressed by way of algebraic formulas, the rational index <math>r\!</math> of the state <math>q\!</math> can be given by the following equivalent formulations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
r(q)<br />
& = &<br />
\displaystyle\sum_k d_k \cdot 2^{-k}<br />
& = &<br />
\displaystyle\sum_k \text{d}^k A(q) \cdot 2^{-k}<br />
\\[8pt]<br />
=<br />
\\[8pt]<br />
\displaystyle\frac{s(q)}{t}<br />
& = &<br />
\displaystyle\frac{\sum_k d_k \cdot 2^{(m-k)}}{2^m}<br />
& = &<br />
\displaystyle\frac{\sum_k \text{d}^k A(q) \cdot 2^{(m-k)}}{2^m}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Applied to the example of <math>4^\text{th}\!</math>-gear curves, this scheme results in the data of Tables&nbsp;17-a and 17-b, which exhibit one period for each orbit. The states in each orbit are listed as ordered pairs <math>(p_i, q_j),\!</math> where <math>p_i\!</math> may be read as a temporal parameter that indicates the present time of the state and where <math>j\!</math> is the decimal equivalent of the binary numeral <math>s.\!</math> Informally and more casually, the Tables exhibit the states <math>q_s\!</math> as subscripted with the numerators of their rational indices, taking for granted the constant denominators of <math>2^m\! = 2^4 = 16.\!</math> In this set-up the temporal successions of states can be reckoned as given by a kind of ''parallel round-up rule''. That is, if <math>(d_k, d_{k+1})\!</math> is any pair of adjacent digits in the state index <math>r,\!</math> then the value of <math>d_k\!</math> in the next state is <math>{d_k}' = d_k + d_{k+1}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:55%"<br />
|+ style="height:30px" | <math>\text{Table 17-a.} ~~ \text{A Couple of Orbits in Fourth Gear : Orbit 1}\!</math><br />
|- style="background:ghostwhite"<br />
| <math>\text{Time}\!</math><br />
| <math>\text{State}\!</math><br />
| <math>A\!</math><br />
| <math>\mathrm{d}A\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| <math>p_i\!</math><br />
| <math>q_j\!</math><br />
| <math>\mathrm{d}^0\!A</math><br />
| <math>\mathrm{d}^1\!A</math><br />
| <math>\mathrm{d}^2\!A</math><br />
| <math>\mathrm{d}^3\!A</math><br />
| <math>\mathrm{d}^4\!A</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
p_0<br />
\\[4pt]<br />
p_1<br />
\\[4pt]<br />
p_2<br />
\\[4pt]<br />
p_3<br />
\\[4pt]<br />
p_4<br />
\\[4pt]<br />
p_5<br />
\\[4pt]<br />
p_6<br />
\\[4pt]<br />
p_7<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
q_{01}<br />
\\[4pt]<br />
q_{03}<br />
\\[4pt]<br />
q_{05}<br />
\\[4pt]<br />
q_{15}<br />
\\[4pt]<br />
q_{17}<br />
\\[4pt]<br />
q_{19}<br />
\\[4pt]<br />
q_{21}<br />
\\[4pt]<br />
q_{31}<br />
\end{matrix}\!</math><br />
| colspan="5" |<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="text-align:center; width:100%"<br />
|<br />
<math>\begin{matrix}<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
1.<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:55%"<br />
|+ style="height:30px" | <math>\text{Table 17-b.} ~~ \text{A Couple of Orbits in Fourth Gear : Orbit 2}\!</math><br />
|- style="background:ghostwhite"<br />
| <math>\text{Time}\!</math><br />
| <math>\text{State}\!</math><br />
| <math>A\!</math><br />
| <math>\mathrm{d}A\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| <math>p_i\!</math><br />
| <math>q_j\!</math><br />
| <math>\mathrm{d}^0\!A</math><br />
| <math>\mathrm{d}^1\!A</math><br />
| <math>\mathrm{d}^2\!A</math><br />
| <math>\mathrm{d}^3\!A</math><br />
| <math>\mathrm{d}^4\!A</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
p_0<br />
\\[4pt]<br />
p_1<br />
\\[4pt]<br />
p_2<br />
\\[4pt]<br />
p_3<br />
\\[4pt]<br />
p_4<br />
\\[4pt]<br />
p_5<br />
\\[4pt]<br />
p_6<br />
\\[4pt]<br />
p_7<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
q_{25}<br />
\\[4pt]<br />
q_{11}<br />
\\[4pt]<br />
q_{29}<br />
\\[4pt]<br />
q_{07}<br />
\\[4pt]<br />
q_{09}<br />
\\[4pt]<br />
q_{27}<br />
\\[4pt]<br />
q_{13}<br />
\\[4pt]<br />
q_{23}<br />
\end{matrix}\!</math><br />
| colspan="5" |<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="text-align:center; width:100%"<br />
|<br />
<math>\begin{matrix}<br />
1.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
==Transformations of Discourse==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
It is understandable that an engineer should be completely absorbed in his speciality, instead of pouring himself out into the freedom and vastness of the world of thought, even though his machines are being sent off to the ends of the earth; for he no more needs to be capable of applying to his own personal soul what is daring and new in the soul of his subject than a machine is in fact capable of applying to itself the differential calculus on which it is based. The same thing cannot, however, be said about mathematics; for here we have the new method of thought, pure intellect, the very well-spring of the times, the ''fons et origo'' of an unfathomable transformation.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 39]<br />
|}<br />
<br />
In this section we take up the general study of logical transformations, or maps that relate one universe of discourse to another. In many ways, and especially as applied to the subject of intelligent dynamic systems, my argument develops the antithesis of the statement just quoted. Along the way, if incidental to my ends, I hope this essay can pose a fittingly irenic epitaph to the frankly ironic epigraph inscribed at its head.<br />
<br />
My goal in this section is to answer a single question: What is a propositional tangent functor? In other words, my aim is to develop a clear conception of what manner of thing would pass in the logical realm for a genuine analogue of the tangent functor, an object conceived to generalize as far as possible in the abstract terms of category theory the ordinary notions of functional differentiation and the all too familiar operations of taking derivatives.<br />
<br />
As a first step I discuss the kinds of transformations that we already know as ''extensions'' and ''projections'', and I use these special cases to illustrate several different styles of logical and visual representation that will figure heavily in the sequel.<br />
<br />
===Foreshadowing Transformations : Extensions and Projections of Discourse===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
And, despite the care which she took to look behind her at every moment, she failed to see a shadow which followed her like her own shadow, which stopped when she stopped, which started again when she did, and which made no more noise than a well-conducted shadow should.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 126]<br />
|}<br />
<br />
Many times in our discussion we have occasion to place one universe of discourse in the context of a larger universe of discourse. An embedding of the general type <math>[\mathcal{X}] \to [\mathcal{Y}]\!</math> is implied any time that we make use of one alphabet <math>[\mathcal{X}]\!</math> that happens to be included in another alphabet <math>[\mathcal{Y}].\!</math> When we are discussing differential issues we usually have in mind that the extended alphabet <math>[\mathcal{Y}]\!</math> has a special construction or a specific lexical relation with respect to the initial alphabet <math>[\mathcal{X}],\!</math> one that is marked by characteristic types of accents, indices, or inflected forms.<br />
<br />
====Extension from 1 to 2 Dimensions====<br />
<br />
Figure 18-a lays out the ''angular form'' of venn diagram for universes of 1 and 2 dimensions, indicating the embedding map of type <math>\mathbb{B}^1 \to \mathbb{B}^2\!</math> and detailing the coordinates that are associated with individual cells. Because all points, cells, or logical interpretations are represented as connected geometric areas, we can say that these pictures provide us with an ''areal view'' of each universe of discourse.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-a -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-a.} ~~ \text{Extension from 1 to 2 Dimensions : Areal}\!</math><br />
|}<br />
<br />
Figure 18-b shows the differential extension from <math>X^\bullet = [x]\!</math> to <math>\mathrm{E}X^\bullet = [x, \mathrm{d}x]\!</math> in a ''bundle of boxes'' form of venn diagram. As awkward as it may seem at first, this type of picture is often the most natural and the most easily available representation when we want to conceptualize the localized information or momentary knowledge of an intelligent dynamic system. It gives a ready picture of a ''proposition at a point'', in the present instance, of a proposition about changing states which is itself associated with a particular dynamic state of a system. It is easy to see how this application might be extended to conceive of more general types of instantaneous knowledge that are possessed by a system.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-b -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-b.} ~~ \text{Extension from 1 to 2 Dimensions : Bundle}\!</math><br />
|}<br />
<br />
Figure 18-c shows the same extension in a ''compact'' style of venn diagram, where the differential features at each position are represented by arrows extending from that position that cross or do not cross, as the case may be, the corresponding feature boundaries.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-c -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-c.} ~~ \text{Extension from 1 to 2 Dimensions : Compact}\!</math><br />
|}<br />
<br />
Figure 18-d compresses the picture of the differential extension even further, yielding a directed graph or ''digraph'' form of representation. (Notice that my definition of a digraph allows for loops or ''slings'' at individual points, in addition to arcs or ''arrows'' between the points.)<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-d -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-d.} ~~ \text{Extension from 1 to 2 Dimensions : Digraph}\!</math><br />
|}<br />
<br />
====Extension from 2 to 4 Dimensions====<br />
<br />
Figure 19-a lays out the ''areal view'' or the ''angular form'' of venn diagram for universes of 2 and 4 dimensions, indicating the embedding map of type <math>\mathbb{B}^2 \to \mathbb{B}^4.\!</math> In many ways these pictures are the best kind there is, giving full canvass to an ideal vista. Their style allows the clearest, the fairest, and the plainest view that we can form of a universe of discourse, affording equal representation to all dispositions and maintaining a balance with respect to ordinary and differential features. If only we could extend this view! Unluckily, an obvious difficulty beclouds this prospect, and that is how precipitately we run into the limits of our plane and visual intuitions. Even within the scope of the spare few dimensions that we have scanned up to this point subtle discrepancies have crept in already. The circumstances that bind us and the frameworks that block us, the flat distortion of the planar projection and the inevitable ineffability that precludes us from wrapping its rhomb figure into rings around a torus, all of these factors disguise the underlying but true connectivity of the universe of discourse.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-a -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-a.} ~~ \text{Extension from 2 to 4 Dimensions : Areal}\!</math><br />
|}<br />
<br />
Figure 19-b shows the differential extension from <math>U^\bullet = [u, v]\!</math> to <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v]\!</math> in the ''bundle of boxes'' form of venn diagram.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-b -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-b.} ~~ \text{Extension from 2 to 4 Dimensions : Bundle}\!</math><br />
|}<br />
<br />
As dimensions increase, this factorization of the extended universe along the lines that are marked out by the bundle picture begins to look more and more like a practical necessity. But whenever we use a propositional model to address a real situation in the context of nature we need to remain aware that this articulation into factors, affecting our description, may be wholly artificial in nature and cleave to nothing, no joint in nature, nor any juncture in time to be in or out of joint.<br />
<br />
Figure 19-c illustrates the extension from 2 to 4 dimensions in the ''compact'' style of venn diagram. Here, just the changes with respect to the center cell are shown.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-c -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-c.} ~~ \text{Extension from 2 to 4 Dimensions : Compact}\!</math><br />
|}<br />
<br />
Figure 19-d gives the ''digraph'' form of representation for the differential extension <math>U^\bullet \to \mathrm{E}U^\bullet,\!</math> where the 4 nodes marked with a circle <math>{}^{\bigcirc}\!</math> are the cells <math>uv,\, u \texttt{(} v \texttt{)},\, \texttt{(} u \texttt{)} v,\, \texttt{(} u \texttt{)(} v \texttt{)},\!</math> respectively, and where a 2-headed arc counts as 2 arcs of the differential digraph.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-d -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-d.} ~~ \text{Extension from 2 to 4 Dimensions : Digraph}\!</math><br />
|}<br />
<br />
===Thematization of Functions : And a Declaration of Independence for Variables===<br />
<br />
{| width="100%"<br />
| align="left" |<br />
''And as imagination bodies forth''<br><br />
''The forms of things unknown, the poet's pen''<br><br />
''Turns them to shapes, and gives to airy nothing''<br><br />
''A local habitation and a name.''<br />
| align="right" valign="bottom" | A Midsummer Night's Dream, 5.1.18<br />
|}<br />
<br />
In the representation of propositions as functions it is possible to notice different degrees of explicitness in the way their functional character is symbolized. To indicate what I mean by this, the next series of Figures illustrates a set of graphic conventions that will be put to frequent use in the remainder of this discussion, both to mark the relevant distinctions and to help us convert between related expressions at different levels of explicitness in their functionality.<br />
<br />
====Thematization : Venn Diagrams====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
The known universe has one complete lover and that is the greatest poet. He consumes an eternal passion and is indifferent which chance happens and which possible contingency of fortune or misfortune and persuades daily and hourly his delicious pay.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 11&ndash;12]<br />
|}<br />
<br />
Figure 20-i traces the first couple of steps in this order of ''thematic'' progression, that will gradually run the gamut through a complete series of degrees of functional explicitness in the expression of logical propositions. The first venn diagram represents a situation where the function is indicated by a shaded figure and a logical expression. At this stage one may be thinking of the proposition only as expressed by a formula in a particular language and its content only as a subset of the universe of discourse, as when considering the proposition <math>u\!\cdot\!v</math> in the universe <math>[u, v].\!</math> The second venn diagram depicts a situation in which two significant steps have been taken. First, one has taken the trouble to give the proposition <math>u\!\cdot\!v</math> a distinctive functional name <math>{}^{\backprime\backprime} J {}^{\prime\prime}.\!</math> Second, one has come to think explicitly about the target domain that contains the functional values of <math>J,\!</math> as when writing <math>J : \langle u, v \rangle \to \mathbb{B}.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 20-i -- Thematization of Conjunction (Stage 1).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 20-i.} ~~ \text{Thematization of Conjunction (Stage 1)}\!</math><br />
|}<br />
<br />
In Figure 20-ii the proposition <math>J\!</math> is viewed explicitly as a transformation from one universe of discourse to another.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 20-ii -- Thematization of Conjunction (Stage 2).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 20-ii.} ~~ \text{Thematization of Conjunction (Stage 2)}\!</math><br />
|}<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
o-------------------------------o o-------------------------------o<br />
| | | |<br />
| o-----o o-----o | | o-----o o-----o |<br />
| / \ / \ | | / \ / \ |<br />
| / o \ | | / o \ |<br />
| / /`\ \ | | / /`\ \ |<br />
| o o```o o | | o o```o o |<br />
| | u |```| v | | | | u |```| v | |<br />
| o o```o o | | o o```o o |<br />
| \ \`/ / | | \ \`/ / |<br />
| \ o / | | \ o / |<br />
| \ / \ / | | \ / \ / |<br />
| o-----o o-----o | | o-----o o-----o |<br />
| | | |<br />
o-------------------------------o o-------------------------------o<br />
\ / \ /<br />
\ / \ /<br />
\ / \ J /<br />
\ / \ /<br />
\ / \ /<br />
o----------\---------/----------o o----------\---------/----------o<br />
| \ / | | \ / |<br />
| \ / | | \ / |<br />
| o-----@-----o | | o-----@-----o |<br />
| /`````````````\ | | /`````````````\ |<br />
| /```````````````\ | | /```````````````\ |<br />
| /`````````````````\ | | /`````````````````\ |<br />
| o```````````````````o | | o```````````````````o |<br />
| |```````````````````| | | |```````````````````| |<br />
| |```````` J ````````| | | |```````` x ````````| |<br />
| |```````````````````| | | |```````````````````| |<br />
| o```````````````````o | | o```````````````````o |<br />
| \`````````````````/ | | \`````````````````/ |<br />
| \```````````````/ | | \```````````````/ |<br />
| \`````````````/ | | \`````````````/ |<br />
| o-----------o | | o-----------o |<br />
| | | |<br />
| | | |<br />
o-------------------------------o o-------------------------------o<br />
J = u v x = J<u, v><br />
<br />
Figure 20-ii. Thematization of Conjunction (Stage 2)<br />
</pre><br />
|}<br />
<br />
In the first venn diagram the name that is assigned to a composite proposition, function, or region in the source universe is delegated to a simple feature in the target universe. This can result in a single character or term exceeding the responsibilities it can carry off well. Allowing the name of a function <math>J : \langle u, v \rangle \to \mathbb{B}\!</math> to serve as the name of its dependent variable <math>J : \mathbb{B}\!</math> does not mean that one has to confuse a function with any of its values, but it does put one at risk for a number of obvious problems, and we should not be surprised, on numerous and limiting occasions, when quibbling arises from the attempts of a too original syntax to serve these two masters.<br />
<br />
The second venn diagram circumvents these difficulties by introducing a new variable name for each basic feature of the target universe, as when writing <math>J : \langle u, v \rangle \to \langle x \rangle,\!</math> and thereby assigns a concrete type <math>\langle x \rangle</math> to the abstract codomain <math>\mathbb{B}.\!</math> To make this induction of variables more formal one can append subscripts, as in <math>x_J,\!</math> to indicate the origin or derivation of the new characters. Or we may use a lexical modifier to convert function names into variable names, for example, associating the function name <math>J\!</math> with the variable name <math>\check{J}.\!</math> Thus we may think of <math>x = x_J = \check{J}\!</math> as the ''cache variable'' corresponding to the function <math>J\!</math> or the symbol <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> considered as a contingent variable.<br />
<br />
In Figure 20-iii we arrive at a stage where the functional equations <math>J = u\!\cdot\!v</math> and <math>x = u\!\cdot\!v</math> are regarded as propositions in their own right, reigning in and ruling over the 3-feature universes of discourse <math>[u, v, J]~\!</math> and <math>[u, v, x],\!</math> respectively. Subject to the cautions already noted, the function name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> can be reinterpreted as the name of a feature <math>\check{J}</math> and the equation <math>J = u\!\cdot\!v</math> can be read as the logical equivalence <math>\texttt{((} J, u ~ v \texttt{))}.\!</math> To give it a generic name let us call this newly expressed, collateral proposition the ''thematization'' or the ''thematic extension'' of the original proposition <math>J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 20-iii -- Thematization of Conjunction (Stage 3).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 20-iii.} ~~ \text{Thematization of Conjunction (Stage 3)}\!</math><br />
|}<br />
<br />
The first venn diagram represents the thematization of the conjunction <math>J\!</math> with shading in the appropriate regions of the universe <math>[u, v, J].\!</math> Also, it illustrates a quick way of constructing a thematic extension. First, draw a line, in practice or the imagination, that bisects every cell of the original universe, placing half of each cell under the aegis of the thematized proposition and the other half under its antithesis. Next, on the scene where the theme applies leave the shade wherever it lies, and off the stage, where it plays otherwise, stagger the pattern in a harlequin guise.<br />
<br />
In the final venn diagram of this sequence the thematic progression comes full circle and completes one round of its development. The ambiguities that were occasioned by the changing role of the name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> are resolved by introducing a new variable name <math>{}^{\backprime\backprime} x {}^{\prime\prime}</math> to take the place of <math>\check{J},\!</math> and the region that represents this fresh featured <math>x\!</math> is circumscribed in a more conventional symmetry of form and placement. Just as we once gave the name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> to the proposition <math>u\!\cdot\!v,</math> we now give the name <math>{}^{\backprime\backprime} \iota {}^{\prime\prime}</math> to its thematization <math>\texttt{((} x, u ~ v \texttt{))}.\!</math> Already, again, at this culminating stage of reflection, we begin to think of the newly named proposition as a distinctive individual, a particular function <math>\iota : \langle u, v, x \rangle \to \mathbb{B}.\!</math><br />
<br />
From now on, the terms ''thematic extension'' and ''thematization'' will be used to describe both the process and degree of explication that progresses through this series of pictures, both the operation of increasingly explicit symbolization and the dimension of variation that is swept out by it. To speak of this change in general, that takes us in our current example from <math>J\!</math> to <math>\iota,\!</math> we introduce a class of operators symbolized by the Greek letter <math>\theta,\!</math> writing <math>\iota = \theta J\!</math> in the present instance. The operator <math>\theta,\!</math> in the present situation bearing the type <math>\theta : [u, v] \to [u, v, x],\!</math> provides us with a convenient way of recapitulating and summarizing the complete cycle of thematic developments.<br />
<br />
Figure 21 shows how the thematic extension operator <math>\theta\!</math> acts on two further examples, the disjunction <math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math> and the equality <math>\texttt{((} u, v \texttt{))}.\!</math> Referring to the disjunction as <math>f(u, v)\!</math> and the equality as <math>f(u, v),\!</math> we may express the thematic extensions as <math>\varphi = \theta f\!</math> and <math>\gamma = \theta g.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 21 -- Thematization of Disjunction and Equality.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 21.} ~~ \text{Thematization of Disjunction and Equality}\!</math><br />
|}<br />
<br />
====Thematization : Truth Tables====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
That which distorts honest shapes or which creates unearthly beings or places or contingencies is a nuisance and a revolt.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 19]<br />
|}<br />
<br />
Tables 22 through 25 outline a method for computing the thematic extensions of propositions in terms of their coordinate values.<br />
<br />
A preliminary step, as illustrated in Table&nbsp;22, is to write out the truth table representations of the propositional forms whose thematic extensions one wants to compute, in the present instance, the functions <math>f(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math> and <math>g(u, v) = \texttt{((} u, v \texttt{))}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:50%"<br />
|+ style="height:30px" | <math>\text{Table 22.} ~~ \text{Disjunction}~ f ~\text{and Equality}~ g\!</math><br />
|- style="height:35px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black" | <math>f\!</math><br />
| <math>g\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Next, each propositional form is individually represented in the fashion shown in Tables&nbsp;23-i and 23-ii, using <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} g {}^{\prime\prime}\!</math> as function names and creating new variables <math>x\!</math> and <math>y\!</math> to hold the associated functional values. This pair of Tables outlines the first stage in the transition from the <math>2\!</math>-dimensional universes of <math>f\!</math> and <math>g\!</math> to the <math>3\!</math>-dimensional universes of <math>\theta f\!</math> and <math>\theta g.\!</math> The top halves of the Tables replicate the truth table patterns for <math>f\!</math> and <math>g\!</math> in the form <math>f : [u, v] \to [x]\!</math> and <math>g : [u, v] \to [y].\!</math> The bottom halves of the tables print the negatives of these pictures, as it were, and paste the truth tables for <math>\texttt{(} f \texttt{)}\!</math> and <math>\texttt{(} g \texttt{)}\!</math> under the copies for <math>f\!</math> and <math>g.\!</math> At this stage, the columns for <math>\theta f\!</math> and <math>\theta g\!</math> are appended almost as afterthoughts, amounting to indicator functions for the sets of ordered triples that make up the functions <math>f\!</math> and <math>g.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 23-i and 23-ii.} ~~ \text{Thematics of Disjunction and Equality (1)}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 23-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black" | <math>f\!</math><br />
| <math>x\!</math><br />
| <math>\varphi\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |<br />
<math>\begin{matrix}\to\\\to\\\to\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" | &nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 23-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black" | <math>g\!</math><br />
| <math>y\!</math><br />
| <math>\gamma\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |<br />
<math>\begin{matrix}\to\\\to\\\to\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" | &nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
All the data are now in place to give the truth tables for <math>\theta f\!</math> and <math>\theta g.\!</math> All that remains to be done is to permute the rows and change the roles of <math>x\!</math> and <math>y\!</math> from dependent to independent variables. In Tables&nbsp;24-i and 24-ii the rows are arranged in such a way as to put the 3-tuples <math>(u, v, x)\!</math> and <math>(u, v, y)\!</math> in binary numerical order, suitable for viewing as the arguments of the maps <math>\theta f = \varphi : [u, v, x] \to \mathbb{B}\!</math> and <math>\theta g = \gamma : [u, v, y] \to \mathbb{B}.\!</math> Moreover, the structure of the tables is altered slightly, allowing the now vestigial functions <math>\theta f\!</math> and <math>\theta g\!</math> to be passed over without further attention and shifting the heavy vertical bars a notch to the right. In effect, this clinches the fact that the thematic variables <math>x := \check{f}\!</math> and <math>y := \check{g}\!</math> are now treated as independent variables.<br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 24-i and 24-ii.} ~~ \text{Thematics of Disjunction and Equality (2)}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 24-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>f\!</math><br />
| <math>x\!</math><br />
| style="border-left:1px solid black" | <math>\varphi\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <font size="+2">&nbsp;<br></font><math>\begin{matrix}\to\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 24-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>g\!</math><br />
| <math>y\!</math><br />
| style="border-left:1px solid black" | <math>\gamma\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | &nbsp;<br><math>\begin{matrix}\to\\\to\end{matrix}\!</math><br>&nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
An optional reshuffling of the rows brings additional features of the thematic extensions to light. Leaving the columns in place for the sake of comparison, Tables&nbsp;25-i and 25-ii sort the rows in a different order, in effect treating <math>x\!</math> and <math>y\!</math> as the primary variables in their respective 3-tuples. Regarding the thematic extensions in the form <math>\varphi : [x, u, v] \to \mathbb{B}\!</math> and <math>\gamma : [y, u, v] \to \mathbb{B}\!</math> makes it easier to see in this tabular setting a property that was graphically obvious in the venn diagrams above. Specifically, when the thematic variable <math>\check{F}\!</math> is true then <math>\theta F\!</math> exhibits the pattern of the original <math>F,\!</math> and when <math>\check{F}\!</math> is false then <math>\theta F\!</math> exhibits the pattern of its negation <math>\texttt{(} F \texttt{)}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 25-i and 25-ii.} ~~ \text{Thematics of Disjunction and Equality (3)}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 25-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>f\!</math><br />
| <math>x\!</math><br />
| style="border-left:1px solid black" | <math>\varphi\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>{\to}\!</math><br><font size="+2">&nbsp;<br>&nbsp;<br>&nbsp;<br></font><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <font size="+2">&nbsp;<br></font><math>\begin{matrix}\to\\\to\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 25-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>g\!</math><br />
| <math>y\!</math><br />
| style="border-left:1px solid black" | <math>\gamma\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | &nbsp;<br><math>\begin{matrix}\to\\\to\end{matrix}\!</math><br>&nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
Finally, Tables&nbsp;26-i and 26-ii compare the tacit extensions <math>\boldsymbol\varepsilon : [u, v] \to [u, v, x]\!</math> and <math>\boldsymbol\varepsilon : [u, v] \to [u, v, y]\!</math> with the thematic extensions of the same types, as applied to the propositions <math>f\!</math> and <math>g,\!</math> respectively.<br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 26-i and 26-ii.} ~~ \text{Tacit Extension and Thematization}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 26-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>x\!</math><br />
| style="border-left:1px solid black" | <math>\boldsymbol\varepsilon f\!</math><br />
| <math>\theta f\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 26-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>y\!</math><br />
| style="border-left:1px solid black" | <math>\boldsymbol\varepsilon g\!</math><br />
| <math>\theta g\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
Table 27 summarizes the thematic extensions of all propositions on two variables. Column&nbsp;4 lists the equations of form <math>\texttt{((} \check{f_i}, f_i (u, v) \texttt{))}\!</math> and Column&nbsp;5 simplifies these equations into the form of algebraic expressions. As always, <math>{}^{\backprime\backprime} + {}^{\prime\prime}\!</math> refers to exclusive disjunction and each <math>{}^{\backprime\backprime} \check{f} {}^{\prime\prime}\!</math> appearing in the last two Columns refers to the corresponding variable name <math>{}^{\backprime\backprime} \check{f_i} {}^{\prime\prime}.~\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 27.} ~~ \text{Thematization of Bivariate Propositions}\!</math><br />
|- style="height:30px; background:ghostwhite"<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>{f}\!</math><br />
| <math>\theta f\!</math><br />
| <math>\theta f\!</math><br />
|- style="background:ghostwhite"<br />
| align="right" | <math>u\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| align="right" | <math>v\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| align="left" | <math>\texttt{((} \check{f} \texttt{,~(~)~))}\!</math><br />
| align="left" | <math>\check{f} + 1\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\\[4pt]<br />
f_{4}<br />
\\[4pt]<br />
f_{8}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
1~0~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} u \texttt{)~} v \texttt{~}<br />
\\[4pt]<br />
\texttt{~} u \texttt{~(} v \texttt{)}<br />
\\[4pt]<br />
\texttt{~} u \texttt{~~} v \texttt{~}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(u)(v)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~(u)~v~~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~u~(v)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~u~~v~~))}<br />
\end{array}</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + u + v + uv<br />
\\[4pt]<br />
\check{f} + v + uv + 1<br />
\\[4pt]<br />
\check{f} + u + uv + 1<br />
\\[4pt]<br />
\check{f} + uv + 1<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{3}<br />
\\[4pt]<br />
f_{12}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{)}<br />
\\[4pt]<br />
\texttt{~} u \texttt{~}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(u)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~u~~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + u<br />
\\[4pt]<br />
\check{f} + u + 1<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[4pt]<br />
f_{9}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[4pt]<br />
1~0~0~1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{,} v \texttt{)}<br />
\\[4pt]<br />
\texttt{((} u \texttt{,} v \texttt{))}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~~(} u \texttt{,} v \texttt{)~~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~((} u \texttt{,} v \texttt{))~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + u + v + 1<br />
\\[4pt]<br />
\check{f} + u + v<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{5}<br />
\\[4pt]<br />
f_{10}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} v \texttt{)}<br />
\\[4pt]<br />
\texttt{~} v \texttt{~}<br />
\end{matrix}</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(} v \texttt{)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~} v \texttt{~~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + v<br />
\\[4pt]<br />
\check{f} + v + 1<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[4pt]<br />
f_{11}<br />
\\[4pt]<br />
f_{13}<br />
\\[4pt]<br />
f_{14}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(~} u \texttt{~~} v \texttt{~)}<br />
\\[4pt]<br />
\texttt{(~} u \texttt{~(} v \texttt{))}<br />
\\[4pt]<br />
\texttt{((} u \texttt{)~} v \texttt{~)}<br />
\\[4pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(~} u \texttt{~~} v \texttt{~)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~(~} u \texttt{~(} v \texttt{))~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~((} u \texttt{)~} v \texttt{~)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~((} u \texttt{)(} v \texttt{))~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + uv<br />
\\[4pt]<br />
\check{f} + u + uv<br />
\\[4pt]<br />
\check{f} + v + uv<br />
\\[4pt]<br />
\check{f} + u + v + uv + 1<br />
\end{array}\!</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| align="left" | <math>\texttt{((} \check{f} \texttt{,~((~))~))}\!</math><br />
| align="left" | <math>\check{f}\!</math><br />
|}<br />
<br />
<br><br />
<br />
In order to show what all of the thematic extensions from two dimensions to three dimensions look like in terms of coordinates, Tables&nbsp;28 and 29 present ordinary truth tables for the functions <math>f_i : \mathbb{B}^2 \to \mathbb{B}\!</math> and for the corresponding thematizations <math>\theta f_i = \varphi_i : \mathbb{B}^3 \to \mathbb{B}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 28.} ~~ \text{Propositions on Two Variables}\!</math><br />
|- style="height:35px; background:ghostwhite"<br />
| style="width:5%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:5%; border-bottom:1px solid black; border-left:1px solid black" | <math>f_{0}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{1}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{2}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{3}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{4}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{5}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{6}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{7}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{8}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{9}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{10}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{11}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{12}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{13}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{14}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{15}\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 29.} ~~ \text{Thematic Extensions of Bivariate Propositions}\!</math><br />
|- style="height:35px; background:ghostwhite"<br />
| style="width:5%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\check{f}\!</math><br />
| style="width:5%; border-bottom:1px solid black; border-left:1px solid black" | <math>\varphi_{0}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{1}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{2}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{3}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{4}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{5}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{6}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{7}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{8}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{9}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{10}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{11}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{12}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{13}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{14}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{15}\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
|-<br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Propositional Transformations===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
If only the word &lsquo;artificial&rsquo; were associated with the idea of ''art'', or expert skill gained through voluntary apprenticeship (instead of suggesting the factitious and unreal), we might say that ''logical'' refers to artificial thought.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 56&ndash;57]<br />
|}<br />
<br />
In this section we develop a comprehensive set of concepts for dealing with transformations between universes of discourse. In this most general setting the source and target universes of a transformation are allowed to be different, but may be the same. When we apply these concepts to dynamic systems we focus on the important special case of transformations that map a universe into itself, regarding them as the state transitions of a discrete dynamical process and placing them among the myriad ways that a universe of discourse might change, and by that change turn into itself.<br />
<br />
====Alias and Alibi Transformations====<br />
<br />
There are customarily two modes of understanding a transformation, at least, when we try to interpret its relevance to something in reality. A transformation always refers to a changing prospect, to say it in a unified but equivocal way, but this can be taken to mean either a subjective change in the interpreting observer's point of view or an objective change in the systematic subject of discussion. In practice these variant uses of the transformation concept are distinguished in the following terms:<br />
<br />
# A ''perspectival'' or ''alias'' transformation refers to a shift in perspective or a change in language that takes place in the observer's frame of reference.<br />
# A ''transitional'' or ''alibi'' transformation refers to a change of position or an alteration of state that occurs in the object system as it falls under study.<br />
<br />
(For a recent discussion of the alias vs. alibi issue, as it relates to linear transformations in vector spaces and to other issues of an algebraic nature, see [MaB, 256, 582-4].)<br />
<br />
Naturally, when we are concerned with the dynamical properties of a system, the transitional aspect of transformation is the factor that comes to the fore, and this involves us in contemplating all of the ways of changing a universe into itself while remaining under the rule of established dynamical laws. In the prospective application to dynamic systems, and to neural networks viewed in this light, our interest lies chiefly with the transformations of a state space into itself that constitute the state transitions of a discrete dynamic process. Nevertheless, many important properties of these transformations, and some constructions that we need to see most clearly, are independent of the transitional interpretation and are likely to be confounded with irrelevant features if presented first and only in that association.<br />
<br />
In addition, and in partial contrast, intelligent systems are exactly that species of dynamic agents that have the capacity to have a point of view, and we cannot do justice to their peculiar properties without examining their ability to form and transform their own frames of reference in exposure to the elements of their own experience. In this setting, the perspectival aspect of transformation is the facet that shines most brightly, perhaps too often leaving us fascinated with mere glimmerings of its actual potential. It needs to be emphasized that nothing of the ordinary sort needs be moved in carrying out a transformation under the alias interpretation, that it may only involve a change in the forms of address, an amendment of the terms which are customed to approach and fashioned to describe the very same things in the very same world. But again, working within a discipline of realistic computation, we know how formidably complex and resource-consuming such transformations of perspective can be to implement in practice, much less to endow in the self-governed form of a nascently intelligent dynamical system.<br />
<br />
====Transformations of General Type====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
''Es ist passiert'', &ldquo;it just sort of happened&rdquo;, people said there when other people in other places thought heaven knows what had occurred. It was a peculiar phrase, not known in this sense to the Germans and with no equivalent in other languages, the very breath of it transforming facts and the bludgeonings of fate into something light as eiderdown, as thought itself.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 34]<br />
|}<br />
<br />
Consider the situation illustrated in Figure&nbsp;30, where the alphabets <math>\mathcal{U} = \{ u, v \}\!</math> and <math>\mathcal{X} = \{ x, y, z \}\!</math> are used to label basic features in two different logical universes, <math>U^\bullet = [u, v]\!</math> and <math>X^\bullet = [x, y, z].\!</math><br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
o-------------------------------------------------------o<br />
| U |<br />
| |<br />
| o-----------o o-----------o |<br />
| / \ / \ |<br />
| / o \ |<br />
| / / \ \ |<br />
| / / \ \ |<br />
| o o o o |<br />
| | | | | |<br />
| | u | | v | |<br />
| | | | | |<br />
| o o o o |<br />
| \ \ / / |<br />
| \ \ / / |<br />
| \ o / |<br />
| \ / \ / |<br />
| o-----------o o-----------o |<br />
| |<br />
| |<br />
o---------------------------o---------------------------o<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
o-------------------------o o-------------------------o o-------------------------o<br />
| U | | U | | U |<br />
| o---o o---o | | o---o o---o | | o---o o---o |<br />
| / \ / \ | | / \ / \ | | / \ / \ |<br />
| / o \ | | / o \ | | / o \ |<br />
| / / \ \ | | / / \ \ | | / / \ \ |<br />
| o o o o | | o o o o | | o o o o |<br />
| | u | | v | | | | u | | v | | | | u | | v | |<br />
| o o o o | | o o o o | | o o o o |<br />
| \ \ / / | | \ \ / / | | \ \ / / |<br />
| \ o / | | \ o / | | \ o / |<br />
| \ / \ / | | \ / \ / | | \ / \ / |<br />
| o---o o---o | | o---o o---o | | o---o o---o |<br />
| | | | | |<br />
o-------------------------o o-------------------------o o-------------------------o<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ g | \ f / | h /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ o----------|-----------\-----/-----------|----------o /<br />
\ | X | \ / | | /<br />
\ | | \ / | | /<br />
\ | | o-----o-----o | | /<br />
\| | / \ | |/<br />
\ | / \ | /<br />
|\ | / \ | /|<br />
| \ | / \ | / |<br />
| \ | / \ | / |<br />
| \ | o x o | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \| | | |/ |<br />
| o--o--------o o--------o--o |<br />
| / \ \ / / \ |<br />
| / \ \ / / \ |<br />
| / \ o / \ |<br />
| / \ / \ / \ |<br />
| / \ / \ / \ |<br />
| o o--o-----o--o o |<br />
| | | | | |<br />
| | | | | |<br />
| | | | | |<br />
| | y | | z | |<br />
| | | | | |<br />
| | | | | |<br />
| o o o o |<br />
| \ \ / / |<br />
| \ \ / / |<br />
| \ o / |<br />
| \ / \ / |<br />
| \ / \ / |<br />
| o-----------o o-----------o |<br />
| |<br />
| |<br />
o---------------------------------------------------o<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ p , q /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
o<br />
<br />
Figure 30. Generic Frame of a Logical Transformation<br />
</pre><br />
|}<br />
<br />
Enter the picture, as we usually do, in the middle of things, with features like <math>x, y , z\!</math> that present themselves to be simple enough in their own right and that form a satisfactory, if temporary foundation to provide a basis for discussion. In this universe and on these terms we find expression for various propositions and questions of principal interest to ourselves, as indicated by the maps <math>p, q : X \to \mathbb{B}.\!</math> Then we discover that the simple features <math>\{ x, y, z \}\!</math> are really more complex than we thought at first, and it becomes useful to regard them as functions <math>\{ f, g, h \}\!</math> of other features <math>\{ u, v \}\!</math> that we place in a preface to our original discourse, or suppose as topics of a preliminary universe of discourse <math>U^\bullet = [u, v].\!</math> It may happen that these late-blooming but pre-ambling features are found to lie closer, in a sense that may be our job to determine, to the central nature of the situation of interest, in which case they earn our regard as being more fundamental, but these functions and features are only required to supply a critical stance on the universe of discourse or an alternate perspective on the nature of things in order to be preserved as useful.<br />
<br />
A particular transformation <math>F : [u, v] \to [x, y, z]\!</math> may be expressed by a system of equations, as shown below. Here, <math>F\!</math> is defined by its component maps <math>F = (F_1, F_2, F_3) = (f, g, h),\!</math> where each component map in <math>\{ f, g, h \}\!</math> is a proposition of type <math>\mathbb{B}^n \to \mathbb{B}^1.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:50%"<br />
|<br />
<math>\begin{matrix}<br />
x & = & f(u, v)<br />
\\[10pt]<br />
y & = & g(u, v)<br />
\\[10pt]<br />
z & = & h(u, v)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Regarded as a logical statement, this system of equations expresses a relation between a collection of freely chosen propositions <math>\{ f, g, h \}\!</math> in one universe of discourse and the special collection of simple propositions <math>\{ x, y, z \}\!</math> on which is founded another universe of discourse. Growing familiarity with a particular transformation of discourse, and the desire to achieve a ready understanding of its implications, requires that we be able to convert this information about generals and simples into information about all the main subtypes of propositions, including the linear and singular propositions.<br />
<br />
===Analytic Expansions : Operators and Functors===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Consider what effects that might ''conceivably'' have practical bearings you ''conceive'' the objects of your ''conception'' to have. Then, your ''conception'' of those effects is the whole of your ''conception'' of the object.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; C.S. Peirce, &ldquo;The Maxim of Pragmatism&rdquo;, CP 5.438<br />
|}<br />
<br />
Given the barest idea of a logical transformation, as suggested by the sketch in Figure&nbsp;30, and having conceptualized the universe of discourse, with all of its points and propositions, as a beginning object of discussion, we are ready to enter the next phase of our investigation.<br />
<br />
====Operators on Propositions and Transformations====<br />
<br />
The next step is naturally inclined toward objects of the next higher order, namely, with operators that take in argument lists of logical transformations and that give back specified types of logical transformations as their results. For our present aims, we do not need to consider the most general class of such operators, nor any one of them for its own sake. Rather, we are interested in the special sorts of operators that arise in the study and analysis of logical transformations. Figuratively speaking, these operators serve as instruments for the live tomography (and hopefully not the vivisection) of the forms of change under view. Beyond that, they open up ways to implement the changes of view that we need to grasp all the variations on a transformational theme, or to appreciate enough of its significant features to &ldquo;get the drift&rdquo; of the change occurring, to form a passing acquaintance or a synthetic comprehension of its general character and disposition.<br />
<br />
The simplest type of operator is one that takes a single transformation as an argument and returns a single transformation as a result, and most of the operators explicitly considered in our discussion will be of this kind. Figure&nbsp;31 illustrates the typical situation.<br />
<br />
{| align="center" border="0" cellpadding="20"<br />
|<br />
<pre><br />
o---------------------------------------o<br />
| |<br />
| |<br />
| U% F X% |<br />
| o------------------>o |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| !W! | | !W! |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| v v |<br />
| o------------------>o |<br />
| !W!U% !W!F !W!X% |<br />
| |<br />
| |<br />
o---------------------------------------o<br />
Figure 31. Operator Diagram (1)<br />
</pre><br />
|}<br />
<br />
In this Figure <math>{}^{\backprime\backprime} \mathsf{W} {}^{\prime\prime}\!</math> stands for a generic operator <math>\mathsf{W},\!</math> in this case one that takes a logical transformation <math>F\!</math> of type <math>(U^\bullet \to X^\bullet)\!</math> into a logical transformation <math>\mathsf{W}F\!</math> of type <math>(\mathsf{W}U^\bullet \to \mathsf{W}X^\bullet).\!</math> Thus, the operator <math>\mathsf{W}\!</math> must be viewed as making assignments for both families of objects we have previously considered, that is, for universes of discourse like <math>{U^\bullet}\!</math> and <math>{X^\bullet}\!</math> and for logical transformations like <math>F.\!</math><br />
<br />
'''Note.''' Strictly speaking, an operator like <math>\mathsf{W}\!</math> works between two whole categories of universes and transformations, which we call the ''source'' and the ''target'' categories of <math>\mathsf{W}.\!</math> Given this setting, <math>\mathsf{W}\!</math> specifies for each universe <math>U^\bullet\!</math> in its source category a definite universe <math>\mathsf{W}U^\bullet\!</math> in its target category, and to each transformation <math>F\!</math> in its source category it assigns a unique transformation <math>\mathsf{W}F\!</math> in its target category. Naturally, this only works if <math>\mathsf{W}\!</math> takes the source <math>U^\bullet</math> and the target <math>X^\bullet</math> of the map <math>F\!</math> over to the source <math>\mathsf{W}U^\bullet\!</math> and the target <math>\mathsf{W}X^\bullet\!</math> of the map <math>\mathsf{W}F.\!</math> With luck or care enough, we can avoid ever having to put anything like that in words again, letting diagrams do the work. In the situations of present concern we are usually focused on a single transformation <math>F,\!</math> and thus we can take it for granted that the assignment of universes under <math>\mathsf{W}\!</math> is defined appropriately at the source and target ends of <math>F.\!</math> It is not always the case, though, that we need to use the particular names (like <math>{}^{\backprime\backprime} \mathsf{W}U^\bullet {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \mathsf{W}X^\bullet {}^{\prime\prime}\!</math>) that <math>\mathsf{W}\!</math> assigns by default to its operative image universes. In most contexts we will usually have a prior acquaintance with these universes under other names and it is necessary only that we can tell from the information associated with an operator <math>\mathsf{W}\!</math> what universes they are.<br />
<br />
In Figure&nbsp;31 the maps <math>F\!</math> and <math>\mathsf{W}F\!</math> are displayed horizontally, the way one normally orients functional arrows in a written text, and <math>\mathsf{W}\!</math> rolls the map <math>F\!</math> downward into the images that are associated with <math>\mathsf{W}F.\!</math> In Figure&nbsp;32 the same information is redrawn so that the maps <math>F\!</math> and <math>\mathsf{W}F\!</math> flow down the page, and <math>\mathsf{W}\!</math> unfurls the map <math>F\!</math> rightward into domains that are the eminent purview of <math>\mathsf{W}F.\!</math><br />
<br />
{| align="center" border="0" cellpadding="20"<br />
|<br />
<pre><br />
o---------------------------------------o<br />
| |<br />
| |<br />
| U% !W! !W!U% |<br />
| o------------------>o |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| F | | !W!F |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| v v |<br />
| o------------------>o |<br />
| X% !W! !W!X% |<br />
| |<br />
| |<br />
o---------------------------------------o<br />
Figure 32. Operator Diagram (2)<br />
</pre><br />
|}<br />
<br />
The latter arrangement, as exhibited in Figure&nbsp;32, is more congruent with the thinking about operators that we shall do in the rest of this discussion, since all logical transformations from here on out will be pictured vertically, after the fashion of Figure&nbsp;30.<br />
<br />
====Differential Analysis of Propositions and Transformations====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" | The resultant metaphysical problem now is this: ''Does the man go round the squirrel or not?''<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 43]<br />
|}<br />
<br />
The approach to the differential analysis of logical propositions and transformations of discourse to be pursued here is carried out in terms of particular operators <math>\mathsf{W}\!</math> that act on propositions <math>F\!</math> or on transformations <math>F\!</math> to yield the corresponding operator maps <math>\mathsf{W}F.\!</math> The operator results then become the subject of a series of further stages of analysis, which take them apart into their propositional components, rendering them as a set of purely logical constituents. After this is done, all the parts are then re-integrated to reconstruct the original object in the light of a more complete understanding, at least in ways that enable one to appreciate certain aspects of it with fresh insight.<br />
<br />
* '''Remark on Strategy.''' At this point we run into a set of conceptual difficulties that force us to make a strategic choice in how we proceed. Part of the problem can be remedied by extending our discussion of tacit extensions to the transformational context. But the troubles that remain are much more obstinate and lead us to try two different types of solution. The approach that we develop first makes use of a variant type of extension operator, the ''trope extension'', to be defined below. This method is more conservative and requires less preparation, but has features which make it seem unsatisfactory in the long run. A more radical approach, but one with a better hope of long term success, makes use of the notion of ''contingency spaces''. These are an even more generous type of extended universe than the kind we currently use, but are defined subject to certain internal constraints. The extra work needed to set up this method forces us to put it off to a later stage. However, as a compromise, and to prepare the ground for the next pass, we call attention to the various conceptual difficulties as they arise along the way and try to give an honest estimate of how well our first approach deals with them.<br />
<br />
We now describe in general terms the particular operators that are instrumental to this form of analysis. The main series of operators all have the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathsf{W}<br />
& : &<br />
( U^\bullet \to X^\bullet )<br />
& \to &<br />
( \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet )<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
If we assume that the source universe <math>U^\bullet</math> and the target universe <math>X^\bullet</math> have finite dimensions <math>n\!</math> and <math>k,\!</math> respectively, then each operator <math>\mathsf{W}\!</math> is encompassed by the same abstract type:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathsf{W}<br />
& : &<br />
( [\mathbb{B}^n] \to [\mathbb{B}^k] )<br />
& \to &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k \times \mathbb{D}^k] )<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Since the range features of the operator result <math>\mathsf{W}F : [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k \times \mathbb{D}^k]</math> can be sorted by their ordinary versus differential qualities and the component maps can be examined independently, the complete operator <math>\mathsf{W}\!</math> can be separated accordingly into two components, in the form <math>\mathsf{W} = (\boldsymbol\varepsilon, \mathrm{W}).\!</math> Given a fixed context of source and target universes, <math>\boldsymbol\varepsilon\!</math> is always the same type of operator, a multiple component version of the tacit extension operators that were described earlier. In this context <math>\boldsymbol\varepsilon\!</math> has the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{array}{lccccc}<br />
\text{Concrete type}<br />
& \boldsymbol\varepsilon<br />
& : &<br />
( U^\bullet \to X^\bullet )<br />
& \to &<br />
( \mathrm{E}U^\bullet \to X^\bullet )<br />
\\[10pt]<br />
\text{Abstract type}<br />
& \boldsymbol\varepsilon<br />
& : &<br />
( [\mathbb{B}^n] \to [\mathbb{B}^k] )<br />
& \to &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k] )<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
On the other hand, the operator <math>\mathrm{W}\!</math> is specific to each <math>\mathsf{W}.\!</math> In this context <math>\mathrm{W}\!</math> always has the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{array}{lccccc}<br />
\text{Concrete type}<br />
& W<br />
& : &<br />
( U^\bullet \to X^\bullet )<br />
& \to &<br />
( \mathrm{E}U^\bullet \to \mathrm{d}X^\bullet )<br />
\\[10pt]<br />
\text{Abstract type}<br />
& W<br />
& : &<br />
( [\mathbb{B}^n] \to [\mathbb{B}^k] )<br />
& \to &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{D}^k] )<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
In the types just assigned to <math>\boldsymbol\varepsilon\!</math> and <math>\mathrm{W}\!</math> and by implication to their results <math>\boldsymbol\varepsilon F\!</math> and <math>\mathrm{W}F,\!</math> we have listed the most restrictive ranges defined for them rather than the more expansive target spaces that subsume these ranges. When there is need to recognize both, we may use type indications like the following:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\boldsymbol\varepsilon F<br />
& : &<br />
( \mathrm{E}U^\bullet \to X^\bullet \subseteq \mathrm{E}X^\bullet )<br />
& \cong &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k] \subseteq [\mathbb{B}^k \times \mathbb{D}^k] )<br />
\\[10pt]<br />
WF<br />
& : &<br />
( \mathrm{E}U^\bullet \to \mathrm{d}X^\bullet \subseteq \mathrm{E}X^\bullet )<br />
& \cong &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{D}^k] \subseteq [\mathbb{B}^k \times \mathbb{D}^k] )<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Hopefully, though, a general appreciation of these subsumptions will prevent us from having to make such declarations more often than absolutely necessary.<br />
<br />
In giving names to these operators we try to preserve as much of the traditional nomenclature and as many of the classical associations as possible. The chief difficulty in doing this is occasioned by the distinction between the &ldquo;sans&nbsp;serif&rdquo; operators <math>\mathsf{W}\!</math> and their &ldquo;serified&rdquo; components <math>\mathrm{W},\!</math> which forces us to find two distinct but parallel sets of terminology. Here is a plan to that purpose. First, the component operators <math>\mathrm{W}\!</math> are named by analogy with the corresponding operators in the classical difference calculus. Next, the complete operators <math>\mathsf{W} = (\boldsymbol\varepsilon, \mathrm{W})</math> are assigned titles according to their roles in a geometric or trigonometric allegory, if only to ensure that the tangent functor, that belongs to this family and whose exposition we are still working toward, comes out fit with its customary name. Finally, the operator results <math>\mathsf{W}F\!</math> and <math>\mathrm{W}F\!</math> can be fixed in our frame of reference by tethering the operative adjective for <math>\mathsf{W}\!</math> or <math>\mathrm{W}\!</math> to the anchoring epithet &ldquo;map&rdquo;, in conformity with an already standard practice.<br />
<br />
=====The Secant Operator : '''E'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Mr. Peirce, after pointing out that our beliefs are really rules for action, said that, to develop a thought's meaning, we need only determine what conduct it is fitted to produce: that conduct is for us its sole significance.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 46]<br />
|}<br />
<br />
Figures&nbsp;33-i and 33-ii depict two stages in the form of analysis that will be applied to transformations throughout the remainder of this study. From now on our interest is staked on an operator denoted <math>{}^{\backprime\backprime} \mathsf{E} {}^{\prime\prime},\!</math> which receives the principal investment of analytic attention, and on the constituent parts of <math>\mathsf{E},\!</math> which derive their shares of significance as developed by the analysis. In the sequel, we refer to <math>\mathsf{E}\!</math> as the ''secant operator'', taking it for granted that a context has been chosen that defines its type. The secant operator has the component description <math>\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}),\!</math> and its active ingredient <math>\mathrm{E}\!</math> is known as the ''enlargement operator''. (Here, we name <math>\mathrm{E}\!</math> after the literal ancestor of the shift operator in the calculus of finite differences, defined so that <math>\mathrm{E}f(x) = f(x+1)\!</math> for any suitable function <math>f,\!</math> though of course the logical analogue that we take up here must have a rather different definition.)<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
U% $E$ $E$U% $E$U% $E$U%<br />
o------------------>o============o============o<br />
| | | |<br />
| | | |<br />
| | | |<br />
| | | |<br />
F | | $E$F = | $d$^0.F + | $r$^0.F<br />
| | | |<br />
| | | |<br />
| | | |<br />
v v v v<br />
o------------------>o============o============o<br />
X% $E$ $E$X% $E$X% $E$X%<br />
<br />
Figure 33-i. Analytic Diagram (1)<br />
</pre><br />
|}<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
U% $E$ $E$U% $E$U% $E$U% $E$U%<br />
o------------------>o============o============o============o<br />
| | | | |<br />
| | | | |<br />
| | | | |<br />
| | | | |<br />
F | | $E$F = | $d$^0.F + | $d$^1.F + | $r$^1.F<br />
| | | | |<br />
| | | | |<br />
| | | | |<br />
v v v v v<br />
o------------------>o============o============o============o<br />
X% $E$ $E$X% $E$X% $E$X% $E$X%<br />
<br />
Figure 33-ii. Analytic Diagram (2)<br />
</pre><br />
|}<br />
<br />
In its action on universes <math>\mathsf{E}\!</math> yields the same result as <math>\mathrm{E},\!</math> a fact that can be expressed in equational form by writing <math>\mathsf{E}U^\bullet = \mathrm{E}U^\bullet\!</math> for any universe <math>U^\bullet.\!</math> Notice that the extended universes across the top and bottom of the diagram are indicated to be strictly identical, rather than requiring a corresponding decomposition for them. In a certain sense, the functional parts of <math>\mathsf{E}F\!</math> are partitioned into separate contexts that have to be re-integrated again, but the best image to use is that of making transparent copies of each universe and then overlapping their functional contents once more at the conclusion of the analysis, as suggested by the graphic conventions that are used at the top of Figure&nbsp;30.<br />
<br />
Acting on a transformation <math>F\!</math> from universe <math>U^\bullet\!</math> to universe <math>X^\bullet,\!</math> the operator <math>\mathsf{E}\!</math> determines a transformation <math>\mathsf{E}F\!</math> from <math>\mathsf{E}U^\bullet\!</math> to <math>\mathsf{E}X^\bullet.\!</math> The map <math>\mathsf{E}F\!</math> forms the main body of evidence to be investigated in performing a differential analysis of <math>F.\!</math> Because we shall frequently be focusing on small pieces of this map for considerable lengths of time, and consequently lose sight of the &ldquo;big picture&rdquo;, it is critically important to emphasize that the map <math>\mathsf{E}F\!</math> is a transformation that determines a relation from one extended universe into another. This means that we should not be satisfied with our understanding of a transformation <math>F\!</math> until we can lay out the full &ldquo;parts diagram&rdquo; of <math>\mathsf{E}F\!</math> along the lines of the generic frame in Figure&nbsp;30.<br />
<br />
Working within the confines of propositional calculus, it is possible to give an elementary definition of <math>\mathsf{E}F\!</math> by means of a system of propositional equations, as we now describe.<br />
<br />
Given a transformation<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>F = (F_1, \ldots, F_k) : \mathbb{B}^n \to \mathbb{B}^k\!</math><br />
|}<br />
<br />
of concrete type<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>F : [u_1, \ldots, u_n] \to [x_1, \ldots, x_k],\!</math><br />
|}<br />
<br />
the transformation<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\mathsf{E}F = (F_1, \ldots, F_k, \mathrm{E}F_1, \ldots, \mathrm{E}F_k) : \mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B}^k \times \mathbb{D}^k\!</math><br />
|}<br />
<br />
of concrete type<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\mathsf{E}F : [u_1, \dots, u_n, \mathrm{d}u_1, \dots, \mathrm{d}u_n] \to [x_1, \ldots, x_k, \mathrm{d}x_1, \ldots, \mathrm{d}x_k]\!</math><br />
|}<br />
<br />
is defined by means of the following system of logical equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\\[16pt]<br />
\mathrm{d}x_1<br />
& = & \mathrm{E}F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1 + \mathrm{d}u_1, \ldots, u_n + \mathrm{d}u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
\mathrm{d}x_k<br />
& = & \mathrm{E}F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1 + \mathrm{d}u_1, \ldots, u_n + \mathrm{d}u_n)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
It is important to note that this system of equations can be read as a conjunction of equational propositions, in effect, as a single proposition in the universe of discourse generated by all the named variables. Specifically, this is the universe of discourse over <math>2(n+k)\!</math> variables denoted by:<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
\mathrm{E}[\mathcal{U} \cup \mathcal{X}]<br />
& = &<br />
[u_1, \ldots, u_n, ~ x_1, \ldots, x_k, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n, ~ \mathrm{d}x_1, \ldots, \mathrm{d}x_k].<br />
\end{matrix}</math><br />
|}<br />
<br />
In this light, it should be clear that the system of equations defining <math>\mathsf{E}F\!</math> embodies, in a higher rank and differentially extended version, an analogy with the process of thematization that we treated earlier for propositions of type <math>F : \mathbb{B}^n \to \mathbb{B}.\!</math><br />
<br />
The entire collection of constraints that is represented in the above system of equations may be abbreviated by writing <math>\mathsf{E}F = (\boldsymbol\varepsilon F, \mathrm{E}F),\!</math> for any map <math>F.\!</math> This is tantamount to regarding <math>\mathsf{E}\!</math> as a complex operator, <math>\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}),\!</math> with a form of application that distributes each component of the operator to work on each component of the operand, as follows:<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
\mathsf{E}F<br />
& = &<br />
(\boldsymbol\varepsilon, \mathrm{E})F<br />
& = &<br />
(\boldsymbol\varepsilon F, \mathrm{E}F)<br />
& = &<br />
(\boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k, ~ \mathrm{E}F_1, \ldots, \mathrm{E}F_k).<br />
\end{matrix}</math><br />
|}<br />
<br />
Quite a lot of &ldquo;thematic infrastructure&rdquo; or interpretive information is being swept under the rug in the use of such abbreviations. When confusion arises about the meaning of such constructions, one always has recourse to the defining system of equations, in its totality a purely propositional expression. This means that the parenthesized argument lists, that were used in this context to build an image of multi-component transformations, should not be expected to determine a well-defined product in themselves but only to serve as reminders of the prior thematic decisions (choices of variable names, etc.) that have to be made in order to determine one. Accordingly, the argument list notation can be regarded as a kind of ''thematic frame'', an interpretive storage device that preserves the proper associations of concrete logical features between the extended universes at the source and target of <math>\mathsf{E}F.\!</math><br />
<br />
The generic notations <math>\mathsf{d}^0\!F, \mathsf{d}^1\!F, \ldots, \mathsf{d}^m\!F\!</math> in Figure&nbsp;33 refer to the increasing orders of differentials that are extracted in the course of analyzing <math>F.\!</math> When the analysis is halted at a partial stage of development, notations like <math>\mathsf{r}^0\!F, \mathsf{r}^1\!F, \ldots, \mathsf{r}^m\!F\!</math> may be used to summarize the contributions to <math>\mathsf{E}F\!</math> that remain to be analyzed. The Figure illustrates a convention that makes <math>\mathsf{r}^m\!F,\!</math> in effect, the sum of all differentials of order strictly greater than <math>m.\!</math><br />
<br />
We next discuss the operators that figure into this form of analysis, describing their effects on transformations. In simplified or specialized contexts these operators tend to take on a variety of different names and notations, some of whose number we introduce along the way.<br />
<br />
=====The Radius Operator : '''e'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
And the tangible fact at the root of all our thought-distinctions, however subtle, is that there is no one of them so fine as to consist in anything but a possible difference of practice.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 46]<br />
|}<br />
<br />
The operator identified as <math>\mathrm{d}^0\!</math> in the analytic diagram (Figure&nbsp;33) has the sole purpose of creating a proxy for <math>F\!</math> in the appropriately extended context. Construed in terms of its broadest components, <math>\mathrm{d}^0\!</math> is equivalent to the doubly tacit extension operator <math>(\boldsymbol\varepsilon, \boldsymbol\varepsilon),\!</math> in recognition of which let us redub it as <math>{}^{\backprime\backprime} \mathsf{e} {}^{\prime\prime}.\!</math> Pursuing a geometric analogy, we may refer to <math>\mathsf{e} =(\boldsymbol\varepsilon, \boldsymbol\varepsilon) = \mathrm{d}^0\!</math> as the ''radius operator''. The operation intended by all of these forms is defined by the following equation:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathsf{e}F<br />
& = &<br />
(\boldsymbol\varepsilon, \boldsymbol\varepsilon)F<br />
\\[4pt]<br />
& = &<br />
(\boldsymbol\varepsilon F, ~ \boldsymbol\varepsilon F)<br />
\\[4pt]<br />
& = &<br />
(\boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k, ~ \boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k).<br />
\end{array}</math><br />
|}<br />
<br />
which is tantamount to the system of equations below.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\\[16pt]<br />
\mathrm{d}x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
\mathrm{d}x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
=====The Phantom of the Operators : '''&eta;'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
I was wondering what the reason could be, when I myself raised my head and everything within me seemed drawn towards the Unseen, ''which was playing the most perfect music''!<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 81]<br />
|}<br />
<br />
We now describe an operator whose persistent but elusive action behind the scenes, whose slightly twisted and ambivalent character, and whose fugitive disposition, caught somewhere in flight between the arrantly negative and the positive but errant intent, has cost us some painstaking trouble to detect. In the end we shall place it among the other extensions and projections, as a shade among shadows, of muted tones and motley hue, that adumbrates its own thematic frame and paradoxically lights the way toward a whole new spectrum of values.<br />
<br />
Given a transformation <math>F : [u_1, \ldots, u_n] \to [x_1, \dots, x_k],\!</math> we often have call to consider a family of related transformations, all having the form:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>F^\dagger : [u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n] \to [\mathrm{d}x_1, \dots, \mathrm{d}x_k].\!</math><br />
|}<br />
<br />
The operator <math>\eta</math> is introduced to deal with the simplest one of these maps:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\eta F : [u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n] \to [\mathrm{d}x_1, \ldots \mathrm{d}x_k],\!</math><br />
|}<br />
<br />
which is defined by the following equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
\mathrm{d}x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
In effect, the operator <math>\eta\!</math> is nothing but the stand-alone version of a procedure that is otherwise invoked subordinate to the work of the radius operator <math>\mathsf{e}.\!</math> Operating independently, <math>\eta\!</math> achieves precisely the same results that the second <math>\boldsymbol\varepsilon\!</math> in <math>(\boldsymbol\varepsilon, \boldsymbol\varepsilon)\!</math> accomplishes by working within the context of its ordered pair thematic frame. From this point on, because the use of <math>\boldsymbol\varepsilon\!</math> and <math>\eta\!</math> in this setting combines the aims of both the tacit and the thematic extensions, and because <math>\eta\!</math> reflects in regard to <math>\boldsymbol\varepsilon\!</math> little more than the application of a differential twist, a mere turn of phrase, we refer to <math>\eta\!</math> as the ''trope extension'' operator.<br />
<br />
=====The Chord Operator : '''D'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
What difference would it practically make to any one if this notion rather than that notion were true? If no practical difference whatever can be traced, then the alternatives mean practically the same thing, and all dispute is idle.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 45]<br />
|}<br />
<br />
Next we discuss an operator that is always immanent in this form of analysis, and remains implicitly present in the entire proceeding. It may appear once as a record: a relic or revenant that reprises the reminders of an earlier stage of development. Or it may appear always as a resource: a reserve or redoubt that caches in advance an echo of what remains to be played out, cleared up, and requited in full at a future stage. And all of this remains true whether or not we recall the key at any time, and whether or not the subtending theme is recited explicitly at any stage of play.<br />
<br />
This is the operator that is referred to as <math>\mathsf{r}^0\!</math> in the initial stage of analysis (Figure&nbsp;33-i) and that is expanded as <math>\mathsf{d}^1 + \mathsf{r}^1\!</math> in the subsequent step (Figure&nbsp;33-ii). In congruence, but not quite harmony with our allusions of analogy that are not quite geometry, we call this the ''chord operator'' and denote it <math>\mathsf{D}.\!</math> In the more casual terms that are here introduced, <math>\mathsf{D}</math> is defined as the remainder of <math>\mathsf{E}\!</math> and <math>\mathsf{e}\!</math> and it assigns a due measure to each undertone of accord or discord that is struck between the note of enterprise <math>\mathsf{E}\!</math> and the bar of exigency <math>\mathsf{e}.\!</math><br />
<br />
The tension between these counterposed notions, in balance transient but regular in stridence, may be refracted along familiar lines, though never by any such fraction resolved. In this style we write <math>\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}),\!</math> calling <math>\mathrm{D}\!</math> the ''difference operator'' and noting that it plays a role in this realm of mutable and diverse discourse that is analogous to the part taken by the discrete difference operator in the ordinary difference calculus. Finally, we should note that the chord <math>\mathsf{D}\!</math> is not one that need be lost at any stage of development. At the <math>m^\text{th}\!</math> stage of play it can always be reconstituted in the following form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathsf{D}<br />
& = & \mathsf{E} - \mathsf{e}<br />
\\[6pt]<br />
& = & \mathsf{r}^0<br />
\\[6pt]<br />
& = & \mathsf{d}^1 + \mathsf{r}^1<br />
\\[6pt]<br />
& = & \mathsf{d}^1 + \ldots + \mathsf{d}^m + \mathsf{r}^m<br />
\\[6pt]<br />
& = & \displaystyle \sum_{i=1}^m \mathsf{d}^i + \mathsf{r}^m<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====The Tangent Operator : '''T'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
They take part in scenes of whose significance they have no inkling. They are merely tangent to curves of history the beginnings and ends and forms of which pass wholly beyond their ken. So we are tangent to the wider life of things.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 300]<br />
|}<br />
<br />
The operator tagged as <math>\mathsf{d}^1\!</math> in the analytic diagram (Figure&nbsp;33) is called the ''tangent operator'' and is usually denoted in this text as <math>\mathsf{d}\!</math> or <math>\mathsf{T}.\!</math> Because it has the properties required to qualify as a functor, namely, preserving the identity element of the composition operation and the articulated form of every composition of transformations, it also earns the title of a ''tangent functor''. According to the custom adopted here, we dissect it as <math>\mathsf{T} = \mathsf{d} = (\boldsymbol\varepsilon, \mathrm{d}),\!</math> where <math>\mathrm{d}\!</math> is the operator that yields the first order differential <math>\mathrm{d}F\!</math> when applied to a transformation <math>F,\!</math> and whose name is legion.<br />
<br />
Figure&nbsp;34 illustrates a stage of analysis where we ignore everything but the tangent functor <math>\mathsf{T}\!</math> and attend to it chiefly as it bears on the first order differential <math>\mathrm{d}F\!</math> in the analytic expansion of <math>F.\!</math> In this situation we often refer to the extended universes <math>\mathrm{E}U^\bullet\!</math> and <math>\mathrm{E}X^\bullet\!</math> under the equivalent designations <math>\mathsf{T}U^\bullet\!</math> and <math>\mathsf{T}X^\bullet,\!</math> respectively. The purpose of the tangent functor <math>\mathsf{T}\!</math> is to extract the tangent map <math>\mathsf{T}F\!</math> at each point of <math>U^\bullet,\!</math> and the tangent map <math>\mathsf{T}F = (\boldsymbol\varepsilon, \mathrm{d})F\!</math> tells us not only what the transformation <math>F\!</math> is doing at each point of the universe <math>U^\bullet\!</math> but also what <math>F\!</math> is doing to states in the neighborhood of that point, approximately, linearly, and relatively speaking.<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
U% $T$ $T$U% $T$U%<br />
o------------------>o============o<br />
| | |<br />
| | |<br />
| | |<br />
| | |<br />
F | | $T$F = | <!e!, d> F<br />
| | |<br />
| | |<br />
| | |<br />
v v v<br />
o------------------>o============o<br />
X% $T$ $T$X% $T$X%<br />
<br />
Figure 34. Tangent Functor Diagram<br />
</pre><br />
|}<br />
<br />
* '''NB.''' There is one aspect of the preceding construction that remains especially problematic. Why did we define the operators <math>\mathrm{W}\!</math> in <math>\{ \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}\!</math> so that the ranges of their resulting maps all fall within the realms of differential quality, even fabricating a variant of the tacit extension operator to have that character? Clearly, not all of the operator maps <math>\mathrm{W}F\!</math> have equally good reasons for placing their values in differential stocks. The reason for it appears to be that, without doing this, we cannot justify the comparison and combination of their functional values in the various analytic steps. By default, only those values in the same functional component can be brought into algebraic modes of interaction. Up till now the only mechanism provided for their broader association has been a purely logical one, their common placement in a target universe of discourse, but the task of converting this logical circumstance into algebraic forms of application has not yet been taken up.<br />
<br />
===Transformations of Type '''B'''<sup>2</sup> &rarr; '''B'''<sup>1</sup>===<br />
<br />
To study the effects of these analytic operators in the simplest possible setting, let us revert to a still more primitive case. Consider the singular proposition <math>J(u, v)= u\!\cdot\!v,\!</math> regarded either as the functional product of the maps <math>u\!</math> and <math>v\!</math> or as the logical conjunction of the features <math>u\!</math> and <math>v,\!</math> a map whose fiber of truth <math>J^{-1}(1)\!</math> picks out the single cell of that logical description in the universe of discourse <math>U^\bullet.\!</math> Thus <math>J,\!</math> or <math>u\!\cdot\!v,\!</math> may be treated as another name for the point whose coordinates are <math>(1, 1)\!</math> in <math>U^\bullet.\!</math><br />
<br />
====Analytic Expansion of Conjunction====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
<p>In her sufferings she read a great deal and discovered that she had lost something, the possession of which she had previously not been much aware of: a&nbsp;soul.</p><br />
<br />
<p>What is that? It is easily defined negatively: it is simply what curls up and hides when there is any mention of algebraic series.</p><br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 118]<br />
|}<br />
<br />
Figure&nbsp;35 pictures the form of conjunction <math>J : \mathbb{B}^2 \to \mathbb{B}\!</math> as a transformation from the <math>2\!</math>-dimensional universe <math>[u, v]\!</math> to the <math>1\!</math>-dimensional universe <math>[x].\!</math> This is a subtle but significant change of viewpoint on the proposition, attaching an arbitrary but concrete quality to its functional value. Using the language introduced earlier, we can express this change by saying that the proposition <math>J : \langle u, v \rangle \to \mathbb{B}\!</math> is being recast into the thematized role of a transformation <math>J : [u, v] \to [x],\!</math> where the new variable <math>x\!</math> takes the part of a thematic variable <math>\check{J}.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 35 -- A Conjunction Viewed as a Transformation.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 35.} ~~ \text{Conjunction as Transformation}\!</math><br />
|}<br />
<br />
=====Tacit Extension of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
I teach straying from me, yet who can stray from me?<br><br />
I follow you whoever you are from the present hour;<br><br />
My words itch at your ears till you understand them.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 83]<br />
|}<br />
<br />
Earlier we defined the tacit extension operators <math>\boldsymbol\varepsilon : X^\bullet \to Y^\bullet\!</math> as maps embedding each proposition of a given universe <math>X^\bullet~\!</math> in a more generously given universe <math>Y^\bullet \supset X^\bullet.\!</math> Of immediate interest are the tacit extensions <math>\boldsymbol\varepsilon : U^\bullet \to \mathrm{E}U^\bullet,\!</math> that locate each proposition of <math>U^\bullet\!</math> in the enlarged context of <math>\mathrm{E}U^\bullet.\!</math> In its application to the propositional conjunction <math>J = u\!\cdot\!v</math> in <math>[u, v],\!</math> the tacit extension operator <math>\boldsymbol\varepsilon\!</math> yields the proposition <math>\boldsymbol\varepsilon J\!</math> in <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v].\!</math> The extended proposition <math>\boldsymbol\varepsilon J\!</math> may be computed according to the scheme in Table&nbsp;36, in effect doing nothing more that conjoining a tautology of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> to <math>J\!</math> in <math>U^\bullet.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 36.} ~~ \text{Computation of}~ \boldsymbol\varepsilon J\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\boldsymbol\varepsilon J & = & J {}_{^\langle} u, v {}_{^\rangle}<br />
\\[4pt]<br />
& = & u \cdot v<br />
\\[4pt]<br />
& = & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{ } \mathrm{d}v \texttt{ }<br />
& + & u \cdot v \cdot \texttt{ } \mathrm{d}u \texttt{ } \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \cdot v \cdot \texttt{ } \mathrm{d}u \texttt{ } \cdot \texttt{ } \mathrm{d}v \texttt{ }<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{array}{*{4}{l}}<br />
\boldsymbol\varepsilon J<br />
& = && u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & u \cdot v \cdot \texttt{~} \mathrm{d}u \texttt{~} \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & u \cdot v \cdot \texttt{~} \mathrm{d}u \texttt{~} \cdot \texttt{~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
The lower portion of the Table contains the dispositional features of <math>\boldsymbol\varepsilon J\!</math> arranged in such a way that the variety of ordinary features spreads across the rows and the variety of differential features runs through the columns. This organization serves to facilitate pattern matching in the remainder of our computations. Again, the tacit extension is usually so trivial a concern that we do not always bother to make an explicit note of it, taking it for granted that any function <math>F\!</math> being employed in a differential context is equivalent to <math>\boldsymbol\varepsilon F\!</math> for a suitable <math>\boldsymbol\varepsilon.\!</math><br />
<br />
Figures&nbsp;37-a through 37-d present several pictures of the proposition <math>J\!</math> and its tacit extension <math>\boldsymbol\varepsilon J.\!</math> Notice in these Figures how <math>\boldsymbol\varepsilon J\!</math> in <math>\mathrm{E}U^\bullet\!</math> visibly extends <math>J\!</math> in <math>U^\bullet\!</math> by annexing to the indicated cells of <math>J\!</math> all the arcs that exit from or flow out of them. In effect, this extension attaches to these cells all the dispositions that spring from them, in other words, it attributes to these cells all the conceivable changes that are their issue.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-a -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-a.} ~~ \text{Tacit Extension of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-b -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-b.} ~~ \text{Tacit Extension of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-c -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-c.} ~~ \text{Tacit Extension of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-d -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-d.} ~~ \text{Tacit Extension of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
The computational scheme shown in Table&nbsp;36 treated <math>J\!</math> as a proposition in <math>U^\bullet\!</math> and formed <math>\boldsymbol\varepsilon J\!</math> as a proposition in <math>\mathrm{E}U^\bullet.\!</math> When <math>J\!</math> is regarded as a mapping <math>J : U^\bullet \to X^\bullet\!</math> then <math>\boldsymbol\varepsilon J\!</math> must be obtained as a mapping <math>\boldsymbol\varepsilon J : \mathrm{E}U^\bullet \to X^\bullet.\!</math> By default, the tacit extension of the map <math>J : [u, v] \to [x]\!</math> is naturally taken to be a particular map,<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\boldsymbol\varepsilon J : [u, v, \mathrm{d}u, \mathrm{d}v] \to [x] \subseteq [x, \mathrm{d}x],\!</math><br />
|}<br />
<br />
namely, the one that looks like <math>J\!</math> when painted in the frame of the extended source universe and that takes the same thematic variable in the extended target universe as the one that <math>J\!</math> already takes.<br />
<br />
But the choice of a particular thematic variable, for example <math>x\!</math> for <math>\check{J},\!</math> is a shade more arbitrary than the choice of original variable names <math>\{ u, v \},\!</math> so the map we are calling the ''trope extension'',<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\eta J : [u, v, \mathrm{d}u, \mathrm{d}v] \to [\mathrm{d}x] \subseteq [x, \mathrm{d}x],\!</math><br />
|}<br />
<br />
since it looks just the same as <math>\boldsymbol\varepsilon J\!</math> in the way its fibers paint the source domain, belongs just as fully to the family of tacit extensions, generically considered.<br />
<br />
These considerations have the practical consequence that all of our computations and illustrations of <math>\boldsymbol\varepsilon J\!</math> perform the double duty of capturing <math>\eta J\!</math> as well. In other words, we are saved the work of carrying out calculations and drawing figures for the trope extension <math>\eta J,\!</math> because it would be identical to the work already done for <math>\boldsymbol\varepsilon J.\!</math> Since the computations given for <math>\boldsymbol\varepsilon J\!</math> are expressed solely in terms of the variables <math>\{ u, v, \mathrm{d}u, \mathrm{d}v \},\!</math> they work equally well for finding <math>\eta J.\!</math> Further, since each of the above Figures shows only how the level sets of <math>\boldsymbol\varepsilon J\!</math> partition the extended source universe <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v],\!</math> all of them serve equally well as portraits of <math>\eta J.\!</math><br />
<br />
=====Enlargement Map of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
No one could have established the existence of any details that might not just as well have existed in earlier times too; but all the relations between things had shifted slightly. Ideas that had once been of lean account grew fat.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 62]<br />
|}<br />
<br />
The enlargement map <math>\mathrm{E}J\!</math> is computed from the proposition <math>J\!</math> by making a particular class of formal substitutions for its variables, in this case <math>u + \mathrm{d}u\!</math> for <math>u\!</math> and <math>v + \mathrm{d}v\!</math> for <math>v,\!</math> and afterwards expanding the result in whatever way is found convenient.<br />
<br />
Table&nbsp;38 shows a typical scheme of computation, following a systematic method of exploiting boolean expansions over selected variables and ultimately developing <math>\mathrm{E}J\!</math> over the cells of <math>[u, v].\!</math> The critical step of this procedure uses the facts that <math>\texttt{(} 0, x \texttt{)} = 0 + x = x\!</math> and <math>\texttt{(} 1, x \texttt{)} = 1 + x = \texttt{(} x \texttt{)}\!</math> for any boolean variable <math>x.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 38.} ~~ \text{Computation of}~ \mathrm{E}J ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{E}J & = & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}<br />
\\[4pt]<br />
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
& = & \texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot J_{(1 + \mathrm{d}u, 1 + \mathrm{d}v)}<br />
& + & \texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot J_{(1 + \mathrm{d}u, \mathrm{d}v)}<br />
& + & \texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot J_{(\mathrm{d}u, 1 + \mathrm{d}v)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot J_{(\mathrm{d}u, \mathrm{d}v)}<br />
\\[4pt]<br />
& = &<br />
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot J_{(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})}<br />
& + &<br />
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot J_{(\texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v)}<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot J_{(\mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)})}<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot J_{(\mathrm{d}u, \mathrm{d}v)}<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{E}J<br />
& = &<br />
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&& + &<br />
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{ } \mathrm{d}v \texttt{ }<br />
\\[4pt]<br />
&&&&& + &<br />
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot \texttt{ } \mathrm{d}u \texttt{ } \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&&&&&& + &<br />
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \texttt{ } \mathrm{d}u \texttt{ }~\texttt{ } \mathrm{d}v \texttt{ }<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;39 exhibits another method that happens to work quickly in this particular case, using distributive laws to multiply things out in an algebraic manner, arranging the notations of feature and fluxion according to a scale of simple character and degree. Proceeding this way leads through an intermediate step which, in chiming the changes of ordinary calculus, should take on a familiar ring. Consequential properties of exclusive disjunction then carry us on to the concluding line.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 39.} ~~ \text{Computation of}~ \mathrm{E}J ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{c}}<br />
\mathrm{E}J<br />
& = & (u + \mathrm{d}u) \cdot (v + \mathrm{d}v)<br />
\\[6pt]<br />
& = & u \cdot v<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = &<br />
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{ } \mathrm{d}v \texttt{ }<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot \texttt{ } \mathrm{d}u \texttt{ } \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \texttt{ } \mathrm{d}u \texttt{ }~\texttt{ } \mathrm{d}v \texttt{ }<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;40-a through 40-d present several views of the enlarged proposition <math>\mathrm{E}J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-a -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-a.} ~~ \text{Enlargement of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-b -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-b.} ~~ \text{Enlargement of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-c -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-c.} ~~ \text{Enlargement of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-d -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-d.} ~~ \text{Enlargement of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
An intuitive reading of the proposition <math>\mathrm{E}J\!</math> becomes available at this point. Recall that propositions in the extended universe <math>\mathrm{E}U^\bullet\!</math> express the ''dispositions'' of a system and the constraints that are placed on them. In other words, a differential proposition in <math>\mathrm{E}U^\bullet\!</math> can be read as referring to various changes that a system might undergo in and from its various states. In particular, we can understand <math>\mathrm{E}J\!</math> as a statement that tells us what changes need to be made with regard to each state in the universe of discourse in order to reach the ''truth'' of <math>J,\!</math> that is, the region of the universe where <math>J\!</math> is true. This interpretation is visibly clear in the Figures above and appeals to the imagination in a satisfying way but it has the added benefit of giving fresh meaning to the original name of the shift operator <math>\mathrm{E}.\!</math> Namely, <math>\mathrm{E}J\!</math> can be read as a proposition that ''enlarges'' on the meaning of <math>J,\!</math> in the sense of explaining its practical bearings and clarifying what it means in terms of actions and effects &mdash; the available options for differential action and the consequential effects that result from each choice.<br />
<br />
Read this way, the enlargement <math>\mathrm{E}J\!</math> has strong ties to the normal use of <math>J,\!</math> no matter whether it is understood as a proposition or a function, namely, to act as a figurative device for indicating the models of <math>J,\!</math> in effect, pointing to the interpretive elements in its fiber of truth <math>J^{-1}(1).\!</math> It is this kind of &ldquo;use&rdquo; that is often contrasted with the &ldquo;mention&rdquo; of a proposition, and thereby hangs a tale.<br />
<br />
=====Digression : Reflection on Use and Mention=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Reflection is turning a topic over in various aspects and in various lights so that nothing significant about it shall be overlooked &mdash; almost as one might turn a stone over to see what its hidden side is like or what is covered by it.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 57]<br />
|}<br />
<br />
The contrast drawn in logic between the ''use'' and the ''mention'' of a proposition corresponds to the difference that we observe in functional terms between using <math>{}^{\backprime\backprime} J \, {}^{\prime\prime}\!</math> to indicate the region <math>J^{-1}(1)\!</math> and using <math>{}^{\backprime\backprime} J \, {}^{\prime\prime}\!</math> to indicate the function <math>J.\!</math> You may think that one of these uses ought to be proscribed, and logicians are quick to prescribe against their confusion. But there seems to be no likelihood in practice that their interactions can be avoided. If the name <math>{}^{\backprime\backprime} J \, {}^{\prime\prime}\!</math> is used as a sign of the function <math>J,\!</math> and if the function <math>J\!</math> has its use in signifying something else, as would constantly be the case when some future theory of signs has given a functional meaning to every sign whatsoever, then is not <math>J,\!</math> by transitivity a sign of the thing itself? There are, of course, two answers to this question. Not every act of signifying or referring need be transitive. Not every warrant or guarantee or certificate is automatically transferable, indeed, not many. Not every feature of a feature is a feature of the featuree. Otherwise, if a buffalo is white, and white is a color, then a buffalo would ''be'' a color.<br />
<br />
The logical or pragmatic distinction between use and mention is cogent and necessary, and so is the analogous functional distinction between determining a value and determining what determines that value, but so are the normal techniques that we use to make these distinctions apply flexibly in practice. The way that the hue and cry about use and mention is raised in logical discussions, you might be led to think that this single dimension of choices embraces the only kinds of use worth mentioning and the only kinds of mention worth using. It will constitute the expeditionary and taxonomic tasks of that future theory of signs to explore and to classify the many other constellations and dimensions of use and mention that are yet to be opened up by the generative potential of full-fledged sign relations.<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
The well-known capacity that thoughts have &mdash; as doctors have discovered &mdash; for dissolving and dispersing those hard lumps of deep, ingrowing, morbidly entangled conflict that arise out of gloomy regions of the self probably rests on nothing other than their social and worldly nature, which links the individual being with other people and things; but unfortunately what gives them their power of healing seems to be the same as what diminishes the quality of personal experience in them.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 130]<br />
|}<br />
<br />
=====Difference Map of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
&ldquo;It doesn't matter what one does,&rdquo; the Man Without Qualities said to himself, shrugging his shoulders. &ldquo;In a tangle of forces like this it doesn't make a scrap of difference.&rdquo; He turned away like a man who has learned renunciation, almost indeed like a sick man who shrinks from any intensity of contact. And then, striding through his adjacent dressing-room, he passed a punching-ball that hung there; he gave it a blow far swifter and harder than is usual in moods of resignation or states of weakness.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 8]<br />
|}<br />
<br />
With the tacit extension map <math>\boldsymbol\varepsilon J\!</math> and the enlargement map <math>\mathrm{E}J\!</math> well in place, the difference map <math>\mathrm{D}J\!</math> can be computed along the lines displayed in Table&nbsp;41, ending up with an expansion of <math>\mathrm{D}J\!</math> over the cells of <math>[u, v].\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 41.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & \mathrm{E}J<br />
& + & \boldsymbol\varepsilon J<br />
\\[6pt]<br />
& = & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}<br />
& + & J_{(u, v)}<br />
\\[6pt]<br />
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \cdot v<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = &<br />
u \cdot v \cdot \qquad 0<br />
\\[6pt]<br />
& + &<br />
u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v<br />
& + &<br />
u \cdot \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v<br />
\\[6pt]<br />
& + &<br />
u \cdot v \cdot \texttt{~} \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
&&& + &<br />
\texttt{(} u \texttt{)} \cdot v \cdot \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
& + &<br />
u \cdot v \cdot \texttt{~} \mathrm{d}u \;\cdot\; \mathrm{d}v \texttt{~}<br />
&&&&& + &<br />
\texttt{(} u \texttt{)} \cdot \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = &<br />
u \cdot v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
u \cdot \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v<br />
& + &<br />
\texttt{(} u \texttt{)} \cdot v \cdot \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)} \cdot \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Alternatively, the difference map <math>\mathrm{D}J\!</math> can be expanded over the cells of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> to arrive at the formulation shown in Table&nbsp;42. The same development would be obtained from the previous Table by collecting terms in an alternate manner, along the rows rather than the columns in the middle portion of the Table.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 42.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & \boldsymbol\varepsilon J<br />
& + & \mathrm{E}J<br />
\\[6pt]<br />
& = & J_{(u, v)}<br />
& + & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}<br />
\\[6pt]<br />
& = & u \cdot v<br />
& + & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
& = & 0<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}J<br />
& = & 0<br />
& + & u \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Even more simply, the same result is reached by matching up the propositional coefficients of <math>\boldsymbol\varepsilon J</math> and <math>\mathrm{E}J\!</math> along the cells of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> and adding the pairs under boolean addition, that is, &ldquo;mod 2&rdquo;, where 1&nbsp;+&nbsp;1&nbsp;=&nbsp;0, as shown in Table&nbsp;43.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 43.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 3)}\!</math><br />
|<br />
<math>\begin{array}{*{5}{l}}<br />
\mathrm{D}J & = & \boldsymbol\varepsilon J & + & \mathrm{E}J<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\boldsymbol\varepsilon J<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ u \,\cdot\, v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~ u \;\cdot\; v \;\cdot\; \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u ~ \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} ~ v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot\, \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & ~~ 0 ~~ \,\cdot\, ~ \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~~ u ~ \,\cdot\, ~~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ ~ v ~~ \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
The difference map <math>\mathrm{D}J\!</math> can also be given a ''dispositional'' interpretation. First, recall that <math>\boldsymbol\varepsilon J\!</math> exhibits the dispositions to change from anywhere in <math>J\!</math> to anywhere at all in the universe of discourse and <math>\mathrm{E}J\!</math> exhibits the dispositions to change from anywhere in the universe to anywhere in <math>J.\!</math> Next, observe that each of these classes of dispositions may be divided in accordance with the case of <math>J\!</math> versus <math>\texttt{(} J \texttt{)}\!</math> that applies to their points of departure and destination, as shown below. Then, since the dispositions corresponding to <math>\boldsymbol\varepsilon J</math> and <math>\mathrm{E}J\!</math> have in common the dispositions to preserve <math>J,\!</math> their symmetric difference <math>\texttt{(} \boldsymbol\varepsilon J, \mathrm{E}J \texttt{)}\!</math> is made up of all the remaining dispositions, which are in fact disposed to cross the boundary of <math>J\!</math> in one direction or the other. In other words, we may conclude that <math>\mathrm{D}J\!</math> expresses the collective disposition to make a definite change with respect to <math>J,\!</math> no matter what value it holds in the current state of affairs.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
\boldsymbol\varepsilon J<br />
& = & \{ \text{Dispositions from}~ J ~\text{to}~ J \}<br />
& + & \{ \text{Dispositions from}~ J ~\text{to}~ \texttt{(} J \texttt{)} \}<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = & \{ \text{Dispositions from}~ J ~\text{to}~ J \}<br />
& + & \{ \text{Dispositions from}~ \texttt{(} J \texttt{)} ~\text{to}~ J \}<br />
\\[6pt]<br />
\mathrm{D}J<br />
& = & \{ \text{Dispositions from}~ J ~\text{to}~ \texttt{(} J \texttt{)} \}<br />
& + & \{ \text{Dispositions from}~ \texttt{(} J \texttt{)} ~\text{to}~ J \}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;44-a through 44-d illustrate the difference proposition <math>\mathrm{D}J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-a -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-a.} ~~ \text{Difference Map of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-b -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-b.} ~~ \text{Difference Map of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-c -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-c.} ~~ \text{Difference Map of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-d -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-d.} ~~ \text{Difference Map of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
=====Differential of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
By deploying discourse throughout a calendar, and by giving a date to each of its elements, one does not obtain a definitive hierarchy of precessions and originalities; this hierarchy is never more than relative to the systems of discourse that it sets out to evaluate.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Archaeology of Knowledge'', [Fou, 143]<br />
|}<br />
<br />
Finally, at long last, the differential proposition <math>\mathrm{d}J\!</math> can be gleaned from the difference proposition <math>\mathrm{D}J\!</math> by ranging over the cells of <math>[u, v]\!</math> and picking out the linear proposition of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> that is &ldquo;closest&rdquo; to the portion of <math>\mathrm{D}J\!</math> that touches on each point. The idea of distance that would give this definition unequivocal sense has been referred to in cautionary quotes, the kind we use to distance ourselves from taking a final position. There are obvious notions of approximation that suggest themselves, but finding one that can be justified as ultimately correct is not as straightforward as it seems.<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
He had drifted into the very heart of the world. From him to the distant beloved was as far as to the next tree.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 144]<br />
|}<br />
<br />
Let us venture a guess as to where these developments might be heading. From the present vantage point it appears that the ultimate answer to the quandary of distances and the question of a fitting measure may be that, rather than having the constitution of an analytic series depend on our familiar notions of approach, proximity, and approximation, it will be found preferable, and perhaps unavoidable, to turn the tables and let the orders of approximation be defined in terms of our favored and operative notions of formal analysis. Only the aftermath of this conversion, if it does converge, could be hoped to prove whether this hortatory form of analysis and the cohort idea of an analytic form &mdash; the limitary concept of a self-corrective process and the coefficient concept of a completable product &mdash; are truly (in practical reality) the more inceptive and persistent of principles and really (for all practical purposes) the more effective and regulative of ideas.<br />
<br />
Awaiting that determination, I proceed with what seems like the obvious course, and compute <math>\mathrm{d}J\!</math> according to the pattern in Table&nbsp;45.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 45.} ~~ \text{Computation of}~ \mathrm{d}J\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}J<br />
& = &<br />
u\!\cdot\!v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
u \, \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \, \mathrm{d}v<br />
& + &<br />
\texttt{(} u \texttt{)} \, v \cdot \mathrm{d}u \, \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \!\cdot\! \mathrm{d}v \texttt{~}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}J<br />
& = &<br />
u\!\cdot\!v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + &<br />
u \, \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + &<br />
\texttt{(} u \texttt{)} \, v \cdot \mathrm{d}u<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;46-a through 46-d illustrate the proposition <math>{\mathrm{d}J},\!</math> rounded out in our usual array of prospects. This proposition of <math>\mathrm{E}U^\bullet\!</math> is what we refer to as the (first order) differential of <math>J,\!</math> and normally regard as ''the'' differential proposition corresponding to <math>J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-a -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-a.} ~~ \text{Differential of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-b -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-b.} ~~ \text{Differential of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-c -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-c.} ~~ \text{Differential of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-d -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-d.} ~~ \text{Differential of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
=====Remainder of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
<p>I bequeath myself to the dirt to grow from the grass I love,<br><br />
If you want me again look for me under your bootsoles.</p><br />
<br />
<p>You will hardly know who I am or what I mean,<br><br />
But I shall be good health to you nevertheless,<br><br />
And filter and fibre your blood.</p><br />
<br />
<p>Failing to fetch me at first keep encouraged,<br><br />
Missing me one place search another,<br><br />
I stop some where waiting for you</p><br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]<br />
|}<br />
<br />
<br><br />
<br />
Let us recapitulate the story so far. We have in effect been carrying out a decomposition of the enlarged proposition <math>\mathrm{E}J\!</math> in a series of stages. First, we considered the equation <math>\mathrm{E}J = \boldsymbol\varepsilon J + \mathrm{D}J,\!</math> which was involved in the definition of <math>\mathrm{D}J\!</math> as the difference <math>\mathrm{E}J - \boldsymbol\varepsilon J.\!</math> Next, we contemplated the equation <math>\mathrm{D}J = \mathrm{d}J + \mathrm{r}J,\!</math> which expresses <math>\mathrm{D}J\!</math> in terms of two components, the differential <math>\mathrm{d}J\!</math> that was just extracted and the residual component <math>\mathrm{r}J = \mathrm{D}J - \mathrm{d}J.~\!</math> This remaining proposition <math>\mathrm{r}J\!</math> can be computed as shown in Table&nbsp;47.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 47.} ~~ \text{Computation of}~ \mathrm{r}J\!</math><br />
|<br />
<math>\begin{array}{*{5}{l}}<br />
\mathrm{r}J & = & \mathrm{D}J & + & \mathrm{d}J<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}J<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{r}J ~<br />
& = & u \!\cdot\! v \cdot ~ \mathrm{d}u \cdot \mathrm{d}v ~ ~ ~ ~ ~<br />
& + & u \texttt{(} v \texttt{)} \cdot \, \mathrm{d}u \cdot \mathrm{d}v \,<br />
& + & \texttt{(} u \texttt{)} v \cdot \, \mathrm{d}u \cdot \mathrm{d}v \,<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \, \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
As it happens, the remainder <math>\mathrm{r}J\!</math> falls under the description of a second order differential <math>\mathrm{r}J = \mathrm{d}^2 J.\!</math> This means that the expansion of <math>\mathrm{E}J\!</math> in the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|<br />
<math>\begin{array}{*{7}{l}}<br />
\mathrm{E}J<br />
& = & \boldsymbol\varepsilon J<br />
& + & \mathrm{D}J<br />
\\[6pt]<br />
& = & \boldsymbol\varepsilon J<br />
& + & \mathrm{d}J<br />
& + & \mathrm{r}J<br />
\\[6pt]<br />
& = & \mathrm{d}^0 J<br />
& + & \mathrm{d}^1 J<br />
& + & \mathrm{d}^2 J<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
which is nothing other than the propositional analogue of a Taylor series, is a decomposition that terminates in a finite number of steps.<br />
<br />
Figures&nbsp;48-a through 48-d illustrate the proposition <math>\mathrm{r}J = \mathrm{d}^2 J,\!</math> which forms the remainder map of <math>J\!</math> and also, in this instance, the second order differential of <math>J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-a -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-a.} ~~ \text{Remainder of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-b -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-b.} ~~ \text{Remainder of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-c -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-c.} ~~ \text{Remainder of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-d -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-d.} ~~ \text{Remainder of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
=====Summary of Conjunction=====<br />
<br />
To establish a convenient reference point for further discussion, Table&nbsp;49 summarizes the operator actions that have been computed for the form of conjunction, as exemplified by the proposition <math>J.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 49.} ~~ \text{Computation Summary for}~ J~\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon J<br />
& = & u \!\cdot\! v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}J<br />
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}J<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{r}J<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Analytic Series : Coordinate Method====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
And if he is told that something ''is'' the way it is, then he thinks: Well, it could probably just as easily be some other way. So the sense of possibility might be defined outright as the capacity to think how everything could &ldquo;just as easily&rdquo; be, and to attach no more importance to what is than to what is not.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 12]<br />
|}<br />
<br />
Table&nbsp;50 exhibits a truth table method for computing the analytic series (or the differential expansion) of a proposition in terms of coordinates.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:70%"<br />
|+ style="height:30px" | <math>\text{Table 50.} ~~ \text{Computation of an Analytic Series in Terms of Coordinates}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:8%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:8%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:8%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}u\!</math><br />
| style="width:8%; border-bottom:1px solid black" | <math>\mathrm{d}v\!</math><br />
| style="width:8%; border-bottom:1px solid black; border-left:1px solid black" | <math>u'\!</math><br />
| style="width:8%; border-bottom:1px solid black" | <math>v'\!</math><br />
| style="width:10%; border-bottom:1px solid black; border-left:4px double black" | <math>\boldsymbol\varepsilon J\!</math><br />
| style="width:10%; border-bottom:1px solid black" | <math>\mathrm{E}J\!</math><br />
| style="width:12%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{D}J\!</math><br />
| style="width:10%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}J\!</math><br />
| style="width:10%; border-bottom:1px solid black" | <math>\mathrm{d}^2\!J\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
The first six columns of the Table, taken as a whole, represent the variables of a construct called the ''contingent universe'' <math>[u, v, \mathrm{d}u, \mathrm{d}v, u', v'],\!</math> or the bundle of ''contingency spaces'' <math>[\mathrm{d}u, \mathrm{d}v, u', v']\!</math> over the universe <math>[u, v].\!</math> Their placement to the left of the double bar indicates that all of them amount to independent variables, but there is a co-dependency among them, as described by the following equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
u' & = & u + \mathrm{d}u & = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\\[8pt]<br />
v' & = & v + \mathrm{d}v & = & \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
These relations correspond to the formal substitutions that are made in defining <math>\mathrm{E}J\!</math> and <math>\mathrm{D}J.\!</math> For now, the whole rigamarole of contingency spaces can be regarded as a technical device for achieving the effect of these substitutions, adapted to a setting where functional compositions and other symbolic manipulations are difficult to contemplate and execute.<br />
<br />
The five columns to the right of the double bar in Table&nbsp;50 contain the values of the dependent variables <math>\{ \boldsymbol\varepsilon J, ~\mathrm{E}J, ~\mathrm{D}J, ~\mathrm{d}J, ~\mathrm{d}^2\!J \}.\!</math> These are normally interpreted as values of functions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B}\!</math> or as values of propositions in the extended universe <math>[u, v, \mathrm{d}u, \mathrm{d}v]\!</math> but the dependencies prevailing in the contingent universe make it possible to regard these same final values as arising via functions on alternative lists of arguments, for example, the set <math>\{ u, v, u', v' \}.\!</math><br />
<br />
The column for <math>\boldsymbol\varepsilon J\!</math> is computed as <math>J(u, v) = uv\!</math> and together with the columns for <math>u\!</math> and <math>v\!</math> illustrates how we &ldquo;share structure&rdquo; in the Table by listing only the first entries of each constant block.<br />
<br />
The column for <math>\mathrm{E}J\!</math> is computed by means of the following chain of identities, where the contingent variables <math>u'\!</math> and <math>v'\!</math> are defined as <math>u' = u + \mathrm{d}u\!</math> and <math>v' = v + \mathrm{d}v.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathrm{E}J(u, v, \mathrm{d}u, \mathrm{d}v)<br />
& = &<br />
J(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& = &<br />
J(u', v')<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
This makes it easy to determine <math>\mathrm{E}J\!</math> by inspection, computing the conjunction <math>J(u', v') = u'v'\!</math> from the columns headed <math>u'\!</math> and <math>v'.\!</math> Since each of these forms expresses the same proposition <math>\mathrm{E}J\!</math> in <math>\mathrm{E}U^\bullet,\!</math> the dependence on <math>\mathrm{d}u\!</math> and <math>\mathrm{d}v\!</math> is still present but merely left implicit in the final variant <math>J(u', v').\!</math><br />
<br />
* '''Note.''' On occasion, it is tempting to use the further notation <math>J'(u, v) = J(u', v'),\!</math> especially to suggest a transformation that acts on whole propositions, for example, taking the proposition <math>J\!</math> into the proposition <math>J' = \mathrm{E}J.\!</math> The prime <math>( {}^{\prime} )\!</math> then signifies an action that is mediated by a field of choices, namely, the values that are picked out for the contingent variables in sweeping through the initial universe. But this heaps an unwieldy lot of construed intentions on a rather slight character and puts too high a premium on the constant correctness of its interpretation. In practice, therefore, it is best to avoid this usage.<br />
<br />
Given the values of <math>\boldsymbol\varepsilon J\!</math> and <math>\mathrm{E}J,\!</math> the columns for the remaining functions can be filled in quickly. The difference map is computed according to the relation <math>\mathrm{D}J = \boldsymbol\varepsilon J + \mathrm{E}J.\!</math> The first order differential <math>\mathrm{d}J\!</math> is found by looking in each block of constant argument pairs <math>u, v\!</math> and choosing the linear function of <math>\mathrm{d}u, \mathrm{d}v\!</math> that best approximates <math>\mathrm{D}J\!</math> in that block. Finally, the remainder is computed as <math>\mathrm{r}J = \mathrm{D}J + \mathrm{d}J,\!</math> in this case yielding the second order differential <math>\mathrm{d}^2\!J.\!</math><br />
<br />
====Analytic Series : Recap====<br />
<br />
Let us now summarize the results of Table&nbsp;50 by writing down for each column and for each block of constant argument pairs <math>u, v\!</math> a reasonably canonical symbolic expression for the function of <math>\mathrm{d}u, \mathrm{d}v\!</math> that appears there. The synopsis formed in this way is presented in Table&nbsp;51. As one has a right to expect, it confirms the results that were obtained previously by operating solely in terms of the formal calculus.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:70%"<br />
|+ style="height:30px" | <math>\text{Table 51.} ~~ \text{Computation of an Analytic Series in Symbolic Terms}\!</math><br />
|- style="height:35px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black" | <math>J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{E}J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{D}J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{d}J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{d}^2\!J\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}u \!\;\cdot\;\! \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}u \!\;\cdot\;\! \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
\mathrm{d}u<br />
\\[4pt]<br />
\mathrm{d}v<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;52 and 53 provide a quick overview of the analysis performed so far, giving the successive decompositions of <math>\mathrm{E}J = J + \mathrm{D}J\!</math> and <math>\mathrm{D}J = \mathrm{d}J + \mathrm{r}J\!</math> in two different styles of diagram.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 52 -- Decomposition of EJ.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 52.} ~~ \text{Decomposition of}~ \mathrm{E}J\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 53 -- Decomposition of DJ.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 53.} ~~ \text{Decomposition of}~ \mathrm{D}J\!</math><br />
|}<br />
<br />
====Terminological Interlude====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Lastly, my attention was especially attracted, not so much to the scene, as to the mirrors that produced it. These mirrors were broken in parts. Yes, they were marked and scratched; they had been &ldquo;starred&rdquo;, in spite of their solidity &hellip;<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 230]<br />
|}<br />
<br />
At this point several issues of terminology have accrued enough substance to intrude on our discussion. The remarks of this Subsection are intended to accomplish two goals. First, we call attention to significant aspects of the previous series of Figures, translating into literal terms what they depict in iconic forms, and we re-stress the most important structural elements they indicate. Next, we prepare the way for taking on more complex examples of transformations, those whose target universes have more than one dimension.<br />
<br />
In talking about the actions of operators it is important to keep in mind the distinctions between the operators per&nbsp;se, their operands, and their results. Furthermore, in working with composite forms of operators <math>\mathrm{W} = (\mathrm{W}_1, \ldots, \mathrm{W}_n),\!</math> transformations <math>\mathrm{F} = (\mathrm{F}_1, \ldots, \mathrm{F}_n),\!</math> and target domains <math>X^\bullet = [x_1, \ldots, x_n],\!</math> we need to preserve a clear distinction between the compound entity of each given type and any one of its separate components. It is curious, given the usefulness of the concepts ''operator'' and ''operand'', that we seem to lack a generic term, formed on the same root, for the corresponding result of an operation. Following the obvious paradigm would lead to words like ''opus'', ''opera'', and ''operant'', but these words are too affected with clang associations to work well at present, though they might be adapted in time. One current usage gets around this problem by using the substantive ''map'' as a systematic epithet to express the result of each operator's action. We will follow this practice as far as possible, for example, using the phrase ''tangent map'' to denote the end product of the tangent functor acting on its operand map.<br />
<br />
* '''Scholium.''' See [JGH, 6-9] for a good account of tangent functors and tangent maps in ordinary analysis and for examples of their use in mechanics. This work as a whole is a model of clarity in applying functorial principles to problems in physical dynamics.<br />
<br />
Whenever we focus on isolated propositions, on single components of composite operators, or on the portions of transformations that have <math>1\!</math>-dimensional ranges, we are free to shift between the native form of a proposition <math>J : U \to \mathbb{B}\!</math> and the thematized form of a mapping <math>J : U^\bullet \to [x]\!</math> without much trouble. In these cases we are able to tolerate a higher degree of ambiguity about the precise nature of an operator's input and output domains than we otherwise might. For example, in the preceding treatment of the example <math>J,\!</math> and for each operator <math>\mathrm{W}\!</math> in the set <math>\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \},\!</math> both the operand <math>J\!</math> and the result <math>\mathrm{W}J\!</math> could be viewed in either one of two ways. On one hand we may treat them as propositions <math>J : U \to \mathbb{B}\!</math> and <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B},\!</math> ignoring the distinction between the range <math>[x] \cong \mathbb{B}\!</math> of <math>\boldsymbol\varepsilon J\!</math> and the range <math>[\mathrm{d}x] \cong \mathbb{D}\!</math> of the other types of <math>\mathrm{W}J.\!</math> This is what we usually do when we content ourselves with simply coloring in regions of venn diagrams. On the other hand we may view these entities as maps <math>J : U^\bullet \to [x] = X^\bullet\!</math> and <math>\boldsymbol\varepsilon J : \mathrm{E}U^\bullet \to [x] \subseteq \mathrm{E}X^\bullet\!</math> or <math>\mathrm{W}J : \mathrm{E}U^\bullet \to [\mathrm{d}x] \subseteq \mathrm{E}X^\bullet,\!</math> in which case the qualitative characters of the output features are not ignored.<br />
<br />
At the beginning of this Section we recast the natural form of a proposition <math>J : U \to \mathbb{B}\!</math> into the thematic role of a transformation <math>J : U^\bullet \to [x],\!</math> where <math>x\!</math> was a variable recruited to express the newly independent <math>\check{J}.\!</math> However, in our computations and representations of operator actions we immediately lapsed back to viewing the results as native elements of the extended universe <math>\mathrm{E}U^\bullet,\!</math> in other words, as propositions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B},\!</math> where <math>\mathrm{W}\!</math> ranged over the set <math>\{ \boldsymbol\varepsilon, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}.\!</math> That is as it should be. We have worked hard to devise a language that gives us these advantages &mdash; the flexibility to exchange terms and types of equal information value and the capacity to reflect as quickly and as wittingly as a controlled reflex on the fibers of our propositions, independently of whether they express amusements, beliefs, or conjectures.<br />
<br />
As we take on target spaces of increasing dimension, however, these types of confusions (and confusions of types) become less and less permissible. For this reason, Tables&nbsp;54 and 55 present a rather detailed summary of the notation and the terminology we are using, as applied to the case <math>J = uv.\!</math> The rationale of these Tables is not so much to train more elephant guns on this poor drosophila of a concrete example but to invest our paradigm with enough solidity to bear the weight of abstraction to come.<br />
<br />
Table&nbsp;54 provides basic notation and descriptive information for the objects and operators used in this Example, giving the generic type (or broadest defined type) for each entity. Here, the sans&nbsp;serif operators <math>\mathsf{W} \in \{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{d}, \mathsf{r} \}\!</math> and their components <math>\mathrm{W} \in \{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}\!</math> both have the same broad type <math>\mathsf{W}, \mathrm{W} : (U^\bullet \to X^\bullet) \to (\mathrm{E}U^\bullet \to \mathrm{E}X^\bullet),\!</math> as appropriate to operators that map transformations <math>J : U^\bullet \to X^\bullet\!</math> to extended transformations <math>\mathsf{W}J, \mathrm{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 54.} ~~ \text{Cast of Characters : Expansive Subtypes of Objects and Operators}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| align="center" | <math>\text{Symbol}\!</math><br />
| align="center" | <math>\text{Notation}\!</math><br />
| align="center" | <math>\text{Description}\!</math><br />
| align="center" | <math>\text{Type}\!</math><br />
|-<br />
| align="center" | <math>U^\bullet\!</math><br />
| <math>= [u, v]\!</math><br />
| <math>\text{Source universe}\!</math><br />
| <math>[\mathbb{B}^2]\!</math><br />
|-<br />
| align="center" | <math>X^\bullet~\!</math><br />
| <math>= [x]\!</math><br />
| <math>\text{Target universe}\!</math><br />
| <math>[\mathbb{B}^1]~\!</math><br />
|-<br />
| align="center" | <math>\mathrm{E}U^\bullet\!</math><br />
| <math>= [u, v, \mathrm{d}u, \mathrm{d}v]\!</math><br />
| <math>\text{Extended source universe}\!</math><br />
| <math>[\mathbb{B}^2 \!\times\! \mathbb{D}^2]</math><br />
|-<br />
| align="center" | <math>\mathrm{E}X^\bullet\!</math><br />
| <math>= [x, \mathrm{d}x]~\!</math><br />
| <math>\text{Extended target universe}\!</math><br />
| <math>[\mathbb{B}^1 \!\times\! \mathbb{D}^1]</math><br />
|-<br />
| align="center" | <math>J\!</math><br />
| <math>J : U \!\to\! \mathbb{B}\!</math><br />
| <math>\text{Proposition}\!</math><br />
| <math>(\mathbb{B}^2 \!\to\! \mathbb{B}) \in [\mathbb{B}^2]\!</math><br />
|-<br />
| align="center" | <math>J\!</math><br />
| <math>J : U^\bullet \!\to\! X^\bullet\!</math><br />
| <math>\text{Transformation or Map}\!</math><br />
| <math>[\mathbb{B}^2] \!\to\! [\mathbb{B}^1]\!</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\boldsymbol\varepsilon<br />
\\<br />
\eta<br />
\\<br />
\mathrm{E}<br />
\\<br />
\mathrm{D}<br />
\\<br />
\mathrm{d}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{W} : U^\bullet \!\to\! \mathrm{E}U^\bullet,<br />
\\<br />
\mathrm{W} : X^\bullet \!\to\! \mathrm{E}X^\bullet,<br />
\\<br />
\mathrm{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\\<br />
\text{for each}~ \mathrm{W} ~\text{in the set:}<br />
\\<br />
\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{ll}<br />
\text{Tacit extension operator} & \boldsymbol\varepsilon<br />
\\<br />
\text{Trope extension operator} & \eta<br />
\\<br />
\text{Enlargement operator} & \mathrm{E}<br />
\\<br />
\text{Difference operator} & \mathrm{D}<br />
\\<br />
\text{Differential operator} & \mathrm{d}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^2] \!\to\! [\mathbb{B}^2 \!\times\! \mathbb{D}^2]},<br />
\\<br />
{[\mathbb{B}^1] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]},<br />
\\\\<br />
([\mathbb{B}^2] \!\to\! [\mathbb{B}^1]) \!\to\!<br />
\\<br />
([\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1])<br />
\end{array}</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\mathsf{e}<br />
\\<br />
\mathsf{E}<br />
\\<br />
\mathsf{D}<br />
\\<br />
\mathsf{T}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{W} : U^\bullet \!\to\! \mathsf{T}U^\bullet = \mathrm{E}U^\bullet,<br />
\\<br />
\mathsf{W} : X^\bullet \!\to\! \mathsf{T}X^\bullet = \mathrm{E}X^\bullet,<br />
\\<br />
\mathsf{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathsf{T}U^\bullet \!\to\! \mathsf{T}X^\bullet)<br />
\\<br />
\text{for each}~ \mathsf{W} ~\text{in the set:}<br />
\\<br />
\{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{T} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{lll}<br />
\text{Radius operator} & \mathsf{e} & = (\boldsymbol\varepsilon, \eta)<br />
\\<br />
\text{Secant operator} & \mathsf{E} & = (\boldsymbol\varepsilon, \mathrm{E})<br />
\\<br />
\text{Chord operator} & \mathsf{D} & = (\boldsymbol\varepsilon, \mathrm{D})<br />
\\<br />
\text{Tangent functor} & \mathsf{T} & = (\boldsymbol\varepsilon, \mathrm{d})<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^2] \!\to\! [\mathbb{B}^2 \!\times\! \mathbb{D}^2]},<br />
\\<br />
{[\mathbb{B}^1] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]},<br />
\\\\<br />
([\mathbb{B}^2] \!\to\! [\mathbb{B}^1]) \!\to\!<br />
\\<br />
([\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1])<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;55 supplies a more detailed outline of terminology for operators and their results. Here, we list the restrictive subtype (or narrowest defined subtype) that applies to each entity and we indicate across the span of the Table the whole spectrum of alternative types that color the interpretation of each symbol. For example, all the component operator maps <math>\mathrm{W}J\!</math> have <math>1\!</math>-dimensional ranges, either <math>\mathbb{B}^1\!</math> or <math>\mathbb{D}^1,\!</math> and so they can be viewed either as propositions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B}\!</math> or as logical transformations <math>\mathrm{W}J : \mathrm{E}U^\bullet \to X^\bullet.\!</math> As a rule, the plan of the Table allows us to name each entry by detaching the underlined adjective at the left of its row and prefixing it to the generic noun at the top of its column. In one case, however, it is customary to depart from this scheme. Because the phrase ''differential proposition'', applied to the result <math>\mathrm{d}J : \mathrm{E}U \to \mathbb{D},\!</math> does not distinguish it from the general run of differential propositions <math>\mathrm{G}: \mathrm{E}U \to \mathbb{B},\!</math> it is usual to single out <math>\mathrm{d}J\!</math> as the ''tangent proposition'' of <math>J.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 55.} ~~ \text{Synopsis of Terminology : Restrictive and Alternative Subtypes}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| &nbsp;<br />
| align="center" | <math>\text{Operator}\!</math><br />
| align="center" | <math>\text{Proposition}\!</math><br />
| align="center" | <math>\text{Map}\!</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tacit}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\boldsymbol\varepsilon : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\boldsymbol\varepsilon : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{B} \\<br />
\boldsymbol\varepsilon J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{B}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x] \\<br />
\boldsymbol\varepsilon J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Trope}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\eta : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\eta : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\eta J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\eta J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Enlargement}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{E} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{E}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{E}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Difference}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{D} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{D}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{D}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Differential}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{d} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{d} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{d}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}\!</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Remainder}}\\\text{operator}\end{matrix}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{r} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{r} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{r}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{r}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Radius}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e} = (\boldsymbol\varepsilon, \eta) \\<br />
\mathsf{e} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{e}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Secant}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}) \\<br />
\mathsf{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{E}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Chord}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}) \\<br />
\mathsf{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{D}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tangent}}\\\text{functor}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T} = (\boldsymbol\varepsilon, \mathrm{d}) \\<br />
\mathsf{T} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{T}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====End of Perfunctory Chatter : Time to Roll the Clip!====<br />
<br />
Two steps remain to finish the analysis of <math>J\!</math> that we began so long ago. First, we need to paste our accumulated heap of flat pictures into the frames of transformations, filling out the shapes of the operator maps <math>\mathsf{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet.~\!</math> This scheme is executed in two styles, using the ''areal views'' in Figures&nbsp;56-a and the ''box views'' in Figures&nbsp;56-b. Finally, in Figures&nbsp;57-1 to 57-4 we put all the pieces together to construct the full operator diagrams for <math>\mathsf{W} : J \to \mathsf{W}J.\!</math> There is a considerable amount of redundancy among the following three series of Figures but that will hopefully provide a fuller picture of the operations under review, enabling these snapshots to serve as successive frames in the animation of logic they are meant to become.<br />
<br />
=====Operator Maps : Areal Views=====<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a1 -- Radius Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a1.} ~~ \text{Radius Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a2 -- Secant Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a2.} ~~ \text{Secant Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a3 -- Chord Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a3.} ~~ \text{Chord Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a4 -- Tangent Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a4.} ~~ \text{Tangent Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
=====Operator Maps : Box Views=====<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b1 -- Radius Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b1.} ~~ \text{Radius Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b2 -- Secant Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b2.} ~~ \text{Secant Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b3 -- Chord Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b3.} ~~ \text{Chord Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b4 -- Tangent Map of J ISW.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b4.} ~~ \text{Tangent Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
=====Operator Diagrams for the Conjunction J = uv=====<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-1 -- Radius Operator Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-1.} ~~ \text{Radius Operator Diagram for the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-2 -- Secant Operator Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-2.} ~~ \text{Secant Operator Diagram for the Conjunction}~ J = uv~\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-3 -- Chord Operator Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-3.} ~~ \text{Chord Operator Diagram for the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-4 -- Tangent Functor Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-4.} ~~ \text{Tangent Functor Diagram for the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
===Taking Aim at Higher Dimensional Targets===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
The past and present wilt . . . . I have filled them and<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;emptied them,<br><br />
And proceed to fill my next fold of the future.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 87]<br />
|}<br />
<br />
In the next Section we consider a transformation <math>F\!</math> of concrete type <math>F : [u, v] \to [x, y]\!</math> and abstract type <math>F : [\mathbb{B}^2] \to [\mathbb{B}^2].\!</math> From the standpoint of propositional calculus we naturally approach the task of understanding such a transformation by parsing it into component maps with <math>1\!</math>-dimensional ranges, as follows:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{ccccccl}<br />
F & = & (F_1, F_2) & = & (f, g) & : & [u, v] \to [x, y],<br />
\\[6pt]<br />
&& F_1 & = & f & : & [u, v] \to [x],<br />
\\[6pt]<br />
&& F_2 & = & g & : & [u, v] \to [y].<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Then we tackle the separate components, now viewed as propositions <math>F_i : U \to \mathbb{B},\!</math> one at a time. At the completion of this analytic phase, we return to the task of synthesizing these partial and transient impressions into an agile form of integrity, a solidly coordinated and deeply integrated comprehension of the ongoing transformation. (Very often, of course, in tangling with refractory cases, we never get as far as the beginning again.)<br />
<br />
Let us now refer to the dimension of the target space or codomain as the ''toll'' (or ''tole'') of a transformation, as distinguished from the dimension of the range or image that is customarily called the ''rank''. When we keep to transformations with a toll of <math>1,\!</math> as <math>J : [u, v] \to [x],\!</math> we tend to get lazy about distinguishing a logical transformation from its component propositions. However, if we deal with transformations of a higher toll, this form of indolence can no longer be tolerated.<br />
<br />
Well, perhaps we can carry it a little further. After all, the operator result <math>\mathrm{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet\!</math> is a map of toll <math>2,\!</math> and cannot be unfolded in one piece as a proposition. But when a map has rank <math>1,\!</math> like <math>\boldsymbol\varepsilon J : \mathrm{E}U \to X \subseteq \mathrm{E}X\!</math> or <math>\mathrm{d}J : \mathrm{E}U \to \mathrm{d}X \subseteq \mathrm{E}X,\!</math> we naturally choose to concentrate on the <math>1\!</math>-dimensional range of the operator result <math>\mathrm{W}J,\!</math> ignoring the final difference in quality between the spaces <math>X\!</math> and <math>\mathrm{d}X,\!</math> and view <math>\mathrm{W}J\!</math> as a proposition about <math>\mathrm{E}U.\!</math><br />
<br />
In this way, an initial ambivalence about the role of the operand <math>J\!</math> conveys a double duty to the result <math>\mathrm{W}J.\!</math> The pivot that is formed by our focus of attention is essential to the linkage that transfers this double moment, as the whole process takes its bearing and wheels around the precise measure of a narrow bead that we can draw on the range of <math>\mathrm{W}J.\!</math> This is the escapement that it takes to get away with what may otherwise seem to be a simple duplicity, and this is the tolerance that is needed to counterbalance a certain arrogance of equivocation, by all of which machinations we make ourselves free to indicate the operator results <math>\mathrm{W}J\!</math> as propositions or as transformations, indifferently.<br />
<br />
But that's it, and no further. Neglect of these distinctions in range and target universes of higher dimensions is bound to cause a hopeless confusion. To guard against these adverse prospects, Tables&nbsp;58 and 59 lay the groundwork for discussing a typical map <math>F : [\mathbb{B}^2] \to [\mathbb{B}^2],\!</math> and begin to pave the way to some extent for discussing any transformation of the form <math>F : [\mathbb{B}^n] \to [\mathbb{B}^k].\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 58.} ~~ \text{Cast of Characters : Expansive Subtypes of Objects and Operators}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| align="center" | <math>\text{Symbol}\!</math><br />
| align="center" | <math>\text{Notation}\!</math><br />
| align="center" | <math>\text{Description}\!</math><br />
| align="center" | <math>\text{Type}\!</math><br />
|-<br />
| align="center" | <math>U^\bullet\!</math><br />
| <math>= [u, v]\!</math><br />
| <math>\text{Source universe}\!</math><br />
| <math>[\mathbb{B}^n]\!</math><br />
|-<br />
| align="center" | <math>X^\bullet~\!</math><br />
| <math>\begin{array}{l}<br />
= [x, y] \\<br />
= [f, g]<br />
\end{array}</math><br />
| <math>\text{Target universe}\!</math><br />
| <math>[\mathbb{B}^k]\!</math><br />
|-<br />
| align="center" | <math>\mathrm{E}U^\bullet\!</math><br />
| <math>= [u, v, \mathrm{d}u, \mathrm{d}v]\!</math><br />
| <math>\text{Extended source universe}\!</math><br />
| <math>[\mathbb{B}^n \!\times\! \mathbb{D}^n]\!</math><br />
|-<br />
| align="center" | <math>\mathrm{E}X^\bullet\!</math><br />
| <math>\begin{array}{l}<br />
= [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
= [f, g, \mathrm{d}f, \mathrm{d}g]<br />
\end{array}</math><br />
| <math>\text{Extended target universe}\!</math><br />
| <math>[\mathbb{B}^k \!\times\! \mathbb{D}^k]\!</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
f \\ g<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{ll}<br />
f : U \!\to\! [x] \cong \mathbb{B} \\<br />
g : U \!\to\! [y] \cong \mathbb{B}<br />
\end{array}</math><br />
| <math>\text{Proposition}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathbb{B}^n \!\to\! \mathbb{B} \\<br />
\in (\mathbb{B}^n, \mathbb{B}^n \!\to\! \mathbb{B}) = [\mathbb{B}^n]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>F\!</math><br />
| <math>F = (f, g) : U^\bullet \!\to\! X^\bullet\!</math><br />
| <math>\text{Transformation of Map}\!</math><br />
| <math>[\mathbb{B}^n] \!\to\! [\mathbb{B}^k]</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\boldsymbol\varepsilon<br />
\\<br />
\eta<br />
\\<br />
\mathrm{E}<br />
\\<br />
\mathrm{D}<br />
\\<br />
\mathrm{d}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{W} : U^\bullet \!\to\! \mathrm{E}U^\bullet,<br />
\\<br />
\mathrm{W} : X^\bullet \!\to\! \mathrm{E}X^\bullet,<br />
\\<br />
\mathrm{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\\<br />
\text{for each}~ \mathrm{W} ~\text{in the set:}<br />
\\<br />
\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{ll}<br />
\text{Tacit extension operator} & \boldsymbol\varepsilon<br />
\\<br />
\text{Trope extension operator} & \eta<br />
\\<br />
\text{Enlargement operator} & \mathrm{E}<br />
\\<br />
\text{Difference operator} & \mathrm{D}<br />
\\<br />
\text{Differential operator} & \mathrm{d}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^n] \!\to\! [\mathbb{B}^n \!\times\! \mathbb{D}^n]},<br />
\\<br />
{[\mathbb{B}^k] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]},<br />
\\\\<br />
([\mathbb{B}^n] \!\to\! [\mathbb{B}^k]) \!\to\!<br />
\\<br />
([\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k])<br />
\end{array}</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\mathsf{e}<br />
\\<br />
\mathsf{E}<br />
\\<br />
\mathsf{D}<br />
\\<br />
\mathsf{T}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{W} : U^\bullet \!\to\! \mathsf{T}U^\bullet = \mathrm{E}U^\bullet,<br />
\\<br />
\mathsf{W} : X^\bullet \!\to\! \mathsf{T}X^\bullet = \mathrm{E}X^\bullet,<br />
\\<br />
\mathsf{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathsf{T}U^\bullet \!\to\! \mathsf{T}X^\bullet)<br />
\\<br />
\text{for each}~ \mathsf{W} ~\text{in the set:}<br />
\\<br />
\{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{T} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{lll}<br />
\text{Radius operator} & \mathsf{e} & = (\boldsymbol\varepsilon, \eta)<br />
\\<br />
\text{Secant operator} & \mathsf{E} & = (\boldsymbol\varepsilon, \mathrm{E})<br />
\\<br />
\text{Chord operator} & \mathsf{D} & = (\boldsymbol\varepsilon, \mathrm{D})<br />
\\<br />
\text{Tangent functor} & \mathsf{T} & = (\boldsymbol\varepsilon, \mathrm{d})<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^n] \!\to\! [\mathbb{B}^n \!\times\! \mathbb{D}^n]},<br />
\\<br />
{[\mathbb{B}^k] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]},<br />
\\\\<br />
([\mathbb{B}^n] \!\to\! [\mathbb{B}^k]) \!\to\!<br />
\\<br />
([\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k])<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 59.} ~~ \text{Synopsis of Terminology : Restrictive and Alternative Subtypes}~\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| &nbsp;<br />
| align="center" | <math>\begin{matrix}\text{Operator}\\\text{or}\\\text{Operand}\end{matrix}</math><br />
| align="center" | <math>\begin{matrix}\text{Proposition}\\\text{or}\\\text{Component}\end{matrix}</math><br />
| align="center" | <math>\begin{matrix}\text{Transformation}\\\text{or}\\\text{Map}\end{matrix}</math><br />
|-<br />
| align="center" | <math>\underline{\text{Operand}}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
F = (F_1, F_2) \\<br />
F = (f, g) : U \!\to\! X<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
F_i : \langle u, v \rangle \!\to\! \mathbb{B} \\<br />
F_i : \mathbb{B}^n \!\to\! \mathbb{B}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
F : [u, v] \!\to\! [x, y] \\<br />
F : [\mathbb{B}^n] \!\to\! [\mathbb{B}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tacit}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\boldsymbol\varepsilon : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\boldsymbol\varepsilon : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{B} \\<br />
\boldsymbol\varepsilon F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{B}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y] \\<br />
\boldsymbol\varepsilon F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Trope}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\eta : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\eta : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\eta F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\eta F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Enlargement}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{E} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{E}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{E}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Difference}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{D} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{D}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{D}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Differential}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{d} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{d} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{d}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}\!</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Remainder}}\\\text{operator}\end{matrix}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{r} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{r} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{r}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{r}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Radius}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e} = (\boldsymbol\varepsilon, \eta) \\<br />
\mathsf{e} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{e}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Secant}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}) \\<br />
\mathsf{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{E}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Chord}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}) \\<br />
\mathsf{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{D}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tangent}}\\\text{functor}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T} = (\boldsymbol\varepsilon, \mathrm{d}) \\<br />
\mathsf{T} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{T}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
===Transformations of Type '''B'''<sup>2</sup> &rarr; '''B'''<sup>2</sup>===<br />
<br />
To take up a slightly more complex example, but one that remains simple enough to pursue through a complete series of developments, consider the transformation from <math>U^\bullet = [u, v]\!</math> to <math>X^\bullet = [x, y]\!</math> that is defined by the following system of equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
x<br />
& = & f(u, v)<br />
& = & \texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[8pt]<br />
y<br />
& = & g(u, v)<br />
& = & \texttt{((} u \texttt{,} v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
The component notation <math>F = (F_1, F_2) = (f, g) : U^\bullet \to X^\bullet\!</math> allows us to give a name and a type to this transformation and permits defining it by the compact description that follows:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
(x, y)<br />
& = & F(u, v)<br />
& = & (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Logical Transformations====<br />
<br />
The information that defines the logical transformation <math>F\!</math> can be represented in the form of a truth table, as shown in Table&nbsp;60. To cut down on subscripts in this example we continue to use plain letter equivalents for all components of spaces and maps.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:60%"<br />
|+ style="height:30px" | <math>\text{Table 60.} ~~ \text{A Propositional Transformation}\!</math><br />
|- style="height:40px; background:ghostwhite; width:100%"<br />
| style="width:25%" | <math>u\!</math><br />
| style="width:25%" | <math>v\!</math><br />
| style="width:25%; border-left:1px solid black" | <math>f\!</math><br />
| style="width:25%" | <math>g\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="border-top:1px solid black" | <math>u\!</math><br />
| style="border-top:1px solid black" | <math>v\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;61 shows how we might paint a picture of the transformation <math>F\!</math> in the manner of Figure&nbsp;30.<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 61 -- Propositional Transformation.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 61.} ~~ \text{A Propositional Transformation}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;62 extracts the gist of Figure&nbsp;61, exhibiting a style of diagram that is adequate for most purposes.<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 62 -- Propositional Transformation (Short Form).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 62.} ~~ \text{A Propositional Transformation (Short Form)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Local Transformations====<br />
<br />
Figure&nbsp;63 gives a more complete picture of the transformation <math>F,\!</math> showing how the points of <math>U^\bullet\!</math> are transformed into points of <math>X^\bullet.\!</math> The bold lines crossing from one universe to the other trace the action that <math>F\!</math> induces on points, in other words, they show how the transformation acts as a mapping from points to points and chart its effects on the elements that are variously called cells, points, positions, or singular propositions.<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 63 -- Transformation of Positions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 63.} ~~ \text{A Transformation of Positions}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;64 shows how the action of <math>F\!</math> on cells or points can be computed in terms of coordinates.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 64.} ~~ \text{A Transformation of Positions}\!</math><br />
|- style="height:40px; background:ghostwhite; width:100%"<br />
| style="width:8%" | <math>u\!</math><br />
| style="width:8%" | <math>v\!</math><br />
| style="width:12%; border-left:1px solid black" | <math>x\!</math><br />
| style="width:12%" | <math>y\!</math><br />
| style="width:10%; border-left:1px solid black" | <math>x~y\!</math><br />
| style="width:10%" | <math>x \texttt{(} y \texttt{)}\!</math><br />
| style="width:10%" | <math>\texttt{(} x \texttt{)} y\!</math><br />
| style="width:10%" | <math>\texttt{(} x \texttt{)(} y \texttt{)}\!</math><br />
| style="width:20%; border-left:1px solid black" | <math>X^\bullet = [x, y]\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\uparrow<br />
\\[4pt]<br />
F =<br />
\\[4pt]<br />
(f, g)<br />
\\[4pt]<br />
\uparrow<br />
\end{matrix}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="border-top:1px solid black" | <math>u\!</math><br />
| style="border-top:1px solid black" | <math>v\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
| style="border-top:1px solid black" | <math>\texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>u~v\!</math><br />
| style="border-top:1px solid black" | <math>\texttt{(} u \texttt{,} v \texttt{)}\!</math><br />
| style="border-top:1px solid black" | <math>\texttt{(} u \texttt{)(} v \texttt{)}\!</math><br />
| style="border-top:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>U^\bullet = [u, v]\!</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;65 extends this scheme from single cells to arbitrary regions, showing how we might compute the action of a logical transformation on arbitrary propositions in the universe of discourse. The effect of a point-transformation on arbitrary propositions, or any other structures erected on points, is referred to as the ''induced action'' of the transformation on the structures in question.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 65-a.} ~~ \text{An Induced Transformation on Propositions}\!</math><br />
|- style="height:50px; background:ghostwhite"<br />
| style="width:20%" | <math>X^\bullet~\!</math><br />
| style="width:20%; border-right:none" | <math>\longleftarrow\!</math><br />
| style="width:20%; border-left:none; border-right:none" | <math>F = (f, g)\!</math><br />
| style="width:20%; border-left:none" | <math>\longleftarrow\!</math><br />
| style="width:20%" | <math>U^\bullet~\!</math><br />
|- style="background:ghostwhite"<br />
| rowspan="2" | <math>f_i (x, y)\!</math><br />
| align="right" | <math>\begin{matrix}u = \\ v =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~0~0\\1~0~1~0\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= u \\ = v\end{matrix}</math><br />
| rowspan="2" style="width:20%" | <math>f_j (u, v)\!</math><br />
|- style="background:ghostwhite"<br />
| align="right" | <math>\begin{matrix}x = \\ y =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~1~0\\1~0~0~1\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= f(u, v) \\ = g(u, v)\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{2}<br />
\\[2pt]<br />
f_{3}<br />
\\[2pt]<br />
f_{4}<br />
\\[2pt]<br />
f_{5}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{7}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)~ ~}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{~ ~(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{,~} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{~~} y \texttt{)}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~0<br />
\\[2pt]<br />
0~0~0~0<br />
\\[2pt]<br />
0~0~0~1<br />
\\[2pt]<br />
0~0~0~1<br />
\\[2pt]<br />
0~1~1~0<br />
\\[2pt]<br />
0~1~1~0<br />
\\[2pt]<br />
0~1~1~1<br />
\\[2pt]<br />
0~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{,~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{,~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{~~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{~~} v \texttt{)}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[2pt]<br />
f_{0}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{7}<br />
\\[2pt]<br />
f_{7}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{8}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{10}<br />
\\[2pt]<br />
f_{11}<br />
\\[2pt]<br />
f_{12}<br />
\\[2pt]<br />
f_{13}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{15}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[2pt]<br />
\texttt{~ ~ ~ ~} y \texttt{~~}<br />
\\[2pt]<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[2pt]<br />
\texttt{~~} x \texttt{~ ~ ~ ~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
1~0~0~0<br />
\\[2pt]<br />
1~0~0~0<br />
\\[2pt]<br />
1~0~0~1<br />
\\[2pt]<br />
1~0~0~1<br />
\\[2pt]<br />
1~1~1~0<br />
\\[2pt]<br />
1~1~1~0<br />
\\[2pt]<br />
1~1~1~1<br />
\\[2pt]<br />
1~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~~} u \texttt{~~} v \texttt{~~}<br />
\\[2pt]<br />
\texttt{~~} u \texttt{~~} v \texttt{~~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{8}<br />
\\[2pt]<br />
f_{8}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{15}<br />
\\[2pt]<br />
f_{15}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 65-b.} ~~ \text{An Induced Transformation on Propositions}\!</math><br />
|- style="height:50px; background:ghostwhite"<br />
| style="width:20%" | <math>X^\bullet~\!</math><br />
| style="width:20%; border-right:none" | <math>\longleftarrow\!</math><br />
| style="width:20%; border-left:none; border-right:none" | <math>F = (f, g)\!</math><br />
| style="width:20%; border-left:none" | <math>\longleftarrow\!</math><br />
| style="width:20%" | <math>U^\bullet~\!</math><br />
|- style="background:ghostwhite"<br />
| rowspan="2" | <math>f_i (x, y)\!</math><br />
| align="right" | <math>\begin{matrix}u = \\ v =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~0~0\\1~0~1~0\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= u \\ = v\end{matrix}</math><br />
| rowspan="2" style="width:20%" | <math>f_j (u, v)\!</math><br />
|- style="background:ghostwhite"<br />
| align="right" | <math>\begin{matrix}x = \\ y =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~1~0\\1~0~0~1\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= f(u, v) \\ = g(u, v)\end{matrix}</math><br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>f_{0}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[2pt]<br />
f_{2}<br />
\\[2pt]<br />
f_{4}<br />
\\[2pt]<br />
f_{8}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~0<br />
\\[2pt]<br />
0~0~0~1<br />
\\[2pt]<br />
0~1~1~0<br />
\\[2pt]<br />
1~0~0~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{,~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{~} u \texttt{~~} v \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{8}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{3}<br />
\\[2pt]<br />
f_{12}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[2pt]<br />
1~1~1~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} u \texttt{)(} v \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[2pt]<br />
f_{14}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[2pt]<br />
f_{9}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[2pt]<br />
1~0~0~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{~~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{~} u \texttt{~~} v \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[2pt]<br />
f_{8}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{5}<br />
\\[2pt]<br />
f_{10}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[2pt]<br />
1~0~0~1<br />
\end{matrix}~\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} u \texttt{,~} v \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[2pt]<br />
f_{9}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[2pt]<br />
f_{11}<br />
\\[2pt]<br />
f_{13}<br />
\\[2pt]<br />
f_{14}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[2pt]<br />
1~0~0~1<br />
\\[2pt]<br />
1~1~1~0<br />
\\[2pt]<br />
1~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} u \texttt{~~} v \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{15}<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>f_{15}\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Difference Operators and Tangent Functors====<br />
<br />
Given the alphabets <math>\mathcal{U} = \{ u, v \}\!</math> and <math>\mathcal{X} = \{ x, y \},\!</math> along with the corresponding universes of discourse <math>U^\bullet, X^\bullet \cong [\mathbb{B}^2],\!</math> how many logical transformations of the general form <math>G = (G_1, G_2) : U^\bullet \to X^\bullet\!</math> are there? Since <math>G_1\!</math> and <math>G_2\!</math> can be any propositions of the type <math>\mathbb{B}^2 \to \mathbb{B},\!</math> there are <math>2^4 = 16\!</math> choices for each of the maps <math>G_1\!</math> and <math>G_2\!</math> and thus there are <math>2^4 \cdot 2^4 = 2^8 = 256\!</math> different mappings altogether of the form <math>G : U^\bullet \to X^\bullet.\!</math> The set of functions of a given type is denoted by placing its type indicator in parentheses, in the present instance writing <math>(U^\bullet \to X^\bullet) = \{ G : U^\bullet \to X^\bullet \},\!</math> and so the cardinality of the ''function space'' <math>(U^\bullet \to X^\bullet)\!</math> is summed up by writing <math>|(U^\bullet \to X^\bullet)| = |(\mathbb{B}^2 \to \mathbb{B}^2)| = 4^4 = 256.\!</math><br />
<br />
Given a transformation <math>G = (G_1, G_2) : U^\bullet \to X^\bullet\!</math> of this type, we proceed to define a pair of further transformations, related to <math>G,\!</math> that operate between the extended universes, <math>\mathrm{E}U^\bullet\!</math> and <math>\mathrm{E}X^\bullet,\!</math> of its source and target domains.<br />
<br />
First, the ''enlargement map'' (or ''secant transformation'') <math>\mathrm{E}G = (\mathrm{E}G_1, \mathrm{E}G_2) : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet\!</math> is defined by the following set of component equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}G_i<br />
& = & G_i (u + \mathrm{d}u, v + \mathrm{d}v)<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Next, the ''difference map'' (or ''chordal transformation'') <math>\mathrm{D}G = (\mathrm{D}G_1, \mathrm{D}G_2) : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet~\!</math> is defined in component-wise fashion as the boolean sum of the initial proposition <math>G_i\!</math> and the enlarged proposition <math>\mathrm{E}G_i,\!</math> for <math>i = 1, 2,\!</math> according to the following set of equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
\mathrm{D}G_i<br />
& = & G_i (u, v)<br />
& + & \mathrm{E}G_i (u, v, \mathrm{d}u, \mathrm{d}v)<br />
\\[8pt]<br />
& = & G_i (u, v)<br />
& + & G_i (u + \mathrm{d}u, v + \mathrm{d}v)<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Maintaining a strict analogy with ordinary difference calculus would perhaps have us write <math>\mathrm{D}G_i = \mathrm{E}G_i - G_i,\!</math> but the sum and difference operations are the same thing in boolean arithmetic. It is more often natural in the logical context to consider an initial proposition <math>q,\!</math> then to compute the enlargement <math>\mathrm{E}q,\!</math> and finally to determine the difference <math>\mathrm{D}q = q + \mathrm{E}q,\!</math> so we let the variant order of terms reflect this sequence of considerations.<br />
<br />
Viewed in this light the difference operator <math>\mathrm{D}\!</math> is imagined to be a function of very wide scope and polymorphic application, one that is able to realize the association between each transformation <math>G\!</math> and its difference map <math>\mathrm{D}G,\!</math> for example, taking the function space <math>(U^\bullet \to X^\bullet)\!</math> into <math>(\mathrm{E}U^\bullet \to \mathrm{E}X^\bullet).\!</math> When we consider the variety of interpretations permitted to propositions over the contexts in which we put them to use, it should be clear that an operator of this scope is not at all a trivial matter to define in general and that it may take some trouble to work out. For the moment we content ourselves with returning to particular cases.<br />
<br />
Acting on the logical transformation <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;),\!</math> the operators <math>\mathrm{E}\!</math> and <math>\mathrm{D}\!</math> yield the enlarged map <math>\mathrm{E}F = (\mathrm{E}f, \mathrm{E}g)\!</math> and the difference map <math>\mathrm{D}F = (\mathrm{D}f, \mathrm{D}g),\!</math> respectively, whose components are given as follows.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}f<br />
& = & \texttt{((} u + \mathrm{d}u \texttt{)(} v + \mathrm{d}v \texttt{))}<br />
\\[8pt]<br />
\mathrm{E}g<br />
& = & \texttt{((} u + \mathrm{d}u \texttt{,~} v + \mathrm{d}v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
\mathrm{D}f<br />
& = & \texttt{((} u \texttt{)(} v \texttt{))}<br />
& + & \texttt{((} u + \mathrm{d}u \texttt{)(} v + \mathrm{d}v \texttt{))}<br />
\\[8pt]<br />
\mathrm{D}g<br />
& = & \texttt{((} u \texttt{,~} v \texttt{))}<br />
& + & \texttt{((} u + \mathrm{d}u \texttt{,~} v + \mathrm{d}v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
But these initial formulas are purely definitional, and help us little in understanding either the purpose of the operators or the meaning of their results. Working symbolically, let us apply the same method to the separate components <math>f\!</math> and <math>g\!</math> that we earlier used on <math>J.\!</math> This work is recorded in Appendix&nbsp;3 and a summary of the results is presented in Tables&nbsp;66-i and 66-ii.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 66-i.} ~~ \text{Computation Summary for}~ f(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f<br />
& = & u \!\cdot\! v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 1<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}f<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \cdot \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{d}f<br />
& = & u \!\cdot\! v \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}f<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 66-ii.} ~~ \text{Computation Summary for}~ g(u, v) = \texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon g<br />
& = & u \!\cdot\! v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}g<br />
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}g<br />
& = & u \!\cdot\! v \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;67 shows how to compute the analytic series for <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math> in terms of coordinates, and Table&nbsp;68 recaps these results in symbolic terms, agreeing with earlier derivations.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 67.} ~~ \text{Computation of an Analytic Series in Terms of Coordinates}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:6%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}u\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>\mathrm{d}v\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>u'\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>v'\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:4px double black" | <math>f\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>g\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{E}f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{E}g}\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{D}f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{D}g}\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{d}f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{d}g}\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{d}^2\!f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{d}^2\!g}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 68.} ~~ \text{Computation of an Analytic Series in Symbolic Terms}\!</math><br />
|- style="height:40px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black" | <math>f\!</math><br />
| <math>g\!</math><br />
| style="border-left:1px solid black" | <math>{\mathrm{D}f}\!</math><br />
| <math>{\mathrm{D}g}\!</math><br />
| style="border-left:1px solid black" | <math>{\mathrm{d}f}\!</math><br />
| <math>{\mathrm{d}g}\!</math><br />
| style="border-left:1px solid black" | <math>{\mathrm{d}^2\!f}\!</math><br />
| <math>{\mathrm{d}^2\!g}\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[4pt]<br />
\texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
\\[4pt]<br />
\texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
\\[4pt]<br />
\texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;69 gives a graphical picture of the difference map <math>\mathrm{D}F = (\mathrm{D}f, \mathrm{D}g)\!</math> for the transformation <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math> This represents the same information about <math>\mathrm{D}f~\!</math> and <math>\mathrm{D}g~\!</math> that was given in the corresponding rows of Tables&nbsp;66-i and 66-ii, for ease of reference repeated below.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[8pt]<br />
\mathrm{D}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 69 -- Difference Map (Short Form).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 69.} ~~ \text{Difference Map of}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;70-a shows a way of visualizing the tangent functor map <math>\mathrm{d}F = (\mathrm{d}f, \mathrm{d}g)\!</math> for the transformation <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math> This amounts to the same information about <math>\mathrm{d}f~\!</math> and <math>\mathrm{d}g~\!</math> that was given in Tables&nbsp;66-i and 66-ii, the corresponding rows of which are repeated below.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{d}f<br />
& = & u \!\cdot\! v \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[8pt]<br />
\mathrm{d}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 70-a -- Tangent Functor Diagram.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 70-a.} ~~ \text{Tangent Functor Diagram for}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math><br />
|}<br />
<br />
<br><br />
<br />
Figure 70-b shows another way to picture the action of the tangent functor on the logical transformation <math>F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 70-b -- Tangent Functor Diagram.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 70-b.} ~~ \text{Tangent Functor Ferris Wheel for}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math><br />
|}<br />
<br />
<br><br />
<br />
* '''Note.''' The original Figure&nbsp;70-b lost some of its labeling in a succession of platform metamorphoses over the years, so we have included an ASCII version below to indicate where the missing labels go.<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| /////\ /////\ | | /XXXX\ /XXXX\ | | /\\\\\ /\\\\\ |<br />
| ///////o//////\ | | /XXXXXXoXXXXXX\ | | /\\\\\\o\\\\\\\ |<br />
| //////// \//////\ | | /XXXXXX/ \XXXXXX\ | | /\\\\\\/ \\\\\\\\ |<br />
| o/////// \//////o | | oXXXXXX/ \XXXXXXo | | o\\\\\\/ \\\\\\\o |<br />
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |<br />
| |/du//| |//dv/| | | |XXXXX| |XXXXX| | | |\du\\| |\\dv\| |<br />
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |<br />
| o//////\ ///////o | | oXXXXXX\ /XXXXXXo | | o\\\\\\\ /\\\\\\o |<br />
| \//////\ //////// | | \XXXXXX\ /XXXXXX/ | | \\\\\\\\ /\\\\\\/ |<br />
| \//////o/////// | | \XXXXXXoXXXXXX/ | | \\\\\\\o\\\\\\/ |<br />
| \///// \///// | | \XXXX/ \XXXX/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ (u)(v) o-----------------------o dv' @ (u)(v) =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| / \ /////\ | | /\\\\\ /XXXX\ | | /\\\\\ /\\\\\ |<br />
| / o//////\ | | /\\\\\\oXXXXXX\ | | /\\\\\\o\\\\\\\ |<br />
| / //\//////\ | | /\\\\\\//\XXXXXX\ | | /\\\\\\/ \\\\\\\\ |<br />
| o ////\//////o | | o\\\\\\////\XXXXXXo | | o\\\\\\/ \\\\\\\o |<br />
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |<br />
| | du |/////|//dv/| | | |\\\\\|/////|XXXXX| | | |\du\\| |\\dv\| |<br />
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |<br />
| o \//////////o | | o\\\\\\\////XXXXXXo | | o\\\\\\\ /\\\\\\o |<br />
| \ \///////// | | \\\\\\\\//XXXXXX/ | | \\\\\\\\ /\\\\\\/ |<br />
| \ o/////// | | \\\\\\\oXXXXXX/ | | \\\\\\\o\\\\\\/ |<br />
| \ / \///// | | \\\\\/ \XXXX/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ (u) v o-----------------------o dv' @ (u) v =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| /////\ / \ | | /XXXX\ /\\\\\ | | /\\\\\ /\\\\\ |<br />
| ///////o \ | | /XXXXXXo\\\\\\\ | | /\\\\\\o\\\\\\\ |<br />
| /////////\ \ | | /XXXXXX//\\\\\\\\ | | /\\\\\\/ \\\\\\\\ |<br />
| o//////////\ o | | oXXXXXX////\\\\\\\o | | o\\\\\\/ \\\\\\\o |<br />
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |<br />
| |/du//|/////| dv | | | |XXXXX|/////|\\\\\| | | |\du\\| |\\dv\| |<br />
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |<br />
| o//////\//// o | | oXXXXXX\////\\\\\\o | | o\\\\\\\ /\\\\\\o |<br />
| \//////\// / | | \XXXXXX\//\\\\\\/ | | \\\\\\\\ /\\\\\\/ |<br />
| \//////o / | | \XXXXXXo\\\\\\/ | | \\\\\\\o\\\\\\/ |<br />
| \///// \ / | | \XXXX/ \\\\\/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ u (v) o-----------------------o dv' @ u (v) =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| / \ / \ | | /\\\\\ /\\\\\ | | /\\\\\ /\\\\\ |<br />
| / o \ | | /\\\\\\o\\\\\\\ | | /\\\\\\o\\\\\\\ |<br />
| / / \ \ | | /\\\\\\/ \\\\\\\\ | | /\\\\\\/ \\\\\\\\ |<br />
| o / \ o | | o\\\\\\/ \\\\\\\o | | o\\\\\\/ \\\\\\\o |<br />
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |<br />
| | du | | dv | | | |\\\\\| |\\\\\| | | |\du\\| |\\dv\| |<br />
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |<br />
| o \ / o | | o\\\\\\\ /\\\\\\o | | o\\\\\\\ /\\\\\\o |<br />
| \ \ / / | | \\\\\\\\ /\\\\\\/ | | \\\\\\\\ /\\\\\\/ |<br />
| \ o / | | \\\\\\\o\\\\\\/ | | \\\\\\\o\\\\\\/ |<br />
| \ / \ / | | \\\\\/ \\\\\/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ u v o-----------------------o dv' @ u v =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| U | |\U\\\\\\\\\\\\\\\\\\\\\| |\U\\\\\\\\\\\\\\\\\\\\\|<br />
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|<br />
| /////\ /////\ | |\\\\\/////\\/////\\\\\\| |\\\\\/ \\/ \\\\\\|<br />
| ///////o//////\ | |\\\\///////o//////\\\\\| |\\\\/ o \\\\\|<br />
| /////////\//////\ | |\\\////////X\//////\\\\| |\\\/ /\\ \\\\|<br />
| o//////////\//////o | |\\o///////XXX\//////o\\| |\\o /\\\\ o\\|<br />
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|<br />
| |//u//|/////|//v//| | |\\|//u//|XXXXX|//v//|\\| |\\| u |\\\\\| v |\\|<br />
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|<br />
| o//////\//////////o | |\\o//////\XXX///////o\\| |\\o \\\\/ o\\|<br />
| \//////\///////// | |\\\\//////\X////////\\\| |\\\\ \\/ /\\\|<br />
| \//////o/////// | |\\\\\//////o///////\\\\| |\\\\\ o /\\\\|<br />
| \///// \///// | |\\\\\\/////\\/////\\\\\| |\\\\\\ /\\ /\\\\\|<br />
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|<br />
| | |\\\\\\\\\\\\\\\\\\\\\\\| |\\\\\\\\\\\\\\\\\\\\\\\|<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= u' o-----------------------o v' =<br />
= | U' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/u'//|XXXXX|\\v'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
Figure 70-b. Tangent Functor Ferris Wheel for F<u, v> = <((u)(v)), ((u, v))><br />
</pre><br />
|}<br />
<br />
==Epilogue, Enchoiry, Exodus==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
It is time to explain myself . . . . let us stand up.<br />
| width="4%" | &nbsp;<br />
|- <br />
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 79]<br />
|}<br />
<br />
==Appendices==<br />
<br />
===Appendix 1. Propositional Forms and Differential Expansions===<br />
<br />
====Table A1. Propositional Forms on Two Variables====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A1.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="background:ghostwhite"<br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}</math><br />
| width="25%" | <math>\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{0}\\f_{1}\\f_{2}\\f_{3}\\f_{4}\\f_{5}\\f_{6}\\f_{7}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0000}\\f_{0001}\\f_{0010}\\f_{0011}\\f_{0100}\\f_{0101}\\f_{0110}\\f_{0111}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~0\\0~0~0~1\\0~0~1~0\\0~0~1~1\\0~1~0~0\\0~1~0~1\\0~1~1~0\\0~1~1~1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)~ ~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~ ~(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{,~} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{~~} y \texttt{)}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{false}<br />
\\<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\<br />
y ~\text{without}~ x<br />
\\<br />
\text{not}~ x<br />
\\<br />
x ~\text{without}~ y<br />
\\<br />
\text{not}~ y<br />
\\<br />
x ~\text{not equal to}~ y<br />
\\<br />
\text{not both}~ x ~\text{and}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\<br />
\lnot x \land \lnot y<br />
\\<br />
\lnot x \land y<br />
\\<br />
\lnot x<br />
\\<br />
x \land \lnot y<br />
\\<br />
\lnot y<br />
\\<br />
x \ne y<br />
\\<br />
\lnot x \lor \lnot y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{8}\\f_{9}\\f_{10}\\f_{11}\\f_{12}\\f_{13}\\f_{14}\\f_{15}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{1000}\\f_{1001}\\f_{1010}\\f_{1011}\\f_{1100}\\f_{1101}\\f_{1110}\\f_{1111}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1~0~0~0\\1~0~0~1\\1~0~1~0\\1~0~1~1\\1~1~0~0\\1~1~0~1\\1~1~1~0\\1~1~1~1<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~ ~ ~ ~} y \texttt{~~}<br />
\\<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{~~} x \texttt{~ ~ ~ ~}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{((~))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{and}~ y<br />
\\<br />
x ~\text{equal to}~ y<br />
\\<br />
y<br />
\\<br />
\text{not}~ x ~\text{without}~ y<br />
\\<br />
x<br />
\\<br />
\text{not}~ y ~\text{without}~ x<br />
\\<br />
x ~\text{or}~ y<br />
\\<br />
\text{true}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x \land y<br />
\\<br />
x = y<br />
\\<br />
y<br />
\\<br />
x \Rightarrow y<br />
\\<br />
x<br />
\\<br />
x \Leftarrow y<br />
\\<br />
x \lor y<br />
\\<br />
1<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A2. Propositional Forms on Two Variables====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A2.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="background:ghostwhite"<br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}</math><br />
| width="25%" | <math>\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>f_{0000}\!</math><br />
| <math>0~0~0~0</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0001}\\f_{0010}\\f_{0100}\\f_{1000}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~1\\0~0~1~0\\0~1~0~0\\1~0~0~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\<br />
y ~\text{without}~ x<br />
\\<br />
x ~\text{without}~ y<br />
\\<br />
x ~\text{and}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot x \land \lnot y<br />
\\<br />
\lnot x \land y<br />
\\<br />
x \land \lnot y<br />
\\<br />
x \land y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{3}\\f_{12}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0011}\\f_{1100}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~1~1\\1~1~0~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ x<br />
\\<br />
x<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot x<br />
\\<br />
x<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{6}\\f_{9}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0110}\\f_{1001}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~0\\1~0~0~1<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{not equal to}~ y<br />
\\<br />
x ~\text{equal to}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x \ne y<br />
\\<br />
x = y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{5}\\f_{10}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0101}\\f_{1010}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~0~1\\1~0~1~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ y<br />
\\<br />
y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot y<br />
\\<br />
y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{7}\\f_{11}\\f_{13}\\f_{14}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0111}\\f_{1011}\\f_{1101}\\f_{1110}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~1\\1~0~1~1\\1~1~0~1\\1~1~1~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not both}~ x ~\text{and}~ y<br />
\\<br />
\text{not}~ x ~\text{without}~ y<br />
\\<br />
\text{not}~ y ~\text{without}~ x<br />
\\<br />
x ~\text{or}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot x \lor \lnot y<br />
\\<br />
x \Rightarrow y<br />
\\<br />
x \Leftarrow y<br />
\\<br />
x \lor y<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>f_{1111}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A3. E''f'' Expanded Over Differential Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A3.} ~~ \mathrm{E}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{T}_{11}f\\\mathrm{E}f|_{\mathrm{d}x ~ \mathrm{d}y}\end{matrix}</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{T}_{10}f\\\mathrm{E}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}\end{matrix}</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{T}_{01}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}\end{matrix}</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{T}_{00}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{3}\\f_{12}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{6}\\f_{9}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{5}\\f_{10}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{7}\\f_{11}\\f_{13}\\f_{14}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
|- style="background:ghostwhite"<br />
| style="border-top:1px solid black" colspan="2" | <math>\text{Fixed Point Total}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>4\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>4\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>4\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>16\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A4. D''f'' Expanded Over Differential Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A4.} ~~ \mathrm{D}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\mathrm{D}f|_{\mathrm{d}x ~ \mathrm{d}y}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}~\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
y<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
y<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
x<br />
\\<br />
x<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
x<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
y<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
y<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <br />
<math>\begin{matrix}<br />
x<br />
\\<br />
x<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A5. E''f'' Expanded Over Ordinary Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A5.} ~~ \mathrm{E}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\mathrm{E}f|_{xy}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{E}f|_{x \texttt{(} y \texttt{)}}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{E}f|_{\texttt{(} x \texttt{)} y}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{E}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{3}\\f_{12}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{6}\\f_{9}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{5}\\f_{10}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{7}\\f_{11}\\f_{13}\\f_{14}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A6. D''f'' Expanded Over Ordinary Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A6.} ~~ \mathrm{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\mathrm{D}f|_{xy}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{x \texttt{(} y \texttt{)}}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} x \texttt{)} y}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\texttt{(} x \texttt{)}\\\texttt{~} x \texttt{~}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Appendix 2. Differential Forms===<br />
<br />
The actions of the difference operator <math>\mathrm{D}\!</math> and the tangent operator <math>\mathrm{d}\!</math> on the 16 bivariate propositions are shown in Tables&nbsp;A7 and A8.<br />
<br />
Table A7 expands the differential forms that result over a ''logical basis'':<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
|<br />
<math>\{~ \texttt{(}\mathrm{d}x\texttt{)(}\mathrm{d}y\texttt{)}, ~\mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}, ~\texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.\!</math><br />
|}<br />
<br />
This set consists of the singular propositions in the first order differential variables, indicating mutually exclusive and exhaustive ''cells'' of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the cell-basis, point-basis, or singular differential basis. In this setting it is frequently convenient to use the following abbreviations:<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
|<br />
<math>\partial x ~=~ \mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}\!</math> &nbsp; &nbsp; and &nbsp; &nbsp; <math>\partial y ~=~ \texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y.\!</math><br />
|}<br />
<br />
Table A8 expands the differential forms that result over an ''algebraic basis'':<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
| <math>\{~ 1, ~\mathrm{d}x, ~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.\!</math><br />
|}<br />
<br />
This set consists of the ''positive propositions'' in the first order differential variables, indicating overlapping positive regions of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the ''positive differential basis''.<br />
<br />
====Table A7. Differential Forms Expanded on a Logical Basis====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A7.} ~~ \text{Differential Forms Expanded on a Logical Basis}\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| &nbsp;<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" | <math>\mathrm{D}f~\!</math><br />
| <math>\mathrm{d}f~\!</math><br />
|-<br />
| <math>f_{0}\!</math><br />
| style="border-right:none" | <math>\texttt{(~)}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x) ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\\<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\\<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\\<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\partial x<br />
\\<br />
\partial x<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
\\<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\partial x & + & \partial y<br />
\\<br />
\partial x & + & \partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\partial y<br />
\\<br />
\partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x) ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\\<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\\<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\\<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| style="border-right:none" | <math>\texttt{((~))}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A8. Differential Forms Expanded on an Algebraic Basis====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A8.} ~~ \text{Differential Forms Expanded on an Algebraic Basis}\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| &nbsp;<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" | <math>\mathrm{D}f~\!</math><br />
| <math>\mathrm{d}f~\!</math><br />
|-<br />
| <math>f_{0}\!</math><br />
| style="border-right:none" | <math>\texttt{(~)}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x<br />
\\<br />
\mathrm{d}x<br />
\end{matrix}\!</math><br />
| <math>\begin{matrix}<br />
\mathrm{d}x<br />
\\<br />
\mathrm{d}x<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\\<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\\<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}y<br />
\\<br />
\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y<br />
\\<br />
\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| style="border-right:none" | <math>\texttt{((~))}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A9. Tangent Proposition as Pointwise Linear Approximation====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A9.} ~~ \text{Tangent Proposition}~ \mathrm{d}f = \text{Pointwise Linear Approximation to the Difference Map}~ \mathrm{D}f\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}f =<br />
\\[2pt]<br />
\partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}^2\!f =<br />
\\[2pt]<br />
\partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\mathrm{d}f|_{x \, y}</math><br />
| <math>\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math><br />
|-<br />
| style="border-right:none" | <math>f_0\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{5}\\f_{10}\end{matrix}\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\end{matrix}\!</math><br />
| <math>\begin{matrix}<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>f_{15}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A10. Taylor Series Expansion Df = d''f'' + d<sup>2</sup>''f''====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" |<br />
<math>\text{Table A10.} ~~ \text{Taylor Series Expansion}~ {\mathrm{D}f = \mathrm{d}f + \mathrm{d}^2\!f}\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{D}f<br />
\\<br />
= & \mathrm{d}f & + & \mathrm{d}^2\!f<br />
\\<br />
= & \partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y & + & \partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\mathrm{d}f|_{x \, y}</math><br />
| <math>\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math><br />
|-<br />
| style="border-right:none" | <math>f_0\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>f_{15}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A11. Partial Differentials and Relative Differentials====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A11.} ~~ \text{Partial Differentials and Relative Differentials}\!</math><br />
|- style="background:ghostwhite; height:50px"<br />
| &nbsp;<br />
| <math>f\!</math><br />
| <math>\frac{\partial f}{\partial x}\!</math><br />
| <math>\frac{\partial f}{\partial y}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}f =<br />
\\[2pt]<br />
\partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\left. \frac{\partial x}{\partial y} \right| f\!</math><br />
| <math>\left. \frac{\partial y}{\partial x} \right| f\!</math><br />
|-<br />
| <math>f_0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A12. Detail of Calculation for the Difference Map====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:4px double black; border-left:4px double black; border-right:4px double black; border-top:4px double black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A12.} ~~ \text{Detail of Calculation for}~ {\mathrm{E}f + f = \mathrm{D}f}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:6%" | &nbsp;<br />
| style="width:14%; border-left:1px solid black" | <math>f\!</math><br />
| style="width:20%; border-left:4px double black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\mathrm{d}x ~ \mathrm{d}y}<br />
\\[4pt]<br />
+ & f|_{\mathrm{d}x ~ \mathrm{d}y}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\mathrm{d}x ~ \mathrm{d}y}<br />
\end{array}</math><br />
| style="width:20%; border-left:1px solid black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}<br />
\\[4pt]<br />
+ & f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}<br />
\end{array}</math><br />
| style="width:20%; border-left:1px solid black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
+ & f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}<br />
\end{array}</math><br />
| style="width:20%; border-left:1px solid black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
+ & f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}<br />
\end{array}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{0}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:4px double black; border-left:4px double black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{1}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{)(} y \texttt{)~}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{2}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{)~} y \texttt{~~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{4}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~~} x \texttt{~(} y \texttt{)~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{8}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~~} x \texttt{~~} y \texttt{~~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{3}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{(} x \texttt{)}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{12}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>x\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{6}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{,~} y \texttt{)~}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{9}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} x \texttt{,~} y \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{5}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{(} y \texttt{)}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{10}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>y\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{7}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{~~} y \texttt{)~}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{11}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{~(} y \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{13}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} x \texttt{)~} y \texttt{)~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{14}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} x \texttt{)(} y \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{15}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:4px double black; border-left:4px double black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Appendix 3. Computational Details===<br />
<br />
====Operator Maps for the Logical Conjunction ''f''<sub>8</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{8}~\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{8}<br />
& = && f_{8}(u, v)<br />
\\[4pt]<br />
& = && uv<br />
\\[4pt]<br />
& = && uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & uv \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & uv \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{8}<br />
& = && uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & uv \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & uv \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & uv \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.2-i} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{E}f_{8}<br />
& = & f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{8}(\mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{8}(\mathrm{d}u, \mathrm{d}v)<br />
\\[4pt]<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[20pt]<br />
\mathrm{E}f_{8}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
\\[4pt]<br />
&&&&& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&&&&&& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.2-ii} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{c}}<br />
\mathrm{E}f_{8}<br />
& = & (u + \mathrm{d}u) \cdot (v + \mathrm{d}v)<br />
\\[6pt]<br />
& = & u \cdot v<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}f_{8}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{8}<br />
& = && \mathrm{E}f_{8}<br />
& + & \boldsymbol\varepsilon f_{8}<br />
\\[4pt]<br />
& = && f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{8}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & uv<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{~}<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}f_{8}<br />
& = & \boldsymbol\varepsilon f_{8}<br />
& + & \mathrm{E}f_{8}<br />
\\[6pt]<br />
& = & f_{8}(u, v)<br />
& + & f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[6pt]<br />
& = & uv<br />
& + & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
& = & 0<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}f_{8}<br />
& = & 0<br />
& + & u \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.3-iii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 3)}\!</math><br />
|<br />
<math>\begin{array}{c*{9}{l}}<br />
\mathrm{D}f_{8}<br />
& = & \boldsymbol\varepsilon f_{8} ~+~ \mathrm{E}f_{8}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{8}<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ u \,\cdot\, v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~ u \;\cdot\; v \;\cdot\; \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}f_{8}<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u ~ \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} ~ v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot\, \mathrm{d}u ~ \mathrm{d}v<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = & ~ ~ 0 ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~ ~ u ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ ~ ~ v ~~ \cdot ~ \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
=====Computation of d''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.4} ~~ \text{Computation of}~ \mathrm{d}f_{8}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{8}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.5} ~~ \text{Computation of}~ \mathrm{r}f_{8}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{8} & = & \mathrm{D}f_{8} ~+~ \mathrm{d}f_{8}<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[20pt]<br />
\mathrm{r}f_{8}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Conjunction=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.6} ~~ \text{Computation Summary for}~ f_{8}(u, v) = uv\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{8}<br />
& = & uv \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}f_{8}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{r}f_{8}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Operator Maps for the Logical Equality ''f''<sub>9</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{9}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{9}<br />
& = && f_{9}(u, v)<br />
\\[4pt]<br />
& = && \texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{9}(1, 1)<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{9}(1, 0)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{9}(0, 1)<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{9}(0, 0)<br />
\\[4pt]<br />
& = && u v & + & 0 & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{9}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.2} ~~ \text{Computation of}~ \mathrm{E}f_{9}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{E}f_{9}<br />
& = && f_{9}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! f_{9}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! f_{9}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{ } \mathrm{d}v \texttt{ })<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! f_{9}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! f_{9}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{ } \mathrm{d}v \texttt{ })<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{ (} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{) }<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{ (} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{) }<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{E}f_{9}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{9}<br />
& = && \mathrm{E}f_{9}<br />
& + & \boldsymbol\varepsilon f_{9}<br />
\\[4pt]<br />
& = && f_{9}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{9}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{9}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[20pt]<br />
\mathrm{D}f_{9}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}f_{9}<br />
& = & 0 \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & 1 \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & 1 \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & 0 \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of d''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.4} ~~ \text{Computation of}~ \mathrm{d}f_{9}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.5} ~~ \text{Computation of}~ \mathrm{r}f_{9}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{9} & = & \mathrm{D}f_{9} ~+~ \mathrm{d}f_{9}<br />
\\[20pt]<br />
\mathrm{D}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[20pt]<br />
\mathrm{r}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Equality=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.6} ~~ \text{Computation Summary for}~ f_{9}(u, v) = \texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{9}<br />
& = & uv \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}f_{9}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f_{9}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{9}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}f_{9}<br />
& = & uv \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Operator Maps for the Logical Implication ''f''<sub>11</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{11}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{11}<br />
& = && f_{11}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(} u \texttt{(} v \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{11}(1, 1)<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{11}(1, 0)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{11}(0, 1)<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{11}(0, 0)<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ }<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ }<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{11}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.2} ~~ \text{Computation of}~ \mathrm{E}f_{11}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{E}f_{11}<br />
& = && f_{11}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = &&<br />
\texttt{(}<br />
\\<br />
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\\<br />
&&& \texttt{(}<br />
\\<br />
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\<br />
&&& \texttt{))}<br />
\\[4pt]<br />
& = &&<br />
u v<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)((} \mathrm{d}v \texttt{)))}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u \texttt{((} \mathrm{d}v \texttt{)))}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[4pt]<br />
& = &&<br />
u v<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{E}f_{11}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{11}<br />
& = && \mathrm{E}f_{11}<br />
& + & \boldsymbol\varepsilon f_{11}<br />
\\[4pt]<br />
& = && f_{11}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{11}(u, v)<br />
\\[4pt]<br />
& = &&<br />
\texttt{(} \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\texttt{(} \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{(} v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & u v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & 0<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & 0<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = && u v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{c*{9}{l}}<br />
\mathrm{D}f_{11}<br />
& = & \boldsymbol\varepsilon f_{11} ~+~ \mathrm{E}f_{11}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{11}<br />
& = & u v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}f_{11}<br />
& = &<br />
u v<br />
\cdot<br />
\texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\cdot<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\cdot<br />
\texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\cdot<br />
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = &<br />
u v<br />
\cdot<br />
\texttt{~(} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{~}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\cdot<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\cdot<br />
\texttt{~} \mathrm{d}u ~ \mathrm{d}v \texttt{~}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\cdot<br />
\texttt{~} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of d''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.4} ~~ \text{Computation of}~ \mathrm{d}f_{11}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{11}<br />
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{11}<br />
& = & u v \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.5} ~~ \text{Computation of}~ \mathrm{r}f_{11}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{11} & = & \mathrm{D}f_{11} ~+~ \mathrm{d}f_{11}<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{11}<br />
& = & u v \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u<br />
\\[20pt]<br />
\mathrm{r}f_{11}<br />
& = & u v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Implication=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.6} ~~ \text{Computation Summary for}~ f_{11}(u, v) = \texttt{(} u \texttt{(} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{11}<br />
& = & u v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}f_{11}<br />
& = & u v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f_{11}<br />
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{11}<br />
& = & u v \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u<br />
\\[6pt]<br />
\mathrm{r}f_{11}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Operator Maps for the Logical Disjunction ''f''<sub>14</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{14}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{14}<br />
& = && f_{14}(u, v)<br />
\\[4pt]<br />
& = && \texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{14}(1, 1)<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{14}(1, 0)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{14}(0, 1)<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{14}(0, 0)<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ }<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ }<br />
& + & 0<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{14}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.2} ~~ \text{Computation of}~ \mathrm{E}f_{14}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{E}f_{14}<br />
& = && f_{14}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = &&<br />
\texttt{((}<br />
\\<br />
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\\<br />
&&& \texttt{)(}<br />
\\<br />
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\<br />
&&& \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! f_{14}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! f_{14}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{ } \mathrm{d}v \texttt{ })<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! f_{14}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! f_{14}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{ } \mathrm{d}v \texttt{ })<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{E}f_{14}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{14}<br />
& = && \mathrm{E}f_{14}<br />
& + & \boldsymbol\varepsilon f_{14}<br />
\\[4pt]<br />
& = && f_{14}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{14}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{))((} v \texttt{,} \mathrm{d}v \texttt{)))}<br />
& + & \texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{14}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
\\[4pt]<br />
&& + & 0<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
\\[4pt]<br />
&& + & uv \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
\\[20pt]<br />
\mathrm{D}f_{14}<br />
& = && uv \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}f_{14}<br />
& = & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & 0 \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of d''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.4} ~~ \text{Computation of}~ \mathrm{d}f_{14}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.5} ~~ \text{Computation of}~ \mathrm{r}f_{14}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{14} & = & \mathrm{D}f_{14} ~+~ \mathrm{d}f_{14}<br />
\\[20pt]<br />
\mathrm{D}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{d}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[20pt]<br />
\mathrm{r}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Disjunction=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.6} ~~ \text{Computation Summary for}~ f_{14}(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{14}<br />
& = & uv \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 1<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}f_{14}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f_{14}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{d}f_{14}<br />
& = & uv \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}f_{14}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
===Appendix 4. Source Materials===<br />
<br />
===Appendix 5. Various Definitions of the Tangent Vector===<br />
<br />
==References==<br />
<br />
===Works Cited===<br />
<br />
{| cellpadding=3<br />
| valign=top | [AuM]<br />
| Auslander, L., and MacKenzie, R.E., ''Introduction to Differentiable Manifolds'', McGraw-Hill, 1963. Reprinted, Dover, New York, NY, 1977.<br />
|-<br />
| valign=top | [BiG]<br />
| Bishop, R.L., and Goldberg, S.I., ''Tensor Analysis on Manifolds'', Macmillan, 1968. Reprinted, Dover, New York, NY, 1980.<br />
|-<br />
| valign=top | [Boo]<br />
| Boole, G., ''An Investigation of The Laws of Thought'', Macmillan, 1854. Reprinted, Dover, New York, NY, 1958.<br />
|-<br />
| valign=top | [BoT]<br />
| Bott, R., and Tu, L.W., ''Differential Forms in Algebraic Topology'', Springer-Verlag, New York, NY, 1982.<br />
|-<br />
| valign=top | [dCa]<br />
| do Carmo, M.P., ''Riemannian Geometry''. Originally published in Portuguese, 1st editiom 1979, 2nd edition 1988. Translated by F. Flaherty, Birkhäuser, Boston, MA, 1992.<br />
|-<br />
| valign=top | [Che46]<br />
| Chevalley, C., ''Theory of Lie Groups'', Princeton University Press, Princeton, NJ, 1946.<br />
|-<br />
| valign=top | [Che56]<br />
| Chevalley, C., ''Fundamental Concepts of Algebra'', Academic Press, 1956.<br />
|-<br />
| valign=top | [Cho86]<br />
| Chomsky, N., ''Knowledge of Language : Its Nature, Origin, and Use'', Praeger, New York, NY, 1986.<br />
|-<br />
| valign=top | [Cho93]<br />
| Chomsky, N., ''Language and Thought'', Moyer Bell, Wakefield, RI, 1993.<br />
|-<br />
| valign=top | [DoM]<br />
| Doolin, B.F., and Martin, C.F., ''Introduction to Differential Geometry for Engineers'', Marcel Dekker, New York, NY, 1990.<br />
|-<br />
| valign=top | [Fuji]<br />
| Fujiwara, H., ''Logic Testing and Design for Testability'', MIT Press, Cambridge, MA, 1985.<br />
|-<br />
| valign=top | [Hic]<br />
| Hicks, N.J., ''Notes on Differential Geometry'', Van Nostrand, Princeton, NJ, 1965.<br />
|-<br />
| valign=top | [Hir]<br />
| Hirsch, M.W., ''Differential Topology'', Springer-Verlag, New York, NY, 1976.<br />
|-<br />
| valign=top | [How]<br />
| Howard, W.A., "The Formulae-as-Types Notion of Construction", Notes circulated from 1969. Reprinted in [SeH, 479-490].<br />
|-<br />
| valign=top | [JGH]<br />
| Jones, A., Gray, A., and Hutton, R., ''Manifolds and Mechanics'', Cambridge University Press, Cambridge, UK, 1987.<br />
|-<br />
| valign=top | [KoA]<br />
| Kosinski, A.A., ''Differential Manifolds'', Academic Press, San Diego, CA, 1993.<br />
|-<br />
| valign=top | [Koh]<br />
| Kohavi, Z., ''Switching and Finite Automata Theory'', 2nd edition, McGraw-Hill, New York, NY, 1978.<br />
|-<br />
| valign=top | [LaS]<br />
| Lambek, J., and Scott, P.J., ''Introduction to Higher Order Categorical Logic'', Cambridge University Press, Cambridge, UK, 1986.<br />
|-<br />
| valign=top | [La83]<br />
| Lang, S., ''Real Analysis'', 2nd edition, Addison-Wesley, Reading, MA, 1983.<br />
|-<br />
| valign=top | [La84]<br />
| Lang, S., ''Algebra'', 2nd edition, Addison-Wesley, Menlo Park, CA, 1984.<br />
|-<br />
| valign=top | [La85]<br />
| Lang, S., ''Differential Manifolds'', Springer-Verlag, New York, NY, 1985.<br />
|-<br />
| valign=top | [La93]<br />
| Lang, S., ''Real and Functional Analysis'', 3rd edition, Springer-Verlag, New York, NY, 1993.<br />
|-<br />
| valign=top | [Lie80]<br />
| Lie, S., "Sophus Lie's 1880 Transformation Group Paper", in ''Lie Groups : History, Frontiers, and Applications, Volume 1'', translated by M. Ackerman, comments by R. Hermann, Math Sci Press, Brookline, MA, 1975. Original paper 1880.<br />
|-<br />
| valign=top | [Lie84]<br />
| Lie, S., "Sophus Lie's 1884 Differential Invariant Paper", in ''Lie Groups : History, Frontiers, and Applications, Volume 3'', translated by M. Ackerman, comments by R. Hermann, Math Sci Press, Brookline, MA, 1976. Original paper 1884.<br />
|-<br />
| valign=top | [LoS]<br />
| Loomis, L.H., and Sternberg, S., ''Advanced Calculus'', Addison-Wesley, Reading, MA, 1968.<br />
|-<br />
| valign=top | [Mel]<br />
| Melzak, Z.A., ''Companion to Concrete Mathematics, Volume 2 : Mathematical Ideas, Modeling, and Applications'', John Wiley amd Sons, New York, NY, 1976.<br />
|-<br />
| valign=top | [Men]<br />
| Menabrea, L.F., "Sketch of the Analytical Engine Invented by Charles Babbage" with Notes by the Translator, Ada Augusta (Byron), Countess of Lovelace'', in [M&M, 225–297]. Originally published 1842.<br />
|-<br />
| valign=top | [M&M]<br />
| Morrison, P., and Morrison, E. (eds.), ''Charles Babbage on the Principles and Development of the Calculator, and Other Seminal Writings by Charles Babbage and Others, With an Introduction by the Editors'', Dover, Mineola, NY, 1961.<br />
|-<br />
| valign=top | [P1]<br />
| Peirce, C.S., ''Collected Papers of Charles Sanders Peirce'', vols. 1–8, C. Hartshorne, P. Weiss, and A.W. Burks (eds.), Harvard University Press, Cambridge, MA, 1931–1960. Cited as CP [volume].[paragraph].<br />
|-<br />
| valign=top | [P2]<br />
| Peirce, C.S., "Qualitative Logic", in ''The New Elements of Mathematics, Volume 4'', C. Eisele (ed.), Mouton, The Hague, 1976. Cited as NE [volume], [page].<br />
|-<br />
| valign=top | [Rob]<br />
| Roberts, D.D., ''The Existential Graphs of Charles S. Peirce'', Mouton, The Hague, 1973.<br />
|-<br />
| valign=top | [SeH]<br />
| Seldin, J.P., and Hindley, J.R. (eds.), ''To H.B. Curry : Essays on Combinatory Logic, Lambda Calculus, and Formalism'', Academic Press, London, UK, 1980.<br />
|-<br />
| valign=top | [SpB]<br />
| Spencer-Brown, G., ''Laws of Form'', George Allen and Unwin, London, UK, 1969.<br />
|-<br />
| valign=top | [Sp65]<br />
| Spivak, M., ''Calculus on Manifolds : A Modern Approach to Classical Theorems of Advanced Calculus'', W.A. Benjamin, New York, NY, 1965.<br />
|-<br />
| valign=top | [Sp79]<br />
| Spivak, M., ''A Comprehensive Introduction to Differential Geometry'', vols. 1–2. 1st edition 1970. 2nd edition, Publish or Perish Inc., Houston, TX, 1979.<br />
|-<br />
| valign=top | [Sty]<br />
| Styazhkin, N.I., ''History of Mathematical Logic from Leibniz to Peano'', 1st published in Russian, Nauka, Moscow, 1964. MIT Press, Cambridge, MA, 1969.<br />
|-<br />
| valign=top | [Wie]<br />
| Wiener, N., ''Cybernetics : or Control and Communication in the Animal and the Machine'', 1st edition 1948. 2nd edition, MIT Press, Cambridge, MA, 1961.<br />
|}<br />
<br />
===Works Consulted===<br />
<br />
{| cellpadding=3<br />
| valign=top | [Ami]<br />
| Amit, D.J., ''Modeling Brain Function : The World of Attractor Neural Networks'', Cambridge University Press, Cambridge, UK, 1989.<br />
|-<br />
| valign=top | [Ed87]<br />
| Edelman, G.M., ''Neural Darwinism : The Theory of Neuronal Group Selection'', Basic Books, New York, NY, 1987.<br />
|-<br />
| valign=top | [Ed88]<br />
| Edelman, G.M., ''Topobiology : An Introduction to Molecular Embryology'', Basic Books, New York, NY, 1988.<br />
|-<br />
| valign=top | [Fla]<br />
| Flanders, H., ''Differential Forms with Applications to the Physical Sciences'', Academic Press, 1963. Reprinted, Dover, Mineola, NY, 1989. <br />
|-<br />
| valign=top | [Has]<br />
| Hassoun, M.H. (ed.), ''Associative Neural Memories : Theory and Implementation'', Oxford University Press, New York, NY, 1993.<br />
|-<br />
| valign=top | [KoB]<br />
| Kosko, B., ''Neural Networks and Fuzzy Systems : A Dynamical Systems Approach to Machine Intelligence'', Prentice-Hall, Englewood Cliffs, NJ, 1992.<br />
|-<br />
| valign=top | [MaB]<br />
| Mac Lane, S., and Birkhoff, G., ''Algebra'', 3rd edition, Chelsea, New York, NY, 1993.<br />
|-<br />
| valign=top | [Mac]<br />
| Mac Lane, S., ''Categories for the Working Mathematician'', Springer-Verlag, New York, NY, 1971.<br />
|-<br />
| valign=top | [McC]<br />
| McCulloch, W.S., ''Embodiments of Mind'', MIT Press, Cambridge, MA, 1965.<br />
|-<br />
| valign=top | [Mc1]<br />
| McCulloch, W.S., "A Heterarchy of Values Determined by the Topology of Nervous Nets", Bulletin of Mathematical Biophysics, vol. 7 (1945), pp. 89–93. Reprinted in [McC].<br />
|-<br />
| valign=top | [MiP]<br />
| Minsky, M.L., and Papert, S.A., ''Perceptrons : An Introduction to Computational Geometry'', MIT Press, Cambridge, MA, 1969. 2nd printing 1972. Expanded edition 1988.<br />
|-<br />
| valign=top | [Rum]<br />
| Rumelhart, D.E., Hinton, G.E., and McClelland, J.L., "A General Framework for Parallel Distributed Processing" = Chapter 2 in Rumelhart, McClelland, and the PDP Research Group, ''Parallel Distributed Processing, Explorations in the Microstructure of Cognition, Volume 1 : Foundations'', MIT Press, Cambridge, MA, 1986.<br />
|}<br />
<br />
===Incidental Works===<br />
<br />
{| cellpadding=3<br />
| valign=top | [Dew]<br />
| Dewey, John, ''How We Think'', D.C. Heath, Lexington, MA, 1910. Reprinted, Prometheus Books, Buffalo, NY, 1991.<br />
|-<br />
| valign=top | [Fou]<br />
| Foucault, Michel, ''The Archaeology of Knowledge and The Discourse on Language'', A.M. Sheridan-Smith and Rupert Swyer (trans.), Pantheon, New York, NY, 1972. Originally published as ''L´Archéologie du Savoir et L´ordre du discours'', Editions Gallimard, 1969 & 1971.<br />
|-<br />
| valign=top | [Hom]<br />
| Homer, ''The Odyssey'', with an English translation by A.T. Murray, Loeb Classical Library, Harvard University Press, Cambridge, MA, 1980. First printed 1919.<br />
|-<br />
| valign=top | [Jam]<br />
| James, William, ''Pragmatism : A New Name for Some Old Ways of Thinking'', Longmans, Green, and Company, New York, NY, 1907.<br />
|-<br />
| valign=top | [Ler]<br />
| Leroux, Gaston, ''The Phantom of the Opera'', foreword by P. Haining, Dorset Press, New York, NY, 1988. Originally published in French, 1911.<br />
|-<br />
| valign=top | [Mus]<br />
| Musil, Robert, ''The Man Without Qualities'', 3 volumes, translated with a foreword by Eithne Wilkins and Ernst Kaiser, Pan Books, London, UK, 1979. English edition first published by Secker and Warburg, 1954. Originally published in German, ''Der Mann ohne Eigenschaften'', 1930 & 1932.<br />
|-<br />
| valign=top | [PlaR]<br />
| Plato, ''The Republic'', with an English translation by Paul Shorey, Loeb Classical Library, Harvard University Press, Cambridge, MA, 1980. First printed 1930 & 1935.<br />
|-<br />
| valign=top | [PlaS]<br />
| Plato, ''The Sophist'', Loeb Classical Library, William Heinemann, London, 1921, 1987.<br />
|-<br />
| valign=top | [Qui]<br />
| Quine, W.V., ''Mathematical Logic'', 1st edition, 1940. Revised edition, 1951. Harvard University Press, Cambridge, MA, 1981.<br />
|-<br />
| valign=top | [SaD]<br />
| de Santillana, Giorgio, and von Dechend, Hertha, ''Hamlet's Mill : An Essay on Myth and the Frame of Time'', David R. Godine, Publisher, Boston, MA, 1977. 1st published 1969.<br />
|-<br />
| valign=top | [Sha]<br />
| Shakespeare, William, '' William Shakespeare : The Complete Works'', Compact Edition, S. Wells and G. Taylor (eds.), Oxford University Press, Oxford, UK, 1988.<br />
|-<br />
| valign=top | [Sh1]<br />
| Shakespeare, William, ''A Midsummer Night's Dream'', Washington Square Press, New York, NY, 1958.<br />
|-<br />
| valign=top | [Sh2]<br />
| Shakespeare, William, ''The Tragedy of Hamlet, Prince of Denmark'', In [Sha], pp. 654&ndash;690.<br />
|-<br />
| valign=top | [Sh3]<br />
| Shakespeare, William, ''Measure for Measure'', Washington Square Press, New York, NY, 1965.<br />
|-<br />
| valign=top | [Web]<br />
| ''Webster's Ninth New Collegiate Dictionary'', Merriam-Webster, Springfield, MA, 1983.<br />
|-<br />
| valign=top | [Whi]<br />
| Whitman, Walt, ''Leaves of Grass'', Vintage Books / The Library of America, New York, NY, 1992. Originally published in numerous editions, 1855&ndash;1892.<br />
|-<br />
| valign=top | [Wil]<br />
| Wilhelm, R., and Baynes, C.F. (trans.), ''The I Ching, or Book of Changes'', foreword by C.G. Jung, preface by H. Wilhelm, 3rd edition, Bollingen Series XIX, Princeton University Press, Princeton, NJ, 1967.<br />
|}<br />
<br />
==Document History==<br />
<br />
<pre><br />
Author: Jon Awbrey<br />
Created: 16 Dec 1993<br />
Relayed: 31 Oct 1994<br />
Revised: 03 Jun 2003<br />
Recoded: 03 Jun 2007<br />
</pre><br />
<br />
[[Category:Adaptive Systems]]<br />
[[Category:Artificial Intelligence]]<br />
[[Category:Boolean Algebra]]<br />
[[Category:Boolean Functions]]<br />
[[Category:Charles Sanders Peirce]]<br />
[[Category:Combinatorics]]<br />
[[Category:Computer Science]]<br />
[[Category:Cybernetics]]<br />
[[Category:Differential Logic]]<br />
[[Category:Discrete Systems]]<br />
[[Category:Dynamical Systems]]<br />
[[Category:Formal Languages]]<br />
[[Category:Formal Sciences]]<br />
[[Category:Formal Systems]]<br />
[[Category:Functional Logic]]<br />
[[Category:Graph Theory]]<br />
[[Category:Group Theory]]<br />
<br />
test<br />
<br />
[[Category:Inquiry]]<br />
[[Category:Knowledge Representation]]<br />
[[Category:Linguistics]]<br />
[[Category:Logic]]<br />
[[Category:Logical Graphs]]<br />
[[Category:Mathematics]]<br />
[[Category:Mathematical Systems Theory]]<br />
[[Category:Philosophy]]<br />
[[Category:Science]]<br />
[[Category:Semiotics]]<br />
[[Category:Systems Science]]<br />
[[Category:Visualization]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Directory:Jon_Awbrey/Differential_Logic_and_Dynamic_Systems_2.0&diff=469878Directory:Jon Awbrey/Differential Logic and Dynamic Systems 2.02021-01-13T18:29:29Z<p>Jon Awbrey: parse okay</p>
<hr />
<div>{{DISPLAYTITLE:Differential Logic and Dynamic Systems 2.0}}<br />
'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''<br />
<br />
{| align="center" cellpadding="10"<br />
| [[Image:Tangent_Functor_Ferris_Wheel.gif]]<br />
|}<br />
<br />
{| style="height:36px; width:100%"<br />
| align="left" | ''Stand and unfold yourself.''<br />
| align="right" | Hamlet: Francsico&mdash;1.1.2<br />
|}<br />
<br />
This article develops a differential extension of propositional calculus and applies it to a context of problems arising in dynamic systems. The work pursued here is coordinated with a parallel application that focuses on neural network systems, but the dependencies are arranged to make the present article the main and the more self-contained work, to serve as a conceptual frame and a technical background for the network project.<br />
<br />
==Review and Transition==<br />
<br />
This note continues a previous discussion on the problem of dealing with change and diversity in logic-based intelligent systems. It is useful to begin by summarizing essential material from previous reports.<br />
<br />
Table 1 outlines a notation for propositional calculus based on two types of logical connectives, both of variable <math>k\!</math>-ary scope.<br />
<br />
* A bracketed list of propositional expressions in the form <math>\texttt{(} e_1, e_2, \ldots, e_{k-1}, e_k \texttt{)}\!</math> indicates that exactly one of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> is false.<br />
<br />
* A concatenation of propositional expressions in the form <math>e_1 ~ e_2 ~ \ldots ~ e_{k-1} ~ e_k</math> indicates that all of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> are true, in other words, that their [[logical conjunction]] is true.<br />
<br />
All other propositional connectives can be obtained in a very efficient style of representation through combinations of these two forms. Strictly speaking, the concatenation form is dispensable in light of the bracketed form, but it is convenient to maintain it as an abbreviation of more complicated bracket expressions.<br />
<br />
This treatment of propositional logic is derived from the work of C.S. Peirce [P1, P2], who gave this approach an extensive development in his graphical systems of predicate, relational, and modal logic [Rob]. More recently, these ideas were revived and supplemented in an alternative interpretation by George Spencer-Brown [SpB]. Both of these authors used other forms of enclosure where I use parentheses, but the structural topologies of expression and the functional varieties of interpretation are fundamentally the same.<br />
<br />
While working with expressions solely in propositional calculus, it is easiest to use plain parentheses for logical connectives. In contexts where parentheses are needed for other purposes &ldquo;teletype&rdquo; parentheses <math>\texttt{(} \ldots \texttt{)}\!</math> or barred parentheses <math>(\!| \ldots |\!)</math> may be used for logical operators.<br />
<br />
The briefest expression for logical truth is the empty word, usually denoted by <math>{}^{\backprime\backprime} \boldsymbol\varepsilon {}^{\prime\prime}\!</math> or <math>{}^{\backprime\backprime} \boldsymbol\lambda {}^{\prime\prime}\!</math> in formal languages, where it forms the identity element for concatenation. To make it visible in this text, it may be denoted by the equivalent expression <math>{}^{\backprime\backprime} \texttt{((~))} {}^{\prime\prime},\!</math> or, especially if operating in an algebraic context, by a simple <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}.\!</math> Also when working in an algebraic mode, the plus sign <math>{}^{\backprime\backprime} + {}^{\prime\prime}\!</math> may be used for [[exclusive disjunction]]. For example, we have the following paraphrases of algebraic expressions by bracket expressions:<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
|<br />
<math>\begin{matrix}<br />
x + y ~=~ \texttt{(} x, y \texttt{)}<br />
\\[6pt]<br />
x + y + z ~=~ \texttt{((} x, y \texttt{)}, z \texttt{)} ~=~ \texttt{(} x, \texttt{(} y, z \texttt{))}<br />
\end{matrix}</math><br />
|}<br />
<br />
It is important to note that the last expressions are not equivalent to the triple bracket <math>\texttt{(} x, y, z \texttt{)}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 1.} ~~ \text{Syntax and Semantics of a Calculus for Propositional Logic}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Expression}~\!</math><br />
| <math>\text{Interpretation}\!</math><br />
| <math>\text{Other Notations}\!</math><br />
|-<br />
| &nbsp;<br />
| <math>\text{True}\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{False}\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>x\!</math><br />
| <math>x\!</math><br />
| <math>x\!</math><br />
|-<br />
| <math>\texttt{(} x \texttt{)}\!</math><br />
| <math>\text{Not}~ x\!</math><br />
|<br />
<math>\begin{matrix}<br />
x'<br />
\\<br />
\tilde{x}<br />
\\<br />
\lnot x<br />
\end{matrix}\!</math><br />
|-<br />
| <math>x~y~z\!</math><br />
| <math>x ~\text{and}~ y ~\text{and}~ z\!</math><br />
| <math>x \land y \land z\!</math><br />
|-<br />
| <math>\texttt{((} x \texttt{)(} y \texttt{)(} z \texttt{))}\!</math><br />
| <math>x ~\text{or}~ y ~\text{or}~ z\!</math><br />
| <math>x \lor y \lor z\!</math><br />
|-<br />
| <math>\texttt{(} x ~ \texttt{(} y \texttt{))}\!</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{implies}~ y<br />
\\<br />
\mathrm{If}~ x ~\text{then}~ y<br />
\end{matrix}</math><br />
| <math>x \Rightarrow y\!</math><br />
|-<br />
| <math>\texttt{(} x \texttt{,} y \texttt{)}\!</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{not equal to}~ y<br />
\\<br />
x ~\text{exclusive or}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x \ne y<br />
\\<br />
x + y<br />
\end{matrix}</math><br />
|-<br />
| <math>\texttt{((} x \texttt{,} y \texttt{))}\!</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{is equal to}~ y<br />
\\<br />
x ~\text{if and only if}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x = y<br />
\\<br />
x \Leftrightarrow y<br />
\end{matrix}</math><br />
|-<br />
| <math>\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Just one of}<br />
\\<br />
x, y, z<br />
\\<br />
\text{is false}.<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x'y~z~ & \lor<br />
\\<br />
x~y'z~ & \lor<br />
\\<br />
x~y~z' &<br />
\end{matrix}</math><br />
|-<br />
| <math>\texttt{((} x \texttt{),(} y \texttt{),(} z \texttt{))}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Just one of}<br />
\\<br />
x, y, z<br />
\\<br />
\text{is true}.<br />
\\<br />
&<br />
\\<br />
\text{Partition all}<br />
\\<br />
\text{into}~ x, y, z.<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x~y'z' & \lor<br />
\\<br />
x'y~z' & \lor<br />
\\<br />
x'y'z~ &<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,} y \texttt{),} z \texttt{)}<br />
\\<br />
&<br />
\\<br />
\texttt{(} x \texttt{,(} y \texttt{,} z \texttt{))}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Oddly many of}<br />
\\<br />
x, y, z<br />
\\<br />
\text{are true}.<br />
\end{matrix}\!</math><br />
|<br />
<p><math>x + y + z\!</math></p><br />
<br><br />
<p><math>\begin{matrix}<br />
x~y~z~ & \lor<br />
\\<br />
x~y'z' & \lor<br />
\\<br />
x'y~z' & \lor<br />
\\<br />
x'y'z~ &<br />
\end{matrix}\!</math></p><br />
|-<br />
| <math>\texttt{(} w \texttt{,(} x \texttt{),(} y \texttt{),(} z \texttt{))}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Partition}~ w<br />
\\<br />
\text{into}~ x, y, z.<br />
\\<br />
&<br />
\\<br />
\text{Genus}~ w ~\text{comprises}<br />
\\<br />
\text{species}~ x, y, z.<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
w'x'y'z' & \lor<br />
\\<br />
w~x~y'z' & \lor<br />
\\<br />
w~x'y~z' & \lor<br />
\\<br />
w~x'y'z~ &<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
'''Note.''' The usage that one often sees, of a plus sign "<math>+\!</math>" to represent inclusive disjunction, and the reference to this operation as ''boolean addition'', is a misnomer on at least two counts. Boole used the plus sign to represent exclusive disjunction (at any rate, an operation of aggregation restricted in its logical interpretation to cases where the represented sets are disjoint (Boole, 32)), as any mathematician with a sensitivity to the ring and field properties of algebra would do:<br />
<br />
<blockquote><br />
The expression <math>x + y\!</math> seems indeed uninterpretable, unless it be assumed that the things represented by <math>x\!</math> and the things represented by <math>y\!</math> are entirely separate; that they embrace no individuals in common. (Boole, 66).<br />
</blockquote><br />
<br />
It was only later that Peirce and Jevons treated inclusive disjunction as a fundamental operation, but these authors, with a respect for the algebraic properties that were already associated with the plus sign, used a variety of other symbols for inclusive disjunction (Sty, 177, 189). It seems to have been Schröder who later reassigned the plus sign to inclusive disjunction (Sty, 208). Additional information, discussion, and references can be found in (Boole) and (Sty, 177&ndash;263). Aside from these historical points, which never really count against a current practice that has gained a life of its own, this usage does have a further disadvantage of cutting or confounding the lines of communication between algebra and logic. For this reason, it will be avoided here.<br />
<br />
==A Functional Conception of Propositional Calculus==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Out of the dimness opposite equals advance . . . .<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Always substance and increase,<br><br />
Always a knit of identity . . . . always distinction . . . .<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;always a breed of life.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]<br />
|}<br />
<br />
In the general case, we start with a set of logical features <math>\{a_1, \ldots, a_n\}</math> that represent properties of objects or propositions about the world. In concrete examples the features <math>\{a_i\!\}</math> commonly appear as capital letters from an ''alphabet'' like <math>\{A, B, C, \ldots\}</math> or as meaningful words from a linguistic ''vocabulary'' of codes. This language can be drawn from any sources, whether natural, technical, or artificial in character and interpretation. In the application to dynamic systems we tend to use the letters <math>\{x_1, \ldots, x_n\}</math> as our coordinate propositions, and to interpret them as denoting properties of a system's ''state'', that is, as propositions about its location in configuration space. Because I have to consider non-deterministic systems from the outset, I often use the word ''state'' in a loose sense, to denote the position or configuration component of a contemplated state vector, whether or not it ever gets a deterministic completion.<br />
<br />
The set of logical features <math>\{a_1, \ldots, a_n\}</math> provides a basis for generating an <math>n\!</math>-dimensional ''[[universe of discourse]]'' that I denote as <math>[a_1, \ldots, a_n].</math> It is useful to consider each universe of discourse as a unified categorical object that incorporates both the set of points <math>\langle a_1, \ldots, a_n \rangle</math> and the set of propositions <math>f : \langle a_1, \ldots, a_n \rangle \to \mathbb{B}</math> that are implicit with the ordinary picture of a venn diagram on <math>n\!</math> features. Thus, we may regard the universe of discourse <math>[a_1, \ldots, a_n]</math> as an ordered pair having the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}),</math> and we may abbreviate this last type designation as <math>\mathbb{B}^n\ +\!\to \mathbb{B},</math> or even more succinctly as <math>[\mathbb{B}^n].</math> (Used this way, the angle brackets <math>\langle\ldots\rangle</math> are referred to as ''generator brackets''.)<br />
<br />
Table 2 exhibits the scheme of notation I use to formalize the domain of propositional calculus, corresponding to the logical content of truth tables and venn diagrams. Although it overworks the square brackets a bit, I also use either one of the equivalent notations <math>[n]\!</math> or <math>\mathbf{n}</math> to denote the data type of a finite set on <math>n\!</math> elements.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 2.} ~~ \text{Propositional Calculus : Basic Notation}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Symbol}\!</math><br />
| <math>\text{Notation}\!</math><br />
| <math>\text{Description}\!</math><br />
| <math>\text{Type}\!</math><br />
|-<br />
| <math>\mathfrak{A}\!</math><br />
| <math>\{ {}^{\backprime\backprime} a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} a_n {}^{\prime\prime} \}\!</math><br />
| <math>\text{Alphabet}\!</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>\mathcal{A}\!</math><br />
| <math>\{ a_1, \ldots, a_n \}\!</math><br />
| <math>\text{Basis}\!</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>A_i\!</math><br />
| <math>\{ \texttt{(} a_i \texttt{)}, a_i \}\!</math><br />
| <math>\text{Dimension}~ i\!</math><br />
| <math>\mathbb{B}\!</math><br />
|-<br />
| <math>A\!</math><br />
|<br />
<math>\begin{matrix}<br />
\langle \mathcal{A} \rangle<br />
\\[2pt]<br />
\langle a_1, \ldots, a_n \rangle<br />
\\[2pt]<br />
\{ (a_1, \ldots, a_n) \}<br />
\\[2pt]<br />
A_1 \times \ldots \times A_n<br />
\\[2pt]<br />
\textstyle \prod_{i=1}^n A_i<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Set of cells},<br />
\\[2pt]<br />
\text{coordinate tuples},<br />
\\[2pt]<br />
\text{points, or vectors}<br />
\\[2pt]<br />
\text{in the universe}<br />
\\[2pt]<br />
\text{of discourse}<br />
\end{matrix}</math><br />
| <math>\mathbb{B}^n\!</math><br />
|-<br />
| <math>A^*\!</math><br />
| <math>(\mathrm{hom} : A \to \mathbb{B})\!</math><br />
| <math>\text{Linear functions}\!</math><br />
| <math>(\mathbb{B}^n)^* \cong \mathbb{B}^n\!</math><br />
|-<br />
| <math>A^\uparrow\!</math><br />
| <math>(A \to \mathbb{B})\!</math><br />
| <math>\text{Boolean functions}\!</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}\!</math><br />
|-<br />
| <math>A^\bullet\!</math><br />
|<br />
<math>\begin{matrix}<br />
[\mathcal{A}]<br />
\\[2pt]<br />
(A, A^\uparrow)<br />
\\[2pt]<br />
(A ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
(A, (A \to \mathbb{B}))<br />
\\[2pt]<br />
[a_1, \ldots, a_n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Universe of discourse}<br />
\\[2pt]<br />
\text{based on the features}<br />
\\[2pt]<br />
\{ a_1, \ldots, a_n \}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))<br />
\\[2pt]<br />
(\mathbb{B}^n ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
[\mathbb{B}^n]<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
===Qualitative Logic and Quantitative Analogy===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
''Logical'', however, is used in a third sense, which is at once more vital and more practical; to denote, namely, the systematic care, negative and positive, taken to safeguard reflection so that it may yield the best results under the given conditions.<br />
| width="4%" | &nbsp;<br />
|- <br />
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 56]<br />
|}<br />
<br />
These concepts and notations may now be explained in greater detail. In order to begin as simply as possible, let us distinguish two levels of analysis and set out initially on the easier path. On the first level of analysis we take spaces like <math>\mathbb{B},</math> <math>\mathbb{B}^n,</math> and <math>(\mathbb{B}^n \to \mathbb{B})</math> at face value and treat them as the primary objects of interest. On the second level of analysis we use these spaces as coordinate charts for talking about points and functions in more fundamental spaces.<br />
<br />
A pair of spaces, of types <math>\mathbb{B}^n</math> and <math>(\mathbb{B}^n \to \mathbb{B}),</math> give typical expression to everything that we commonly associate with the ordinary picture of a venn diagram. The dimension, <math>n,\!</math> counts the number of "circles" or simple closed curves that are inscribed in the universe of discourse, corresponding to its relevant logical features or basic propositions. Elements of type <math>\mathbb{B}^n</math> correspond to what are often called propositional ''interpretations'' in logic, that is, the different assignments of truth values to sentence letters. Relative to a given universe of discourse, these interpretations are visualized as its ''cells'', in other words, the smallest enclosed areas or undivided regions of the venn diagram. The functions <math>f : \mathbb{B}^n \to \mathbb{B}</math> correspond to the different ways of shading the venn diagram to indicate arbitrary propositions, regions, or sets. Regions included under a shading indicate the ''models'', and regions excluded represent the ''non-models'' of a proposition. To recognize and formalize the natural cohesion of these two layers of concepts into a single universe of discourse, we introduce the type notations <math>[\mathbb{B}^n] = \mathbb{B}^n\ +\!\to \mathbb{B}</math> to stand for the pair of types <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})).</math> The resulting "stereotype" serves to frame the universe of discourse as a unified categorical object, and makes it subject to prescribed sets of evaluations and transformations (categorical morphisms or ''arrows'') that affect the universe of discourse as an integrated whole.<br />
<br />
Most of the time we can serve the algebraic, geometric, and logical interests of our study without worrying about their occasional conflicts and incidental divergences. The conventions and definitions already set down will continue to cover most of the algebraic and functional aspects of our discussion, but to handle the logical and qualitative aspects we will need to add a few more. In general, abstract sets may be denoted by gothic, greek, or script capital variants of <math>A, B, C,\!</math> and so on, with their elements being denoted by a corresponding set of subscripted letters in plain lower case, for example, <math>\mathcal{A} = \{a_i\}.</math> Most of the time, a set such as <math>\mathcal{A} = \{a_i\}\!</math> will be employed as the ''alphabet'' of a [[formal language]]. These alphabet letters serve to name the logical features (properties or propositions) that generate a particular universe of discourse. When we want to discuss the particular features of a universe of discourse, beyond the abstract designation of a type like <math>(\mathbb{B}^n\ +\!\to \mathbb{B}),</math> then we may use the following notations. If <math>\mathcal{A} = \{a_1, \ldots, a_n\}</math> is an alphabet of logical features, then <math>A = \langle \mathcal{A} \rangle = \langle a_1, \ldots, a_n \rangle</math> is the set of interpretations, <math>A^\uparrow = (A \to \mathbb{B})</math> is the set of propositions, and <math>A^\bullet = [\mathcal{A}] = [a_1, \ldots, a_n]</math> is the combination of these interpretations and propositions into the universe of discourse that is based on the features <math>\{a_1, \ldots, a_n\}.</math><br />
<br />
As always, especially in concrete examples, these rules may be dropped whenever necessary, reverting to a free assortment of feature labels. However, when we need to talk about the logical aspects of a space that is already named as a vector space, it will be necessary to make special provisions. At any rate, these elaborations can be deferred until actually needed.<br />
<br />
===Philosophy of Notation : Formal Terms and Flexible Types===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Where number is irrelevant, regimented mathematical technique has hitherto tended to be lacking. Thus it is that the progress of natural science has depended so largely upon the discernment of measurable quantity of one sort or another.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]<br />
|}<br />
<br />
For much of our discussion propositions and boolean functions are treated as the same formal objects, or as different interpretations of the same formal calculus. This rule of interpretation has exceptions, though. There is a distinctively logical interest in the use of propositional calculus that is not exhausted by its functional interpretation. It is part of our task in this study to deal with these uniquely logical characteristics as they present themselves both in our subject matter and in our formal calculus. Just to provide a hint of what's at stake: In logic, as opposed to the more imaginative realms of mathematics, we consider it a good thing to always know what we are talking about. Where mathematics encourages tolerance for uninterpreted symbols as intermediate terms, logic exerts a keener effort to interpret directly each oblique carrier of meaning, no matter how slight, and to unfold the complicities of every indirection in the flow of information. Translated into functional terms, this means that we want to maintain a continual, immediate, and persistent sense of both the converse relation <math>f^{-1} \subseteq \mathbb{B} \times \mathbb{B}^n,</math> or what is the same thing, <math>f^{-1} : \mathbb{B} \to \mathcal{P}(\mathbb{B}^n),</math> and the ''fibers'' or inverse images <math>f^{-1}(0)\!</math> and <math>f^{-1}(1),\!</math> associated with each boolean function <math>f : \mathbb{B}^n \to \mathbb{B}</math> that we use. In practical terms, the desired implementation of a propositional interpreter should incorporate our intuitive recognition that the induced partition of the functional domain into level sets <math>f^{-1}(b),\!</math> for <math>b \in \mathbb{B},</math> is part and parcel of understanding the denotative uses of each propositional function <math>f.\!</math><br />
<br />
===Special Classes of Propositions===<br />
<br />
It is important to remember that the coordinate propositions <math>\{a_i\},\!</math> besides being projection maps <math>a_i : \mathbb{B}^n \to \mathbb{B},</math> are propositions on an equal footing with all others, even though employed as a basis in a particular moment. This set of <math>n\!</math> propositions may sometimes be referred to as the ''basic propositions'', the ''coordinate propositions'', or the ''simple propositions'' that found a universe of discourse. Either one of the equivalent notations, <math>\{a_i : \mathbb{B}^n \to \mathbb{B}\}</math> or <math>(\mathbb{B}^n \xrightarrow{i} \mathbb{B}),</math> may be used to indicate the adoption of the propositions <math>a_i\!</math> as a basis for describing a universe of discourse.<br />
<br />
Among the <math>2^{2^n}</math> propositions in <math>[a_1, \ldots, a_n]</math> are several families of <math>2^n\!</math> propositions each that take on special forms with respect to the basis <math>\{ a_1, \ldots, a_n \}.</math> Three of these families are especially prominent in the present context, the ''linear'', the ''positive'', and the ''singular'' propositions. Each family is naturally parameterized by the coordinate <math>n\!</math>-tuples in <math>\mathbb{B}^n</math> and falls into <math>n + 1\!</math> ranks, with a binomial coefficient <math>\tbinom{n}{k}</math> giving the number of propositions that have rank or weight <math>k.\!</math><br />
<br />
<ul><br />
<br />
<li><br />
<p>The ''linear propositions'', <math>\{ \ell : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{\ell} \mathbb{B}),\!</math> may be written as sums:</p><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\sum_{i=1}^n e_i ~=~ e_1 + \ldots + e_n<br />
~\text{where}~<br />
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 0 \end{matrix}\right\}<br />
~\text{for}~ i = 1 ~\text{to}~ n.\!</math><br />
|}<br />
</li><br />
<br />
<li><br />
<p>The ''positive propositions'', <math>\{ p : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{p} \mathbb{B}),\!</math> may be written as products:</p><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\prod_{i=1}^n e_i ~=~ e_1 \cdot \ldots \cdot e_n<br />
~\text{where}~<br />
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 1 \end{matrix}\right\}<br />
~\text{for}~ i = 1 ~\text{to}~ n.\!</math><br />
|}<br />
</li><br />
<br />
<li><br />
<p>The ''singular propositions'', <math>\{ \mathbf{x} : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{s} \mathbb{B}),\!</math> may be written as products:</p><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\prod_{i=1}^n e_i ~=~ e_1 \cdot \ldots \cdot e_n<br />
~\text{where}~<br />
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = \texttt{(} a_i \texttt{)} \end{matrix}\right\}<br />
~\text{for}~ i = 1 ~\text{to}~ n.\!</math><br />
|}<br />
</li><br />
<br />
</ul><br />
<br />
In each case the rank <math>k\!</math> ranges from <math>0\!</math> to <math>n\!</math> and counts the number of positive appearances of the coordinate propositions <math>a_1, \ldots, a_n\!</math> in the resulting expression. For example, for <math>{n = 3},\!</math> the linear proposition of rank <math>0\!</math> is <math>0,\!</math> the positive proposition of rank <math>0\!</math> is <math>1,\!</math> and the singular proposition of rank <math>0\!</math> is <math>\texttt{(} a_1 \texttt{)(} a_2 \texttt{)(} a_3\texttt{)}.\!</math><br />
<br />
The basic propositions <math>a_i : \mathbb{B}^n \to \mathbb{B}</math> are both linear and positive. So these two kinds of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions.<br />
<br />
Linear propositions and positive propositions are generated by taking boolean sums and products, respectively, over selected subsets of basic propositions, so both families of propositions are parameterized by the powerset <math>\mathcal{P}(\mathcal{I}),</math> that is, the set of all subsets <math>J\!</math> of the basic index set <math>\mathcal{I} = \{1, \ldots, n\}.\!</math><br />
<br />
Let us define <math>\mathcal{A}_J</math> as the subset of <math>\mathcal{A}</math> that is given by <math>\{a_i : i \in J\}.\!</math> Then we may comprehend the action of the linear and the positive propositions in the following terms:<br />
<br />
* The linear proposition <math>\ell_J : \mathbb{B}^n \to \mathbb{B}</math> evaluates each cell <math>\mathbf{x}</math> of <math>\mathbb{B}^n</math> by looking at the coefficients of <math>\mathbf{x}</math> with respect to the features that <math>\ell_J</math> "likes", namely those in <math>\mathcal{A}_J,</math> and then adds them up in <math>\mathbb{B}.</math> Thus, <math>\ell_J(\mathbf{x})</math> computes the parity of the number of features that <math>\mathbf{x}</math> has in <math>\mathcal{A}_J,</math> yielding one for odd and zero for even. Expressed in this idiom, <math>\ell_J(\mathbf{x}) = 1</math> says that <math>\mathbf{x}</math> seems ''odd'' (or ''oddly true'') to <math>\mathcal{A}_J,</math> whereas <math>\ell_J(\mathbf{x}) = 0</math> says that <math>\mathbf{x}</math> seems ''even'' (or ''evenly true'') to <math>\mathcal{A}_J,</math> so long as we recall that ''zero times'' is evenly often, too.<br />
<br />
* The positive proposition <math>p_J : \mathbb{B}^n \to \mathbb{B}</math> evaluates each cell <math>\mathbf{x}</math> of <math>\mathbb{B}^n</math> by looking at the coefficients of <math>\mathbf{x}</math> with regard to the features that <math>p_J\!</math> "likes", namely those in <math>\mathcal{A}_J,</math> and then takes their product in <math>\mathbb{B}.</math> Thus, <math>p_J(\mathbf{x})</math> assesses the unanimity of the multitude of features that <math>\mathbf{x}</math> has in <math>\mathcal{A}_J,</math> yielding one for all and aught for else. In these consensual or contractual terms, <math>p_J(\mathbf{x}) = 1</math> means that <math>\mathbf{x}</math> is ''AOK'' or congruent with all of the conditions of <math>\mathcal{A}_J,</math> while <math>p_J(\mathbf{x}) = 0</math> means that <math>\mathbf{x}</math> defaults or dissents from some condition of <math>\mathcal{A}_J.</math><br />
<br />
===Basis Relativity and Type Ambiguity===<br />
<br />
Finally, two things are important to keep in mind with regard to the simplicity, linearity, positivity, and singularity of propositions.<br />
<br />
First, all of these properties are relative to a particular basis. For example, a singular proposition with respect to a basis <math>\mathcal{A}</math> will not remain singular if <math>\mathcal{A}</math> is extended by a number of new and independent features. Even if we stick to the original set of pairwise options <math>\{a_i\} \cup \{ \texttt{(} a_i \texttt{)} \}\!</math> to select a new basis, the sets of linear and positive propositions are determined by the choice of simple propositions, and this determination is tantamount to the conventional choice of a cell as origin.<br />
<br />
Second, the singular propositions <math>\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B},</math> picking out as they do a single cell or a coordinate tuple <math>\mathbf{x}</math> of <math>\mathbb{B}^n,</math> become the carriers or the vehicles of a certain type-ambiguity that vacillates between the dual forms <math>\mathbb{B}^n</math> and <math>(\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B})</math> and infects the whole hierarchy of types built on them. In other words, the terms that signify the interpretations <math>\mathbf{x} : \mathbb{B}^n</math> and the singular propositions <math>\mathbf{x} : \mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B}</math> are fully equivalent in information, and this means that every token of the type <math>\mathbb{B}^n</math> can be reinterpreted as an appearance of the subtype <math>\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B}.</math> And vice versa, the two types can be exchanged with each other everywhere that they turn up. In practical terms, this allows the use of singular propositions as a way of denoting points, forming an alternative to coordinate tuples.<br />
<br />
For example, relative to the universe of discourse <math>[a_1, a_2, a_3]\!</math> the singular proposition <math>a_1 a_2 a_3 : \mathbb{B}^3 \xrightarrow{s} \mathbb{B}</math> could be explicitly retyped as <math>a_1 a_2 a_3 : \mathbb{B}^3</math> to indicate the point <math>(1, 1, 1)\!</math> but in most cases the proper interpretation could be gathered from context. Both notations remain dependent on a particular basis, but the code that is generated under the singular option has the advantage in its self-commenting features, in other words, it constantly reminds us of its basis in the process of denoting points. When the time comes to put a multiplicity of different bases into play, and to search for objects and properties that remain invariant under the transformations between them, this infinitesimal potential advantage may well evolve into an overwhelming practical necessity.<br />
<br />
===The Analogy Between Real and Boolean Types===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Measurement consists in correlating our subject matter with the series of real numbers; and such correlations are desirable because, once they are set up, all the well-worked theory of numerical mathematics lies ready at hand as a tool for our further reasoning.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]<br />
|}<br />
<br />
There are two further reasons why it useful to spend time on a careful treatment of types, and they both have to do with our being able to take full computational advantage of certain dimensions of flexibility in the types that apply to terms. First, the domains of differential geometry and logic programming are connected by analogies between real and boolean types of the same pattern. Second, the types involved in these patterns have important isomorphisms connecting them that apply on both the real and the boolean sides of the picture.<br />
<br />
Amazingly enough, these isomorphisms are themselves schematized by the axioms and theorems of propositional logic. This fact is known as the ''propositions as types'' analogy or the Curry&ndash;Howard isomorphism [How]. In another formulation it says that terms are to types as proofs are to propositions. See [LaS, 42&ndash;46] and [SeH] for a good discussion and further references. To anticipate the bearing of these issues on our immediate topic, Table&nbsp;3 sketches a partial overview of the Real to Boolean analogy that may serve to illustrate the paradigm in question.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 3.} ~~ \text{Analogy Between Real and Boolean Types}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Real Domain} ~ \mathbb{R}\!</math><br />
| <math>\longleftrightarrow\!</math><br />
| <math>\text{Boolean Domain} ~ \mathbb{B}\!</math><br />
|-<br />
| <math>\mathbb{R}^n\!</math><br />
| <math>\text{Basic Space}\!</math><br />
| <math>\mathbb{B}^n\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}\!</math><br />
| <math>\text{Function Space}\!</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R}\!</math><br />
| <math>\text{Tangent Vector}\!</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to ((\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R})\!</math><br />
| <math>\text{Vector Field}\!</math><br />
| <math>\mathbb{B}^n \to ((\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B})\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \times (\mathbb{R}^n \to \mathbb{R})) \to \mathbb{R}\!</math><br />
| '''<font size="4">"</font>'''<br />
| <math>(\mathbb{B}^n \times (\mathbb{B}^n \to \mathbb{B})) \to \mathbb{B}\!</math><br />
|-<br />
| <math>((\mathbb{R}^n \to \mathbb{R}) \times \mathbb{R}^n) \to \mathbb{R}\!</math><br />
| '''<font size="4">"</font>'''<br />
| <math>((\mathbb{B}^n \to \mathbb{B}) \times \mathbb{B}^n) \to \mathbb{B}\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^n \to \mathbb{R})~\!</math><br />
| <math>\text{Derivation}\!</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^n \to \mathbb{B})\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}^m\!</math><br />
| <math>\begin{matrix}\text{Basic}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}^m\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^m \to \mathbb{R})\!</math><br />
| <math>\begin{matrix}\text{Function}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^m \to \mathbb{B})\!</math><br />
|}<br />
<br />
<br><br />
<br />
The Table exhibits a sample of likely parallels between the real and boolean domains. The central column gives a selection of terminology that is borrowed from differential geometry and extended in its meaning to the logical side of the Table. These are the varieties of spaces that come up when we turn to analyzing the dynamics of processes that pursue their courses through the states of an arbitrary space <math>X.\!</math> Moreover, when it becomes necessary to approach situations of overwhelming dynamic complexity in a succession of qualitative reaches, then the methods of logic that are afforded by the boolean domains, with their declarative means of synthesis and deductive modes of analysis, supply a natural battery of tools for the task.<br />
<br />
It is usually expedient to take these spaces two at a time, in dual pairs of the form <math>X\!</math> and <math>(X \to \mathbb{K}).</math> In general, one creates pairs of type schemas by replacing any space <math>X\!</math> with its dual <math>(X \to \mathbb{K}),</math> for example, pairing the type <math>X \to Y</math> with the type <math>(X \to \mathbb{K}) \to (Y \to \mathbb{K}),</math> and <math>X \times Y</math> with <math>(X \to \mathbb{K}) \times (Y \to \mathbb{K}).</math> The word ''dual'' is used here in its broader sense to mean all of the functionals, not just the linear ones. Given any function <math>f : X \to \mathbb{K},</math> the ''converse'' or inverse relation corresponding to <math>f\!</math> is denoted <math>f^{-1},\!</math> and the subsets of <math>X\!</math> that are defined by <math>f^{-1}(k),\!</math> taken over <math>k\!</math> in <math>\mathbb{K},</math> are called the ''fibers'' or the ''level sets'' of the function <math>f.\!</math><br />
<br />
===Theory of Control and Control of Theory===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
You will hardly know who I am or what I mean,<br><br />
But I shall be good health to you nevertheless,<br><br />
And filter and fibre your blood.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]<br />
|}<br />
<br />
In the boolean context a function <math>f : X \to \mathbb{B}</math> is tantamount to a ''proposition'' about elements of <math>X,\!</math> and the elements of <math>X\!</math> constitute the ''interpretations'' of that proposition. The fiber <math>f^{-1}(1)\!</math> comprises the set of ''models'' of <math>f,\!</math> or examples of elements in <math>X\!</math> satisfying the proposition <math>f.\!</math> The fiber <math>f^{-1}(0)\!</math> collects the complementary set of ''anti-models'', or the exceptions to the proposition <math>f\!</math> that exist in <math>X.\!</math> Of course, the space of functions <math>(X \to \mathbb{B})\!</math> is isomorphic to the set of all subsets of <math>X,\!</math> called the ''power set'' of <math>X,\!</math> and often denoted <math>\mathcal{P}(X)</math> or <math>2^X.\!</math><br />
<br />
The operation of replacing <math>X\!</math> by <math>(X \to \mathbb{B})\!</math> in a type schema corresponds to a certain shift of attitude towards the space <math>X,\!</math> in which one passes from a focus on the ostensibly individual elements of <math>X\!</math> to a concern with the states of information and uncertainty that one possesses about objects and situations in <math>X.\!</math> The conceptual obstacles in the path of this transition can be smoothed over by using singular functions <math>(\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B})</math> as stepping stones. First of all, it's an easy step from an element <math>\mathbf{x}</math> of type <math>\mathbb{B}^n</math> to the equivalent information of a singular proposition <math>\mathbf{x} : X \xrightarrow{s} \mathbb{B}, </math> and then only a small jump of generalization remains to reach the type of an arbitrary proposition <math>f : X \to \mathbb{B},</math> perhaps understood to indicate a relaxed constraint on the singularity of points or a neighborhood circumscribing the original <math>\mathbf{x}.</math> This is frequently a useful transformation, communicating between the ''objective'' and the ''intentional'' perspectives, in spite perhaps of the open objection that this distinction is transient in the mean time and ultimately superficial.<br />
<br />
It is hoped that this measure of flexibility, allowing us to stretch a point into a proposition, can be useful in the examination of inquiry driven systems, where the differences between empirical, intentional, and theoretical propositions constitute the discrepancies and the distributions that drive experimental activity. I can give this model of inquiry a cybernetic cast by realizing that theory change and theory evolution, as well as the choice and the evaluation of experiments, are actions that are taken by a system or its agent in response to the differences that are detected between observational contents and theoretical coverage.<br />
<br />
All of the above notwithstanding, there are several points that distinguish these two tasks, namely, the ''theory of control'' and the ''control of theory'', features that are often obscured by too much precipitation in the quickness with which we understand their similarities. In the control of uncertainty through inquiry, some of the actuators that we need to be concerned with are axiom changers and theory modifiers, operators with the power to compile and to revise the theories that generate expectations and predictions, effectors that form and edit our grammars for the languages of observational data, and agencies that rework the proposed model to fit the actual sequences of events and the realized relationships of values that are observed in the environment. Moreover, when steps must be taken to carry out an experimental action, there must be something about the particular shape of our uncertainty that guides us in choosing what directions to explore, and this impression is more than likely influenced by previous accumulations of experience. Thus it must be anticipated that much of what goes into scientific progress, or any sustainable effort toward a goal of knowledge, is necessarily predicated on long term observation and modal expectations, not only on the more local or short term prediction and correction.<br />
<br />
===Propositions as Types and Higher Order Types===<br />
<br />
The types collected in Table&nbsp;3 (repeated below) serve to illustrate the themes of ''higher order propositional expressions'' and the ''propositions as types'' (PAT) analogy.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 3.} ~~ \text{Analogy Between Real and Boolean Types}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Real Domain} ~ \mathbb{R}\!</math><br />
| <math>\longleftrightarrow\!</math><br />
| <math>\text{Boolean Domain} ~ \mathbb{B}\!</math><br />
|-<br />
| <math>\mathbb{R}^n\!</math><br />
| <math>\text{Basic Space}\!</math><br />
| <math>\mathbb{B}^n\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}\!</math><br />
| <math>\text{Function Space}\!</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R}\!</math><br />
| <math>\text{Tangent Vector}\!</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to ((\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R})\!</math><br />
| <math>\text{Vector Field}\!</math><br />
| <math>\mathbb{B}^n \to ((\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B})\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \times (\mathbb{R}^n \to \mathbb{R})) \to \mathbb{R}\!</math><br />
| '''<font size="4">"</font>'''<br />
| <math>(\mathbb{B}^n \times (\mathbb{B}^n \to \mathbb{B})) \to \mathbb{B}\!</math><br />
|-<br />
| <math>((\mathbb{R}^n \to \mathbb{R}) \times \mathbb{R}^n) \to \mathbb{R}\!</math><br />
| '''<font size="4">"</font>'''<br />
| <math>((\mathbb{B}^n \to \mathbb{B}) \times \mathbb{B}^n) \to \mathbb{B}\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^n \to \mathbb{R})~\!</math><br />
| <math>\text{Derivation}\!</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^n \to \mathbb{B})\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}^m\!</math><br />
| <math>\begin{matrix}\text{Basic}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}^m\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^m \to \mathbb{R})\!</math><br />
| <math>\begin{matrix}\text{Function}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^m \to \mathbb{B})\!</math><br />
|}<br />
<br />
<br><br />
<br />
First, observe that the type of a ''tangent vector at a point'', also known as a ''directional derivative'' at that point, has the form <math>(\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K},</math> where <math>\mathbb{K}</math> is the chosen ground field, in the present case either <math>\mathbb{R}</math> or <math>\mathbb{B}.</math> At a point in a space of type <math>\mathbb{K}^n,</math> a directional derivative operator <math>\vartheta\!</math> takes a function on that space, an <math>f\!</math> of type <math>(\mathbb{K}^n \to \mathbb{K}),</math> and maps it to a ground field value of type <math>\mathbb{K}.</math> This value is known as the ''derivative'' of <math>f\!</math> in the direction <math>\vartheta\!</math> [Che46, 76&ndash;77]. In the boolean case <math>\vartheta : (\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math> has the form of a proposition about propositions, in other words, a proposition of the next higher type.<br />
<br />
Next, by way of illustrating the propositions as types idea, consider a proposition of the form <math>X \Rightarrow (Y \Rightarrow Z).</math> One knows from propositional calculus that this is logically equivalent to a proposition of the form <math>(X \land Y) \Rightarrow Z.</math> But this equivalence should remind us of the functional isomorphism that exists between a construction of the type <math>X \to (Y \to Z)</math> and a construction of the type <math>(X \times Y) \to Z.</math> The propositions as types analogy permits us to take a functional type like this and, under the right conditions, replace the functional arrows &ldquo;<math>\to~\!</math>&rdquo; and products &ldquo;<math>\times\!</math>&rdquo; with the respective logical arrows &ldquo;<math>\Rightarrow\!</math>&rdquo; and products &ldquo;<math>\land\!</math>&rdquo;. Accordingly, viewing the result as a proposition, we can employ axioms and theorems of propositional calculus to suggest appropriate isomorphisms among the categorical and functional constructions.<br />
<br />
Finally, examine the middle four rows of Table&nbsp;3. These display a series of isomorphic types that stretch from the categories that are labeled ''Vector Field'' to those that are labeled ''Derivation''. A ''vector field'', also known as an ''infinitesimal transformation'', associates a tangent vector at a point with each point of a space. In symbols, a vector field is a function of the form <math>\textstyle \xi : X \to \bigcup_{x \in X} \xi_x\!</math> that assigns to each point <math>x\!</math> of the space <math>X\!</math> a tangent vector to <math>X\!</math> at that point, namely, the tangent vector <math>\xi_x\!</math> [Che46, 82&ndash;83]. If <math>X\!</math> is of the type <math>\mathbb{K}^n,</math> then <math>\xi\!</math> is of the type <math>\mathbb{K}^n \to ((\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K}).</math> This has the pattern <math>X \to (Y \to Z),</math> with <math>X = \mathbb{K}^n,</math> <math>Y = (\mathbb{K}^n \to \mathbb{K}),</math> and <math>Z = \mathbb{K}.</math><br />
<br />
Applying the propositions as types analogy, one can follow this pattern through a series of metamorphoses from the type of a vector field to the type of a derivation, as traced out in Table&nbsp;4. Observe how the function <math>f : X \to \mathbb{K},</math> associated with the place of <math>Y\!</math> in the pattern, moves through its paces from the second to the first position. In this way, the vector field <math>\xi,\!</math> initially viewed as attaching each tangent vector <math>\xi_x\!</math> to the site <math>x\!</math> where it acts in <math>X,\!</math> now comes to be seen as acting on each scalar potential <math>f : X \to \mathbb{K}</math> like a generalized species of differentiation, producing another function <math>\xi f : X \to \mathbb{K}</math> of the same type.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 4.} ~~ \text{An Equivalence Based on the Propositions as Types Analogy}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Pattern}\!</math><br />
| <math>\text{Construct}\!</math><br />
| <math>\text{Instance}\!</math><br />
|-<br />
| <math>X \to (Y \to Z)\!</math><br />
| <math>\text{Vector Field}\!</math><br />
| <math>\mathbb{K}^n \to ((\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K})\!</math><br />
|-<br />
| <math>(X \times Y) \to Z\!</math><br />
| <math>\Uparrow\!</math><br />
| <math>(\mathbb{K}^n \times (\mathbb{K}^n \to \mathbb{K})) \to \mathbb{K}\!</math><br />
|-<br />
| <math>(Y \times X) \to Z\!</math><br />
| <math>\Downarrow\!</math><br />
| <math>((\mathbb{K}^n \to \mathbb{K}) \times \mathbb{K}^n) \to \mathbb{K}\!</math><br />
|-<br />
| <math>Y \to (X \to Z)\!</math><br />
| <math>\text{Derivation}\!</math><br />
| <math>(\mathbb{K}^n \to \mathbb{K}) \to (\mathbb{K}^n \to \mathbb{K})\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Reality at the Threshold of Logic===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
But no science can rest entirely on measurement, and many scientific investigations are quite out of reach of that device. To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]<br />
|}<br />
<br />
Table 5 accumulates an array of notation that I hope will not be too distracting. Some of it is rarely needed, but has been filled in for the sake of completeness. Its purpose is simple, to give literal expression to the visual intuitions that come with venn diagrams, and to help build a bridge between our qualitative and quantitative outlooks on dynamic systems.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 5.} ~~ \text{A Bridge Over Troubled Waters}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| align="center" | <math>\text{Linear Space}\!</math><br />
| align="center" | <math>\text{Liminal Space}\!</math><br />
| align="center" | <math>\text{Logical Space}\!</math><br />
|-<br />
| <math>\begin{matrix}\mathcal{X} & = & \{ x_1, \ldots, x_n \}\end{matrix}</math><br />
| <math>\begin{matrix}\underline{\mathcal{X}} & = & \{ \underline{x}_1, \ldots, \underline{x}_n \}\end{matrix}</math><br />
| <math>\begin{matrix}\mathcal{A} & = & \{ a_1, \ldots, a_n \}\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X_i & = & \langle x_i \rangle<br />
\\<br />
& \cong & \mathbb{K}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}_i & = & \{ \texttt{(} \underline{x}_i \texttt{)}, \underline{x}_i \}<br />
\\<br />
& \cong & \mathbb{B}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A_i & = & \{ \texttt{(} a_i \texttt{)}, a_i \}<br />
\\<br />
& \cong & \mathbb{B}<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X<br />
\\<br />
= & \langle \mathcal{X} \rangle<br />
\\<br />
= & \langle x_1, \ldots, x_n \rangle<br />
\\<br />
= & X_1 \times \ldots \times X_n<br />
\\<br />
= & \prod_{i=1}^n X_i<br />
\\<br />
\cong & \mathbb{K}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}<br />
\\<br />
= & \langle \underline{\mathcal{X}} \rangle<br />
\\<br />
= & \langle \underline{x}_1, \ldots, \underline{x}_n \rangle<br />
\\<br />
= & \underline{X}_1 \times \ldots \times \underline{X}_n<br />
\\<br />
= & \prod_{i=1}^n \underline{X}_i<br />
\\<br />
\cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A<br />
\\<br />
= & \langle \mathcal{A} \rangle<br />
\\<br />
= & \langle a_1, \ldots, a_n \rangle<br />
\\<br />
= & A_1 \times \ldots \times A_n<br />
\\<br />
= & \prod_{i=1}^n A_i<br />
\\<br />
\cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X^* & = & (\ell : X \to \mathbb{K})<br />
\\<br />
& \cong & \mathbb{K}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}^* & = & (\ell : \underline{X} \to \mathbb{B})<br />
\\<br />
& \cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A^* & = & (\ell : A \to \mathbb{B})<br />
\\<br />
& \cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X^\uparrow & = & (X \to \mathbb{K})<br />
\\<br />
& \cong & (\mathbb{K}^n \to \mathbb{K})<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}^\uparrow & = & (\underline{X} \to \mathbb{B})<br />
\\<br />
& \cong & (\mathbb{B}^n \to \mathbb{B})<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A^\uparrow & = & (A \to \mathbb{B})<br />
\\<br />
& \cong & (\mathbb{B}^n \to \mathbb{B})<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X^\bullet<br />
\\<br />
= & [\mathcal{X}]<br />
\\<br />
= & [x_1, \ldots, x_n]<br />
\\<br />
= & (X, X^\uparrow)<br />
\\<br />
= & (X ~+\!\to \mathbb{K})<br />
\\<br />
= & (X, (X \to \mathbb{K}))<br />
\\<br />
\cong & (\mathbb{K}^n, (\mathbb{K}^n \to \mathbb{K}))<br />
\\<br />
= & (\mathbb{K}^n ~+\!\to \mathbb{K})<br />
\\<br />
= & [\mathbb{K}^n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}^\bullet<br />
\\<br />
= & [\underline{\mathcal{X}}]<br />
\\<br />
= & [\underline{x}_1, \ldots, \underline{x}_n]<br />
\\<br />
= & (\underline{X}, \underline{X}^\uparrow)<br />
\\<br />
= & (\underline{X} ~+\!\to \mathbb{B})<br />
\\<br />
= & (\underline{X}, (\underline{X} \to \mathbb{B}))<br />
\\<br />
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))<br />
\\<br />
= & (\mathbb{B}^n ~+\!\to \mathbb{B})<br />
\\<br />
= & [\mathbb{B}^n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A^\bullet<br />
\\<br />
= & [\mathcal{A}]<br />
\\<br />
= & [a_1, \ldots, a_n]<br />
\\<br />
= & (A, A^\uparrow)<br />
\\<br />
= & (A ~+\!\to \mathbb{B})<br />
\\<br />
= & (A, (A \to \mathbb{B}))<br />
\\<br />
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))<br />
\\<br />
= & (\mathbb{B}^n ~+\!\to \mathbb{B})<br />
\\<br />
= & [\mathbb{B}^n]<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
The left side of the Table collects mostly standard notation for an <math>n\!</math>-dimensional vector space over a field <math>\mathbb{K}.</math> The right side of the table repeats the first elements of a notation that I sketched above, to be used in further developments of propositional calculus. (I plan to use this notation in the logical analysis of neural network systems.) The middle column of the table is designed as a transitional step from the case of an arbitrary field <math>\mathbb{K},</math> with a special interest in the continuous line <math>\mathbb{R},</math> to the qualitative and discrete situations that are instanced and typified by <math>\mathbb{B}.</math><br />
<br />
I now proceed to explain these concepts in more detail. The most important ideas developed in Table&nbsp;5 are these:<br />
<br />
* The idea of a universe of discourse, which includes both a space of ''points'' and a space of ''maps'' on those points.<br />
<br />
* The idea of passing from a more complex universe to a simpler universe by a process of ''thresholding'' each dimension of variation down to a single bit of information.<br />
<br />
For the sake of concreteness, let us suppose that we start with a continuous <math>n\!</math>-dimensional vector space like <math>X = \langle x_1, \ldots, x_n \rangle \cong \mathbb{R}^n.</math> The coordinate system <math>\mathcal{X} = \{x_i\}</math> is a set of maps <math>x_i : \mathbb{R}^n \to \mathbb{R},</math> also known as the ''coordinate projections''. Given a "dataset" of points <math>\mathbf{x}</math> in <math>\mathbb{R}^n,</math> there are numerous ways of sensibly reducing the data down to one bit for each dimension. One strategy that is general enough for our present purposes is as follows. For each <math>i\!</math> we choose an <math>n\!</math>-ary relation <math>L_i\!</math> on <math>\mathbb{R}^n,</math> that is, a subset of <math>\mathbb{R}^n,</math> and then we define the <math>i^\mathrm{th}\!</math> threshold map, or ''limen'' <math>\underline{x}_i</math> as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\underline{x}_i : \mathbb{R}^n \to \mathbb{B}\ \text{such that:}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
\underline{x}_i(\mathbf{x}) = 1 & \text{if} & \mathbf{x} \in L_i,<br />
\\[4pt]<br />
\underline{x}_i(\mathbf{x}) = 0 & \text{if} & \mathbf{x} \not\in L_i.<br />
\end{matrix}</math><br />
|}<br />
<br />
In other notations that are sometimes used, the operator <math>\chi (\ldots)</math> or the corner brackets <math>\lceil\ldots\rceil</math> can be used to denote a ''characteristic function'', that is, a mapping from statements to their truth values in <math>\mathbb{B}.</math> Finally, it is not uncommon to use the name of the relation itself as a predicate that maps <math>n\!</math>-tuples into truth values. Thus we have the following notational variants of the above definition:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
\underline{x}_i (\mathbf{x}) & = & \chi (\mathbf{x} \in L_i) & = & \lceil \mathbf{x} \in L_i \rceil & = & L_i (\mathbf{x}).<br />
\end{matrix}</math><br />
|}<br />
<br />
Notice that, as defined here, there need be no actual relation between the <math>n\!</math>-dimensional subsets <math>\{L_i\}\!</math> and the coordinate axes corresponding to <math>\{x_i\},\!</math> aside from the circumstance that the two sets have the same cardinality. In concrete cases, though, one usually has some reason for associating these "volumes" with these "lines", for instance, <math>L_i\!</math> is bounded by some hyperplane that intersects the <math>i^\text{th}\!</math> axis at a unique threshold value <math>r_i \in \mathbb{R}.\!</math> Often, the hyperplane is chosen normal to the axis. In recognition of this motive, let us make the following convention. When the set <math>L_i\!</math> has points on the <math>i^\text{th}\!</math> axis, that is, points of the form <math>(0, \ldots, 0, r_i, 0, \ldots, 0)</math> where only the <math>x_i\!</math> coordinate is possibly non-zero, we may pick any one of these coordinate values as a parametric index of the relation. In this case we say that the indexing is ''real'', otherwise the indexing is ''imaginary''. For a knowledge based system <math>X,\!</math> this should serve once again to mark the distinction between ''acquaintance'' and ''opinion''.<br />
<br />
States of knowledge about the location of a system or about the distribution of a population of systems in a state space <math>X = \mathbb{R}^n</math> can now be expressed by taking the set <math>\underline{\mathcal{X}} = \{\underline{x}_i\}</math> as a basis of logical features. In picturesque terms, one may think of the underscore and the subscript as combining to form a subtextual spelling for the <math>i^\text{th}\!</math> threshold map. This can help to remind us that the ''threshold operator'' <math>(\underline{~})_i</math> acts on <math>\mathbf{x}</math> by setting up a kind of a &ldquo;hurdle&rdquo; for it. In this interpretation the coordinate proposition <math>\underline{x}_i</math> asserts that the representative point <math>\mathbf{x}</math> resides ''above'' the <math>i^\mathrm{th}\!</math> threshold.<br />
<br />
Primitive assertions of the form <math>\underline{x}_i (\mathbf{x})</math> may then be negated and joined by means of propositional connectives in the usual ways to provide information about the state <math>\mathbf{x}</math> of a contemplated system or a statistical ensemble of systems. Parentheses <math>\texttt{(} \ldots \texttt{)}\!</math> may be used to indicate logical negation. Eventually one discovers the usefulness of the <math>k\!</math>-ary ''just one false'' operators of the form <math>\texttt{(} a_1 \texttt{,} \ldots \texttt{,} a_k \texttt{)},\!</math> as treated in earlier reports. This much tackle generates a space of points (cells, interpretations), <math>\underline{X} \cong \mathbb{B}^n,</math> and a space of functions (regions, propositions), <math>\underline{X}^\uparrow \cong (\mathbb{B}^n \to \mathbb{B}).</math> Together these form a new universe of discourse <math>\underline{X}^\bullet</math> of the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})),</math> which we may abbreviate as <math>\mathbb{B}^n\ +\!\to \mathbb{B}</math> or most succinctly as <math>[\mathbb{B}^n].</math><br />
<br />
The square brackets have been chosen to recall the rectangular frame of a venn diagram. In thinking about a universe of discourse it is a good idea to keep this picture in mind, graphically illustrating the links among the elementary cells <math>\underline{\mathbf{x}},</math> the defining features <math>\underline{x}_i,</math> and the potential shadings <math>f : \underline{X} \to \mathbb{B}</math> all at the same time, not to mention the arbitrariness of the way we choose to inscribe our distinctions in the medium of a continuous space.<br />
<br />
Finally, let <math>X^*\!</math> denote the space of linear functions, <math>(\ell : X \to \mathbb{K}),</math> which has in the finite case the same dimensionality as <math>X,\!</math> and let the same notation be extended across the Table.<br />
<br />
We have just gone through a lot of work, apparently doing nothing more substantial than spinning a complex spell of notational devices through a labyrinth of baffled spaces and baffling maps. The reason for doing this was to bind together and to constitute the intuitive concept of a universe of discourse into a coherent categorical object, the kind of thing, once grasped, that can be turned over in the mind and considered in all its manifold changes and facets. The effort invested in these preliminary measures is intended to pay off later, when we need to consider the state transformations and the time evolution of neural network systems.<br />
<br />
===Tables of Propositional Forms===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope. It provides explicit techniques for manipulating the most basic ingredients of discourse.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7&ndash;8]<br />
|}<br />
<br />
To prepare for the next phase of discussion, Tables&nbsp;6 and 7 collect and summarize all of the propositional forms on one and two variables. These propositional forms are represented over bases of boolean variables as complete sets of boolean-valued functions. Adjacent to their names and specifications are listed what are roughly the simplest expressions in the ''[[cactus language]]'', the particular syntax for propositional calculus that I use in formal and computational contexts. For the sake of orientation, the English paraphrases and the more common notations are listed in the last two columns. As simple and circumscribed as these low-dimensional universes may appear to be, a careful exploration of their differential extensions will involve us in complexities sufficient to demand our attention for some time to come.<br />
<br />
Propositional forms on one variable correspond to boolean functions <math>f : \mathbb{B}^1 \to \mathbb{B}.</math> In Table&nbsp;6 these functions are listed in a variant form of [[truth table]], one in which the axes of the usual arrangement are rotated through a right angle. Each function <math>f_i\!</math> is indexed by the string of values that it takes on the points of the universe <math>X^\bullet = [x] \cong \mathbb{B}^1.</math> The binary index generated in this way is converted to its decimal equivalent and these are used as conventional names for the <math>f_i,\!</math> as shown in the first column of the Table. In their own right the <math>2^1\!</math> points of the universe <math>X^\bullet</math> are coordinated as a space of type <math>\mathbb{B}^1,</math> this in light of the universe <math>X^\bullet</math> being a functional domain where the coordinate projection <math>x\!</math> takes on its values in <math>\mathbb{B}.</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 6.} ~~ \text{Propositional Forms on One Variable}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_0\!</math><br />
| <math>f_{00}\!</math><br />
| <math>0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>f_1\!</math><br />
| <math>f_{01}\!</math><br />
| <math>0~1\!</math><br />
| <math>\texttt{(} x \texttt{)}\!</math><br />
| <math>\text{not}~ x\!</math><br />
| <math>\lnot x\!</math><br />
|-<br />
| <math>f_2\!</math><br />
| <math>f_{10}\!</math><br />
| <math>1~0\!</math><br />
| <math>x\!</math><br />
| <math>x\!</math><br />
| <math>x\!</math><br />
|-<br />
| <math>f_3\!</math><br />
| <math>f_{11}\!</math><br />
| <math>1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
Propositional forms on two variables correspond to boolean functions <math>f : \mathbb{B}^2 \to \mathbb{B}.</math> In Table&nbsp;7 each function <math>f_i\!</math> is indexed by the values that it takes on the points of the universe <math>X^\bullet = [x, y] \cong \mathbb{B}^2.</math> Converting the binary index thus generated to a decimal equivalent, we obtain the functional nicknames that are listed in the first column. The <math>2^2\!</math> points of the universe <math>X^\bullet</math> are coordinated as a space of type <math>\mathbb{B}^2,</math> as indicated under the heading of the Table, where the coordinate projections <math>x\!</math> and <math>y\!</math> run through the various combinations of their values in <math>\mathbb{B}.</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 7-a.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}\!</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}\!</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}\!</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}\!</math><br />
| style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}\!</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}\!</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[4pt]<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\\[4pt]<br />
f_{3}<br />
\\[4pt]<br />
f_{4}<br />
\\[4pt]<br />
f_{5}<br />
\\[4pt]<br />
f_{6}<br />
\\[4pt]<br />
f_{7}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0000}<br />
\\[4pt]<br />
f_{0001}<br />
\\[4pt]<br />
f_{0010}<br />
\\[4pt]<br />
f_{0011}<br />
\\[4pt]<br />
f_{0100}<br />
\\[4pt]<br />
f_{0101}<br />
\\[4pt]<br />
f_{0110}<br />
\\[4pt]<br />
f_{0111}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~0<br />
\\[4pt]<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~0~1~1<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
0~1~0~1<br />
\\[4pt]<br />
0~1~1~0<br />
\\[4pt]<br />
0~1~1~1<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{)} ~ y ~<br />
\\[4pt]<br />
\texttt{(} x \texttt{)}<br />
\\[4pt]<br />
~ x ~ \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{,} ~ y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x ~ y \texttt{)}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{false}<br />
\\[4pt]<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\[4pt]<br />
y ~\text{without}~ x<br />
\\[4pt]<br />
\text{not}~ x<br />
\\[4pt]<br />
x ~\text{without}~ y<br />
\\[4pt]<br />
\text{not}~ y<br />
\\[4pt]<br />
x ~\text{not equal to}~ y<br />
\\[4pt]<br />
\text{not both}~ x ~\text{and}~ y<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
\lnot x \land \lnot y<br />
\\[4pt]<br />
\lnot x \land y<br />
\\[4pt]<br />
\lnot x<br />
\\[4pt]<br />
x \land \lnot y<br />
\\[4pt]<br />
\lnot y<br />
\\[4pt]<br />
x \ne y<br />
\\[4pt]<br />
\lnot x \lor \lnot y<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{8}<br />
\\[4pt]<br />
f_{9}<br />
\\[4pt]<br />
f_{10}<br />
\\[4pt]<br />
f_{11}<br />
\\[4pt]<br />
f_{12}<br />
\\[4pt]<br />
f_{13}<br />
\\[4pt]<br />
f_{14}<br />
\\[4pt]<br />
f_{15}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1000}<br />
\\[4pt]<br />
f_{1001}<br />
\\[4pt]<br />
f_{1010}<br />
\\[4pt]<br />
f_{1011}<br />
\\[4pt]<br />
f_{1100}<br />
\\[4pt]<br />
f_{1101}<br />
\\[4pt]<br />
f_{1110}<br />
\\[4pt]<br />
f_{1111}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
1~0~0~0<br />
\\[4pt]<br />
1~0~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\\[4pt]<br />
1~1~1~1<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x ~ y<br />
\\[4pt]<br />
\texttt{((} x \texttt{,} ~ y \texttt{))}<br />
\\[4pt]<br />
y<br />
\\[4pt]<br />
\texttt{(} x ~ \texttt{(} y \texttt{))}<br />
\\[4pt]<br />
x<br />
\\[4pt]<br />
\texttt{((} x \texttt{)} ~ y \texttt{)}<br />
\\[4pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
\texttt{((~))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x ~\text{and}~ y<br />
\\[4pt]<br />
x ~\text{equal to}~ y<br />
\\[4pt]<br />
y<br />
\\[4pt]<br />
\text{not}~ x ~\text{without}~ y<br />
\\[4pt]<br />
x<br />
\\[4pt]<br />
\text{not}~ y ~\text{without}~ x<br />
\\[4pt]<br />
x ~\text{or}~ y<br />
\\[4pt]<br />
\text{true}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x \land y<br />
\\[4pt]<br />
x = y<br />
\\[4pt]<br />
y<br />
\\[4pt]<br />
x \Rightarrow y<br />
\\[4pt]<br />
x<br />
\\[4pt]<br />
x \Leftarrow y<br />
\\[4pt]<br />
x \lor y<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 7-b.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}</math><br />
| style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>f_{0000}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\\[4pt]<br />
f_{4}<br />
\\[4pt]<br />
f_{8}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0001}<br />
\\[4pt]<br />
f_{0010}<br />
\\[4pt]<br />
f_{0100}<br />
\\[4pt]<br />
f_{1000}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
1~0~0~0<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{)} ~ y ~<br />
\\[4pt]<br />
~ x ~ \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
~ x ~~ y ~<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\[4pt]<br />
y ~\text{without}~ x<br />
\\[4pt]<br />
x ~\text{without}~ y<br />
\\[4pt]<br />
x ~\text{and}~ y<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot x \land \lnot y<br />
\\[4pt]<br />
\lnot x \land y<br />
\\[4pt]<br />
x \land \lnot y<br />
\\[4pt]<br />
x \land y<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{3}<br />
\\[4pt]<br />
f_{12}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0011}<br />
\\[4pt]<br />
f_{1100}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\[4pt]<br />
x<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{not}~ x<br />
\\[4pt]<br />
x<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot x<br />
\\[4pt]<br />
x<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[4pt]<br />
f_{9}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0110}<br />
\\[4pt]<br />
f_{1001}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[4pt]<br />
1~0~0~1<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{,} y \texttt{)}<br />
\\[4pt]<br />
\texttt{((} x \texttt{,} y \texttt{))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x ~\text{not equal to}~ y<br />
\\[4pt]<br />
x ~\text{equal to}~ y<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x \ne y<br />
\\[4pt]<br />
x = y<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{5}<br />
\\[4pt]<br />
f_{10}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0101}<br />
\\[4pt]<br />
f_{1010}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\[4pt]<br />
y<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{not}~ y<br />
\\[4pt]<br />
y<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot y<br />
\\[4pt]<br />
y<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[4pt]<br />
f_{11}<br />
\\[4pt]<br />
f_{13}<br />
\\[4pt]<br />
f_{14}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0111}<br />
\\[4pt]<br />
f_{1011}<br />
\\[4pt]<br />
f_{1101}<br />
\\[4pt]<br />
f_{1110}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} ~ x ~~ y ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{(} ~ x ~ \texttt{(} y \texttt{))}<br />
\\[4pt]<br />
\texttt{((} x \texttt{)} ~ y ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{not both}~ x ~\text{and}~ y<br />
\\[4pt]<br />
\text{not}~ x ~\text{without}~ y<br />
\\[4pt]<br />
\text{not}~ y ~\text{without}~ x<br />
\\[4pt]<br />
x ~\text{or}~ y<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot x \lor \lnot y<br />
\\[4pt]<br />
x \Rightarrow y<br />
\\[4pt]<br />
x \Leftarrow y<br />
\\[4pt]<br />
x \lor y<br />
\end{matrix}\!</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>f_{1111}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
==A Differential Extension of Propositional Calculus==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Fire over water:<br><br />
The image of the condition before transition.<br><br />
Thus the superior man is careful<br><br />
In the differentiation of things,<br><br />
So that each finds its place.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; ''I Ching'', Hexagram 64, [Wil, 249]<br />
|}<br />
<br />
This much preparation is enough to begin introducing my subject, if I excuse myself from giving full arguments for my definitional choices until some later stage. I am trying to develop a ''differential theory of qualitative equations'' that parallels the application of differential geometry to dynamic systems. The idea of a tangent vector is key to this work and a major goal is to find the right logical analogues of tangent spaces, bundles, and functors. The strategy is taken of looking for the simplest versions of these constructions that can be discovered within the realm of propositional calculus, so long as they serve to fill out the general theme.<br />
<br />
===Differential Propositions : Qualitative Analogues of Differential Equations===<br />
<br />
In order to define the differential extension of a universe of discourse <math>[\mathcal{A}],</math> the initial alphabet <math>\mathcal{A}</math> must be extended to include a collection of symbols for ''differential features'', or basic ''changes'' that are capable of occurring in <math>[\mathcal{A}].</math> Intuitively, these symbols may be construed as denoting primitive features of change, qualitative attributes of motion, or propositions about how things or points in the universe may be changing or moving with respect to the features that are noted in the initial alphabet.<br />
<br />
Therefore, let us define the corresponding ''differential alphabet'' or ''tangent alphabet'' as <math>\mathrm{d}\mathcal{A}\!</math> <math>=\!</math> <math>\{\mathrm{d}a_1, \ldots, \mathrm{d}a_n\},</math> in principle just an arbitrary alphabet of symbols, disjoint from the initial alphabet <math>\mathcal{A}\!</math> <math>=\!</math> <math>\{ a_1, \ldots, a_n \},\!</math> that is intended to be interpreted in the way just indicated. It only remains to be understood that the precise interpretation of the symbols in <math>\mathrm{d}\mathcal{A}\!</math> is often conceived to be changeable from point to point of the underlying space <math>A.\!</math> Indeed, for all we know, the state space <math>A\!</math> might well be the state space of a language interpreter, one that is concerned, among other things, with the idiomatic meanings of the dialect generated by <math>\mathcal{A}</math> and <math>\mathrm{d}\mathcal{A}.\!</math><br />
<br />
The ''tangent space'' to <math>A\!</math> at one of its points <math>x,\!</math> sometimes written <math>\mathrm{T}_x(A),</math> takes the form <math>\mathrm{d}A</math> <math>=\!</math> <math>\langle \mathrm{d}\mathcal{A} \rangle</math> <math>=\!</math> <math>\langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle.\!</math> Strictly speaking, the name ''cotangent space'' is probably more correct for this construction, but the fact that we take up spaces and their duals in pairs to form our universes of discourse allows our language to be pliable here.<br />
<br />
Proceeding as we did with the base space <math>A,\!</math> the tangent space <math>\mathrm{d}A</math> at a point of <math>A\!</math> can be analyzed as a product of distinct and independent factors:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\mathrm{d}A ~=~ \prod_{i=1}^n \mathrm{d}A_i ~=~ \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n.\!</math><br />
|}<br />
<br />
Here, <math>\mathrm{d}A_i\!</math> is a set of two differential propositions, <math>\mathrm{d}A_i = \{ (\mathrm{d}a_i), \mathrm{d}a_i \},\!</math> where <math>\texttt{(} \mathrm{d}a_i \texttt{)}\!</math> is a proposition with the logical value of <math>\text{not} ~ \mathrm{d}a_i.\!</math> Each component <math>\mathrm{d}A_i\!</math> has the type <math>\mathbb{B},\!</math> operating under the ordered correspondence <math>\{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \} \cong \{ 0, 1 \}.\!</math> However, clarity is often served by acknowledging this differential usage with a superficially distinct type <math>\mathbb{D},\!</math> whose intension may be indicated as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\mathbb{D} = \{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \} = \{ \text{same}, \text{different} \} = \{ \text{stay}, \text{change} \} = \{ \text{stop}, \text{step} \}.\!</math><br />
|}<br />
<br />
Viewed within a coordinate representation, spaces of type <math>\mathbb{B}^n\!</math> and <math>\mathbb{D}^n\!</math> may appear to be identical sets of binary vectors, but taking a view at this level of abstraction would be like ignoring the qualitative units and the diverse dimensions that distinguish position and momentum, or the different roles of quantity and impulse.<br />
<br />
===An Interlude on the Path===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
There would have been no beginnings: instead, speech would proceed from me, while I stood in its path &ndash; a slender gap &ndash; the point of its possible disappearance.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
A sense of the relation between <math>\mathbb{B}</math> and <math>\mathbb{D}</math> may be obtained by considering the ''path classifier'' (or the ''equivalence class of curves'') approach to tangent vectors. Consider a universe <math>[\mathcal{X}].\!</math> Given the boolean value system, a path in the space <math>X = \langle \mathcal{X} \rangle</math> is a map <math>q : \mathbb{B} \to X.</math> In this context the set of paths <math>(\mathbb{B} \to X)</math> is isomorphic to the cartesian square <math>X^2 = X \times X,</math> or the set of ordered pairs chosen from <math>X.\!</math><br />
<br />
We may analyze <math>X^2 = \{ (u, v) : u, v \in X \}</math> into two parts, specifically, the ordered pairs <math>(u, v)\!</math> that lie on and off the diagonal:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}X^2 & = & \{ (u, v) : u = v \} & \cup & \{ (u, v) : u \ne v \}.\end{matrix}</math><br />
|}<br />
<br />
This partition may also be expressed in the following symbolic form:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}X^2 & \cong & \operatorname{diag} (X) & + & 2 \binom{X}{2}.\end{matrix}</math><br />
|}<br />
<br />
The separate terms of this formula are defined as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}\operatorname{diag} (X) & = & \{ (x, x) : x \in X \}.\end{matrix}\!</math><br />
|}<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}\binom{X}{k} & = & X ~\text{choose}~ k & = & \{ k\text{-sets from}~ X \}.\end{matrix}\!</math><br />
|}<br />
<br />
Thus we have:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}\binom{X}{2} & = & \{ \{ u, v \} : u, v \in X \}.\end{matrix}</math><br />
|}<br />
<br />
We may now use the features in <math>\mathrm{d}\mathcal{X} = \{ \mathrm{d}x_i \} = \{ \mathrm{d}x_1, \ldots, \mathrm{d}x_n \}</math> to classify the paths of <math>(\mathbb{B} \to X)</math> by way of the pairs in <math>X^2.\!</math> If <math>X \cong \mathbb{B}^n,</math> then a path <math>q\!</math> in <math>X\!</math> has the following form:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
q : (\mathbb{B} \to \mathbb{B}^n) & \cong & \mathbb{B}^n \times \mathbb{B}^n & \cong & \mathbb{B}^{2n} & \cong & (\mathbb{B}^2)^n.<br />
\end{matrix}</math><br />
|}<br />
<br />
Intuitively, we want to map this <math>(\mathbb{B}^2)^n</math> onto <math>\mathbb{D}^n</math> by mapping each component <math>\mathbb{B}^2</math> onto a copy of <math>\mathbb{D}.</math> But in the presenting context <math>{}^{\backprime\backprime} \mathbb{D} {}^{\prime\prime}</math> is just a name associated with, or an incidental quality attributed to, coefficient values in <math>\mathbb{B}</math> when they are attached to features in <math>\mathrm{d}\mathcal{X}.</math><br />
<br />
Taking these intentions into account, define <math>\mathrm{d}x_i : X^2 \to \mathbb{B}</math> in the following manner:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcrcl}<br />
\mathrm{d}x_i(u, v)<br />
& = & \texttt{(} ~ x_i(u) & \texttt{,} & x_i(v) ~ \texttt{)}<br />
\\<br />
& = & x_i(u) & + & x_i(v)<br />
\\<br />
& = & x_i(v) & - & x_i(u).<br />
\end{array}</math><br />
|}<br />
<br />
In the above transcription, the operator bracket of the form <math>\texttt{(} \ldots \texttt{,} \ldots \texttt{)}\!</math> is a ''cactus lobe'', in general signifying that just one of the arguments listed is false. In the case of two arguments this is the same thing as saying that the arguments are not equal. The plus sign signifies boolean addition, in the sense of addition in <math>\mathrm{GF}(2),</math> and thus means the same thing in this context as the minus sign, in the sense of adding the additive inverse.<br />
<br />
The above definition of <math>\mathrm{d}x_i : X^2 \to \mathbb{B}</math> is equivalent to defining <math>\mathrm{d}x_i : (\mathbb{B} \to X) \to \mathbb{B}\!</math> in the following way:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcrcl}<br />
\mathrm{d}x_i (q)<br />
& = & \texttt{(} ~ x_i(q_0) & \texttt{,} & x_i(q_1) ~ \texttt{)}<br />
\\<br />
& = & x_i(q_0) & + & x_i(q_1)<br />
\\<br />
& = & x_i(q_1) & - & x_i(q_0).<br />
\end{array}</math><br />
|}<br />
<br />
In this definition <math>q_b = q(b),\!</math> for each <math>b\!</math> in <math>\mathbb{B}.</math> Thus, the proposition <math>\mathrm{d}x_i</math> is true of the path <math>q = (u, v)\!</math> exactly if the terms of <math>q,\!</math> the endpoints <math>u\!</math> and <math>v,\!</math> lie on different sides of the question <math>x_i.\!</math><br />
<br />
The language of features in <math>\langle \mathrm{d}\mathcal{X} \rangle,</math> indeed the whole calculus of propositions in <math>[\mathrm{d}\mathcal{X}],</math> may now be used to classify paths and sets of paths. In other words, the paths can be taken as models of the propositions <math>g : \mathrm{d}X \to \mathbb{B}.</math> For example, the paths corresponding to <math>\mathrm{diag}(X)</math> fall under the description <math>\texttt{(} \mathrm{d}x_1 \texttt{)} \cdots \texttt{(} \mathrm{d}x_n \texttt{)},\!</math> which says that nothing changes against the backdrop of the coordinate frame <math>\{ x_1, \ldots, x_n \}.\!</math><br />
<br />
Finally, a few words of explanation may be in order. If this concept of a path appears to be described in a roundabout fashion, it is because I am trying to avoid using any assumption of vector space properties for the space <math>X\!</math> that contains its range. In many ways the treatment is still unsatisfactory, but improvements will have to wait for the introduction of substitution operators acting on singular propositions.<br />
<br />
===The Extended Universe of Discourse===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
At the moment of speaking, I would like to have perceived a nameless voice, long preceding me, leaving me merely to enmesh myself in it, taking up its cadence, and to lodge myself, when no one was looking, in its interstices as if it had paused an instant, in suspense, to beckon to me.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
Next we define the ''extended alphabet'' or ''bundled alphabet'' <math>\mathrm{E}\mathcal{A}</math> as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lclcl}<br />
\mathrm{E}\mathcal{A}<br />
& = & \mathcal{A} \cup \mathrm{d}\mathcal{A}<br />
& = & \{ a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}.<br />
\end{array}</math><br />
|}<br />
<br />
This supplies enough material to construct the ''differential extension'' <math>\mathrm{E}A,</math> or the ''tangent bundle'' over the initial space <math>A,\!</math> in the following fashion:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcl}<br />
\mathrm{E}A<br />
& = & \langle \mathrm{E}\mathcal{A} \rangle<br />
\\[4pt]<br />
& = & \langle \mathcal{A} \cup \mathrm{d}\mathcal{A} \rangle<br />
\\[4pt]<br />
& = & \langle a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle,<br />
\end{array}</math><br />
|}<br />
<br />
and also:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcl}<br />
\mathrm{E}A<br />
& = & A \times \mathrm{d}A<br />
\\[4pt]<br />
& = & A_1 \times \ldots \times A_n \times \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n.<br />
\end{array}</math><br />
|}<br />
<br />
This gives <math>\mathrm{E}A</math> the type <math>\mathbb{B}^n \times \mathbb{D}^n.</math><br />
<br />
Finally, the tangent universe <math>\mathrm{E}A^\bullet = [\mathrm{E}\mathcal{A}]\!</math> is constituted from the totality of points and maps, or interpretations and propositions, that are based on the extended set of features <math>\mathrm{E}\mathcal{A},</math> and this fact is summed up in the following notation:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lclcl}<br />
\mathrm{E}A^\bullet<br />
& = & [\mathrm{E}\mathcal{A}]<br />
& = & [a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n].<br />
\end{array}</math><br />
|}<br />
<br />
This gives the tangent universe <math>\mathrm{E}A^\bullet\!</math> the type:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcl}<br />
(\mathbb{B}^n \times \mathbb{D}^n\ +\!\to \mathbb{B})<br />
& = & (\mathbb{B}^n \times \mathbb{D}^n, (\mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B})).<br />
\end{array}</math><br />
|}<br />
<br />
A proposition in the tangent universe <math>[\mathrm{E}\mathcal{A}]</math> is called a ''differential proposition'' and forms the analogue of a system of differential equations, constraints, or relations in ordinary calculus.<br />
<br />
With these constructions, the differential extension <math>\mathrm{E}A</math> and the space of differential propositions <math>(\mathrm{E}A \to \mathbb{B}),\!</math> we have arrived, in main outline, at one of the major subgoals of this study. Table&nbsp;8 summarizes the concepts that have been introduced for working with differentially extended universes of discourse.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 8.} ~~ \text{Differential Extension : Basic Notation}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Symbol}\!</math><br />
| <math>\text{Notation}\!</math><br />
| <math>\text{Description}\!</math><br />
| <math>\text{Type}\!</math><br />
|-<br />
| <math>\mathrm{d}\mathfrak{A}\!</math><br />
| <math>\{ {}^{\backprime\backprime} \mathrm{d}a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} \mathrm{d}a_n {}^{\prime\prime} \}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Alphabet of}<br />
\\[2pt]<br />
\text{differential symbols}<br />
\end{matrix}</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>\mathrm{d}\mathcal{A}\!</math><br />
| <math>\{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Basis of}<br />
\\[2pt]<br />
\text{differential features}<br />
\end{matrix}</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>\mathrm{d}A_i\!</math><br />
| <math>\{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \}\!</math><br />
| <math>\text{Differential dimension}~ i\!</math><br />
| <math>\mathbb{D}\!</math><br />
|-<br />
| <math>\mathrm{d}A\!</math><br />
|<br />
<math>\begin{matrix}<br />
\langle \mathrm{d}\mathcal{A} \rangle<br />
\\[2pt]<br />
\langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle<br />
\\[2pt]<br />
\{ (\mathrm{d}a_1, \ldots, \mathrm{d}a_n) \}<br />
\\[2pt]<br />
\mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n<br />
\\[2pt]<br />
\textstyle \prod_i \mathrm{d}A_i<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Tangent space at a point:}<br />
\\[2pt]<br />
\text{Set of changes, motions,}<br />
\\[2pt]<br />
\text{steps, tangent vectors}<br />
\\[2pt]<br />
\text{at a point}<br />
\end{matrix}</math><br />
| <math>\mathbb{D}^n\!</math><br />
|-<br />
| <math>\mathrm{d}A^*\!</math><br />
| <math>(\mathrm{hom} : \mathrm{d}A \to \mathbb{B})\!</math><br />
| <math>\text{Linear functions on}~ \mathrm{d}A\!</math><br />
| <math>(\mathbb{D}^n)^* \cong \mathbb{D}^n\!</math><br />
|-<br />
| <math>\mathrm{d}A^\uparrow\!</math><br />
| <math>(\mathrm{d}A \to \mathbb{B})\!</math><br />
| <math>\text{Boolean functions on}~ \mathrm{d}A\!</math><br />
| <math>\mathbb{D}^n \to \mathbb{B}\!</math><br />
|-<br />
| <math>\mathrm{d}A^\bullet\!</math><br />
|<br />
<math>\begin{matrix}<br />
[\mathrm{d}\mathcal{A}]<br />
\\[2pt]<br />
(\mathrm{d}A, \mathrm{d}A^\uparrow)<br />
\\[2pt]<br />
(\mathrm{d}A ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
(\mathrm{d}A, (\mathrm{d}A \to \mathbb{B}))<br />
\\[2pt]<br />
[\mathrm{d}a_1, \ldots, \mathrm{d}a_n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Tangent universe at a point of}~ A^\bullet,<br />
\\[2pt]<br />
\text{based on the tangent features}<br />
\\[2pt]<br />
\{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
(\mathbb{D}^n, (\mathbb{D}^n \to \mathbb{B}))<br />
\\[2pt]<br />
(\mathbb{D}^n ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
[\mathbb{D}^n]<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
The adjectives ''differential'' or ''tangent'' are systematically attached to every construct based on the differential alphabet <math>\mathrm{d}\mathfrak{A},</math> taken by itself. Strictly speaking, we probably ought to call <math>\mathrm{d}\mathcal{A}</math> the set of ''cotangent'' features derived from <math>\mathcal{A},</math> but the only time this distinction really seems to matter is when we need to distinguish the tangent vectors as maps of type <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math> from cotangent vectors as elements of type <math>\mathbb{D}^n.</math> In like fashion, having defined <math>\mathrm{E}\mathcal{A} = \mathcal{A} \cup \mathrm{d}\mathcal{A},</math> we can systematically attach the adjective ''extended'' or the substantive ''bundle'' to all of the constructs associated with this full complement of <math>{2n}\!</math> features.<br />
<br />
It eventually becomes necessary to extend the initial alphabet even further, to allow for the discussion of higher order differential expressions. Table&nbsp;9 provides a suggestion of how these further extensions can be carried out.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 9.} ~~ \text{Higher Order Differential Features}\!</math><br />
|<br />
<p><math>\begin{array}{lllll}<br />
\mathrm{d}^0 \mathcal{A} & = & \{ a_1, \ldots, a_n \} & = & \mathcal{A}<br />
\\<br />
\mathrm{d}^1 \mathcal{A} & = & \{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \} & = & \mathrm{d}\mathcal{A}<br />
\end{array}</math></p><br />
<br />
<p><math>\begin{array}{lll}<br />
\mathrm{d}^k \mathcal{A} & = & \{ \mathrm{d}^k a_1, \ldots, \mathrm{d}^k a_n \}<br />
\\<br />
\mathrm{d}^* \mathcal{A} & = & \{ \mathrm{d}^0 \mathcal{A}, \ldots, \mathrm{d}^k \mathcal{A}, \ldots \}<br />
\end{array}</math></p><br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}^0 \mathcal{A} & = & \mathrm{d}^0 \mathcal{A}<br />
\\<br />
\mathrm{E}^1 \mathcal{A} & = & \mathrm{d}^0 \mathcal{A} ~\cup~ \mathrm{d}^1 \mathcal{A}<br />
\\<br />
\mathrm{E}^k \mathcal{A} & = & \mathrm{d}^0 \mathcal{A} ~\cup~ \ldots ~\cup~ \mathrm{d}^k \mathcal{A}<br />
\\<br />
\mathrm{E}^\infty \mathcal{A} & = & \bigcup~ \mathrm{d}^* \mathcal{A}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
===Intentional Propositions===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Do you guess I have some intricate purpose?<br><br />
Well I have . . . . for the April rain has, and the mica on<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the side of a rock has.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 45]<br />
|}<br />
<br />
In order to analyze the behavior of a system at successive moments in time, while staying within the limitations of propositional logic, it is necessary to create independent alphabets of logical features for each moment of time that we contemplate using in our discussion. These moments have reference to typical instances and relative intervals, not actual or absolute times. For example, to discuss ''velocities'' (first order rates of change) we need to consider points of time in pairs. There are a number of natural ways of doing this. Given an initial alphabet, we could use its symbols as a lexical basis to generate successive alphabets of compound symbols, say, with temporal markers appended as suffixes.<br />
<br />
As a standard way of dealing with these situations, the following scheme of notation suggests a way of extending any alphabet of logical features through as many temporal moments as a particular order of analysis may demand. The lexical operators <math>\mathrm{p}^k</math> and <math>\mathrm{Q}^k</math> are convenient in many contexts where the accumulation of prime symbols and union symbols would otherwise be cumbersome.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 10.} ~~ \text{A Realm of Intentional Features}\!</math><br />
|<br />
<p><math>\begin{array}{lllll}<br />
\mathrm{p}^0 \mathcal{A} & = & \{ a_1, \ldots, a_n \} & = & \mathcal{A}<br />
\\<br />
\mathrm{p}^1 \mathcal{A} & = & \{ a_1^\prime, \ldots, a_n^\prime \} & = & \mathcal{A}^\prime<br />
\\<br />
\mathrm{p}^2 \mathcal{A} & = & \{ a_1^{\prime\prime}, \ldots, a_n^{\prime\prime} \} & = & \mathcal{A}^{\prime\prime}<br />
\\<br />
\cdots & & \cdots &<br />
\end{array}</math></p><br />
<br />
<p><math>\begin{array}{lll}<br />
\mathrm{p}^k \mathcal{A} & = & \{ \mathrm{p}^k a_1, \ldots, \mathrm{p}^k a_n \}<br />
\end{array}</math></p><br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{Q}^0 \mathcal{A} & = & \mathcal{A}<br />
\\<br />
\mathrm{Q}^1 \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}'<br />
\\<br />
\mathrm{Q}^2 \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}' \cup \mathcal{A}''<br />
\\<br />
\cdots & & \cdots<br />
\\<br />
\mathrm{Q}^k \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}' \cup \ldots \cup \mathrm{p}^k \mathcal{A}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
The resulting augmentations of our logical basis determine a series of discursive universes that may be called the ''intentional extension'' of propositional calculus. This extension follows a pattern analogous to the differential extension, which was developed in terms of the operators <math>\mathrm{d}^k</math> and <math>\mathrm{E}^k,</math> and there is a natural relation between these two extensions that bears further examination. In contexts displaying this pattern, where a sequence of domains stretches from an anchoring domain <math>X\!</math> through an indefinite number of higher reaches, a particular collection of domains based on <math>X\!</math> will be referred to as a ''realm'' of <math>X,\!</math> and when the succession exhibits a temporal aspect, as a ''reign'' of <math>X.\!</math><br />
<br />
For the purposes of this discussion, an ''intentional proposition'' is defined as a proposition in the universe of discourse <math>\mathrm{Q}X^\bullet = [\mathrm{Q}\mathcal{X}],</math> in other words, a map <math>q : \mathrm{Q}X \to \mathbb{B}.</math> The sense of this definition may be seen if we consider the following facts. First, the equivalence <math>\mathrm{Q}X = X \times X'</math> motivates the following chain of isomorphisms between spaces:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lllcl}<br />
(\mathrm{Q}X \to \mathbb{B})<br />
& \cong & (X & \times & ~X' \to \mathbb{B})<br />
\\[4pt]<br />
& \cong & (X & \to & (X' \to \mathbb{B}))<br />
\\[4pt]<br />
& \cong & (X' & \to & (X~ \to \mathbb{B})).<br />
\end{array}</math><br />
|}<br />
<br />
Viewed in this light, an intentional proposition <math>q\!</math> may be rephrased as a map <math>q : X \times X' \to \mathbb{B},</math> which judges the juxtaposition of states in <math>X\!</math> from one moment to the next. Alternatively, <math>q\!</math> may be parsed in two stages in two different ways, as <math>q : X \to (X' \to \mathbb{B})</math> and as <math>q : X' \to (X \to \mathbb{B}),</math> which associate to each point of <math>X\!</math> or <math>X'\!</math> a proposition about states in <math>X'\!</math> or <math>X,\!</math> respectively. In this way, an intentional proposition embodies a type of value system, in effect, a proposal that places a value on a collection of ends-in-view, or a project that evaluates a set of goals as regarded from each point of view in the state space of a system.<br />
<br />
In sum, the intentional proposition <math>q\!</math> indicates a method for the systematic selection of local goals. As a general form of description, a map of the type <math>q : \mathrm{Q}^i X \to \mathbb{B}\!</math> may be referred to as an "<math>i^\text{th}</math> order intentional proposition". Naturally, when we speak of intentional propositions without qualification, we usually mean first order intentions.<br />
<br />
Many different realms of discourse have the same structure as the extensions that have been indicated here. From a strictly logical point of view, each new layer of terms is composed of independent logical variables that are no different in kind from those that go before, and each further course of logical atoms is treated like so many individual, but otherwise indifferent bricks by the prototype computer program that I use as a propositional interpreter. Thus, the names that I use to single out the differential and the intentional extensions, and the lexical paradigms that I follow to construct them, are meant to suggest the interpretations that I have in mind, but they can only hint at the extra meanings that human communicators may pack into their terms and inflections.<br />
<br />
As applied here, the word ''intentional'' is drawn from common use and may have little bearing on its technical use in other, more properly philosophical, contexts. I am merely using the complex of intentional concepts &mdash; aims, ends, goals, objectives, purposes, and so on &mdash; metaphorically to flesh out and vividly to represent any situation where one needs to contemplate a system in multiple aspects of state and destination, that is, its being in certain states and at the same time acting as if headed through certain states. If confusion arises, more neutral words like ''conative'', ''contingent'', ''discretionary'', ''experimental'', ''kinetic'', ''progressive'', ''tentative'', or ''trial'' would probably serve as well.<br />
<br />
===Life on Easy Street===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Failing to fetch me at first keep encouraged,<br><br />
Missing me one place search another,<br><br />
I stop some where waiting for you<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]<br />
|}<br />
<br />
The finite character of the extended universe <math>[\mathrm{E}\mathcal{A}]</math> makes the problem of solving differential propositions relatively straightforward, at least, in principle. The solution set of the differential proposition <math>q : \mathrm{E}A \to \mathbb{B}</math> is the set of models <math>q^{-1}(1)\!</math> in <math>\mathrm{E}A.</math> Finding all the models of <math>q,\!</math> the extended interpretations in <math>\mathrm{E}A</math> that satisfy <math>q,\!</math> can be carried out by a finite search. Being in possession of complete algorithms for propositional calculus modeling, theorem checking, or theorem proving makes the analytic task fairly simple in principle, though the question of efficiency in the face of arbitrary complexity may always remain another matter entirely. While the fact that propositional satisfiability is NP-complete may be discouraging for the prospects of a single efficient algorithm that covers the whole space <math>[\mathrm{E}\mathcal{A}]</math> with equal facility, there appears to be much room for improvement in classifying special forms and in developing algorithms that are tailored to their practical processing.<br />
<br />
In view of these constraints and contingencies, our focus shifts to the tasks of approximation and interpretation that support intuition, especially in dealing with the natural kinds of differential propositions that arise in applications, and in the effort to understand, in succinct and adaptive forms, their dynamic implications. In the absence of direct insights, these tasks are partially carried out by forging analogies with the familiar situations and customary routines of ordinary calculus. But the indirect approach, going by way of specious analogy and intuitive habit, forces us to remain on guard against the circumstance that occurs when the word ''forging'' takes on its shadier nuance, indicting the constant risk of a counterfeit in the proportion.<br />
<br />
==Back to the Beginning : Exemplary Universes==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
I would have preferred to be enveloped in words, borne way beyond all possible beginnings.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
To anchor our understanding of differential logic, let us look at how the various concepts apply in the simplest possible concrete cases, where the initial dimension is only 1 or 2. In spite of the obvious simplicity of these cases, it is possible to observe how central difficulties of the subject begin to arise already at this stage.<br />
<br />
===A One-Dimensional Universe===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
There was never any more inception than there is now,<br><br />
Nor any more youth or age than there is now;<br><br />
And will never be any more perfection than there is now,<br><br />
Nor any more heaven or hell than there is now.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]<br />
|}<br />
<br />
Let <math>\mathcal{X} = \{ x_1 \} = \{ A \}</math> be an alphabet that represents one boolean variable or a single logical feature. In this example the capital letter <math>{}^{\backprime\backprime} A {}^{\prime\prime}\!</math> is used usual informally, to name a feature and not a space, in departure from our formerly stated formal conventions. At any rate, the basis element <math>A = x_1\!</math> may be interpreted as a simple proposition or a coordinate projection <math>A = x_1 : \mathbb{B} \xrightarrow{i} \mathbb{B}.</math> The space <math>X = \langle A \rangle = \{ \texttt{(} A \texttt{)}, A \}</math> of points (cells, vectors, interpretations) has cardinality <math>2^n = 2^1 = 2\!</math> and is isomorphic to <math>\mathbb{B} = \{ 0, 1 \}.</math> Moreover, <math>X\!</math> may be identified with the set of singular propositions <math>\{ x : \mathbb{B} \xrightarrow{s} \mathbb{B} \}.</math> The space of linear propositions <math>X^* = \{ \mathrm{hom} : \mathbb{B} \xrightarrow{\ell} \mathbb{B} \} = \{ 0, A \}</math> is algebraically dual to <math>X\!</math> and also has cardinality <math>2.\!</math> Here, <math>{}^{\backprime\backprime} 0 {}^{\prime\prime}\!</math> is interpreted as denoting the constant function <math>0 : \mathbb{B} \to \mathbb{B},</math> amounting to the linear proposition of rank <math>0,\!</math> while <math>A\!</math> is the linear proposition of rank <math>1.\!</math> Last but not least we have the positive propositions <math>\{ \mathrm{pos} : \mathbb{B} \xrightarrow{p} \mathbb{B} \} = \{ A, 1 \},\!</math> of rank <math>1\!</math> and <math>0,\!</math> respectively, where <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}\!</math> is understood as denoting the constant function <math>1 : \mathbb{B} \to \mathbb{B}.</math> In sum, there are <math>2^{2^n} = 2^{2^1} = 4</math> propositions altogether in the universe of discourse, comprising the set <math>X^\uparrow = \{ f : X \to \mathbb{B} \} = \{ 0, \texttt{(} A \texttt{)}, A, 1 \} \cong (\mathbb{B} \to \mathbb{B}).</math><br />
<br />
The first order differential extension of <math>\mathcal{X}</math> is <math>\mathrm{E}\mathcal{X} = \{ x_1, \mathrm{d}x_1 \} = \{ A, \mathrm{d}A \}.</math> If the feature <math>A\!</math> is understood as applying to some object or state, then the feature <math>\mathrm{d}A</math> may be interpreted as an attribute of the same object or state that says that it is changing ''significantly'' with respect to the property <math>A,\!</math> or that it has an ''escape velocity'' with respect to the state <math>A.\!</math> In practice, differential features acquire their logical meaning through a class of ''temporal inference rules''.<br />
<br />
For example, relative to a frame of observation that is left implicit for now, one is permitted to make the following sorts of inference: From the fact that <math>A\!</math> and <math>\mathrm{d}A</math> are true at a given moment one may infer that <math>\texttt{(} A \texttt{)}\!</math> will be true in the next moment of observation. Altogether in the present instance, there is the fourfold scheme of inference that is shown below:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:50%"<br />
|<br />
<math>\begin{matrix}<br />
\text{From} & \texttt{(} A \texttt{)}<br />
& \text{and} & \texttt{(} \mathrm{d}A \texttt{)}<br />
& \text{infer} & \texttt{(} A \texttt{)} & \text{next.}<br />
\\[8pt]<br />
\text{From} & \texttt{(} A \texttt{)}<br />
& \text{and} & \mathrm{d}A<br />
& \text{infer} & A & \text{next.}<br />
\\[8pt]<br />
\text{From} & A<br />
& \text{and} & \texttt{(} \mathrm{d}A \texttt{)}<br />
& \text{infer} & A & \text{next.}<br />
\\[8pt]<br />
\text{From} & A<br />
& \text{and} & \mathrm{d}A<br />
& \text{infer} & \texttt{(} A \texttt{)} & \text{next.}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
It might be thought that an independent time variable needs to be brought in at this point, but it is an insight of fundamental importance that the idea of process is logically prior to the notion of time. A time variable is a reference to a ''clock'' &mdash; a canonical, conventional process that is accepted or established as a standard of measurement, but in essence no different than any other process. This raises the question of how different subsystems in a more global process can be brought into comparison, and what it means for one process to serve the function of a local standard for others. But these inquiries only wrap up puzzles in further riddles, and are obviously too involved to be handled at our current level of approximation.<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
The clock indicates the moment . . . . but what does<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;eternity indicate?<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 79]<br />
|}<br />
<br />
Observe that the secular inference rules, used by themselves, involve a loss of information, since nothing in them can tell us whether the momenta <math>\{ \texttt{(} \mathrm{d}A \texttt{)}, \mathrm{d}A \}\!</math> are changed or unchanged in the next instance. In order to know this, one would have to determine <math>\mathrm{d}^2 A,\!</math> and so on, pursuing an infinite regress. Ultimately, in order to rest with a finitely determinate system, it is necessary to make an infinite assumption, for example, that <math>\mathrm{d}^k A = 0\!</math> for all <math>k\!</math> greater than some fixed value <math>M.\!</math> Another way to escape the regress is through the provision of a dynamic law, in typical form making higher order differentials dependent on lower degrees and estates.<br />
<br />
===Example 1. A Square Rigging===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Urge and urge and urge,<br><br />
Always the procreant urge of the world.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]<br />
|}<br />
<br />
By way of example, suppose that we are given the initial condition <math>A = \mathrm{d}A\!</math> and the law <math>\mathrm{d}^2 A = \texttt{(} A \texttt{)}.\!</math> Since the equation <math>A = \mathrm{d}A\!</math> is logically equivalent to the disjunction <math>A ~ \mathrm{d}A ~\text{or}~ \texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)},\!</math> we may infer two possible trajectories, as displayed in Table&nbsp;11. In either case the state <math>A ~ \texttt{(} \mathrm{d}A \texttt{)(} \mathrm{d}^2 A \texttt{)}\!</math> is a stable attractor or a terminal condition for both starting points.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 11.} ~~ \text{A Pair of Commodious Trajectories}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Time}\!</math><br />
| <math>\text{Trajectory 1}\!</math><br />
| <math>\text{Trajectory 2}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
2<br />
\\[4pt]<br />
3<br />
\\[4pt]<br />
4<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A & \mathrm{d}A & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
\texttt{(} A \texttt{)} & \mathrm{d}A & \mathrm{d}^2 A<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
{}^{\shortparallel} & {}^{\shortparallel} & {}^{\shortparallel}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{)} & \texttt{(} \mathrm{d}A \texttt{)} & \mathrm{d}^2 A<br />
\\[4pt]<br />
\texttt{(} A \texttt{)} & \mathrm{d}A & \mathrm{d}^2 A<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
{}^{\shortparallel} & {}^{\shortparallel} & {}^{\shortparallel}<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Because the initial space <math>X = \langle A \rangle\!</math> is one-dimensional, we can easily fit the second order extension <math>\mathrm{E}^2 X = \langle A, \mathrm{d}A, \mathrm{d}^2 A \rangle\!</math> within the compass of a single venn diagram, charting the couple of converging trajectories as shown in Figure&nbsp;12.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 12 -- The Anchor.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 12.} ~~ \text{The Anchor}\!</math><br />
|}<br />
<br />
If we eliminate from view the regions of <math>\mathrm{E}^2 X\!</math> that are ruled out by the dynamic law <math>\mathrm{d}^2 A = \texttt{(} A \texttt{)},\!</math> then what remains is the quotient structure that is shown in Figure&nbsp;13. This picture makes it easy to see that the dynamically allowable portion of the universe is partitioned between the properties <math>A\!</math> and <math>\mathrm{d}^2 A\!.</math> As it happens, this fact might have been expressed &ldquo;right off the bat&rdquo; by an equivalent formulation of the differential law, one that uses the exclusive disjunction to state the law as <math>\texttt{(} A \texttt{,} \mathrm{d}^2 A \texttt{)}\!.</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 13 -- The Tiller.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 13.} ~~ \text{The Tiller}\!</math><br />
|}<br />
<br />
What we have achieved in this example is to give a differential description of a simple dynamic process. In effect, we did this by embedding a directed graph, which can be taken to represent the state transitions of a finite automaton, in a dynamically allotted quotient structure that is created from a boolean lattice or an <math>n\!</math>-cube by nullifying all of the regions that the dynamics outlaws. With growth in the dimensions of our contemplated universes, it becomes essential, both for human comprehension and for computer implementation, that the dynamic structures of interest to us be represented not actually, by acquaintance, but virtually, by description. In our present study, we are using the language of propositional calculus to express the relevant descriptions, and to comprehend the structure that is implicit in the subsets of a <math>n\!</math>-cube without necessarily being forced to actualize all of its points.<br />
<br />
One of the reasons for engaging in this kind of extremely reduced, but explicitly controlled case study is to throw light on the general study of languages, formal and natural, in their full array of syntactic, semantic, and pragmatic aspects. Propositional calculus is one of the last points of departure where we can view these three aspects interacting in a non-trivial way without being immediately and totally overwhelmed by the complexity they generate. Often this complexity causes investigators of formal and natural languages to adopt the strategy of focusing on a single aspect and to abandon all hope of understanding the whole, whether it's the still living natural language or the dynamics of inquiry that lies crystallized in formal logic.<br />
<br />
From the perspective that I find most useful here, a language is a syntactic system that is designed or evolved in part to express a set of descriptions. When the explicit symbols of a language have extensions in its object world that are actually infinite, or when the implicit categories and generative devices of a linguistic theory have extensions in its subject matter that are potentially infinite, then the finite characters of terms, statements, arguments, grammars, logics, and rhetorics force an excess of intension to reside in all these symbols and functions, across the spectrum from the object language to the metalinguistic uses. In the aphorism from W. von Humboldt that Chomsky often cites, for example, in [Cho86, 30] and [Cho93, 49], language requires &ldquo;the infinite use of finite means&rdquo;. This is necessarily true when the extensions are infinite, when the referential symbols and grammatical categories of a language possess infinite sets of models and instances. But it also voices a practical truth when the extensions, though finite at every stage, tend to grow at exponential rates.<br />
<br />
This consequence of dealing with extensions that are &ldquo;practically infinite&rdquo; becomes crucial when one tries to build neural network systems that learn, since the learning competence of any intelligent system is limited to the objects and domains that it is able to represent. If we want to design systems that operate intelligently with the full deck of propositions dealt by intact universes of discourse, then we must supply them with succinct representations and efficient transformations in this domain. Furthermore, in the project of constructing inquiry driven systems, we find ourselves forced to contemplate the level of generality that is embodied in propositions, because the dynamic evolution of these systems is driven by the measurable discrepancies that occur among their expectations, intentions, and observations, and because each of these subsystems or components of knowledge constitutes a propositional modality that can take on the fully generic character of an empirical summary or an axiomatic theory.<br />
<br />
A compression scheme by any other name is a symbolic representation, and this is what the differential extension of propositional calculus, through all of its many universes of discourse, is intended to supply. Why is this particular program of mental calisthenics worth carrying out in general? By providing a uniform logical medium for describing dynamic systems we can make the task of understanding complex systems much easier, both in looking for invariant representations of individual cases and in finding points of comparison among diverse structures that would otherwise appear as isolated systems. All of this goes to facilitate the search for compact knowledge and to adapt what is learned from individual cases to the general realm.<br />
<br />
===Back to the Feature===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
I guess it must be the flag of my disposition, out of hopeful<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;green stuff woven.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 31]<br />
|}<br />
<br />
Let us assume that the sense intended for differential features is well enough established in the intuition, for now, that we may continue with outlining the structure of the differential extension <math>[\mathrm{E}\mathcal{X}] = [A, \mathrm{d}A].\!</math> Over the extended alphabet <math>\mathrm{E}\mathcal{X} = \{ x_1, \mathrm{d}x_1 \} = \{ A, \mathrm{d}A \}\!</math> of cardinality <math>2^n = 2\!</math> we generate the set of points <math>\mathrm{E}X\!</math> of cardinality <math>2^{2n} = 4\!</math> that bears the following chain of equivalent descriptions:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}X & = & \langle A, \mathrm{d}A \rangle<br />
\\[4pt]<br />
& = & \{ \texttt{(} A \texttt{)}, A \} ~\times~ \{ \texttt{(} \mathrm{d}A \texttt{)}, \mathrm{d}A \}<br />
\\[4pt]<br />
& = &<br />
\{<br />
\texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)},~<br />
\texttt{(} A \texttt{)} \mathrm{d}A,~<br />
A \texttt{(} \mathrm{d}A \texttt{)},~<br />
A ~ \mathrm{d}A<br />
\}.<br />
\end{array}</math><br />
|}<br />
<br />
The space <math>\mathrm{E}X\!</math> may be assigned the mnemonic type <math>\mathbb{B} \times \mathbb{D},\!</math> which is really no different than <math>\mathbb{B} \times \mathbb{B} = \mathbb{B}^2.\!</math> An individual element of <math>\mathrm{E}X\!</math> may be regarded as a ''disposition at a point'' or a ''situated direction'', in effect, a singular mode of change occurring at a single point in the universe of discourse. In applications, the modality of this change can be interpreted in various ways, for example, as an expectation, an intention, or an observation with respect to the behavior of a system.<br />
<br />
To complete the construction of the extended universe of discourse <math>\mathrm{E}X^\bullet = [x_1, \mathrm{d}x_1] = [A, \mathrm{d}A]\!</math> one must add the set of differential propositions <math>\mathrm{E}X^\uparrow = \{ g : \mathrm{E}X \to \mathbb{B} \} \cong (\mathbb{B} \times \mathbb{D} \to \mathbb{B})\!</math> to the set of dispositions in <math>\mathrm{E}X.\!</math> There are <math>2^{2^{2n}} = 16\!</math> propositions in <math>\mathrm{E}X^\uparrow,\!</math> as detailed in Table&nbsp;14.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 14.} ~~ \text{Differential Propositions}\!</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>A\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>\mathrm{d}A\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>g_{0}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{1}<br />
\\[4pt]<br />
g_{2}<br />
\\[4pt]<br />
g_{4}<br />
\\[4pt]<br />
g_{8}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
1~0~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
\texttt{(} A \texttt{)} ~ \mathrm{d}A ~<br />
\\[4pt]<br />
~ A ~ \texttt{(} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
~ A ~~ \mathrm{d}A ~<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{neither}~ A ~\text{nor}~ \mathrm{d}A<br />
\\[4pt]<br />
\mathrm{d}A ~\text{and not}~ A<br />
\\[4pt]<br />
A ~\text{and not}~ \mathrm{d}A<br />
\\[4pt]<br />
A ~\text{and}~ \mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot A \land \lnot \mathrm{d}A<br />
\\[4pt]<br />
\lnot A \land \mathrm{d}A<br />
\\[4pt]<br />
A \land \lnot \mathrm{d}A<br />
\\[4pt]<br />
A \land \mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
g_{3}<br />
\\[4pt]<br />
g_{12}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{)}<br />
\\[4pt]<br />
A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ A<br />
\\[4pt]<br />
A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot A<br />
\\[4pt]<br />
A<br />
\end{matrix}\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{6}<br />
\\[4pt]<br />
g_{9}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[4pt]<br />
1~0~0~1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{,} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
\texttt{((} A \texttt{,} \mathrm{d}A \texttt{))}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
A ~\text{not equal to}~ \mathrm{d}A<br />
\\[4pt]<br />
A ~\text{equal to}~ \mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
A \ne \mathrm{d}A<br />
\\[4pt]<br />
A = \mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{5}<br />
\\[4pt]<br />
g_{10}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
\mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ \mathrm{d}A<br />
\\[4pt]<br />
\mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot \mathrm{d}A<br />
\\[4pt]<br />
\mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{7}<br />
\\[4pt]<br />
g_{11}<br />
\\[4pt]<br />
g_{13}<br />
\\[4pt]<br />
g_{14}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} ~ A ~~ \mathrm{d}A ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{(} ~ A ~ \texttt{(} \mathrm{d}A \texttt{))}<br />
\\[4pt]<br />
\texttt{((} A \texttt{)} ~ \mathrm{d}A ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{((} A \texttt{)(} \mathrm{d}A \texttt{))}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not both}~ A ~\text{and}~ \mathrm{d}A<br />
\\[4pt]<br />
\text{not}~ A ~\text{without}~ \mathrm{d}A<br />
\\[4pt]<br />
\text{not}~ \mathrm{d}A ~\text{without}~ A<br />
\\[4pt]<br />
A ~\text{or}~ \mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot A \lor \lnot \mathrm{d}A<br />
\\[4pt]<br />
A \Rightarrow \mathrm{d}A<br />
\\[4pt]<br />
A \Leftarrow \mathrm{d}A<br />
\\[4pt]<br />
A \lor \mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
| <math>f_{3}\!</math><br />
| <math>g_{15}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
Aside from changing the names of variables and shuffling the order of rows, this Table follows the format that was used previously for boolean functions of two variables. The rows are grouped to reflect natural similarity classes among the propositions. In a future discussion, these classes will be given additional explanation and motivation as the orbits of a certain transformation group acting on the set of 16 propositions. Notice that four of the propositions, in their logical expressions, resemble those given in the table for <math>X^\uparrow.\!</math> Thus the first set of propositions <math>\{ f_i \}\!</math> is automatically embedded in the present set <math>\{ g_j \}\!</math> and the corresponding inclusions are indicated at the far left margin of the Table.<br />
<br />
===Tacit Extensions===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
I would really like to have slipped imperceptibly into this lecture, as into all the others I shall be delivering, perhaps over the years ahead.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
Strictly speaking, however, there is a subtle distinction in type between the function <math>f_i : X \to \mathbb{B}</math> and the corresponding function <math>g_j : \mathrm{E}X \to \mathbb{B},</math> even though they share the same logical expression. Naturally, we want to maintain the logical equivalence of expressions that represent the same proposition while appreciating the full diversity of that proposition's functional and typical representatives. Both perspectives, and all the levels of abstraction extending through them, have their reasons, as will develop in time.<br />
<br />
Because this special circumstance points up an important general theme, it is a good idea to discuss it more carefully. Whenever there arises a situation like this, where one alphabet <math>\mathcal{X}</math> is a subset of another alphabet <math>\mathcal{Y},</math> then we say that any proposition <math>f : \langle \mathcal{X} \rangle \to \mathbb{B}</math> has a ''tacit extension'' to a proposition <math>\boldsymbol\varepsilon f : \langle \mathcal{Y} \rangle \to \mathbb{B},\!</math> and that the space <math>(\langle \mathcal{X} \rangle \to \mathbb{B})</math> has an ''automatic embedding'' within the space <math>(\langle \mathcal{Y} \rangle \to \mathbb{B}).</math> The extension is defined in such a way that <math>\boldsymbol\varepsilon f\!</math> puts the same constraint on the variables of <math>\mathcal{X}</math> that are contained in <math>\mathcal{Y}</math> as the proposition <math>f\!</math> initially did, while it puts no constraint on the variables of <math>\mathcal{Y}</math> outside of <math>\mathcal{X},</math> in effect, conjoining the two constraints.<br />
<br />
If the variables in question are indexed as <math>\mathcal{X} = \{ x_1, \ldots, x_n \}</math> and <math>\mathcal{Y} = \{ x_1, \ldots, x_n, \ldots, x_{n+k} \},</math> then the definition of the tacit extension from <math>\mathcal{X}</math> to <math>\mathcal{Y}</math> may be expressed in the form of an equation:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\boldsymbol\varepsilon f(x_1, \ldots, x_n, \ldots, x_{n+k}) ~=~ f(x_1, \ldots, x_n).\!</math><br />
|}<br />
<br />
On formal occasions, such as the present context of definition, the tacit extension from <math>\mathcal{X}</math> to <math>\mathcal{Y}</math> is explicitly symbolized by the operator <math>\boldsymbol\varepsilon : (\langle \mathcal{X} \rangle \to \mathbb{B}) \to (\langle \mathcal{Y} \rangle \to \mathbb{B}),</math> where the appropriate alphabets <math>\mathcal{X}</math> and <math>\mathcal{Y}</math> are understood from context, but normally one may leave the "<math>\boldsymbol\varepsilon\!</math>" silent.<br />
<br />
Let's explore what this means for the present Example. Here, <math>\mathcal{X} = \{ A \}</math> and <math>\mathcal{Y} = \mathrm{E}\mathcal{X} = \{ A, \mathrm{d}A \}.</math> For each of the propositions <math>f_i\!</math> over <math>X\!,</math> specifically, those whose expression <math>e_i\!</math> lies in the collection <math>\{ 0, \texttt{(} A \texttt{)}, A, 1 \},\!</math> the tacit extension <math>\boldsymbol\varepsilon f\!</math> of <math>f\!</math> to <math>\mathrm{E}X</math> can be phrased as a logical conjunction of two factors, <math>f_i = e_i \cdot \tau ~ ,\!</math> where <math>\tau\!</math> is a logical tautology that uses all the variables of <math>\mathcal{Y} - \mathcal{X}.</math> Working in these terms, the tacit extensions <math>\boldsymbol\varepsilon f\!</math> of <math>f\!</math> to <math>\mathrm{E}X</math> may be explicated as shown in Table&nbsp;15.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:60%"<br />
|+ style="height:30px" | <math>\text{Table 15.} ~~ \text{Tacit Extension of}~ [A] ~\text{to}~ [A, \mathrm{d}A]\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0<br />
& = & 0 & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))} & = & & 0<br />
\\[8pt]<br />
\texttt{(} A \texttt{)}<br />
& = & \texttt{(} A \texttt{)} & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))}<br />
& = & \texttt{(} A \texttt{)} \, \mathrm{d}A ~ & + & \texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)}<br />
\\[8pt]<br />
A<br />
& = & ~A~ & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))}<br />
& = & ~A~ ~\mathrm{d}A~ & + & ~A~ \texttt{(} \mathrm{d}A \texttt{)}<br />
\\[8pt]<br />
1<br />
& = & 1 & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))} & = & & 1<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
In its effect on the singular propositions over <math>X,\!</math> this analysis has an interesting interpretation. The tacit extension takes us from thinking about a particular state, like <math>A\!</math> or <math>\texttt{(} A \texttt{)},\!</math> to considering the collection of outcomes, the outgoing changes or the singular dispositions, that spring from that state.<br />
<br />
===Example 2. Drives and Their Vicissitudes===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
I open my scuttle at night and see the far-sprinkled systems,<br><br />
And all I see, multiplied as high as I can cipher, edge but<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the rim of the farther systems.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 81]<br />
|}<br />
<br />
Before we leave the one-feature case let's look at a more substantial example, one that illustrates a general class of curves that can be charted through the extended feature spaces and that provides an opportunity to discuss a number of important themes concerning their structure and dynamics.<br />
<br />
Again, let <math>\mathcal{X} = \{ x_1 \} = \{ A \}.\!</math> In the discussion that follows we will consider a class of trajectories having the property that <math>\mathrm{d}^k A = 0\!</math> for all <math>k\!</math> greater than some fixed <math>m\!</math> and we may indulge in the use of some picturesque terms that describe salient classes of such curves. Given the finite order condition, there is a highest order non-zero difference <math>\mathrm{d}^m A\!</math> exhibited at each point in the course of any determinate trajectory that one may wish to consider. With respect to any point of the corresponding orbit or curve let us call this highest order differential feature <math>\mathrm{d}^m A\!</math> the ''drive'' at that point. Curves of constant drive <math>\mathrm{d}^m A\!</math> are then referred to as ''<math>m^\text{th}\!</math>-gear curves''.<br />
<br />
* '''Scholium.''' The fact that a difference calculus can be developed for boolean functions is well known [Fuji], [Koh, &sect; 8-4] and was probably familiar to Boole, who was an expert in difference equations before he turned to logic. And of course there is the strange but true story of how the Turin machines of the 1840s prefigured the Turing machines of the 1940s [Men, 225-297]. At the very outset of general purpose, mechanized computing we find that the motive power driving the Analytical Engine of Babbage, the kernel of an idea behind all of his wheels, was exactly his notion that difference operations, suitably trained, can serve as universal joints for any conceivable computation [M&M], [Mel, ch. 4].<br />
<br />
Given this language, the Example we take up here can be described as the family of <math>4^\text{th}\!</math>-gear curves through <math>\mathrm{E}^4 X\!</math> <math>=\!</math> <math>\langle A, ~\mathrm{d}A, ~\mathrm{d}^2\!A, ~\mathrm{d}^3\!A, ~\mathrm{d}^4\!A \rangle.</math> These are the trajectories generated subject to the dynamic law <math>\mathrm{d}^4 A = 1,\!</math> where it is understood in such a statement that all higher order differences are equal to <math>0.\!</math> Since <math>\mathrm{d}^4 A\!</math> and all higher <math>\mathrm{d}^k A\!</math> are fixed, the temporal or transitional conditions (initial, mediate, terminal &mdash; transient or stable states) vary only with respect to their projections as points of <math>\mathrm{E}^3 X = \langle A, ~\mathrm{d}A, ~\mathrm{d}^2\!A, ~\mathrm{d}^3\!A \rangle.</math> Thus, there is just enough space in a planar venn diagram to plot all of these orbits and to show how they partition the points of <math>\mathrm{E}^3 X.\!</math> It turns out that there are exactly two possible orbits, of eight points each, as illustrated in Figure&nbsp;16.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 16 -- A Couple of Fourth Gear Orbits.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 16.} ~~ \text{A Couple of Fourth Gear Orbits}\!</math><br />
|}<br />
<br />
With a little thought it is possible to devise an indexing scheme for the general run of dynamic states that allows for comparing universes of discourse that weigh in on different scales of observation. With this end in sight, let us index the states <math>q \in \mathrm{E}^m X\!</math> with the dyadic rationals (or the binary fractions) in the half-open interval <math>[0, 2).\!</math> Formally and canonically, a state <math>q_r\!</math> is indexed by a fraction <math>r = \tfrac{s}{t}\!</math> whose denominator is the power of two <math>t = 2^m\!</math> and whose numerator is a binary numeral formed from the coefficients of state in a manner to be described next. The ''differential coefficients'' of the state <math>q\!</math> are just the values <math>\mathrm{d}^k\!A(q)</math> for <math>k = 0 ~\text{to}~ m,\!</math> where <math>\mathrm{d}^0\!A</math> is defined as being identical to <math>A.\!</math> To form the binary index <math>d_0.d_1 \ldots d_m\!</math> of the state <math>q\!</math> the coefficient <math>\mathrm{d}^k\!A(q)</math> is read off as the binary digit <math>d_k\!</math> associated with the place value <math>2^{-k}.\!</math> Expressed by way of algebraic formulas, the rational index <math>r\!</math> of the state <math>q\!</math> can be given by the following equivalent formulations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
r(q)<br />
& = &<br />
\displaystyle\sum_k d_k \cdot 2^{-k}<br />
& = &<br />
\displaystyle\sum_k \text{d}^k A(q) \cdot 2^{-k}<br />
\\[8pt]<br />
=<br />
\\[8pt]<br />
\displaystyle\frac{s(q)}{t}<br />
& = &<br />
\displaystyle\frac{\sum_k d_k \cdot 2^{(m-k)}}{2^m}<br />
& = &<br />
\displaystyle\frac{\sum_k \text{d}^k A(q) \cdot 2^{(m-k)}}{2^m}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Applied to the example of <math>4^\text{th}\!</math>-gear curves, this scheme results in the data of Tables&nbsp;17-a and 17-b, which exhibit one period for each orbit. The states in each orbit are listed as ordered pairs <math>(p_i, q_j),\!</math> where <math>p_i\!</math> may be read as a temporal parameter that indicates the present time of the state and where <math>j\!</math> is the decimal equivalent of the binary numeral <math>s.\!</math> Informally and more casually, the Tables exhibit the states <math>q_s\!</math> as subscripted with the numerators of their rational indices, taking for granted the constant denominators of <math>2^m\! = 2^4 = 16.\!</math> In this set-up the temporal successions of states can be reckoned as given by a kind of ''parallel round-up rule''. That is, if <math>(d_k, d_{k+1})\!</math> is any pair of adjacent digits in the state index <math>r,\!</math> then the value of <math>d_k\!</math> in the next state is <math>{d_k}' = d_k + d_{k+1}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:55%"<br />
|+ style="height:30px" | <math>\text{Table 17-a.} ~~ \text{A Couple of Orbits in Fourth Gear : Orbit 1}\!</math><br />
|- style="background:ghostwhite"<br />
| <math>\text{Time}\!</math><br />
| <math>\text{State}\!</math><br />
| <math>A\!</math><br />
| <math>\mathrm{d}A\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| <math>p_i\!</math><br />
| <math>q_j\!</math><br />
| <math>\mathrm{d}^0\!A</math><br />
| <math>\mathrm{d}^1\!A</math><br />
| <math>\mathrm{d}^2\!A</math><br />
| <math>\mathrm{d}^3\!A</math><br />
| <math>\mathrm{d}^4\!A</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
p_0<br />
\\[4pt]<br />
p_1<br />
\\[4pt]<br />
p_2<br />
\\[4pt]<br />
p_3<br />
\\[4pt]<br />
p_4<br />
\\[4pt]<br />
p_5<br />
\\[4pt]<br />
p_6<br />
\\[4pt]<br />
p_7<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
q_{01}<br />
\\[4pt]<br />
q_{03}<br />
\\[4pt]<br />
q_{05}<br />
\\[4pt]<br />
q_{15}<br />
\\[4pt]<br />
q_{17}<br />
\\[4pt]<br />
q_{19}<br />
\\[4pt]<br />
q_{21}<br />
\\[4pt]<br />
q_{31}<br />
\end{matrix}\!</math><br />
| colspan="5" |<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="text-align:center; width:100%"<br />
|<br />
<math>\begin{matrix}<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
1.<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:55%"<br />
|+ style="height:30px" | <math>\text{Table 17-b.} ~~ \text{A Couple of Orbits in Fourth Gear : Orbit 2}\!</math><br />
|- style="background:ghostwhite"<br />
| <math>\text{Time}\!</math><br />
| <math>\text{State}\!</math><br />
| <math>A\!</math><br />
| <math>\mathrm{d}A\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| <math>p_i\!</math><br />
| <math>q_j\!</math><br />
| <math>\mathrm{d}^0\!A</math><br />
| <math>\mathrm{d}^1\!A</math><br />
| <math>\mathrm{d}^2\!A</math><br />
| <math>\mathrm{d}^3\!A</math><br />
| <math>\mathrm{d}^4\!A</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
p_0<br />
\\[4pt]<br />
p_1<br />
\\[4pt]<br />
p_2<br />
\\[4pt]<br />
p_3<br />
\\[4pt]<br />
p_4<br />
\\[4pt]<br />
p_5<br />
\\[4pt]<br />
p_6<br />
\\[4pt]<br />
p_7<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
q_{25}<br />
\\[4pt]<br />
q_{11}<br />
\\[4pt]<br />
q_{29}<br />
\\[4pt]<br />
q_{07}<br />
\\[4pt]<br />
q_{09}<br />
\\[4pt]<br />
q_{27}<br />
\\[4pt]<br />
q_{13}<br />
\\[4pt]<br />
q_{23}<br />
\end{matrix}\!</math><br />
| colspan="5" |<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="text-align:center; width:100%"<br />
|<br />
<math>\begin{matrix}<br />
1.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
==Transformations of Discourse==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
It is understandable that an engineer should be completely absorbed in his speciality, instead of pouring himself out into the freedom and vastness of the world of thought, even though his machines are being sent off to the ends of the earth; for he no more needs to be capable of applying to his own personal soul what is daring and new in the soul of his subject than a machine is in fact capable of applying to itself the differential calculus on which it is based. The same thing cannot, however, be said about mathematics; for here we have the new method of thought, pure intellect, the very well-spring of the times, the ''fons et origo'' of an unfathomable transformation.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 39]<br />
|}<br />
<br />
In this section we take up the general study of logical transformations, or maps that relate one universe of discourse to another. In many ways, and especially as applied to the subject of intelligent dynamic systems, my argument develops the antithesis of the statement just quoted. Along the way, if incidental to my ends, I hope this essay can pose a fittingly irenic epitaph to the frankly ironic epigraph inscribed at its head.<br />
<br />
My goal in this section is to answer a single question: What is a propositional tangent functor? In other words, my aim is to develop a clear conception of what manner of thing would pass in the logical realm for a genuine analogue of the tangent functor, an object conceived to generalize as far as possible in the abstract terms of category theory the ordinary notions of functional differentiation and the all too familiar operations of taking derivatives.<br />
<br />
As a first step I discuss the kinds of transformations that we already know as ''extensions'' and ''projections'', and I use these special cases to illustrate several different styles of logical and visual representation that will figure heavily in the sequel.<br />
<br />
===Foreshadowing Transformations : Extensions and Projections of Discourse===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
And, despite the care which she took to look behind her at every moment, she failed to see a shadow which followed her like her own shadow, which stopped when she stopped, which started again when she did, and which made no more noise than a well-conducted shadow should.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 126]<br />
|}<br />
<br />
Many times in our discussion we have occasion to place one universe of discourse in the context of a larger universe of discourse. An embedding of the general type <math>[\mathcal{X}] \to [\mathcal{Y}]\!</math> is implied any time that we make use of one alphabet <math>[\mathcal{X}]\!</math> that happens to be included in another alphabet <math>[\mathcal{Y}].\!</math> When we are discussing differential issues we usually have in mind that the extended alphabet <math>[\mathcal{Y}]\!</math> has a special construction or a specific lexical relation with respect to the initial alphabet <math>[\mathcal{X}],\!</math> one that is marked by characteristic types of accents, indices, or inflected forms.<br />
<br />
====Extension from 1 to 2 Dimensions====<br />
<br />
Figure 18-a lays out the ''angular form'' of venn diagram for universes of 1 and 2 dimensions, indicating the embedding map of type <math>\mathbb{B}^1 \to \mathbb{B}^2\!</math> and detailing the coordinates that are associated with individual cells. Because all points, cells, or logical interpretations are represented as connected geometric areas, we can say that these pictures provide us with an ''areal view'' of each universe of discourse.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-a -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-a.} ~~ \text{Extension from 1 to 2 Dimensions : Areal}\!</math><br />
|}<br />
<br />
Figure 18-b shows the differential extension from <math>X^\bullet = [x]\!</math> to <math>\mathrm{E}X^\bullet = [x, \mathrm{d}x]\!</math> in a ''bundle of boxes'' form of venn diagram. As awkward as it may seem at first, this type of picture is often the most natural and the most easily available representation when we want to conceptualize the localized information or momentary knowledge of an intelligent dynamic system. It gives a ready picture of a ''proposition at a point'', in the present instance, of a proposition about changing states which is itself associated with a particular dynamic state of a system. It is easy to see how this application might be extended to conceive of more general types of instantaneous knowledge that are possessed by a system.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-b -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-b.} ~~ \text{Extension from 1 to 2 Dimensions : Bundle}\!</math><br />
|}<br />
<br />
Figure 18-c shows the same extension in a ''compact'' style of venn diagram, where the differential features at each position are represented by arrows extending from that position that cross or do not cross, as the case may be, the corresponding feature boundaries.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-c -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-c.} ~~ \text{Extension from 1 to 2 Dimensions : Compact}\!</math><br />
|}<br />
<br />
Figure 18-d compresses the picture of the differential extension even further, yielding a directed graph or ''digraph'' form of representation. (Notice that my definition of a digraph allows for loops or ''slings'' at individual points, in addition to arcs or ''arrows'' between the points.)<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-d -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-d.} ~~ \text{Extension from 1 to 2 Dimensions : Digraph}\!</math><br />
|}<br />
<br />
====Extension from 2 to 4 Dimensions====<br />
<br />
Figure 19-a lays out the ''areal view'' or the ''angular form'' of venn diagram for universes of 2 and 4 dimensions, indicating the embedding map of type <math>\mathbb{B}^2 \to \mathbb{B}^4.\!</math> In many ways these pictures are the best kind there is, giving full canvass to an ideal vista. Their style allows the clearest, the fairest, and the plainest view that we can form of a universe of discourse, affording equal representation to all dispositions and maintaining a balance with respect to ordinary and differential features. If only we could extend this view! Unluckily, an obvious difficulty beclouds this prospect, and that is how precipitately we run into the limits of our plane and visual intuitions. Even within the scope of the spare few dimensions that we have scanned up to this point subtle discrepancies have crept in already. The circumstances that bind us and the frameworks that block us, the flat distortion of the planar projection and the inevitable ineffability that precludes us from wrapping its rhomb figure into rings around a torus, all of these factors disguise the underlying but true connectivity of the universe of discourse.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-a -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-a.} ~~ \text{Extension from 2 to 4 Dimensions : Areal}\!</math><br />
|}<br />
<br />
Figure 19-b shows the differential extension from <math>U^\bullet = [u, v]\!</math> to <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v]\!</math> in the ''bundle of boxes'' form of venn diagram.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-b -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-b.} ~~ \text{Extension from 2 to 4 Dimensions : Bundle}\!</math><br />
|}<br />
<br />
As dimensions increase, this factorization of the extended universe along the lines that are marked out by the bundle picture begins to look more and more like a practical necessity. But whenever we use a propositional model to address a real situation in the context of nature we need to remain aware that this articulation into factors, affecting our description, may be wholly artificial in nature and cleave to nothing, no joint in nature, nor any juncture in time to be in or out of joint.<br />
<br />
Figure 19-c illustrates the extension from 2 to 4 dimensions in the ''compact'' style of venn diagram. Here, just the changes with respect to the center cell are shown.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-c -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-c.} ~~ \text{Extension from 2 to 4 Dimensions : Compact}\!</math><br />
|}<br />
<br />
Figure 19-d gives the ''digraph'' form of representation for the differential extension <math>U^\bullet \to \mathrm{E}U^\bullet,\!</math> where the 4 nodes marked with a circle <math>{}^{\bigcirc}\!</math> are the cells <math>uv,\, u \texttt{(} v \texttt{)},\, \texttt{(} u \texttt{)} v,\, \texttt{(} u \texttt{)(} v \texttt{)},\!</math> respectively, and where a 2-headed arc counts as 2 arcs of the differential digraph.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-d -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-d.} ~~ \text{Extension from 2 to 4 Dimensions : Digraph}\!</math><br />
|}<br />
<br />
===Thematization of Functions : And a Declaration of Independence for Variables===<br />
<br />
{| width="100%"<br />
| align="left" |<br />
''And as imagination bodies forth''<br><br />
''The forms of things unknown, the poet's pen''<br><br />
''Turns them to shapes, and gives to airy nothing''<br><br />
''A local habitation and a name.''<br />
| align="right" valign="bottom" | A Midsummer Night's Dream, 5.1.18<br />
|}<br />
<br />
In the representation of propositions as functions it is possible to notice different degrees of explicitness in the way their functional character is symbolized. To indicate what I mean by this, the next series of Figures illustrates a set of graphic conventions that will be put to frequent use in the remainder of this discussion, both to mark the relevant distinctions and to help us convert between related expressions at different levels of explicitness in their functionality.<br />
<br />
====Thematization : Venn Diagrams====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
The known universe has one complete lover and that is the greatest poet. He consumes an eternal passion and is indifferent which chance happens and which possible contingency of fortune or misfortune and persuades daily and hourly his delicious pay.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 11&ndash;12]<br />
|}<br />
<br />
Figure 20-i traces the first couple of steps in this order of ''thematic'' progression, that will gradually run the gamut through a complete series of degrees of functional explicitness in the expression of logical propositions. The first venn diagram represents a situation where the function is indicated by a shaded figure and a logical expression. At this stage one may be thinking of the proposition only as expressed by a formula in a particular language and its content only as a subset of the universe of discourse, as when considering the proposition <math>u\!\cdot\!v</math> in the universe <math>[u, v].\!</math> The second venn diagram depicts a situation in which two significant steps have been taken. First, one has taken the trouble to give the proposition <math>u\!\cdot\!v</math> a distinctive functional name <math>{}^{\backprime\backprime} J {}^{\prime\prime}.\!</math> Second, one has come to think explicitly about the target domain that contains the functional values of <math>J,\!</math> as when writing <math>J : \langle u, v \rangle \to \mathbb{B}.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 20-i -- Thematization of Conjunction (Stage 1).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 20-i.} ~~ \text{Thematization of Conjunction (Stage 1)}\!</math><br />
|}<br />
<br />
In Figure 20-ii the proposition <math>J\!</math> is viewed explicitly as a transformation from one universe of discourse to another.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 20-ii -- Thematization of Conjunction (Stage 2).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 20-ii.} ~~ \text{Thematization of Conjunction (Stage 2)}\!</math><br />
|}<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
o-------------------------------o o-------------------------------o<br />
| | | |<br />
| o-----o o-----o | | o-----o o-----o |<br />
| / \ / \ | | / \ / \ |<br />
| / o \ | | / o \ |<br />
| / /`\ \ | | / /`\ \ |<br />
| o o```o o | | o o```o o |<br />
| | u |```| v | | | | u |```| v | |<br />
| o o```o o | | o o```o o |<br />
| \ \`/ / | | \ \`/ / |<br />
| \ o / | | \ o / |<br />
| \ / \ / | | \ / \ / |<br />
| o-----o o-----o | | o-----o o-----o |<br />
| | | |<br />
o-------------------------------o o-------------------------------o<br />
\ / \ /<br />
\ / \ /<br />
\ / \ J /<br />
\ / \ /<br />
\ / \ /<br />
o----------\---------/----------o o----------\---------/----------o<br />
| \ / | | \ / |<br />
| \ / | | \ / |<br />
| o-----@-----o | | o-----@-----o |<br />
| /`````````````\ | | /`````````````\ |<br />
| /```````````````\ | | /```````````````\ |<br />
| /`````````````````\ | | /`````````````````\ |<br />
| o```````````````````o | | o```````````````````o |<br />
| |```````````````````| | | |```````````````````| |<br />
| |```````` J ````````| | | |```````` x ````````| |<br />
| |```````````````````| | | |```````````````````| |<br />
| o```````````````````o | | o```````````````````o |<br />
| \`````````````````/ | | \`````````````````/ |<br />
| \```````````````/ | | \```````````````/ |<br />
| \`````````````/ | | \`````````````/ |<br />
| o-----------o | | o-----------o |<br />
| | | |<br />
| | | |<br />
o-------------------------------o o-------------------------------o<br />
J = u v x = J<u, v><br />
<br />
Figure 20-ii. Thematization of Conjunction (Stage 2)<br />
</pre><br />
|}<br />
<br />
In the first venn diagram the name that is assigned to a composite proposition, function, or region in the source universe is delegated to a simple feature in the target universe. This can result in a single character or term exceeding the responsibilities it can carry off well. Allowing the name of a function <math>J : \langle u, v \rangle \to \mathbb{B}\!</math> to serve as the name of its dependent variable <math>J : \mathbb{B}\!</math> does not mean that one has to confuse a function with any of its values, but it does put one at risk for a number of obvious problems, and we should not be surprised, on numerous and limiting occasions, when quibbling arises from the attempts of a too original syntax to serve these two masters.<br />
<br />
The second venn diagram circumvents these difficulties by introducing a new variable name for each basic feature of the target universe, as when writing <math>J : \langle u, v \rangle \to \langle x \rangle,\!</math> and thereby assigns a concrete type <math>\langle x \rangle</math> to the abstract codomain <math>\mathbb{B}.\!</math> To make this induction of variables more formal one can append subscripts, as in <math>x_J,\!</math> to indicate the origin or derivation of the new characters. Or we may use a lexical modifier to convert function names into variable names, for example, associating the function name <math>J\!</math> with the variable name <math>\check{J}.\!</math> Thus we may think of <math>x = x_J = \check{J}\!</math> as the ''cache variable'' corresponding to the function <math>J\!</math> or the symbol <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> considered as a contingent variable.<br />
<br />
In Figure 20-iii we arrive at a stage where the functional equations <math>J = u\!\cdot\!v</math> and <math>x = u\!\cdot\!v</math> are regarded as propositions in their own right, reigning in and ruling over the 3-feature universes of discourse <math>[u, v, J]~\!</math> and <math>[u, v, x],\!</math> respectively. Subject to the cautions already noted, the function name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> can be reinterpreted as the name of a feature <math>\check{J}</math> and the equation <math>J = u\!\cdot\!v</math> can be read as the logical equivalence <math>\texttt{((} J, u ~ v \texttt{))}.\!</math> To give it a generic name let us call this newly expressed, collateral proposition the ''thematization'' or the ''thematic extension'' of the original proposition <math>J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 20-iii -- Thematization of Conjunction (Stage 3).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 20-iii.} ~~ \text{Thematization of Conjunction (Stage 3)}\!</math><br />
|}<br />
<br />
The first venn diagram represents the thematization of the conjunction <math>J\!</math> with shading in the appropriate regions of the universe <math>[u, v, J].\!</math> Also, it illustrates a quick way of constructing a thematic extension. First, draw a line, in practice or the imagination, that bisects every cell of the original universe, placing half of each cell under the aegis of the thematized proposition and the other half under its antithesis. Next, on the scene where the theme applies leave the shade wherever it lies, and off the stage, where it plays otherwise, stagger the pattern in a harlequin guise.<br />
<br />
In the final venn diagram of this sequence the thematic progression comes full circle and completes one round of its development. The ambiguities that were occasioned by the changing role of the name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> are resolved by introducing a new variable name <math>{}^{\backprime\backprime} x {}^{\prime\prime}</math> to take the place of <math>\check{J},\!</math> and the region that represents this fresh featured <math>x\!</math> is circumscribed in a more conventional symmetry of form and placement. Just as we once gave the name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> to the proposition <math>u\!\cdot\!v,</math> we now give the name <math>{}^{\backprime\backprime} \iota {}^{\prime\prime}</math> to its thematization <math>\texttt{((} x, u ~ v \texttt{))}.\!</math> Already, again, at this culminating stage of reflection, we begin to think of the newly named proposition as a distinctive individual, a particular function <math>\iota : \langle u, v, x \rangle \to \mathbb{B}.\!</math><br />
<br />
From now on, the terms ''thematic extension'' and ''thematization'' will be used to describe both the process and degree of explication that progresses through this series of pictures, both the operation of increasingly explicit symbolization and the dimension of variation that is swept out by it. To speak of this change in general, that takes us in our current example from <math>J\!</math> to <math>\iota,\!</math> we introduce a class of operators symbolized by the Greek letter <math>\theta,\!</math> writing <math>\iota = \theta J\!</math> in the present instance. The operator <math>\theta,\!</math> in the present situation bearing the type <math>\theta : [u, v] \to [u, v, x],\!</math> provides us with a convenient way of recapitulating and summarizing the complete cycle of thematic developments.<br />
<br />
Figure 21 shows how the thematic extension operator <math>\theta\!</math> acts on two further examples, the disjunction <math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math> and the equality <math>\texttt{((} u, v \texttt{))}.\!</math> Referring to the disjunction as <math>f(u, v)\!</math> and the equality as <math>f(u, v),\!</math> we may express the thematic extensions as <math>\varphi = \theta f\!</math> and <math>\gamma = \theta g.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 21 -- Thematization of Disjunction and Equality.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 21.} ~~ \text{Thematization of Disjunction and Equality}\!</math><br />
|}<br />
<br />
====Thematization : Truth Tables====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
That which distorts honest shapes or which creates unearthly beings or places or contingencies is a nuisance and a revolt.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 19]<br />
|}<br />
<br />
Tables 22 through 25 outline a method for computing the thematic extensions of propositions in terms of their coordinate values.<br />
<br />
A preliminary step, as illustrated in Table&nbsp;22, is to write out the truth table representations of the propositional forms whose thematic extensions one wants to compute, in the present instance, the functions <math>f(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math> and <math>g(u, v) = \texttt{((} u, v \texttt{))}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:50%"<br />
|+ style="height:30px" | <math>\text{Table 22.} ~~ \text{Disjunction}~ f ~\text{and Equality}~ g\!</math><br />
|- style="height:35px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black" | <math>f\!</math><br />
| <math>g\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Next, each propositional form is individually represented in the fashion shown in Tables&nbsp;23-i and 23-ii, using <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} g {}^{\prime\prime}\!</math> as function names and creating new variables <math>x\!</math> and <math>y\!</math> to hold the associated functional values. This pair of Tables outlines the first stage in the transition from the <math>2\!</math>-dimensional universes of <math>f\!</math> and <math>g\!</math> to the <math>3\!</math>-dimensional universes of <math>\theta f\!</math> and <math>\theta g.\!</math> The top halves of the Tables replicate the truth table patterns for <math>f\!</math> and <math>g\!</math> in the form <math>f : [u, v] \to [x]\!</math> and <math>g : [u, v] \to [y].\!</math> The bottom halves of the tables print the negatives of these pictures, as it were, and paste the truth tables for <math>\texttt{(} f \texttt{)}\!</math> and <math>\texttt{(} g \texttt{)}\!</math> under the copies for <math>f\!</math> and <math>g.\!</math> At this stage, the columns for <math>\theta f\!</math> and <math>\theta g\!</math> are appended almost as afterthoughts, amounting to indicator functions for the sets of ordered triples that make up the functions <math>f\!</math> and <math>g.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 23-i and 23-ii.} ~~ \text{Thematics of Disjunction and Equality (1)}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 23-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black" | <math>f\!</math><br />
| <math>x\!</math><br />
| <math>\varphi\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |<br />
<math>\begin{matrix}\to\\\to\\\to\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" | &nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 23-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black" | <math>g\!</math><br />
| <math>y\!</math><br />
| <math>\gamma\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |<br />
<math>\begin{matrix}\to\\\to\\\to\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" | &nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
All the data are now in place to give the truth tables for <math>\theta f\!</math> and <math>\theta g.\!</math> All that remains to be done is to permute the rows and change the roles of <math>x\!</math> and <math>y\!</math> from dependent to independent variables. In Tables&nbsp;24-i and 24-ii the rows are arranged in such a way as to put the 3-tuples <math>(u, v, x)\!</math> and <math>(u, v, y)\!</math> in binary numerical order, suitable for viewing as the arguments of the maps <math>\theta f = \varphi : [u, v, x] \to \mathbb{B}\!</math> and <math>\theta g = \gamma : [u, v, y] \to \mathbb{B}.\!</math> Moreover, the structure of the tables is altered slightly, allowing the now vestigial functions <math>\theta f\!</math> and <math>\theta g\!</math> to be passed over without further attention and shifting the heavy vertical bars a notch to the right. In effect, this clinches the fact that the thematic variables <math>x := \check{f}\!</math> and <math>y := \check{g}\!</math> are now treated as independent variables.<br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 24-i and 24-ii.} ~~ \text{Thematics of Disjunction and Equality (2)}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 24-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>f\!</math><br />
| <math>x\!</math><br />
| style="border-left:1px solid black" | <math>\varphi\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <font size="+2">&nbsp;<br></font><math>\begin{matrix}\to\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 24-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>g\!</math><br />
| <math>y\!</math><br />
| style="border-left:1px solid black" | <math>\gamma\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | &nbsp;<br><math>\begin{matrix}\to\\\to\end{matrix}\!</math><br>&nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
An optional reshuffling of the rows brings additional features of the thematic extensions to light. Leaving the columns in place for the sake of comparison, Tables&nbsp;25-i and 25-ii sort the rows in a different order, in effect treating <math>x\!</math> and <math>y\!</math> as the primary variables in their respective 3-tuples. Regarding the thematic extensions in the form <math>\varphi : [x, u, v] \to \mathbb{B}\!</math> and <math>\gamma : [y, u, v] \to \mathbb{B}\!</math> makes it easier to see in this tabular setting a property that was graphically obvious in the venn diagrams above. Specifically, when the thematic variable <math>\check{F}\!</math> is true then <math>\theta F\!</math> exhibits the pattern of the original <math>F,\!</math> and when <math>\check{F}\!</math> is false then <math>\theta F\!</math> exhibits the pattern of its negation <math>\texttt{(} F \texttt{)}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 25-i and 25-ii.} ~~ \text{Thematics of Disjunction and Equality (3)}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 25-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>f\!</math><br />
| <math>x\!</math><br />
| style="border-left:1px solid black" | <math>\varphi\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>{\to}\!</math><br><font size="+2">&nbsp;<br>&nbsp;<br>&nbsp;<br></font><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <font size="+2">&nbsp;<br></font><math>\begin{matrix}\to\\\to\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 25-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>g\!</math><br />
| <math>y\!</math><br />
| style="border-left:1px solid black" | <math>\gamma\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | &nbsp;<br><math>\begin{matrix}\to\\\to\end{matrix}\!</math><br>&nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
Finally, Tables&nbsp;26-i and 26-ii compare the tacit extensions <math>\boldsymbol\varepsilon : [u, v] \to [u, v, x]\!</math> and <math>\boldsymbol\varepsilon : [u, v] \to [u, v, y]\!</math> with the thematic extensions of the same types, as applied to the propositions <math>f\!</math> and <math>g,\!</math> respectively.<br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 26-i and 26-ii.} ~~ \text{Tacit Extension and Thematization}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 26-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>x\!</math><br />
| style="border-left:1px solid black" | <math>\boldsymbol\varepsilon f\!</math><br />
| <math>\theta f\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 26-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>y\!</math><br />
| style="border-left:1px solid black" | <math>\boldsymbol\varepsilon g\!</math><br />
| <math>\theta g\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
Table 27 summarizes the thematic extensions of all propositions on two variables. Column&nbsp;4 lists the equations of form <math>\texttt{((} \check{f_i}, f_i (u, v) \texttt{))}\!</math> and Column&nbsp;5 simplifies these equations into the form of algebraic expressions. As always, <math>{}^{\backprime\backprime} + {}^{\prime\prime}\!</math> refers to exclusive disjunction and each <math>{}^{\backprime\backprime} \check{f} {}^{\prime\prime}\!</math> appearing in the last two Columns refers to the corresponding variable name <math>{}^{\backprime\backprime} \check{f_i} {}^{\prime\prime}.~\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 27.} ~~ \text{Thematization of Bivariate Propositions}\!</math><br />
|- style="height:30px; background:ghostwhite"<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>{f}\!</math><br />
| <math>\theta f\!</math><br />
| <math>\theta f\!</math><br />
|- style="background:ghostwhite"<br />
| align="right" | <math>u\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| align="right" | <math>v\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| align="left" | <math>\texttt{((} \check{f} \texttt{,~(~)~))}\!</math><br />
| align="left" | <math>\check{f} + 1\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\\[4pt]<br />
f_{4}<br />
\\[4pt]<br />
f_{8}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
1~0~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} u \texttt{)~} v \texttt{~}<br />
\\[4pt]<br />
\texttt{~} u \texttt{~(} v \texttt{)}<br />
\\[4pt]<br />
\texttt{~} u \texttt{~~} v \texttt{~}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(u)(v)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~(u)~v~~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~u~(v)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~u~~v~~))}<br />
\end{array}</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + u + v + uv<br />
\\[4pt]<br />
\check{f} + v + uv + 1<br />
\\[4pt]<br />
\check{f} + u + uv + 1<br />
\\[4pt]<br />
\check{f} + uv + 1<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{3}<br />
\\[4pt]<br />
f_{12}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{)}<br />
\\[4pt]<br />
\texttt{~} u \texttt{~}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(u)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~u~~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + u<br />
\\[4pt]<br />
\check{f} + u + 1<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[4pt]<br />
f_{9}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[4pt]<br />
1~0~0~1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{,} v \texttt{)}<br />
\\[4pt]<br />
\texttt{((} u \texttt{,} v \texttt{))}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~~(} u \texttt{,} v \texttt{)~~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~((} u \texttt{,} v \texttt{))~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + u + v + 1<br />
\\[4pt]<br />
\check{f} + u + v<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{5}<br />
\\[4pt]<br />
f_{10}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} v \texttt{)}<br />
\\[4pt]<br />
\texttt{~} v \texttt{~}<br />
\end{matrix}</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(} v \texttt{)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~} v \texttt{~~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + v<br />
\\[4pt]<br />
\check{f} + v + 1<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[4pt]<br />
f_{11}<br />
\\[4pt]<br />
f_{13}<br />
\\[4pt]<br />
f_{14}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(~} u \texttt{~~} v \texttt{~)}<br />
\\[4pt]<br />
\texttt{(~} u \texttt{~(} v \texttt{))}<br />
\\[4pt]<br />
\texttt{((} u \texttt{)~} v \texttt{~)}<br />
\\[4pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(~} u \texttt{~~} v \texttt{~)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~(~} u \texttt{~(} v \texttt{))~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~((} u \texttt{)~} v \texttt{~)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~((} u \texttt{)(} v \texttt{))~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + uv<br />
\\[4pt]<br />
\check{f} + u + uv<br />
\\[4pt]<br />
\check{f} + v + uv<br />
\\[4pt]<br />
\check{f} + u + v + uv + 1<br />
\end{array}\!</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| align="left" | <math>\texttt{((} \check{f} \texttt{,~((~))~))}\!</math><br />
| align="left" | <math>\check{f}\!</math><br />
|}<br />
<br />
<br><br />
<br />
In order to show what all of the thematic extensions from two dimensions to three dimensions look like in terms of coordinates, Tables&nbsp;28 and 29 present ordinary truth tables for the functions <math>f_i : \mathbb{B}^2 \to \mathbb{B}\!</math> and for the corresponding thematizations <math>\theta f_i = \varphi_i : \mathbb{B}^3 \to \mathbb{B}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 28.} ~~ \text{Propositions on Two Variables}\!</math><br />
|- style="height:35px; background:ghostwhite"<br />
| style="width:5%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:5%; border-bottom:1px solid black; border-left:1px solid black" | <math>f_{0}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{1}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{2}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{3}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{4}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{5}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{6}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{7}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{8}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{9}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{10}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{11}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{12}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{13}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{14}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{15}\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 29.} ~~ \text{Thematic Extensions of Bivariate Propositions}\!</math><br />
|- style="height:35px; background:ghostwhite"<br />
| style="width:5%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\check{f}\!</math><br />
| style="width:5%; border-bottom:1px solid black; border-left:1px solid black" | <math>\varphi_{0}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{1}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{2}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{3}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{4}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{5}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{6}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{7}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{8}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{9}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{10}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{11}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{12}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{13}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{14}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{15}\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
|-<br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Propositional Transformations===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
If only the word &lsquo;artificial&rsquo; were associated with the idea of ''art'', or expert skill gained through voluntary apprenticeship (instead of suggesting the factitious and unreal), we might say that ''logical'' refers to artificial thought.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 56&ndash;57]<br />
|}<br />
<br />
In this section we develop a comprehensive set of concepts for dealing with transformations between universes of discourse. In this most general setting the source and target universes of a transformation are allowed to be different, but may be the same. When we apply these concepts to dynamic systems we focus on the important special case of transformations that map a universe into itself, regarding them as the state transitions of a discrete dynamical process and placing them among the myriad ways that a universe of discourse might change, and by that change turn into itself.<br />
<br />
====Alias and Alibi Transformations====<br />
<br />
There are customarily two modes of understanding a transformation, at least, when we try to interpret its relevance to something in reality. A transformation always refers to a changing prospect, to say it in a unified but equivocal way, but this can be taken to mean either a subjective change in the interpreting observer's point of view or an objective change in the systematic subject of discussion. In practice these variant uses of the transformation concept are distinguished in the following terms:<br />
<br />
# A ''perspectival'' or ''alias'' transformation refers to a shift in perspective or a change in language that takes place in the observer's frame of reference.<br />
# A ''transitional'' or ''alibi'' transformation refers to a change of position or an alteration of state that occurs in the object system as it falls under study.<br />
<br />
(For a recent discussion of the alias vs. alibi issue, as it relates to linear transformations in vector spaces and to other issues of an algebraic nature, see [MaB, 256, 582-4].)<br />
<br />
Naturally, when we are concerned with the dynamical properties of a system, the transitional aspect of transformation is the factor that comes to the fore, and this involves us in contemplating all of the ways of changing a universe into itself while remaining under the rule of established dynamical laws. In the prospective application to dynamic systems, and to neural networks viewed in this light, our interest lies chiefly with the transformations of a state space into itself that constitute the state transitions of a discrete dynamic process. Nevertheless, many important properties of these transformations, and some constructions that we need to see most clearly, are independent of the transitional interpretation and are likely to be confounded with irrelevant features if presented first and only in that association.<br />
<br />
In addition, and in partial contrast, intelligent systems are exactly that species of dynamic agents that have the capacity to have a point of view, and we cannot do justice to their peculiar properties without examining their ability to form and transform their own frames of reference in exposure to the elements of their own experience. In this setting, the perspectival aspect of transformation is the facet that shines most brightly, perhaps too often leaving us fascinated with mere glimmerings of its actual potential. It needs to be emphasized that nothing of the ordinary sort needs be moved in carrying out a transformation under the alias interpretation, that it may only involve a change in the forms of address, an amendment of the terms which are customed to approach and fashioned to describe the very same things in the very same world. But again, working within a discipline of realistic computation, we know how formidably complex and resource-consuming such transformations of perspective can be to implement in practice, much less to endow in the self-governed form of a nascently intelligent dynamical system.<br />
<br />
====Transformations of General Type====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
''Es ist passiert'', &ldquo;it just sort of happened&rdquo;, people said there when other people in other places thought heaven knows what had occurred. It was a peculiar phrase, not known in this sense to the Germans and with no equivalent in other languages, the very breath of it transforming facts and the bludgeonings of fate into something light as eiderdown, as thought itself.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 34]<br />
|}<br />
<br />
Consider the situation illustrated in Figure&nbsp;30, where the alphabets <math>\mathcal{U} = \{ u, v \}\!</math> and <math>\mathcal{X} = \{ x, y, z \}\!</math> are used to label basic features in two different logical universes, <math>U^\bullet = [u, v]\!</math> and <math>X^\bullet = [x, y, z].\!</math><br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
o-------------------------------------------------------o<br />
| U |<br />
| |<br />
| o-----------o o-----------o |<br />
| / \ / \ |<br />
| / o \ |<br />
| / / \ \ |<br />
| / / \ \ |<br />
| o o o o |<br />
| | | | | |<br />
| | u | | v | |<br />
| | | | | |<br />
| o o o o |<br />
| \ \ / / |<br />
| \ \ / / |<br />
| \ o / |<br />
| \ / \ / |<br />
| o-----------o o-----------o |<br />
| |<br />
| |<br />
o---------------------------o---------------------------o<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
o-------------------------o o-------------------------o o-------------------------o<br />
| U | | U | | U |<br />
| o---o o---o | | o---o o---o | | o---o o---o |<br />
| / \ / \ | | / \ / \ | | / \ / \ |<br />
| / o \ | | / o \ | | / o \ |<br />
| / / \ \ | | / / \ \ | | / / \ \ |<br />
| o o o o | | o o o o | | o o o o |<br />
| | u | | v | | | | u | | v | | | | u | | v | |<br />
| o o o o | | o o o o | | o o o o |<br />
| \ \ / / | | \ \ / / | | \ \ / / |<br />
| \ o / | | \ o / | | \ o / |<br />
| \ / \ / | | \ / \ / | | \ / \ / |<br />
| o---o o---o | | o---o o---o | | o---o o---o |<br />
| | | | | |<br />
o-------------------------o o-------------------------o o-------------------------o<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ g | \ f / | h /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ o----------|-----------\-----/-----------|----------o /<br />
\ | X | \ / | | /<br />
\ | | \ / | | /<br />
\ | | o-----o-----o | | /<br />
\| | / \ | |/<br />
\ | / \ | /<br />
|\ | / \ | /|<br />
| \ | / \ | / |<br />
| \ | / \ | / |<br />
| \ | o x o | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \| | | |/ |<br />
| o--o--------o o--------o--o |<br />
| / \ \ / / \ |<br />
| / \ \ / / \ |<br />
| / \ o / \ |<br />
| / \ / \ / \ |<br />
| / \ / \ / \ |<br />
| o o--o-----o--o o |<br />
| | | | | |<br />
| | | | | |<br />
| | | | | |<br />
| | y | | z | |<br />
| | | | | |<br />
| | | | | |<br />
| o o o o |<br />
| \ \ / / |<br />
| \ \ / / |<br />
| \ o / |<br />
| \ / \ / |<br />
| \ / \ / |<br />
| o-----------o o-----------o |<br />
| |<br />
| |<br />
o---------------------------------------------------o<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ p , q /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
o<br />
<br />
Figure 30. Generic Frame of a Logical Transformation<br />
</pre><br />
|}<br />
<br />
Enter the picture, as we usually do, in the middle of things, with features like <math>x, y , z\!</math> that present themselves to be simple enough in their own right and that form a satisfactory, if temporary foundation to provide a basis for discussion. In this universe and on these terms we find expression for various propositions and questions of principal interest to ourselves, as indicated by the maps <math>p, q : X \to \mathbb{B}.\!</math> Then we discover that the simple features <math>\{ x, y, z \}\!</math> are really more complex than we thought at first, and it becomes useful to regard them as functions <math>\{ f, g, h \}\!</math> of other features <math>\{ u, v \}\!</math> that we place in a preface to our original discourse, or suppose as topics of a preliminary universe of discourse <math>U^\bullet = [u, v].\!</math> It may happen that these late-blooming but pre-ambling features are found to lie closer, in a sense that may be our job to determine, to the central nature of the situation of interest, in which case they earn our regard as being more fundamental, but these functions and features are only required to supply a critical stance on the universe of discourse or an alternate perspective on the nature of things in order to be preserved as useful.<br />
<br />
A particular transformation <math>F : [u, v] \to [x, y, z]\!</math> may be expressed by a system of equations, as shown below. Here, <math>F\!</math> is defined by its component maps <math>F = (F_1, F_2, F_3) = (f, g, h),\!</math> where each component map in <math>\{ f, g, h \}\!</math> is a proposition of type <math>\mathbb{B}^n \to \mathbb{B}^1.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:50%"<br />
|<br />
<math>\begin{matrix}<br />
x & = & f(u, v)<br />
\\[10pt]<br />
y & = & g(u, v)<br />
\\[10pt]<br />
z & = & h(u, v)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Regarded as a logical statement, this system of equations expresses a relation between a collection of freely chosen propositions <math>\{ f, g, h \}\!</math> in one universe of discourse and the special collection of simple propositions <math>\{ x, y, z \}\!</math> on which is founded another universe of discourse. Growing familiarity with a particular transformation of discourse, and the desire to achieve a ready understanding of its implications, requires that we be able to convert this information about generals and simples into information about all the main subtypes of propositions, including the linear and singular propositions.<br />
<br />
===Analytic Expansions : Operators and Functors===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Consider what effects that might ''conceivably'' have practical bearings you ''conceive'' the objects of your ''conception'' to have. Then, your ''conception'' of those effects is the whole of your ''conception'' of the object.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; C.S. Peirce, &ldquo;The Maxim of Pragmatism&rdquo;, CP 5.438<br />
|}<br />
<br />
Given the barest idea of a logical transformation, as suggested by the sketch in Figure&nbsp;30, and having conceptualized the universe of discourse, with all of its points and propositions, as a beginning object of discussion, we are ready to enter the next phase of our investigation.<br />
<br />
====Operators on Propositions and Transformations====<br />
<br />
The next step is naturally inclined toward objects of the next higher order, namely, with operators that take in argument lists of logical transformations and that give back specified types of logical transformations as their results. For our present aims, we do not need to consider the most general class of such operators, nor any one of them for its own sake. Rather, we are interested in the special sorts of operators that arise in the study and analysis of logical transformations. Figuratively speaking, these operators serve as instruments for the live tomography (and hopefully not the vivisection) of the forms of change under view. Beyond that, they open up ways to implement the changes of view that we need to grasp all the variations on a transformational theme, or to appreciate enough of its significant features to &ldquo;get the drift&rdquo; of the change occurring, to form a passing acquaintance or a synthetic comprehension of its general character and disposition.<br />
<br />
The simplest type of operator is one that takes a single transformation as an argument and returns a single transformation as a result, and most of the operators explicitly considered in our discussion will be of this kind. Figure&nbsp;31 illustrates the typical situation.<br />
<br />
{| align="center" border="0" cellpadding="20"<br />
|<br />
<pre><br />
o---------------------------------------o<br />
| |<br />
| |<br />
| U% F X% |<br />
| o------------------>o |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| !W! | | !W! |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| v v |<br />
| o------------------>o |<br />
| !W!U% !W!F !W!X% |<br />
| |<br />
| |<br />
o---------------------------------------o<br />
Figure 31. Operator Diagram (1)<br />
</pre><br />
|}<br />
<br />
In this Figure <math>{}^{\backprime\backprime} \mathsf{W} {}^{\prime\prime}\!</math> stands for a generic operator <math>\mathsf{W},\!</math> in this case one that takes a logical transformation <math>F\!</math> of type <math>(U^\bullet \to X^\bullet)\!</math> into a logical transformation <math>\mathsf{W}F\!</math> of type <math>(\mathsf{W}U^\bullet \to \mathsf{W}X^\bullet).\!</math> Thus, the operator <math>\mathsf{W}\!</math> must be viewed as making assignments for both families of objects we have previously considered, that is, for universes of discourse like <math>{U^\bullet}\!</math> and <math>{X^\bullet}\!</math> and for logical transformations like <math>F.\!</math><br />
<br />
'''Note.''' Strictly speaking, an operator like <math>\mathsf{W}\!</math> works between two whole categories of universes and transformations, which we call the ''source'' and the ''target'' categories of <math>\mathsf{W}.\!</math> Given this setting, <math>\mathsf{W}\!</math> specifies for each universe <math>U^\bullet\!</math> in its source category a definite universe <math>\mathsf{W}U^\bullet\!</math> in its target category, and to each transformation <math>F\!</math> in its source category it assigns a unique transformation <math>\mathsf{W}F\!</math> in its target category. Naturally, this only works if <math>\mathsf{W}\!</math> takes the source <math>U^\bullet</math> and the target <math>X^\bullet</math> of the map <math>F\!</math> over to the source <math>\mathsf{W}U^\bullet\!</math> and the target <math>\mathsf{W}X^\bullet\!</math> of the map <math>\mathsf{W}F.\!</math> With luck or care enough, we can avoid ever having to put anything like that in words again, letting diagrams do the work. In the situations of present concern we are usually focused on a single transformation <math>F,\!</math> and thus we can take it for granted that the assignment of universes under <math>\mathsf{W}\!</math> is defined appropriately at the source and target ends of <math>F.\!</math> It is not always the case, though, that we need to use the particular names (like <math>{}^{\backprime\backprime} \mathsf{W}U^\bullet {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \mathsf{W}X^\bullet {}^{\prime\prime}\!</math>) that <math>\mathsf{W}\!</math> assigns by default to its operative image universes. In most contexts we will usually have a prior acquaintance with these universes under other names and it is necessary only that we can tell from the information associated with an operator <math>\mathsf{W}\!</math> what universes they are.<br />
<br />
In Figure&nbsp;31 the maps <math>F\!</math> and <math>\mathsf{W}F\!</math> are displayed horizontally, the way one normally orients functional arrows in a written text, and <math>\mathsf{W}\!</math> rolls the map <math>F\!</math> downward into the images that are associated with <math>\mathsf{W}F.\!</math> In Figure&nbsp;32 the same information is redrawn so that the maps <math>F\!</math> and <math>\mathsf{W}F\!</math> flow down the page, and <math>\mathsf{W}\!</math> unfurls the map <math>F\!</math> rightward into domains that are the eminent purview of <math>\mathsf{W}F.\!</math><br />
<br />
{| align="center" border="0" cellpadding="20"<br />
|<br />
<pre><br />
o---------------------------------------o<br />
| |<br />
| |<br />
| U% !W! !W!U% |<br />
| o------------------>o |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| F | | !W!F |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| v v |<br />
| o------------------>o |<br />
| X% !W! !W!X% |<br />
| |<br />
| |<br />
o---------------------------------------o<br />
Figure 32. Operator Diagram (2)<br />
</pre><br />
|}<br />
<br />
The latter arrangement, as exhibited in Figure&nbsp;32, is more congruent with the thinking about operators that we shall do in the rest of this discussion, since all logical transformations from here on out will be pictured vertically, after the fashion of Figure&nbsp;30.<br />
<br />
====Differential Analysis of Propositions and Transformations====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" | The resultant metaphysical problem now is this: ''Does the man go round the squirrel or not?''<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 43]<br />
|}<br />
<br />
The approach to the differential analysis of logical propositions and transformations of discourse to be pursued here is carried out in terms of particular operators <math>\mathsf{W}\!</math> that act on propositions <math>F\!</math> or on transformations <math>F\!</math> to yield the corresponding operator maps <math>\mathsf{W}F.\!</math> The operator results then become the subject of a series of further stages of analysis, which take them apart into their propositional components, rendering them as a set of purely logical constituents. After this is done, all the parts are then re-integrated to reconstruct the original object in the light of a more complete understanding, at least in ways that enable one to appreciate certain aspects of it with fresh insight.<br />
<br />
* '''Remark on Strategy.''' At this point we run into a set of conceptual difficulties that force us to make a strategic choice in how we proceed. Part of the problem can be remedied by extending our discussion of tacit extensions to the transformational context. But the troubles that remain are much more obstinate and lead us to try two different types of solution. The approach that we develop first makes use of a variant type of extension operator, the ''trope extension'', to be defined below. This method is more conservative and requires less preparation, but has features which make it seem unsatisfactory in the long run. A more radical approach, but one with a better hope of long term success, makes use of the notion of ''contingency spaces''. These are an even more generous type of extended universe than the kind we currently use, but are defined subject to certain internal constraints. The extra work needed to set up this method forces us to put it off to a later stage. However, as a compromise, and to prepare the ground for the next pass, we call attention to the various conceptual difficulties as they arise along the way and try to give an honest estimate of how well our first approach deals with them.<br />
<br />
We now describe in general terms the particular operators that are instrumental to this form of analysis. The main series of operators all have the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathsf{W}<br />
& : &<br />
( U^\bullet \to X^\bullet )<br />
& \to &<br />
( \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet )<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
If we assume that the source universe <math>U^\bullet</math> and the target universe <math>X^\bullet</math> have finite dimensions <math>n\!</math> and <math>k,\!</math> respectively, then each operator <math>\mathsf{W}\!</math> is encompassed by the same abstract type:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathsf{W}<br />
& : &<br />
( [\mathbb{B}^n] \to [\mathbb{B}^k] )<br />
& \to &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k \times \mathbb{D}^k] )<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Since the range features of the operator result <math>\mathsf{W}F : [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k \times \mathbb{D}^k]</math> can be sorted by their ordinary versus differential qualities and the component maps can be examined independently, the complete operator <math>\mathsf{W}\!</math> can be separated accordingly into two components, in the form <math>\mathsf{W} = (\boldsymbol\varepsilon, \mathrm{W}).\!</math> Given a fixed context of source and target universes, <math>\boldsymbol\varepsilon\!</math> is always the same type of operator, a multiple component version of the tacit extension operators that were described earlier. In this context <math>\boldsymbol\varepsilon\!</math> has the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{array}{lccccc}<br />
\text{Concrete type}<br />
& \boldsymbol\varepsilon<br />
& : &<br />
( U^\bullet \to X^\bullet )<br />
& \to &<br />
( \mathrm{E}U^\bullet \to X^\bullet )<br />
\\[10pt]<br />
\text{Abstract type}<br />
& \boldsymbol\varepsilon<br />
& : &<br />
( [\mathbb{B}^n] \to [\mathbb{B}^k] )<br />
& \to &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k] )<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
On the other hand, the operator <math>\mathrm{W}\!</math> is specific to each <math>\mathsf{W}.\!</math> In this context <math>\mathrm{W}\!</math> always has the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{array}{lccccc}<br />
\text{Concrete type}<br />
& W<br />
& : &<br />
( U^\bullet \to X^\bullet )<br />
& \to &<br />
( \mathrm{E}U^\bullet \to \mathrm{d}X^\bullet )<br />
\\[10pt]<br />
\text{Abstract type}<br />
& W<br />
& : &<br />
( [\mathbb{B}^n] \to [\mathbb{B}^k] )<br />
& \to &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{D}^k] )<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
In the types just assigned to <math>\boldsymbol\varepsilon\!</math> and <math>\mathrm{W}\!</math> and by implication to their results <math>\boldsymbol\varepsilon F\!</math> and <math>\mathrm{W}F,\!</math> we have listed the most restrictive ranges defined for them rather than the more expansive target spaces that subsume these ranges. When there is need to recognize both, we may use type indications like the following:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\boldsymbol\varepsilon F<br />
& : &<br />
( \mathrm{E}U^\bullet \to X^\bullet \subseteq \mathrm{E}X^\bullet )<br />
& \cong &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k] \subseteq [\mathbb{B}^k \times \mathbb{D}^k] )<br />
\\[10pt]<br />
WF<br />
& : &<br />
( \mathrm{E}U^\bullet \to \mathrm{d}X^\bullet \subseteq \mathrm{E}X^\bullet )<br />
& \cong &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{D}^k] \subseteq [\mathbb{B}^k \times \mathbb{D}^k] )<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Hopefully, though, a general appreciation of these subsumptions will prevent us from having to make such declarations more often than absolutely necessary.<br />
<br />
In giving names to these operators we try to preserve as much of the traditional nomenclature and as many of the classical associations as possible. The chief difficulty in doing this is occasioned by the distinction between the &ldquo;sans&nbsp;serif&rdquo; operators <math>\mathsf{W}\!</math> and their &ldquo;serified&rdquo; components <math>\mathrm{W},\!</math> which forces us to find two distinct but parallel sets of terminology. Here is a plan to that purpose. First, the component operators <math>\mathrm{W}\!</math> are named by analogy with the corresponding operators in the classical difference calculus. Next, the complete operators <math>\mathsf{W} = (\boldsymbol\varepsilon, \mathrm{W})</math> are assigned titles according to their roles in a geometric or trigonometric allegory, if only to ensure that the tangent functor, that belongs to this family and whose exposition we are still working toward, comes out fit with its customary name. Finally, the operator results <math>\mathsf{W}F\!</math> and <math>\mathrm{W}F\!</math> can be fixed in our frame of reference by tethering the operative adjective for <math>\mathsf{W}\!</math> or <math>\mathrm{W}\!</math> to the anchoring epithet &ldquo;map&rdquo;, in conformity with an already standard practice.<br />
<br />
=====The Secant Operator : '''E'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Mr. Peirce, after pointing out that our beliefs are really rules for action, said that, to develop a thought's meaning, we need only determine what conduct it is fitted to produce: that conduct is for us its sole significance.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 46]<br />
|}<br />
<br />
Figures&nbsp;33-i and 33-ii depict two stages in the form of analysis that will be applied to transformations throughout the remainder of this study. From now on our interest is staked on an operator denoted <math>{}^{\backprime\backprime} \mathsf{E} {}^{\prime\prime},\!</math> which receives the principal investment of analytic attention, and on the constituent parts of <math>\mathsf{E},\!</math> which derive their shares of significance as developed by the analysis. In the sequel, we refer to <math>\mathsf{E}\!</math> as the ''secant operator'', taking it for granted that a context has been chosen that defines its type. The secant operator has the component description <math>\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}),\!</math> and its active ingredient <math>\mathrm{E}\!</math> is known as the ''enlargement operator''. (Here, we name <math>\mathrm{E}\!</math> after the literal ancestor of the shift operator in the calculus of finite differences, defined so that <math>\mathrm{E}f(x) = f(x+1)\!</math> for any suitable function <math>f,\!</math> though of course the logical analogue that we take up here must have a rather different definition.)<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
U% $E$ $E$U% $E$U% $E$U%<br />
o------------------>o============o============o<br />
| | | |<br />
| | | |<br />
| | | |<br />
| | | |<br />
F | | $E$F = | $d$^0.F + | $r$^0.F<br />
| | | |<br />
| | | |<br />
| | | |<br />
v v v v<br />
o------------------>o============o============o<br />
X% $E$ $E$X% $E$X% $E$X%<br />
<br />
Figure 33-i. Analytic Diagram (1)<br />
</pre><br />
|}<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
U% $E$ $E$U% $E$U% $E$U% $E$U%<br />
o------------------>o============o============o============o<br />
| | | | |<br />
| | | | |<br />
| | | | |<br />
| | | | |<br />
F | | $E$F = | $d$^0.F + | $d$^1.F + | $r$^1.F<br />
| | | | |<br />
| | | | |<br />
| | | | |<br />
v v v v v<br />
o------------------>o============o============o============o<br />
X% $E$ $E$X% $E$X% $E$X% $E$X%<br />
<br />
Figure 33-ii. Analytic Diagram (2)<br />
</pre><br />
|}<br />
<br />
In its action on universes <math>\mathsf{E}\!</math> yields the same result as <math>\mathrm{E},\!</math> a fact that can be expressed in equational form by writing <math>\mathsf{E}U^\bullet = \mathrm{E}U^\bullet\!</math> for any universe <math>U^\bullet.\!</math> Notice that the extended universes across the top and bottom of the diagram are indicated to be strictly identical, rather than requiring a corresponding decomposition for them. In a certain sense, the functional parts of <math>\mathsf{E}F\!</math> are partitioned into separate contexts that have to be re-integrated again, but the best image to use is that of making transparent copies of each universe and then overlapping their functional contents once more at the conclusion of the analysis, as suggested by the graphic conventions that are used at the top of Figure&nbsp;30.<br />
<br />
Acting on a transformation <math>F\!</math> from universe <math>U^\bullet\!</math> to universe <math>X^\bullet,\!</math> the operator <math>\mathsf{E}\!</math> determines a transformation <math>\mathsf{E}F\!</math> from <math>\mathsf{E}U^\bullet\!</math> to <math>\mathsf{E}X^\bullet.\!</math> The map <math>\mathsf{E}F\!</math> forms the main body of evidence to be investigated in performing a differential analysis of <math>F.\!</math> Because we shall frequently be focusing on small pieces of this map for considerable lengths of time, and consequently lose sight of the &ldquo;big picture&rdquo;, it is critically important to emphasize that the map <math>\mathsf{E}F\!</math> is a transformation that determines a relation from one extended universe into another. This means that we should not be satisfied with our understanding of a transformation <math>F\!</math> until we can lay out the full &ldquo;parts diagram&rdquo; of <math>\mathsf{E}F\!</math> along the lines of the generic frame in Figure&nbsp;30.<br />
<br />
Working within the confines of propositional calculus, it is possible to give an elementary definition of <math>\mathsf{E}F\!</math> by means of a system of propositional equations, as we now describe.<br />
<br />
Given a transformation<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>F = (F_1, \ldots, F_k) : \mathbb{B}^n \to \mathbb{B}^k\!</math><br />
|}<br />
<br />
of concrete type<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>F : [u_1, \ldots, u_n] \to [x_1, \ldots, x_k],\!</math><br />
|}<br />
<br />
the transformation<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\mathsf{E}F = (F_1, \ldots, F_k, \mathrm{E}F_1, \ldots, \mathrm{E}F_k) : \mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B}^k \times \mathbb{D}^k\!</math><br />
|}<br />
<br />
of concrete type<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\mathsf{E}F : [u_1, \dots, u_n, \mathrm{d}u_1, \dots, \mathrm{d}u_n] \to [x_1, \ldots, x_k, \mathrm{d}x_1, \ldots, \mathrm{d}x_k]\!</math><br />
|}<br />
<br />
is defined by means of the following system of logical equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\\[16pt]<br />
\mathrm{d}x_1<br />
& = & \mathrm{E}F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1 + \mathrm{d}u_1, \ldots, u_n + \mathrm{d}u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
\mathrm{d}x_k<br />
& = & \mathrm{E}F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1 + \mathrm{d}u_1, \ldots, u_n + \mathrm{d}u_n)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
It is important to note that this system of equations can be read as a conjunction of equational propositions, in effect, as a single proposition in the universe of discourse generated by all the named variables. Specifically, this is the universe of discourse over <math>2(n+k)\!</math> variables denoted by:<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
\mathrm{E}[\mathcal{U} \cup \mathcal{X}]<br />
& = &<br />
[u_1, \ldots, u_n, ~ x_1, \ldots, x_k, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n, ~ \mathrm{d}x_1, \ldots, \mathrm{d}x_k].<br />
\end{matrix}</math><br />
|}<br />
<br />
In this light, it should be clear that the system of equations defining <math>\mathsf{E}F\!</math> embodies, in a higher rank and differentially extended version, an analogy with the process of thematization that we treated earlier for propositions of type <math>F : \mathbb{B}^n \to \mathbb{B}.\!</math><br />
<br />
The entire collection of constraints that is represented in the above system of equations may be abbreviated by writing <math>\mathsf{E}F = (\boldsymbol\varepsilon F, \mathrm{E}F),\!</math> for any map <math>F.\!</math> This is tantamount to regarding <math>\mathsf{E}\!</math> as a complex operator, <math>\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}),\!</math> with a form of application that distributes each component of the operator to work on each component of the operand, as follows:<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
\mathsf{E}F<br />
& = &<br />
(\boldsymbol\varepsilon, \mathrm{E})F<br />
& = &<br />
(\boldsymbol\varepsilon F, \mathrm{E}F)<br />
& = &<br />
(\boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k, ~ \mathrm{E}F_1, \ldots, \mathrm{E}F_k).<br />
\end{matrix}</math><br />
|}<br />
<br />
Quite a lot of &ldquo;thematic infrastructure&rdquo; or interpretive information is being swept under the rug in the use of such abbreviations. When confusion arises about the meaning of such constructions, one always has recourse to the defining system of equations, in its totality a purely propositional expression. This means that the parenthesized argument lists, that were used in this context to build an image of multi-component transformations, should not be expected to determine a well-defined product in themselves but only to serve as reminders of the prior thematic decisions (choices of variable names, etc.) that have to be made in order to determine one. Accordingly, the argument list notation can be regarded as a kind of ''thematic frame'', an interpretive storage device that preserves the proper associations of concrete logical features between the extended universes at the source and target of <math>\mathsf{E}F.\!</math><br />
<br />
The generic notations <math>\mathsf{d}^0\!F, \mathsf{d}^1\!F, \ldots, \mathsf{d}^m\!F\!</math> in Figure&nbsp;33 refer to the increasing orders of differentials that are extracted in the course of analyzing <math>F.\!</math> When the analysis is halted at a partial stage of development, notations like <math>\mathsf{r}^0\!F, \mathsf{r}^1\!F, \ldots, \mathsf{r}^m\!F\!</math> may be used to summarize the contributions to <math>\mathsf{E}F\!</math> that remain to be analyzed. The Figure illustrates a convention that makes <math>\mathsf{r}^m\!F,\!</math> in effect, the sum of all differentials of order strictly greater than <math>m.\!</math><br />
<br />
We next discuss the operators that figure into this form of analysis, describing their effects on transformations. In simplified or specialized contexts these operators tend to take on a variety of different names and notations, some of whose number we introduce along the way.<br />
<br />
=====The Radius Operator : '''e'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
And the tangible fact at the root of all our thought-distinctions, however subtle, is that there is no one of them so fine as to consist in anything but a possible difference of practice.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 46]<br />
|}<br />
<br />
The operator identified as <math>\mathrm{d}^0\!</math> in the analytic diagram (Figure&nbsp;33) has the sole purpose of creating a proxy for <math>F\!</math> in the appropriately extended context. Construed in terms of its broadest components, <math>\mathrm{d}^0\!</math> is equivalent to the doubly tacit extension operator <math>(\boldsymbol\varepsilon, \boldsymbol\varepsilon),\!</math> in recognition of which let us redub it as <math>{}^{\backprime\backprime} \mathsf{e} {}^{\prime\prime}.\!</math> Pursuing a geometric analogy, we may refer to <math>\mathsf{e} =(\boldsymbol\varepsilon, \boldsymbol\varepsilon) = \mathrm{d}^0\!</math> as the ''radius operator''. The operation intended by all of these forms is defined by the following equation:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathsf{e}F<br />
& = &<br />
(\boldsymbol\varepsilon, \boldsymbol\varepsilon)F<br />
\\[4pt]<br />
& = &<br />
(\boldsymbol\varepsilon F, ~ \boldsymbol\varepsilon F)<br />
\\[4pt]<br />
& = &<br />
(\boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k, ~ \boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k).<br />
\end{array}</math><br />
|}<br />
<br />
which is tantamount to the system of equations below.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\\[16pt]<br />
\mathrm{d}x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
\mathrm{d}x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
=====The Phantom of the Operators : '''&eta;'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
I was wondering what the reason could be, when I myself raised my head and everything within me seemed drawn towards the Unseen, ''which was playing the most perfect music''!<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 81]<br />
|}<br />
<br />
We now describe an operator whose persistent but elusive action behind the scenes, whose slightly twisted and ambivalent character, and whose fugitive disposition, caught somewhere in flight between the arrantly negative and the positive but errant intent, has cost us some painstaking trouble to detect. In the end we shall place it among the other extensions and projections, as a shade among shadows, of muted tones and motley hue, that adumbrates its own thematic frame and paradoxically lights the way toward a whole new spectrum of values.<br />
<br />
Given a transformation <math>F : [u_1, \ldots, u_n] \to [x_1, \dots, x_k],\!</math> we often have call to consider a family of related transformations, all having the form:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>F^\dagger : [u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n] \to [\mathrm{d}x_1, \dots, \mathrm{d}x_k].\!</math><br />
|}<br />
<br />
The operator <math>\eta</math> is introduced to deal with the simplest one of these maps:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\eta F : [u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n] \to [\mathrm{d}x_1, \ldots \mathrm{d}x_k],\!</math><br />
|}<br />
<br />
which is defined by the following equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
\mathrm{d}x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
In effect, the operator <math>\eta\!</math> is nothing but the stand-alone version of a procedure that is otherwise invoked subordinate to the work of the radius operator <math>\mathsf{e}.\!</math> Operating independently, <math>\eta\!</math> achieves precisely the same results that the second <math>\boldsymbol\varepsilon\!</math> in <math>(\boldsymbol\varepsilon, \boldsymbol\varepsilon)\!</math> accomplishes by working within the context of its ordered pair thematic frame. From this point on, because the use of <math>\boldsymbol\varepsilon\!</math> and <math>\eta\!</math> in this setting combines the aims of both the tacit and the thematic extensions, and because <math>\eta\!</math> reflects in regard to <math>\boldsymbol\varepsilon\!</math> little more than the application of a differential twist, a mere turn of phrase, we refer to <math>\eta\!</math> as the ''trope extension'' operator.<br />
<br />
=====The Chord Operator : '''D'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
What difference would it practically make to any one if this notion rather than that notion were true? If no practical difference whatever can be traced, then the alternatives mean practically the same thing, and all dispute is idle.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 45]<br />
|}<br />
<br />
Next we discuss an operator that is always immanent in this form of analysis, and remains implicitly present in the entire proceeding. It may appear once as a record: a relic or revenant that reprises the reminders of an earlier stage of development. Or it may appear always as a resource: a reserve or redoubt that caches in advance an echo of what remains to be played out, cleared up, and requited in full at a future stage. And all of this remains true whether or not we recall the key at any time, and whether or not the subtending theme is recited explicitly at any stage of play.<br />
<br />
This is the operator that is referred to as <math>\mathsf{r}^0\!</math> in the initial stage of analysis (Figure&nbsp;33-i) and that is expanded as <math>\mathsf{d}^1 + \mathsf{r}^1\!</math> in the subsequent step (Figure&nbsp;33-ii). In congruence, but not quite harmony with our allusions of analogy that are not quite geometry, we call this the ''chord operator'' and denote it <math>\mathsf{D}.\!</math> In the more casual terms that are here introduced, <math>\mathsf{D}</math> is defined as the remainder of <math>\mathsf{E}\!</math> and <math>\mathsf{e}\!</math> and it assigns a due measure to each undertone of accord or discord that is struck between the note of enterprise <math>\mathsf{E}\!</math> and the bar of exigency <math>\mathsf{e}.\!</math><br />
<br />
The tension between these counterposed notions, in balance transient but regular in stridence, may be refracted along familiar lines, though never by any such fraction resolved. In this style we write <math>\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}),\!</math> calling <math>\mathrm{D}\!</math> the ''difference operator'' and noting that it plays a role in this realm of mutable and diverse discourse that is analogous to the part taken by the discrete difference operator in the ordinary difference calculus. Finally, we should note that the chord <math>\mathsf{D}\!</math> is not one that need be lost at any stage of development. At the <math>m^\text{th}\!</math> stage of play it can always be reconstituted in the following form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathsf{D}<br />
& = & \mathsf{E} - \mathsf{e}<br />
\\[6pt]<br />
& = & \mathsf{r}^0<br />
\\[6pt]<br />
& = & \mathsf{d}^1 + \mathsf{r}^1<br />
\\[6pt]<br />
& = & \mathsf{d}^1 + \ldots + \mathsf{d}^m + \mathsf{r}^m<br />
\\[6pt]<br />
& = & \displaystyle \sum_{i=1}^m \mathsf{d}^i + \mathsf{r}^m<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====The Tangent Operator : '''T'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
They take part in scenes of whose significance they have no inkling. They are merely tangent to curves of history the beginnings and ends and forms of which pass wholly beyond their ken. So we are tangent to the wider life of things.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 300]<br />
|}<br />
<br />
The operator tagged as <math>\mathsf{d}^1\!</math> in the analytic diagram (Figure&nbsp;33) is called the ''tangent operator'' and is usually denoted in this text as <math>\mathsf{d}\!</math> or <math>\mathsf{T}.\!</math> Because it has the properties required to qualify as a functor, namely, preserving the identity element of the composition operation and the articulated form of every composition of transformations, it also earns the title of a ''tangent functor''. According to the custom adopted here, we dissect it as <math>\mathsf{T} = \mathsf{d} = (\boldsymbol\varepsilon, \mathrm{d}),\!</math> where <math>\mathrm{d}\!</math> is the operator that yields the first order differential <math>\mathrm{d}F\!</math> when applied to a transformation <math>F,\!</math> and whose name is legion.<br />
<br />
Figure&nbsp;34 illustrates a stage of analysis where we ignore everything but the tangent functor <math>\mathsf{T}\!</math> and attend to it chiefly as it bears on the first order differential <math>\mathrm{d}F\!</math> in the analytic expansion of <math>F.\!</math> In this situation we often refer to the extended universes <math>\mathrm{E}U^\bullet\!</math> and <math>\mathrm{E}X^\bullet\!</math> under the equivalent designations <math>\mathsf{T}U^\bullet\!</math> and <math>\mathsf{T}X^\bullet,\!</math> respectively. The purpose of the tangent functor <math>\mathsf{T}\!</math> is to extract the tangent map <math>\mathsf{T}F\!</math> at each point of <math>U^\bullet,\!</math> and the tangent map <math>\mathsf{T}F = (\boldsymbol\varepsilon, \mathrm{d})F\!</math> tells us not only what the transformation <math>F\!</math> is doing at each point of the universe <math>U^\bullet\!</math> but also what <math>F\!</math> is doing to states in the neighborhood of that point, approximately, linearly, and relatively speaking.<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
U% $T$ $T$U% $T$U%<br />
o------------------>o============o<br />
| | |<br />
| | |<br />
| | |<br />
| | |<br />
F | | $T$F = | <!e!, d> F<br />
| | |<br />
| | |<br />
| | |<br />
v v v<br />
o------------------>o============o<br />
X% $T$ $T$X% $T$X%<br />
<br />
Figure 34. Tangent Functor Diagram<br />
</pre><br />
|}<br />
<br />
* '''NB.''' There is one aspect of the preceding construction that remains especially problematic. Why did we define the operators <math>\mathrm{W}\!</math> in <math>\{ \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}\!</math> so that the ranges of their resulting maps all fall within the realms of differential quality, even fabricating a variant of the tacit extension operator to have that character? Clearly, not all of the operator maps <math>\mathrm{W}F\!</math> have equally good reasons for placing their values in differential stocks. The reason for it appears to be that, without doing this, we cannot justify the comparison and combination of their functional values in the various analytic steps. By default, only those values in the same functional component can be brought into algebraic modes of interaction. Up till now the only mechanism provided for their broader association has been a purely logical one, their common placement in a target universe of discourse, but the task of converting this logical circumstance into algebraic forms of application has not yet been taken up.<br />
<br />
===Transformations of Type '''B'''<sup>2</sup> &rarr; '''B'''<sup>1</sup>===<br />
<br />
To study the effects of these analytic operators in the simplest possible setting, let us revert to a still more primitive case. Consider the singular proposition <math>J(u, v)= u\!\cdot\!v,\!</math> regarded either as the functional product of the maps <math>u\!</math> and <math>v\!</math> or as the logical conjunction of the features <math>u\!</math> and <math>v,\!</math> a map whose fiber of truth <math>J^{-1}(1)\!</math> picks out the single cell of that logical description in the universe of discourse <math>U^\bullet.\!</math> Thus <math>J,\!</math> or <math>u\!\cdot\!v,\!</math> may be treated as another name for the point whose coordinates are <math>(1, 1)\!</math> in <math>U^\bullet.\!</math><br />
<br />
====Analytic Expansion of Conjunction====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
<p>In her sufferings she read a great deal and discovered that she had lost something, the possession of which she had previously not been much aware of: a&nbsp;soul.</p><br />
<br />
<p>What is that? It is easily defined negatively: it is simply what curls up and hides when there is any mention of algebraic series.</p><br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 118]<br />
|}<br />
<br />
Figure&nbsp;35 pictures the form of conjunction <math>J : \mathbb{B}^2 \to \mathbb{B}\!</math> as a transformation from the <math>2\!</math>-dimensional universe <math>[u, v]\!</math> to the <math>1\!</math>-dimensional universe <math>[x].\!</math> This is a subtle but significant change of viewpoint on the proposition, attaching an arbitrary but concrete quality to its functional value. Using the language introduced earlier, we can express this change by saying that the proposition <math>J : \langle u, v \rangle \to \mathbb{B}\!</math> is being recast into the thematized role of a transformation <math>J : [u, v] \to [x],\!</math> where the new variable <math>x\!</math> takes the part of a thematic variable <math>\check{J}.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 35 -- A Conjunction Viewed as a Transformation.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 35.} ~~ \text{Conjunction as Transformation}\!</math><br />
|}<br />
<br />
=====Tacit Extension of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
I teach straying from me, yet who can stray from me?<br><br />
I follow you whoever you are from the present hour;<br><br />
My words itch at your ears till you understand them.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 83]<br />
|}<br />
<br />
Earlier we defined the tacit extension operators <math>\boldsymbol\varepsilon : X^\bullet \to Y^\bullet\!</math> as maps embedding each proposition of a given universe <math>X^\bullet~\!</math> in a more generously given universe <math>Y^\bullet \supset X^\bullet.\!</math> Of immediate interest are the tacit extensions <math>\boldsymbol\varepsilon : U^\bullet \to \mathrm{E}U^\bullet,\!</math> that locate each proposition of <math>U^\bullet\!</math> in the enlarged context of <math>\mathrm{E}U^\bullet.\!</math> In its application to the propositional conjunction <math>J = u\!\cdot\!v</math> in <math>[u, v],\!</math> the tacit extension operator <math>\boldsymbol\varepsilon\!</math> yields the proposition <math>\boldsymbol\varepsilon J\!</math> in <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v].\!</math> The extended proposition <math>\boldsymbol\varepsilon J\!</math> may be computed according to the scheme in Table&nbsp;36, in effect doing nothing more that conjoining a tautology of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> to <math>J\!</math> in <math>U^\bullet.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 36.} ~~ \text{Computation of}~ \boldsymbol\varepsilon J\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\boldsymbol\varepsilon J & = & J {}_{^\langle} u, v {}_{^\rangle}<br />
\\[4pt]<br />
& = & u \cdot v<br />
\\[4pt]<br />
& = & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{ } \mathrm{d}v \texttt{ }<br />
& + & u \cdot v \cdot \texttt{ } \mathrm{d}u \texttt{ } \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \cdot v \cdot \texttt{ } \mathrm{d}u \texttt{ } \cdot \texttt{ } \mathrm{d}v \texttt{ }<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{array}{*{4}{l}}<br />
\boldsymbol\varepsilon J<br />
& = && u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & u \cdot v \cdot \texttt{~} \mathrm{d}u \texttt{~} \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & u \cdot v \cdot \texttt{~} \mathrm{d}u \texttt{~} \cdot \texttt{~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
The lower portion of the Table contains the dispositional features of <math>\boldsymbol\varepsilon J\!</math> arranged in such a way that the variety of ordinary features spreads across the rows and the variety of differential features runs through the columns. This organization serves to facilitate pattern matching in the remainder of our computations. Again, the tacit extension is usually so trivial a concern that we do not always bother to make an explicit note of it, taking it for granted that any function <math>F\!</math> being employed in a differential context is equivalent to <math>\boldsymbol\varepsilon F\!</math> for a suitable <math>\boldsymbol\varepsilon.\!</math><br />
<br />
Figures&nbsp;37-a through 37-d present several pictures of the proposition <math>J\!</math> and its tacit extension <math>\boldsymbol\varepsilon J.\!</math> Notice in these Figures how <math>\boldsymbol\varepsilon J\!</math> in <math>\mathrm{E}U^\bullet\!</math> visibly extends <math>J\!</math> in <math>U^\bullet\!</math> by annexing to the indicated cells of <math>J\!</math> all the arcs that exit from or flow out of them. In effect, this extension attaches to these cells all the dispositions that spring from them, in other words, it attributes to these cells all the conceivable changes that are their issue.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-a -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-a.} ~~ \text{Tacit Extension of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-b -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-b.} ~~ \text{Tacit Extension of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-c -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-c.} ~~ \text{Tacit Extension of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-d -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-d.} ~~ \text{Tacit Extension of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
The computational scheme shown in Table&nbsp;36 treated <math>J\!</math> as a proposition in <math>U^\bullet\!</math> and formed <math>\boldsymbol\varepsilon J\!</math> as a proposition in <math>\mathrm{E}U^\bullet.\!</math> When <math>J\!</math> is regarded as a mapping <math>J : U^\bullet \to X^\bullet\!</math> then <math>\boldsymbol\varepsilon J\!</math> must be obtained as a mapping <math>\boldsymbol\varepsilon J : \mathrm{E}U^\bullet \to X^\bullet.\!</math> By default, the tacit extension of the map <math>J : [u, v] \to [x]\!</math> is naturally taken to be a particular map,<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\boldsymbol\varepsilon J : [u, v, \mathrm{d}u, \mathrm{d}v] \to [x] \subseteq [x, \mathrm{d}x],\!</math><br />
|}<br />
<br />
namely, the one that looks like <math>J\!</math> when painted in the frame of the extended source universe and that takes the same thematic variable in the extended target universe as the one that <math>J\!</math> already takes.<br />
<br />
But the choice of a particular thematic variable, for example <math>x\!</math> for <math>\check{J},\!</math> is a shade more arbitrary than the choice of original variable names <math>\{ u, v \},\!</math> so the map we are calling the ''trope extension'',<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\eta J : [u, v, \mathrm{d}u, \mathrm{d}v] \to [\mathrm{d}x] \subseteq [x, \mathrm{d}x],\!</math><br />
|}<br />
<br />
since it looks just the same as <math>\boldsymbol\varepsilon J\!</math> in the way its fibers paint the source domain, belongs just as fully to the family of tacit extensions, generically considered.<br />
<br />
These considerations have the practical consequence that all of our computations and illustrations of <math>\boldsymbol\varepsilon J\!</math> perform the double duty of capturing <math>\eta J\!</math> as well. In other words, we are saved the work of carrying out calculations and drawing figures for the trope extension <math>\eta J,\!</math> because it would be identical to the work already done for <math>\boldsymbol\varepsilon J.\!</math> Since the computations given for <math>\boldsymbol\varepsilon J\!</math> are expressed solely in terms of the variables <math>\{ u, v, \mathrm{d}u, \mathrm{d}v \},\!</math> they work equally well for finding <math>\eta J.\!</math> Further, since each of the above Figures shows only how the level sets of <math>\boldsymbol\varepsilon J\!</math> partition the extended source universe <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v],\!</math> all of them serve equally well as portraits of <math>\eta J.\!</math><br />
<br />
=====Enlargement Map of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
No one could have established the existence of any details that might not just as well have existed in earlier times too; but all the relations between things had shifted slightly. Ideas that had once been of lean account grew fat.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 62]<br />
|}<br />
<br />
The enlargement map <math>\mathrm{E}J\!</math> is computed from the proposition <math>J\!</math> by making a particular class of formal substitutions for its variables, in this case <math>u + \mathrm{d}u\!</math> for <math>u\!</math> and <math>v + \mathrm{d}v\!</math> for <math>v,\!</math> and afterwards expanding the result in whatever way is found convenient.<br />
<br />
Table&nbsp;38 shows a typical scheme of computation, following a systematic method of exploiting boolean expansions over selected variables and ultimately developing <math>\mathrm{E}J\!</math> over the cells of <math>[u, v].\!</math> The critical step of this procedure uses the facts that <math>\texttt{(} 0, x \texttt{)} = 0 + x = x\!</math> and <math>\texttt{(} 1, x \texttt{)} = 1 + x = \texttt{(} x \texttt{)}\!</math> for any boolean variable <math>x.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 38.} ~~ \text{Computation of}~ \mathrm{E}J ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{E}J & = & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}<br />
\\[4pt]<br />
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
& = & \texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot J_{(1 + \mathrm{d}u, 1 + \mathrm{d}v)}<br />
& + & \texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot J_{(1 + \mathrm{d}u, \mathrm{d}v)}<br />
& + & \texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot J_{(\mathrm{d}u, 1 + \mathrm{d}v)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot J_{(\mathrm{d}u, \mathrm{d}v)}<br />
\\[4pt]<br />
& = &<br />
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot J_{(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})}<br />
& + &<br />
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot J_{(\texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v)}<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot J_{(\mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)})}<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot J_{(\mathrm{d}u, \mathrm{d}v)}<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{E}J<br />
& = &<br />
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&& + &<br />
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{ } \mathrm{d}v \texttt{ }<br />
\\[4pt]<br />
&&&&& + &<br />
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot \texttt{ } \mathrm{d}u \texttt{ } \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&&&&&& + &<br />
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \texttt{ } \mathrm{d}u \texttt{ }~\texttt{ } \mathrm{d}v \texttt{ }<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;39 exhibits another method that happens to work quickly in this particular case, using distributive laws to multiply things out in an algebraic manner, arranging the notations of feature and fluxion according to a scale of simple character and degree. Proceeding this way leads through an intermediate step which, in chiming the changes of ordinary calculus, should take on a familiar ring. Consequential properties of exclusive disjunction then carry us on to the concluding line.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 39.} ~~ \text{Computation of}~ \mathrm{E}J ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{c}}<br />
\mathrm{E}J<br />
& = & (u + \mathrm{d}u) \cdot (v + \mathrm{d}v)<br />
\\[6pt]<br />
& = & u \cdot v<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = &<br />
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{ } \mathrm{d}v \texttt{ }<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot \texttt{ } \mathrm{d}u \texttt{ } \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \texttt{ } \mathrm{d}u \texttt{ }~\texttt{ } \mathrm{d}v \texttt{ }<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;40-a through 40-d present several views of the enlarged proposition <math>\mathrm{E}J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-a -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-a.} ~~ \text{Enlargement of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-b -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-b.} ~~ \text{Enlargement of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-c -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-c.} ~~ \text{Enlargement of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-d -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-d.} ~~ \text{Enlargement of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
An intuitive reading of the proposition <math>\mathrm{E}J\!</math> becomes available at this point. Recall that propositions in the extended universe <math>\mathrm{E}U^\bullet\!</math> express the ''dispositions'' of a system and the constraints that are placed on them. In other words, a differential proposition in <math>\mathrm{E}U^\bullet\!</math> can be read as referring to various changes that a system might undergo in and from its various states. In particular, we can understand <math>\mathrm{E}J\!</math> as a statement that tells us what changes need to be made with regard to each state in the universe of discourse in order to reach the ''truth'' of <math>J,\!</math> that is, the region of the universe where <math>J\!</math> is true. This interpretation is visibly clear in the Figures above and appeals to the imagination in a satisfying way but it has the added benefit of giving fresh meaning to the original name of the shift operator <math>\mathrm{E}.\!</math> Namely, <math>\mathrm{E}J\!</math> can be read as a proposition that ''enlarges'' on the meaning of <math>J,\!</math> in the sense of explaining its practical bearings and clarifying what it means in terms of actions and effects &mdash; the available options for differential action and the consequential effects that result from each choice.<br />
<br />
Read this way, the enlargement <math>\mathrm{E}J\!</math> has strong ties to the normal use of <math>J,\!</math> no matter whether it is understood as a proposition or a function, namely, to act as a figurative device for indicating the models of <math>J,\!</math> in effect, pointing to the interpretive elements in its fiber of truth <math>J^{-1}(1).\!</math> It is this kind of &ldquo;use&rdquo; that is often contrasted with the &ldquo;mention&rdquo; of a proposition, and thereby hangs a tale.<br />
<br />
=====Digression : Reflection on Use and Mention=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Reflection is turning a topic over in various aspects and in various lights so that nothing significant about it shall be overlooked &mdash; almost as one might turn a stone over to see what its hidden side is like or what is covered by it.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 57]<br />
|}<br />
<br />
The contrast drawn in logic between the ''use'' and the ''mention'' of a proposition corresponds to the difference that we observe in functional terms between using <math>{}^{\backprime\backprime} J \, {}^{\prime\prime}\!</math> to indicate the region <math>J^{-1}(1)\!</math> and using <math>{}^{\backprime\backprime} J \, {}^{\prime\prime}\!</math> to indicate the function <math>J.\!</math> You may think that one of these uses ought to be proscribed, and logicians are quick to prescribe against their confusion. But there seems to be no likelihood in practice that their interactions can be avoided. If the name <math>{}^{\backprime\backprime} J \, {}^{\prime\prime}\!</math> is used as a sign of the function <math>J,\!</math> and if the function <math>J\!</math> has its use in signifying something else, as would constantly be the case when some future theory of signs has given a functional meaning to every sign whatsoever, then is not <math>J,\!</math> by transitivity a sign of the thing itself? There are, of course, two answers to this question. Not every act of signifying or referring need be transitive. Not every warrant or guarantee or certificate is automatically transferable, indeed, not many. Not every feature of a feature is a feature of the featuree. Otherwise, if a buffalo is white, and white is a color, then a buffalo would ''be'' a color.<br />
<br />
The logical or pragmatic distinction between use and mention is cogent and necessary, and so is the analogous functional distinction between determining a value and determining what determines that value, but so are the normal techniques that we use to make these distinctions apply flexibly in practice. The way that the hue and cry about use and mention is raised in logical discussions, you might be led to think that this single dimension of choices embraces the only kinds of use worth mentioning and the only kinds of mention worth using. It will constitute the expeditionary and taxonomic tasks of that future theory of signs to explore and to classify the many other constellations and dimensions of use and mention that are yet to be opened up by the generative potential of full-fledged sign relations.<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
The well-known capacity that thoughts have &mdash; as doctors have discovered &mdash; for dissolving and dispersing those hard lumps of deep, ingrowing, morbidly entangled conflict that arise out of gloomy regions of the self probably rests on nothing other than their social and worldly nature, which links the individual being with other people and things; but unfortunately what gives them their power of healing seems to be the same as what diminishes the quality of personal experience in them.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 130]<br />
|}<br />
<br />
=====Difference Map of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
&ldquo;It doesn't matter what one does,&rdquo; the Man Without Qualities said to himself, shrugging his shoulders. &ldquo;In a tangle of forces like this it doesn't make a scrap of difference.&rdquo; He turned away like a man who has learned renunciation, almost indeed like a sick man who shrinks from any intensity of contact. And then, striding through his adjacent dressing-room, he passed a punching-ball that hung there; he gave it a blow far swifter and harder than is usual in moods of resignation or states of weakness.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 8]<br />
|}<br />
<br />
With the tacit extension map <math>\boldsymbol\varepsilon J\!</math> and the enlargement map <math>\mathrm{E}J\!</math> well in place, the difference map <math>\mathrm{D}J\!</math> can be computed along the lines displayed in Table&nbsp;41, ending up with an expansion of <math>\mathrm{D}J\!</math> over the cells of <math>[u, v].\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 41.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & \mathrm{E}J<br />
& + & \boldsymbol\varepsilon J<br />
\\[6pt]<br />
& = & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}<br />
& + & J_{(u, v)}<br />
\\[6pt]<br />
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \cdot v<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = &<br />
u \cdot v \cdot \qquad 0<br />
\\[6pt]<br />
& + &<br />
u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v<br />
& + &<br />
u \cdot \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v<br />
\\[6pt]<br />
& + &<br />
u \cdot v \cdot \texttt{~} \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
&&& + &<br />
\texttt{(} u \texttt{)} \cdot v \cdot \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
& + &<br />
u \cdot v \cdot \texttt{~} \mathrm{d}u \;\cdot\; \mathrm{d}v \texttt{~}<br />
&&&&& + &<br />
\texttt{(} u \texttt{)} \cdot \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = &<br />
u \cdot v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
u \cdot \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v<br />
& + &<br />
\texttt{(} u \texttt{)} \cdot v \cdot \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)} \cdot \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Alternatively, the difference map <math>\mathrm{D}J\!</math> can be expanded over the cells of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> to arrive at the formulation shown in Table&nbsp;42. The same development would be obtained from the previous Table by collecting terms in an alternate manner, along the rows rather than the columns in the middle portion of the Table.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 42.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & \boldsymbol\varepsilon J<br />
& + & \mathrm{E}J<br />
\\[6pt]<br />
& = & J_{(u, v)}<br />
& + & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}<br />
\\[6pt]<br />
& = & u \cdot v<br />
& + & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
& = & 0<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}J<br />
& = & 0<br />
& + & u \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Even more simply, the same result is reached by matching up the propositional coefficients of <math>\boldsymbol\varepsilon J</math> and <math>\mathrm{E}J\!</math> along the cells of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> and adding the pairs under boolean addition, that is, &ldquo;mod 2&rdquo;, where 1&nbsp;+&nbsp;1&nbsp;=&nbsp;0, as shown in Table&nbsp;43.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 43.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 3)}\!</math><br />
|<br />
<math>\begin{array}{*{5}{l}}<br />
\mathrm{D}J & = & \boldsymbol\varepsilon J & + & \mathrm{E}J<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\boldsymbol\varepsilon J<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ u \,\cdot\, v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~ u \;\cdot\; v \;\cdot\; \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u ~ \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} ~ v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot\, \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & ~~ 0 ~~ \,\cdot\, ~ \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~~ u ~ \,\cdot\, ~~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ ~ v ~~ \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
The difference map <math>\mathrm{D}J\!</math> can also be given a ''dispositional'' interpretation. First, recall that <math>\boldsymbol\varepsilon J\!</math> exhibits the dispositions to change from anywhere in <math>J\!</math> to anywhere at all in the universe of discourse and <math>\mathrm{E}J\!</math> exhibits the dispositions to change from anywhere in the universe to anywhere in <math>J.\!</math> Next, observe that each of these classes of dispositions may be divided in accordance with the case of <math>J\!</math> versus <math>\texttt{(} J \texttt{)}\!</math> that applies to their points of departure and destination, as shown below. Then, since the dispositions corresponding to <math>\boldsymbol\varepsilon J</math> and <math>\mathrm{E}J\!</math> have in common the dispositions to preserve <math>J,\!</math> their symmetric difference <math>\texttt{(} \boldsymbol\varepsilon J, \mathrm{E}J \texttt{)}\!</math> is made up of all the remaining dispositions, which are in fact disposed to cross the boundary of <math>J\!</math> in one direction or the other. In other words, we may conclude that <math>\mathrm{D}J\!</math> expresses the collective disposition to make a definite change with respect to <math>J,\!</math> no matter what value it holds in the current state of affairs.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
\boldsymbol\varepsilon J<br />
& = & \{ \text{Dispositions from}~ J ~\text{to}~ J \}<br />
& + & \{ \text{Dispositions from}~ J ~\text{to}~ \texttt{(} J \texttt{)} \}<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = & \{ \text{Dispositions from}~ J ~\text{to}~ J \}<br />
& + & \{ \text{Dispositions from}~ \texttt{(} J \texttt{)} ~\text{to}~ J \}<br />
\\[6pt]<br />
\mathrm{D}J<br />
& = & \{ \text{Dispositions from}~ J ~\text{to}~ \texttt{(} J \texttt{)} \}<br />
& + & \{ \text{Dispositions from}~ \texttt{(} J \texttt{)} ~\text{to}~ J \}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;44-a through 44-d illustrate the difference proposition <math>\mathrm{D}J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-a -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-a.} ~~ \text{Difference Map of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-b -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-b.} ~~ \text{Difference Map of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-c -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-c.} ~~ \text{Difference Map of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-d -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-d.} ~~ \text{Difference Map of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
=====Differential of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
By deploying discourse throughout a calendar, and by giving a date to each of its elements, one does not obtain a definitive hierarchy of precessions and originalities; this hierarchy is never more than relative to the systems of discourse that it sets out to evaluate.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Archaeology of Knowledge'', [Fou, 143]<br />
|}<br />
<br />
Finally, at long last, the differential proposition <math>\mathrm{d}J\!</math> can be gleaned from the difference proposition <math>\mathrm{D}J\!</math> by ranging over the cells of <math>[u, v]\!</math> and picking out the linear proposition of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> that is &ldquo;closest&rdquo; to the portion of <math>\mathrm{D}J\!</math> that touches on each point. The idea of distance that would give this definition unequivocal sense has been referred to in cautionary quotes, the kind we use to distance ourselves from taking a final position. There are obvious notions of approximation that suggest themselves, but finding one that can be justified as ultimately correct is not as straightforward as it seems.<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
He had drifted into the very heart of the world. From him to the distant beloved was as far as to the next tree.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 144]<br />
|}<br />
<br />
Let us venture a guess as to where these developments might be heading. From the present vantage point it appears that the ultimate answer to the quandary of distances and the question of a fitting measure may be that, rather than having the constitution of an analytic series depend on our familiar notions of approach, proximity, and approximation, it will be found preferable, and perhaps unavoidable, to turn the tables and let the orders of approximation be defined in terms of our favored and operative notions of formal analysis. Only the aftermath of this conversion, if it does converge, could be hoped to prove whether this hortatory form of analysis and the cohort idea of an analytic form &mdash; the limitary concept of a self-corrective process and the coefficient concept of a completable product &mdash; are truly (in practical reality) the more inceptive and persistent of principles and really (for all practical purposes) the more effective and regulative of ideas.<br />
<br />
Awaiting that determination, I proceed with what seems like the obvious course, and compute <math>\mathrm{d}J\!</math> according to the pattern in Table&nbsp;45.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 45.} ~~ \text{Computation of}~ \mathrm{d}J\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}J<br />
& = &<br />
u\!\cdot\!v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
u \, \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \, \mathrm{d}v<br />
& + &<br />
\texttt{(} u \texttt{)} \, v \cdot \mathrm{d}u \, \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \!\cdot\! \mathrm{d}v \texttt{~}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}J<br />
& = &<br />
u\!\cdot\!v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + &<br />
u \, \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + &<br />
\texttt{(} u \texttt{)} \, v \cdot \mathrm{d}u<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;46-a through 46-d illustrate the proposition <math>{\mathrm{d}J},\!</math> rounded out in our usual array of prospects. This proposition of <math>\mathrm{E}U^\bullet\!</math> is what we refer to as the (first order) differential of <math>J,\!</math> and normally regard as ''the'' differential proposition corresponding to <math>J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-a -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-a.} ~~ \text{Differential of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-b -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-b.} ~~ \text{Differential of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-c -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-c.} ~~ \text{Differential of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-d -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-d.} ~~ \text{Differential of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
=====Remainder of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
<p>I bequeath myself to the dirt to grow from the grass I love,<br><br />
If you want me again look for me under your bootsoles.</p><br />
<br />
<p>You will hardly know who I am or what I mean,<br><br />
But I shall be good health to you nevertheless,<br><br />
And filter and fibre your blood.</p><br />
<br />
<p>Failing to fetch me at first keep encouraged,<br><br />
Missing me one place search another,<br><br />
I stop some where waiting for you</p><br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]<br />
|}<br />
<br />
<br><br />
<br />
Let us recapitulate the story so far. We have in effect been carrying out a decomposition of the enlarged proposition <math>\mathrm{E}J\!</math> in a series of stages. First, we considered the equation <math>\mathrm{E}J = \boldsymbol\varepsilon J + \mathrm{D}J,\!</math> which was involved in the definition of <math>\mathrm{D}J\!</math> as the difference <math>\mathrm{E}J - \boldsymbol\varepsilon J.\!</math> Next, we contemplated the equation <math>\mathrm{D}J = \mathrm{d}J + \mathrm{r}J,\!</math> which expresses <math>\mathrm{D}J\!</math> in terms of two components, the differential <math>\mathrm{d}J\!</math> that was just extracted and the residual component <math>\mathrm{r}J = \mathrm{D}J - \mathrm{d}J.~\!</math> This remaining proposition <math>\mathrm{r}J\!</math> can be computed as shown in Table&nbsp;47.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 47.} ~~ \text{Computation of}~ \mathrm{r}J\!</math><br />
|<br />
<math>\begin{array}{*{5}{l}}<br />
\mathrm{r}J & = & \mathrm{D}J & + & \mathrm{d}J<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}J<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{r}J ~<br />
& = & u \!\cdot\! v \cdot ~ \mathrm{d}u \cdot \mathrm{d}v ~ ~ ~ ~ ~<br />
& + & u \texttt{(} v \texttt{)} \cdot \, \mathrm{d}u \cdot \mathrm{d}v \,<br />
& + & \texttt{(} u \texttt{)} v \cdot \, \mathrm{d}u \cdot \mathrm{d}v \,<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \, \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
As it happens, the remainder <math>\mathrm{r}J\!</math> falls under the description of a second order differential <math>\mathrm{r}J = \mathrm{d}^2 J.\!</math> This means that the expansion of <math>\mathrm{E}J\!</math> in the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|<br />
<math>\begin{array}{*{7}{l}}<br />
\mathrm{E}J<br />
& = & \boldsymbol\varepsilon J<br />
& + & \mathrm{D}J<br />
\\[6pt]<br />
& = & \boldsymbol\varepsilon J<br />
& + & \mathrm{d}J<br />
& + & \mathrm{r}J<br />
\\[6pt]<br />
& = & \mathrm{d}^0 J<br />
& + & \mathrm{d}^1 J<br />
& + & \mathrm{d}^2 J<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
which is nothing other than the propositional analogue of a Taylor series, is a decomposition that terminates in a finite number of steps.<br />
<br />
Figures&nbsp;48-a through 48-d illustrate the proposition <math>\mathrm{r}J = \mathrm{d}^2 J,\!</math> which forms the remainder map of <math>J\!</math> and also, in this instance, the second order differential of <math>J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-a -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-a.} ~~ \text{Remainder of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-b -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-b.} ~~ \text{Remainder of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-c -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-c.} ~~ \text{Remainder of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-d -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-d.} ~~ \text{Remainder of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
=====Summary of Conjunction=====<br />
<br />
To establish a convenient reference point for further discussion, Table&nbsp;49 summarizes the operator actions that have been computed for the form of conjunction, as exemplified by the proposition <math>J.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 49.} ~~ \text{Computation Summary for}~ J~\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon J<br />
& = & u \!\cdot\! v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}J<br />
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}J<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{r}J<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Analytic Series : Coordinate Method====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
And if he is told that something ''is'' the way it is, then he thinks: Well, it could probably just as easily be some other way. So the sense of possibility might be defined outright as the capacity to think how everything could &ldquo;just as easily&rdquo; be, and to attach no more importance to what is than to what is not.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 12]<br />
|}<br />
<br />
Table&nbsp;50 exhibits a truth table method for computing the analytic series (or the differential expansion) of a proposition in terms of coordinates.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:70%"<br />
|+ style="height:30px" | <math>\text{Table 50.} ~~ \text{Computation of an Analytic Series in Terms of Coordinates}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:8%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:8%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:8%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}u\!</math><br />
| style="width:8%; border-bottom:1px solid black" | <math>\mathrm{d}v\!</math><br />
| style="width:8%; border-bottom:1px solid black; border-left:1px solid black" | <math>u'\!</math><br />
| style="width:8%; border-bottom:1px solid black" | <math>v'\!</math><br />
| style="width:10%; border-bottom:1px solid black; border-left:4px double black" | <math>\boldsymbol\varepsilon J\!</math><br />
| style="width:10%; border-bottom:1px solid black" | <math>\mathrm{E}J\!</math><br />
| style="width:12%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{D}J\!</math><br />
| style="width:10%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}J\!</math><br />
| style="width:10%; border-bottom:1px solid black" | <math>\mathrm{d}^2\!J\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
The first six columns of the Table, taken as a whole, represent the variables of a construct called the ''contingent universe'' <math>[u, v, \mathrm{d}u, \mathrm{d}v, u', v'],\!</math> or the bundle of ''contingency spaces'' <math>[\mathrm{d}u, \mathrm{d}v, u', v']\!</math> over the universe <math>[u, v].\!</math> Their placement to the left of the double bar indicates that all of them amount to independent variables, but there is a co-dependency among them, as described by the following equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
u' & = & u + \mathrm{d}u & = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\\[8pt]<br />
v' & = & v + \mathrm{d}v & = & \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
These relations correspond to the formal substitutions that are made in defining <math>\mathrm{E}J\!</math> and <math>\mathrm{D}J.\!</math> For now, the whole rigamarole of contingency spaces can be regarded as a technical device for achieving the effect of these substitutions, adapted to a setting where functional compositions and other symbolic manipulations are difficult to contemplate and execute.<br />
<br />
The five columns to the right of the double bar in Table&nbsp;50 contain the values of the dependent variables <math>\{ \boldsymbol\varepsilon J, ~\mathrm{E}J, ~\mathrm{D}J, ~\mathrm{d}J, ~\mathrm{d}^2\!J \}.\!</math> These are normally interpreted as values of functions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B}\!</math> or as values of propositions in the extended universe <math>[u, v, \mathrm{d}u, \mathrm{d}v]\!</math> but the dependencies prevailing in the contingent universe make it possible to regard these same final values as arising via functions on alternative lists of arguments, for example, the set <math>\{ u, v, u', v' \}.\!</math><br />
<br />
The column for <math>\boldsymbol\varepsilon J\!</math> is computed as <math>J(u, v) = uv\!</math> and together with the columns for <math>u\!</math> and <math>v\!</math> illustrates how we &ldquo;share structure&rdquo; in the Table by listing only the first entries of each constant block.<br />
<br />
The column for <math>\mathrm{E}J\!</math> is computed by means of the following chain of identities, where the contingent variables <math>u'\!</math> and <math>v'\!</math> are defined as <math>u' = u + \mathrm{d}u\!</math> and <math>v' = v + \mathrm{d}v.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathrm{E}J(u, v, \mathrm{d}u, \mathrm{d}v)<br />
& = &<br />
J(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& = &<br />
J(u', v')<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
This makes it easy to determine <math>\mathrm{E}J\!</math> by inspection, computing the conjunction <math>J(u', v') = u'v'\!</math> from the columns headed <math>u'\!</math> and <math>v'.\!</math> Since each of these forms expresses the same proposition <math>\mathrm{E}J\!</math> in <math>\mathrm{E}U^\bullet,\!</math> the dependence on <math>\mathrm{d}u\!</math> and <math>\mathrm{d}v\!</math> is still present but merely left implicit in the final variant <math>J(u', v').\!</math><br />
<br />
* '''Note.''' On occasion, it is tempting to use the further notation <math>J'(u, v) = J(u', v'),\!</math> especially to suggest a transformation that acts on whole propositions, for example, taking the proposition <math>J\!</math> into the proposition <math>J' = \mathrm{E}J.\!</math> The prime <math>( {}^{\prime} )\!</math> then signifies an action that is mediated by a field of choices, namely, the values that are picked out for the contingent variables in sweeping through the initial universe. But this heaps an unwieldy lot of construed intentions on a rather slight character and puts too high a premium on the constant correctness of its interpretation. In practice, therefore, it is best to avoid this usage.<br />
<br />
Given the values of <math>\boldsymbol\varepsilon J\!</math> and <math>\mathrm{E}J,\!</math> the columns for the remaining functions can be filled in quickly. The difference map is computed according to the relation <math>\mathrm{D}J = \boldsymbol\varepsilon J + \mathrm{E}J.\!</math> The first order differential <math>\mathrm{d}J\!</math> is found by looking in each block of constant argument pairs <math>u, v\!</math> and choosing the linear function of <math>\mathrm{d}u, \mathrm{d}v\!</math> that best approximates <math>\mathrm{D}J\!</math> in that block. Finally, the remainder is computed as <math>\mathrm{r}J = \mathrm{D}J + \mathrm{d}J,\!</math> in this case yielding the second order differential <math>\mathrm{d}^2\!J.\!</math><br />
<br />
====Analytic Series : Recap====<br />
<br />
Let us now summarize the results of Table&nbsp;50 by writing down for each column and for each block of constant argument pairs <math>u, v\!</math> a reasonably canonical symbolic expression for the function of <math>\mathrm{d}u, \mathrm{d}v\!</math> that appears there. The synopsis formed in this way is presented in Table&nbsp;51. As one has a right to expect, it confirms the results that were obtained previously by operating solely in terms of the formal calculus.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:70%"<br />
|+ style="height:30px" | <math>\text{Table 51.} ~~ \text{Computation of an Analytic Series in Symbolic Terms}\!</math><br />
|- style="height:35px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black" | <math>J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{E}J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{D}J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{d}J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{d}^2\!J\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}u \!\;\cdot\;\! \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}u \!\;\cdot\;\! \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
\mathrm{d}u<br />
\\[4pt]<br />
\mathrm{d}v<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;52 and 53 provide a quick overview of the analysis performed so far, giving the successive decompositions of <math>\mathrm{E}J = J + \mathrm{D}J\!</math> and <math>\mathrm{D}J = \mathrm{d}J + \mathrm{r}J\!</math> in two different styles of diagram.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 52 -- Decomposition of EJ.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 52.} ~~ \text{Decomposition of}~ \mathrm{E}J\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 53 -- Decomposition of DJ.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 53.} ~~ \text{Decomposition of}~ \mathrm{D}J\!</math><br />
|}<br />
<br />
====Terminological Interlude====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Lastly, my attention was especially attracted, not so much to the scene, as to the mirrors that produced it. These mirrors were broken in parts. Yes, they were marked and scratched; they had been &ldquo;starred&rdquo;, in spite of their solidity &hellip;<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 230]<br />
|}<br />
<br />
At this point several issues of terminology have accrued enough substance to intrude on our discussion. The remarks of this Subsection are intended to accomplish two goals. First, we call attention to significant aspects of the previous series of Figures, translating into literal terms what they depict in iconic forms, and we re-stress the most important structural elements they indicate. Next, we prepare the way for taking on more complex examples of transformations, those whose target universes have more than one dimension.<br />
<br />
In talking about the actions of operators it is important to keep in mind the distinctions between the operators per&nbsp;se, their operands, and their results. Furthermore, in working with composite forms of operators <math>\mathrm{W} = (\mathrm{W}_1, \ldots, \mathrm{W}_n),\!</math> transformations <math>\mathrm{F} = (\mathrm{F}_1, \ldots, \mathrm{F}_n),\!</math> and target domains <math>X^\bullet = [x_1, \ldots, x_n],\!</math> we need to preserve a clear distinction between the compound entity of each given type and any one of its separate components. It is curious, given the usefulness of the concepts ''operator'' and ''operand'', that we seem to lack a generic term, formed on the same root, for the corresponding result of an operation. Following the obvious paradigm would lead to words like ''opus'', ''opera'', and ''operant'', but these words are too affected with clang associations to work well at present, though they might be adapted in time. One current usage gets around this problem by using the substantive ''map'' as a systematic epithet to express the result of each operator's action. We will follow this practice as far as possible, for example, using the phrase ''tangent map'' to denote the end product of the tangent functor acting on its operand map.<br />
<br />
* '''Scholium.''' See [JGH, 6-9] for a good account of tangent functors and tangent maps in ordinary analysis and for examples of their use in mechanics. This work as a whole is a model of clarity in applying functorial principles to problems in physical dynamics.<br />
<br />
Whenever we focus on isolated propositions, on single components of composite operators, or on the portions of transformations that have <math>1\!</math>-dimensional ranges, we are free to shift between the native form of a proposition <math>J : U \to \mathbb{B}\!</math> and the thematized form of a mapping <math>J : U^\bullet \to [x]\!</math> without much trouble. In these cases we are able to tolerate a higher degree of ambiguity about the precise nature of an operator's input and output domains than we otherwise might. For example, in the preceding treatment of the example <math>J,\!</math> and for each operator <math>\mathrm{W}\!</math> in the set <math>\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \},\!</math> both the operand <math>J\!</math> and the result <math>\mathrm{W}J\!</math> could be viewed in either one of two ways. On one hand we may treat them as propositions <math>J : U \to \mathbb{B}\!</math> and <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B},\!</math> ignoring the distinction between the range <math>[x] \cong \mathbb{B}\!</math> of <math>\boldsymbol\varepsilon J\!</math> and the range <math>[\mathrm{d}x] \cong \mathbb{D}\!</math> of the other types of <math>\mathrm{W}J.\!</math> This is what we usually do when we content ourselves with simply coloring in regions of venn diagrams. On the other hand we may view these entities as maps <math>J : U^\bullet \to [x] = X^\bullet\!</math> and <math>\boldsymbol\varepsilon J : \mathrm{E}U^\bullet \to [x] \subseteq \mathrm{E}X^\bullet\!</math> or <math>\mathrm{W}J : \mathrm{E}U^\bullet \to [\mathrm{d}x] \subseteq \mathrm{E}X^\bullet,\!</math> in which case the qualitative characters of the output features are not ignored.<br />
<br />
At the beginning of this Section we recast the natural form of a proposition <math>J : U \to \mathbb{B}\!</math> into the thematic role of a transformation <math>J : U^\bullet \to [x],\!</math> where <math>x\!</math> was a variable recruited to express the newly independent <math>\check{J}.\!</math> However, in our computations and representations of operator actions we immediately lapsed back to viewing the results as native elements of the extended universe <math>\mathrm{E}U^\bullet,\!</math> in other words, as propositions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B},\!</math> where <math>\mathrm{W}\!</math> ranged over the set <math>\{ \boldsymbol\varepsilon, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}.\!</math> That is as it should be. We have worked hard to devise a language that gives us these advantages &mdash; the flexibility to exchange terms and types of equal information value and the capacity to reflect as quickly and as wittingly as a controlled reflex on the fibers of our propositions, independently of whether they express amusements, beliefs, or conjectures.<br />
<br />
As we take on target spaces of increasing dimension, however, these types of confusions (and confusions of types) become less and less permissible. For this reason, Tables&nbsp;54 and 55 present a rather detailed summary of the notation and the terminology we are using, as applied to the case <math>J = uv.\!</math> The rationale of these Tables is not so much to train more elephant guns on this poor drosophila of a concrete example but to invest our paradigm with enough solidity to bear the weight of abstraction to come.<br />
<br />
Table&nbsp;54 provides basic notation and descriptive information for the objects and operators used in this Example, giving the generic type (or broadest defined type) for each entity. Here, the sans&nbsp;serif operators <math>\mathsf{W} \in \{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{d}, \mathsf{r} \}\!</math> and their components <math>\mathrm{W} \in \{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}\!</math> both have the same broad type <math>\mathsf{W}, \mathrm{W} : (U^\bullet \to X^\bullet) \to (\mathrm{E}U^\bullet \to \mathrm{E}X^\bullet),\!</math> as appropriate to operators that map transformations <math>J : U^\bullet \to X^\bullet\!</math> to extended transformations <math>\mathsf{W}J, \mathrm{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 54.} ~~ \text{Cast of Characters : Expansive Subtypes of Objects and Operators}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| align="center" | <math>\text{Symbol}\!</math><br />
| align="center" | <math>\text{Notation}\!</math><br />
| align="center" | <math>\text{Description}\!</math><br />
| align="center" | <math>\text{Type}\!</math><br />
|-<br />
| align="center" | <math>U^\bullet\!</math><br />
| <math>= [u, v]\!</math><br />
| <math>\text{Source universe}\!</math><br />
| <math>[\mathbb{B}^2]\!</math><br />
|-<br />
| align="center" | <math>X^\bullet~\!</math><br />
| <math>= [x]\!</math><br />
| <math>\text{Target universe}\!</math><br />
| <math>[\mathbb{B}^1]~\!</math><br />
|-<br />
| align="center" | <math>\mathrm{E}U^\bullet\!</math><br />
| <math>= [u, v, \mathrm{d}u, \mathrm{d}v]\!</math><br />
| <math>\text{Extended source universe}\!</math><br />
| <math>[\mathbb{B}^2 \!\times\! \mathbb{D}^2]</math><br />
|-<br />
| align="center" | <math>\mathrm{E}X^\bullet\!</math><br />
| <math>= [x, \mathrm{d}x]~\!</math><br />
| <math>\text{Extended target universe}\!</math><br />
| <math>[\mathbb{B}^1 \!\times\! \mathbb{D}^1]</math><br />
|-<br />
| align="center" | <math>J\!</math><br />
| <math>J : U \!\to\! \mathbb{B}\!</math><br />
| <math>\text{Proposition}\!</math><br />
| <math>(\mathbb{B}^2 \!\to\! \mathbb{B}) \in [\mathbb{B}^2]\!</math><br />
|-<br />
| align="center" | <math>J\!</math><br />
| <math>J : U^\bullet \!\to\! X^\bullet\!</math><br />
| <math>\text{Transformation or Map}\!</math><br />
| <math>[\mathbb{B}^2] \!\to\! [\mathbb{B}^1]\!</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\boldsymbol\varepsilon<br />
\\<br />
\eta<br />
\\<br />
\mathrm{E}<br />
\\<br />
\mathrm{D}<br />
\\<br />
\mathrm{d}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{W} : U^\bullet \!\to\! \mathrm{E}U^\bullet,<br />
\\<br />
\mathrm{W} : X^\bullet \!\to\! \mathrm{E}X^\bullet,<br />
\\<br />
\mathrm{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\\<br />
\text{for each}~ \mathrm{W} ~\text{in the set:}<br />
\\<br />
\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{ll}<br />
\text{Tacit extension operator} & \boldsymbol\varepsilon<br />
\\<br />
\text{Trope extension operator} & \eta<br />
\\<br />
\text{Enlargement operator} & \mathrm{E}<br />
\\<br />
\text{Difference operator} & \mathrm{D}<br />
\\<br />
\text{Differential operator} & \mathrm{d}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^2] \!\to\! [\mathbb{B}^2 \!\times\! \mathbb{D}^2]},<br />
\\<br />
{[\mathbb{B}^1] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]},<br />
\\\\<br />
([\mathbb{B}^2] \!\to\! [\mathbb{B}^1]) \!\to\!<br />
\\<br />
([\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1])<br />
\end{array}</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\mathsf{e}<br />
\\<br />
\mathsf{E}<br />
\\<br />
\mathsf{D}<br />
\\<br />
\mathsf{T}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{W} : U^\bullet \!\to\! \mathsf{T}U^\bullet = \mathrm{E}U^\bullet,<br />
\\<br />
\mathsf{W} : X^\bullet \!\to\! \mathsf{T}X^\bullet = \mathrm{E}X^\bullet,<br />
\\<br />
\mathsf{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathsf{T}U^\bullet \!\to\! \mathsf{T}X^\bullet)<br />
\\<br />
\text{for each}~ \mathsf{W} ~\text{in the set:}<br />
\\<br />
\{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{T} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{lll}<br />
\text{Radius operator} & \mathsf{e} & = (\boldsymbol\varepsilon, \eta)<br />
\\<br />
\text{Secant operator} & \mathsf{E} & = (\boldsymbol\varepsilon, \mathrm{E})<br />
\\<br />
\text{Chord operator} & \mathsf{D} & = (\boldsymbol\varepsilon, \mathrm{D})<br />
\\<br />
\text{Tangent functor} & \mathsf{T} & = (\boldsymbol\varepsilon, \mathrm{d})<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^2] \!\to\! [\mathbb{B}^2 \!\times\! \mathbb{D}^2]},<br />
\\<br />
{[\mathbb{B}^1] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]},<br />
\\\\<br />
([\mathbb{B}^2] \!\to\! [\mathbb{B}^1]) \!\to\!<br />
\\<br />
([\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1])<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;55 supplies a more detailed outline of terminology for operators and their results. Here, we list the restrictive subtype (or narrowest defined subtype) that applies to each entity and we indicate across the span of the Table the whole spectrum of alternative types that color the interpretation of each symbol. For example, all the component operator maps <math>\mathrm{W}J\!</math> have <math>1\!</math>-dimensional ranges, either <math>\mathbb{B}^1\!</math> or <math>\mathbb{D}^1,\!</math> and so they can be viewed either as propositions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B}\!</math> or as logical transformations <math>\mathrm{W}J : \mathrm{E}U^\bullet \to X^\bullet.\!</math> As a rule, the plan of the Table allows us to name each entry by detaching the underlined adjective at the left of its row and prefixing it to the generic noun at the top of its column. In one case, however, it is customary to depart from this scheme. Because the phrase ''differential proposition'', applied to the result <math>\mathrm{d}J : \mathrm{E}U \to \mathbb{D},\!</math> does not distinguish it from the general run of differential propositions <math>\mathrm{G}: \mathrm{E}U \to \mathbb{B},\!</math> it is usual to single out <math>\mathrm{d}J\!</math> as the ''tangent proposition'' of <math>J.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 55.} ~~ \text{Synopsis of Terminology : Restrictive and Alternative Subtypes}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| &nbsp;<br />
| align="center" | <math>\text{Operator}\!</math><br />
| align="center" | <math>\text{Proposition}\!</math><br />
| align="center" | <math>\text{Map}\!</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tacit}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\boldsymbol\varepsilon : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\boldsymbol\varepsilon : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{B} \\<br />
\boldsymbol\varepsilon J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{B}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x] \\<br />
\boldsymbol\varepsilon J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Trope}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\eta : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\eta : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\eta J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\eta J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Enlargement}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{E} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{E}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{E}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Difference}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{D} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{D}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{D}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Differential}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{d} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{d} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{d}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}\!</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Remainder}}\\\text{operator}\end{matrix}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{r} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{r} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{r}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{r}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Radius}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e} = (\boldsymbol\varepsilon, \eta) \\<br />
\mathsf{e} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{e}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Secant}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}) \\<br />
\mathsf{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{E}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Chord}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}) \\<br />
\mathsf{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{D}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tangent}}\\\text{functor}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T} = (\boldsymbol\varepsilon, \mathrm{d}) \\<br />
\mathsf{T} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{T}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====End of Perfunctory Chatter : Time to Roll the Clip!====<br />
<br />
Two steps remain to finish the analysis of <math>J\!</math> that we began so long ago. First, we need to paste our accumulated heap of flat pictures into the frames of transformations, filling out the shapes of the operator maps <math>\mathsf{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet.~\!</math> This scheme is executed in two styles, using the ''areal views'' in Figures&nbsp;56-a and the ''box views'' in Figures&nbsp;56-b. Finally, in Figures&nbsp;57-1 to 57-4 we put all the pieces together to construct the full operator diagrams for <math>\mathsf{W} : J \to \mathsf{W}J.\!</math> There is a considerable amount of redundancy among the following three series of Figures but that will hopefully provide a fuller picture of the operations under review, enabling these snapshots to serve as successive frames in the animation of logic they are meant to become.<br />
<br />
=====Operator Maps : Areal Views=====<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a1 -- Radius Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a1.} ~~ \text{Radius Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a2 -- Secant Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a2.} ~~ \text{Secant Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a3 -- Chord Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a3.} ~~ \text{Chord Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a4 -- Tangent Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a4.} ~~ \text{Tangent Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
=====Operator Maps : Box Views=====<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b1 -- Radius Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b1.} ~~ \text{Radius Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b2 -- Secant Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b2.} ~~ \text{Secant Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b3 -- Chord Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b3.} ~~ \text{Chord Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b4 -- Tangent Map of J ISW.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b4.} ~~ \text{Tangent Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
=====Operator Diagrams for the Conjunction J = uv=====<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-1 -- Radius Operator Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-1.} ~~ \text{Radius Operator Diagram for the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-2 -- Secant Operator Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-2.} ~~ \text{Secant Operator Diagram for the Conjunction}~ J = uv~\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-3 -- Chord Operator Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-3.} ~~ \text{Chord Operator Diagram for the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-4 -- Tangent Functor Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-4.} ~~ \text{Tangent Functor Diagram for the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
===Taking Aim at Higher Dimensional Targets===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
The past and present wilt . . . . I have filled them and<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;emptied them,<br><br />
And proceed to fill my next fold of the future.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 87]<br />
|}<br />
<br />
In the next Section we consider a transformation <math>F\!</math> of concrete type <math>F : [u, v] \to [x, y]\!</math> and abstract type <math>F : [\mathbb{B}^2] \to [\mathbb{B}^2].\!</math> From the standpoint of propositional calculus we naturally approach the task of understanding such a transformation by parsing it into component maps with <math>1\!</math>-dimensional ranges, as follows:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{ccccccl}<br />
F & = & (F_1, F_2) & = & (f, g) & : & [u, v] \to [x, y],<br />
\\[6pt]<br />
&& F_1 & = & f & : & [u, v] \to [x],<br />
\\[6pt]<br />
&& F_2 & = & g & : & [u, v] \to [y].<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Then we tackle the separate components, now viewed as propositions <math>F_i : U \to \mathbb{B},\!</math> one at a time. At the completion of this analytic phase, we return to the task of synthesizing these partial and transient impressions into an agile form of integrity, a solidly coordinated and deeply integrated comprehension of the ongoing transformation. (Very often, of course, in tangling with refractory cases, we never get as far as the beginning again.)<br />
<br />
Let us now refer to the dimension of the target space or codomain as the ''toll'' (or ''tole'') of a transformation, as distinguished from the dimension of the range or image that is customarily called the ''rank''. When we keep to transformations with a toll of <math>1,\!</math> as <math>J : [u, v] \to [x],\!</math> we tend to get lazy about distinguishing a logical transformation from its component propositions. However, if we deal with transformations of a higher toll, this form of indolence can no longer be tolerated.<br />
<br />
Well, perhaps we can carry it a little further. After all, the operator result <math>\mathrm{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet\!</math> is a map of toll <math>2,\!</math> and cannot be unfolded in one piece as a proposition. But when a map has rank <math>1,\!</math> like <math>\boldsymbol\varepsilon J : \mathrm{E}U \to X \subseteq \mathrm{E}X\!</math> or <math>\mathrm{d}J : \mathrm{E}U \to \mathrm{d}X \subseteq \mathrm{E}X,\!</math> we naturally choose to concentrate on the <math>1\!</math>-dimensional range of the operator result <math>\mathrm{W}J,\!</math> ignoring the final difference in quality between the spaces <math>X\!</math> and <math>\mathrm{d}X,\!</math> and view <math>\mathrm{W}J\!</math> as a proposition about <math>\mathrm{E}U.\!</math><br />
<br />
In this way, an initial ambivalence about the role of the operand <math>J\!</math> conveys a double duty to the result <math>\mathrm{W}J.\!</math> The pivot that is formed by our focus of attention is essential to the linkage that transfers this double moment, as the whole process takes its bearing and wheels around the precise measure of a narrow bead that we can draw on the range of <math>\mathrm{W}J.\!</math> This is the escapement that it takes to get away with what may otherwise seem to be a simple duplicity, and this is the tolerance that is needed to counterbalance a certain arrogance of equivocation, by all of which machinations we make ourselves free to indicate the operator results <math>\mathrm{W}J\!</math> as propositions or as transformations, indifferently.<br />
<br />
But that's it, and no further. Neglect of these distinctions in range and target universes of higher dimensions is bound to cause a hopeless confusion. To guard against these adverse prospects, Tables&nbsp;58 and 59 lay the groundwork for discussing a typical map <math>F : [\mathbb{B}^2] \to [\mathbb{B}^2],\!</math> and begin to pave the way to some extent for discussing any transformation of the form <math>F : [\mathbb{B}^n] \to [\mathbb{B}^k].\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 58.} ~~ \text{Cast of Characters : Expansive Subtypes of Objects and Operators}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| align="center" | <math>\text{Symbol}\!</math><br />
| align="center" | <math>\text{Notation}\!</math><br />
| align="center" | <math>\text{Description}\!</math><br />
| align="center" | <math>\text{Type}\!</math><br />
|-<br />
| align="center" | <math>U^\bullet\!</math><br />
| <math>= [u, v]\!</math><br />
| <math>\text{Source universe}\!</math><br />
| <math>[\mathbb{B}^n]\!</math><br />
|-<br />
| align="center" | <math>X^\bullet~\!</math><br />
| <math>\begin{array}{l}<br />
= [x, y] \\<br />
= [f, g]<br />
\end{array}</math><br />
| <math>\text{Target universe}\!</math><br />
| <math>[\mathbb{B}^k]\!</math><br />
|-<br />
| align="center" | <math>\mathrm{E}U^\bullet\!</math><br />
| <math>= [u, v, \mathrm{d}u, \mathrm{d}v]\!</math><br />
| <math>\text{Extended source universe}\!</math><br />
| <math>[\mathbb{B}^n \!\times\! \mathbb{D}^n]\!</math><br />
|-<br />
| align="center" | <math>\mathrm{E}X^\bullet\!</math><br />
| <math>\begin{array}{l}<br />
= [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
= [f, g, \mathrm{d}f, \mathrm{d}g]<br />
\end{array}</math><br />
| <math>\text{Extended target universe}\!</math><br />
| <math>[\mathbb{B}^k \!\times\! \mathbb{D}^k]\!</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
f \\ g<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{ll}<br />
f : U \!\to\! [x] \cong \mathbb{B} \\<br />
g : U \!\to\! [y] \cong \mathbb{B}<br />
\end{array}</math><br />
| <math>\text{Proposition}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathbb{B}^n \!\to\! \mathbb{B} \\<br />
\in (\mathbb{B}^n, \mathbb{B}^n \!\to\! \mathbb{B}) = [\mathbb{B}^n]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>F\!</math><br />
| <math>F = (f, g) : U^\bullet \!\to\! X^\bullet\!</math><br />
| <math>\text{Transformation of Map}\!</math><br />
| <math>[\mathbb{B}^n] \!\to\! [\mathbb{B}^k]</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\boldsymbol\varepsilon<br />
\\<br />
\eta<br />
\\<br />
\mathrm{E}<br />
\\<br />
\mathrm{D}<br />
\\<br />
\mathrm{d}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{W} : U^\bullet \!\to\! \mathrm{E}U^\bullet,<br />
\\<br />
\mathrm{W} : X^\bullet \!\to\! \mathrm{E}X^\bullet,<br />
\\<br />
\mathrm{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\\<br />
\text{for each}~ \mathrm{W} ~\text{in the set:}<br />
\\<br />
\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{ll}<br />
\text{Tacit extension operator} & \boldsymbol\varepsilon<br />
\\<br />
\text{Trope extension operator} & \eta<br />
\\<br />
\text{Enlargement operator} & \mathrm{E}<br />
\\<br />
\text{Difference operator} & \mathrm{D}<br />
\\<br />
\text{Differential operator} & \mathrm{d}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^n] \!\to\! [\mathbb{B}^n \!\times\! \mathbb{D}^n]},<br />
\\<br />
{[\mathbb{B}^k] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]},<br />
\\\\<br />
([\mathbb{B}^n] \!\to\! [\mathbb{B}^k]) \!\to\!<br />
\\<br />
([\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k])<br />
\end{array}</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\mathsf{e}<br />
\\<br />
\mathsf{E}<br />
\\<br />
\mathsf{D}<br />
\\<br />
\mathsf{T}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{W} : U^\bullet \!\to\! \mathsf{T}U^\bullet = \mathrm{E}U^\bullet,<br />
\\<br />
\mathsf{W} : X^\bullet \!\to\! \mathsf{T}X^\bullet = \mathrm{E}X^\bullet,<br />
\\<br />
\mathsf{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathsf{T}U^\bullet \!\to\! \mathsf{T}X^\bullet)<br />
\\<br />
\text{for each}~ \mathsf{W} ~\text{in the set:}<br />
\\<br />
\{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{T} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{lll}<br />
\text{Radius operator} & \mathsf{e} & = (\boldsymbol\varepsilon, \eta)<br />
\\<br />
\text{Secant operator} & \mathsf{E} & = (\boldsymbol\varepsilon, \mathrm{E})<br />
\\<br />
\text{Chord operator} & \mathsf{D} & = (\boldsymbol\varepsilon, \mathrm{D})<br />
\\<br />
\text{Tangent functor} & \mathsf{T} & = (\boldsymbol\varepsilon, \mathrm{d})<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^n] \!\to\! [\mathbb{B}^n \!\times\! \mathbb{D}^n]},<br />
\\<br />
{[\mathbb{B}^k] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]},<br />
\\\\<br />
([\mathbb{B}^n] \!\to\! [\mathbb{B}^k]) \!\to\!<br />
\\<br />
([\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k])<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 59.} ~~ \text{Synopsis of Terminology : Restrictive and Alternative Subtypes}~\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| &nbsp;<br />
| align="center" | <math>\begin{matrix}\text{Operator}\\\text{or}\\\text{Operand}\end{matrix}</math><br />
| align="center" | <math>\begin{matrix}\text{Proposition}\\\text{or}\\\text{Component}\end{matrix}</math><br />
| align="center" | <math>\begin{matrix}\text{Transformation}\\\text{or}\\\text{Map}\end{matrix}</math><br />
|-<br />
| align="center" | <math>\underline{\text{Operand}}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
F = (F_1, F_2) \\<br />
F = (f, g) : U \!\to\! X<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
F_i : \langle u, v \rangle \!\to\! \mathbb{B} \\<br />
F_i : \mathbb{B}^n \!\to\! \mathbb{B}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
F : [u, v] \!\to\! [x, y] \\<br />
F : [\mathbb{B}^n] \!\to\! [\mathbb{B}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tacit}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\boldsymbol\varepsilon : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\boldsymbol\varepsilon : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{B} \\<br />
\boldsymbol\varepsilon F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{B}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y] \\<br />
\boldsymbol\varepsilon F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Trope}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\eta : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\eta : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\eta F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\eta F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Enlargement}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{E} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{E}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{E}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Difference}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{D} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{D}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{D}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Differential}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{d} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{d} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{d}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}\!</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Remainder}}\\\text{operator}\end{matrix}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{r} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{r} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{r}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{r}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Radius}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e} = (\boldsymbol\varepsilon, \eta) \\<br />
\mathsf{e} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{e}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Secant}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}) \\<br />
\mathsf{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{E}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Chord}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}) \\<br />
\mathsf{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{D}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tangent}}\\\text{functor}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T} = (\boldsymbol\varepsilon, \mathrm{d}) \\<br />
\mathsf{T} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{T}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
===Transformations of Type '''B'''<sup>2</sup> &rarr; '''B'''<sup>2</sup>===<br />
<br />
To take up a slightly more complex example, but one that remains simple enough to pursue through a complete series of developments, consider the transformation from <math>U^\bullet = [u, v]\!</math> to <math>X^\bullet = [x, y]\!</math> that is defined by the following system of equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
x<br />
& = & f(u, v)<br />
& = & \texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[8pt]<br />
y<br />
& = & g(u, v)<br />
& = & \texttt{((} u \texttt{,} v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
The component notation <math>F = (F_1, F_2) = (f, g) : U^\bullet \to X^\bullet\!</math> allows us to give a name and a type to this transformation and permits defining it by the compact description that follows:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
(x, y)<br />
& = & F(u, v)<br />
& = & (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Logical Transformations====<br />
<br />
The information that defines the logical transformation <math>F\!</math> can be represented in the form of a truth table, as shown in Table&nbsp;60. To cut down on subscripts in this example we continue to use plain letter equivalents for all components of spaces and maps.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:60%"<br />
|+ style="height:30px" | <math>\text{Table 60.} ~~ \text{A Propositional Transformation}\!</math><br />
|- style="height:40px; background:ghostwhite; width:100%"<br />
| style="width:25%" | <math>u\!</math><br />
| style="width:25%" | <math>v\!</math><br />
| style="width:25%; border-left:1px solid black" | <math>f\!</math><br />
| style="width:25%" | <math>g\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="border-top:1px solid black" | <math>u\!</math><br />
| style="border-top:1px solid black" | <math>v\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;61 shows how we might paint a picture of the transformation <math>F\!</math> in the manner of Figure&nbsp;30.<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 61 -- Propositional Transformation.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 61.} ~~ \text{A Propositional Transformation}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;62 extracts the gist of Figure&nbsp;61, exhibiting a style of diagram that is adequate for most purposes.<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 62 -- Propositional Transformation (Short Form).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 62.} ~~ \text{A Propositional Transformation (Short Form)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Local Transformations====<br />
<br />
Figure&nbsp;63 gives a more complete picture of the transformation <math>F,\!</math> showing how the points of <math>U^\bullet\!</math> are transformed into points of <math>X^\bullet.\!</math> The bold lines crossing from one universe to the other trace the action that <math>F\!</math> induces on points, in other words, they show how the transformation acts as a mapping from points to points and chart its effects on the elements that are variously called cells, points, positions, or singular propositions.<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 63 -- Transformation of Positions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 63.} ~~ \text{A Transformation of Positions}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;64 shows how the action of <math>F\!</math> on cells or points can be computed in terms of coordinates.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 64.} ~~ \text{A Transformation of Positions}\!</math><br />
|- style="height:40px; background:ghostwhite; width:100%"<br />
| style="width:8%" | <math>u\!</math><br />
| style="width:8%" | <math>v\!</math><br />
| style="width:12%; border-left:1px solid black" | <math>x\!</math><br />
| style="width:12%" | <math>y\!</math><br />
| style="width:10%; border-left:1px solid black" | <math>x~y\!</math><br />
| style="width:10%" | <math>x \texttt{(} y \texttt{)}\!</math><br />
| style="width:10%" | <math>\texttt{(} x \texttt{)} y\!</math><br />
| style="width:10%" | <math>\texttt{(} x \texttt{)(} y \texttt{)}\!</math><br />
| style="width:20%; border-left:1px solid black" | <math>X^\bullet = [x, y]\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\uparrow<br />
\\[4pt]<br />
F =<br />
\\[4pt]<br />
(f, g)<br />
\\[4pt]<br />
\uparrow<br />
\end{matrix}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="border-top:1px solid black" | <math>u\!</math><br />
| style="border-top:1px solid black" | <math>v\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
| style="border-top:1px solid black" | <math>\texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>u~v\!</math><br />
| style="border-top:1px solid black" | <math>\texttt{(} u \texttt{,} v \texttt{)}\!</math><br />
| style="border-top:1px solid black" | <math>\texttt{(} u \texttt{)(} v \texttt{)}\!</math><br />
| style="border-top:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>U^\bullet = [u, v]\!</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;65 extends this scheme from single cells to arbitrary regions, showing how we might compute the action of a logical transformation on arbitrary propositions in the universe of discourse. The effect of a point-transformation on arbitrary propositions, or any other structures erected on points, is referred to as the ''induced action'' of the transformation on the structures in question.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 65-a.} ~~ \text{An Induced Transformation on Propositions}\!</math><br />
|- style="height:50px; background:ghostwhite"<br />
| style="width:20%" | <math>X^\bullet~\!</math><br />
| style="width:20%; border-right:none" | <math>\longleftarrow\!</math><br />
| style="width:20%; border-left:none; border-right:none" | <math>F = (f, g)\!</math><br />
| style="width:20%; border-left:none" | <math>\longleftarrow\!</math><br />
| style="width:20%" | <math>U^\bullet~\!</math><br />
|- style="background:ghostwhite"<br />
| rowspan="2" | <math>f_i (x, y)\!</math><br />
| align="right" | <math>\begin{matrix}u = \\ v =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~0~0\\1~0~1~0\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= u \\ = v\end{matrix}</math><br />
| rowspan="2" style="width:20%" | <math>f_j (u, v)\!</math><br />
|- style="background:ghostwhite"<br />
| align="right" | <math>\begin{matrix}x = \\ y =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~1~0\\1~0~0~1\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= f(u, v) \\ = g(u, v)\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{2}<br />
\\[2pt]<br />
f_{3}<br />
\\[2pt]<br />
f_{4}<br />
\\[2pt]<br />
f_{5}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{7}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)~ ~}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{~ ~(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{,~} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{~~} y \texttt{)}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~0<br />
\\[2pt]<br />
0~0~0~0<br />
\\[2pt]<br />
0~0~0~1<br />
\\[2pt]<br />
0~0~0~1<br />
\\[2pt]<br />
0~1~1~0<br />
\\[2pt]<br />
0~1~1~0<br />
\\[2pt]<br />
0~1~1~1<br />
\\[2pt]<br />
0~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{,~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{,~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{~~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{~~} v \texttt{)}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[2pt]<br />
f_{0}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{7}<br />
\\[2pt]<br />
f_{7}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{8}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{10}<br />
\\[2pt]<br />
f_{11}<br />
\\[2pt]<br />
f_{12}<br />
\\[2pt]<br />
f_{13}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{15}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[2pt]<br />
\texttt{~ ~ ~ ~} y \texttt{~~}<br />
\\[2pt]<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[2pt]<br />
\texttt{~~} x \texttt{~ ~ ~ ~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
1~0~0~0<br />
\\[2pt]<br />
1~0~0~0<br />
\\[2pt]<br />
1~0~0~1<br />
\\[2pt]<br />
1~0~0~1<br />
\\[2pt]<br />
1~1~1~0<br />
\\[2pt]<br />
1~1~1~0<br />
\\[2pt]<br />
1~1~1~1<br />
\\[2pt]<br />
1~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~~} u \texttt{~~} v \texttt{~~}<br />
\\[2pt]<br />
\texttt{~~} u \texttt{~~} v \texttt{~~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{8}<br />
\\[2pt]<br />
f_{8}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{15}<br />
\\[2pt]<br />
f_{15}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 65-b.} ~~ \text{An Induced Transformation on Propositions}\!</math><br />
|- style="height:50px; background:ghostwhite"<br />
| style="width:20%" | <math>X^\bullet~\!</math><br />
| style="width:20%; border-right:none" | <math>\longleftarrow\!</math><br />
| style="width:20%; border-left:none; border-right:none" | <math>F = (f, g)\!</math><br />
| style="width:20%; border-left:none" | <math>\longleftarrow\!</math><br />
| style="width:20%" | <math>U^\bullet~\!</math><br />
|- style="background:ghostwhite"<br />
| rowspan="2" | <math>f_i (x, y)\!</math><br />
| align="right" | <math>\begin{matrix}u = \\ v =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~0~0\\1~0~1~0\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= u \\ = v\end{matrix}</math><br />
| rowspan="2" style="width:20%" | <math>f_j (u, v)\!</math><br />
|- style="background:ghostwhite"<br />
| align="right" | <math>\begin{matrix}x = \\ y =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~1~0\\1~0~0~1\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= f(u, v) \\ = g(u, v)\end{matrix}</math><br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>f_{0}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[2pt]<br />
f_{2}<br />
\\[2pt]<br />
f_{4}<br />
\\[2pt]<br />
f_{8}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~0<br />
\\[2pt]<br />
0~0~0~1<br />
\\[2pt]<br />
0~1~1~0<br />
\\[2pt]<br />
1~0~0~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{,~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{~} u \texttt{~~} v \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{8}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{3}<br />
\\[2pt]<br />
f_{12}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[2pt]<br />
1~1~1~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} u \texttt{)(} v \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[2pt]<br />
f_{14}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[2pt]<br />
f_{9}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[2pt]<br />
1~0~0~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{~~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{~} u \texttt{~~} v \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[2pt]<br />
f_{8}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{5}<br />
\\[2pt]<br />
f_{10}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[2pt]<br />
1~0~0~1<br />
\end{matrix}~\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} u \texttt{,~} v \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[2pt]<br />
f_{9}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[2pt]<br />
f_{11}<br />
\\[2pt]<br />
f_{13}<br />
\\[2pt]<br />
f_{14}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[2pt]<br />
1~0~0~1<br />
\\[2pt]<br />
1~1~1~0<br />
\\[2pt]<br />
1~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} u \texttt{~~} v \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{15}<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>f_{15}\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Difference Operators and Tangent Functors====<br />
<br />
Given the alphabets <math>\mathcal{U} = \{ u, v \}\!</math> and <math>\mathcal{X} = \{ x, y \},\!</math> along with the corresponding universes of discourse <math>U^\bullet, X^\bullet \cong [\mathbb{B}^2],\!</math> how many logical transformations of the general form <math>G = (G_1, G_2) : U^\bullet \to X^\bullet\!</math> are there? Since <math>G_1\!</math> and <math>G_2\!</math> can be any propositions of the type <math>\mathbb{B}^2 \to \mathbb{B},\!</math> there are <math>2^4 = 16\!</math> choices for each of the maps <math>G_1\!</math> and <math>G_2\!</math> and thus there are <math>2^4 \cdot 2^4 = 2^8 = 256\!</math> different mappings altogether of the form <math>G : U^\bullet \to X^\bullet.\!</math> The set of functions of a given type is denoted by placing its type indicator in parentheses, in the present instance writing <math>(U^\bullet \to X^\bullet) = \{ G : U^\bullet \to X^\bullet \},\!</math> and so the cardinality of the ''function space'' <math>(U^\bullet \to X^\bullet)\!</math> is summed up by writing <math>|(U^\bullet \to X^\bullet)| = |(\mathbb{B}^2 \to \mathbb{B}^2)| = 4^4 = 256.\!</math><br />
<br />
Given a transformation <math>G = (G_1, G_2) : U^\bullet \to X^\bullet\!</math> of this type, we proceed to define a pair of further transformations, related to <math>G,\!</math> that operate between the extended universes, <math>\mathrm{E}U^\bullet\!</math> and <math>\mathrm{E}X^\bullet,\!</math> of its source and target domains.<br />
<br />
First, the ''enlargement map'' (or ''secant transformation'') <math>\mathrm{E}G = (\mathrm{E}G_1, \mathrm{E}G_2) : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet\!</math> is defined by the following set of component equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}G_i<br />
& = & G_i (u + \mathrm{d}u, v + \mathrm{d}v)<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Next, the ''difference map'' (or ''chordal transformation'') <math>\mathrm{D}G = (\mathrm{D}G_1, \mathrm{D}G_2) : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet~\!</math> is defined in component-wise fashion as the boolean sum of the initial proposition <math>G_i\!</math> and the enlarged proposition <math>\mathrm{E}G_i,\!</math> for <math>i = 1, 2,\!</math> according to the following set of equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
\mathrm{D}G_i<br />
& = & G_i (u, v)<br />
& + & \mathrm{E}G_i (u, v, \mathrm{d}u, \mathrm{d}v)<br />
\\[8pt]<br />
& = & G_i (u, v)<br />
& + & G_i (u + \mathrm{d}u, v + \mathrm{d}v)<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Maintaining a strict analogy with ordinary difference calculus would perhaps have us write <math>\mathrm{D}G_i = \mathrm{E}G_i - G_i,\!</math> but the sum and difference operations are the same thing in boolean arithmetic. It is more often natural in the logical context to consider an initial proposition <math>q,\!</math> then to compute the enlargement <math>\mathrm{E}q,\!</math> and finally to determine the difference <math>\mathrm{D}q = q + \mathrm{E}q,\!</math> so we let the variant order of terms reflect this sequence of considerations.<br />
<br />
Viewed in this light the difference operator <math>\mathrm{D}\!</math> is imagined to be a function of very wide scope and polymorphic application, one that is able to realize the association between each transformation <math>G\!</math> and its difference map <math>\mathrm{D}G,\!</math> for example, taking the function space <math>(U^\bullet \to X^\bullet)\!</math> into <math>(\mathrm{E}U^\bullet \to \mathrm{E}X^\bullet).\!</math> When we consider the variety of interpretations permitted to propositions over the contexts in which we put them to use, it should be clear that an operator of this scope is not at all a trivial matter to define in general and that it may take some trouble to work out. For the moment we content ourselves with returning to particular cases.<br />
<br />
Acting on the logical transformation <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;),\!</math> the operators <math>\mathrm{E}\!</math> and <math>\mathrm{D}\!</math> yield the enlarged map <math>\mathrm{E}F = (\mathrm{E}f, \mathrm{E}g)\!</math> and the difference map <math>\mathrm{D}F = (\mathrm{D}f, \mathrm{D}g),\!</math> respectively, whose components are given as follows.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}f<br />
& = & \texttt{((} u + \mathrm{d}u \texttt{)(} v + \mathrm{d}v \texttt{))}<br />
\\[8pt]<br />
\mathrm{E}g<br />
& = & \texttt{((} u + \mathrm{d}u \texttt{,~} v + \mathrm{d}v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
\mathrm{D}f<br />
& = & \texttt{((} u \texttt{)(} v \texttt{))}<br />
& + & \texttt{((} u + \mathrm{d}u \texttt{)(} v + \mathrm{d}v \texttt{))}<br />
\\[8pt]<br />
\mathrm{D}g<br />
& = & \texttt{((} u \texttt{,~} v \texttt{))}<br />
& + & \texttt{((} u + \mathrm{d}u \texttt{,~} v + \mathrm{d}v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
But these initial formulas are purely definitional, and help us little in understanding either the purpose of the operators or the meaning of their results. Working symbolically, let us apply the same method to the separate components <math>f\!</math> and <math>g\!</math> that we earlier used on <math>J.\!</math> This work is recorded in Appendix&nbsp;3 and a summary of the results is presented in Tables&nbsp;66-i and 66-ii.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 66-i.} ~~ \text{Computation Summary for}~ f(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f<br />
& = & u \!\cdot\! v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 1<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}f<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \cdot \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{d}f<br />
& = & u \!\cdot\! v \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}f<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 66-ii.} ~~ \text{Computation Summary for}~ g(u, v) = \texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon g<br />
& = & u \!\cdot\! v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}g<br />
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}g<br />
& = & u \!\cdot\! v \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;67 shows how to compute the analytic series for <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math> in terms of coordinates, and Table&nbsp;68 recaps these results in symbolic terms, agreeing with earlier derivations.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 67.} ~~ \text{Computation of an Analytic Series in Terms of Coordinates}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:6%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}u\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>\mathrm{d}v\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>u'\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>v'\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:4px double black" | <math>f\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>g\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{E}f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{E}g}\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{D}f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{D}g}\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{d}f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{d}g}\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{d}^2\!f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{d}^2\!g}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 68.} ~~ \text{Computation of an Analytic Series in Symbolic Terms}\!</math><br />
|- style="height:40px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black" | <math>f\!</math><br />
| <math>g\!</math><br />
| style="border-left:1px solid black" | <math>{\mathrm{D}f}\!</math><br />
| <math>{\mathrm{D}g}\!</math><br />
| style="border-left:1px solid black" | <math>{\mathrm{d}f}\!</math><br />
| <math>{\mathrm{d}g}\!</math><br />
| style="border-left:1px solid black" | <math>{\mathrm{d}^2\!f}\!</math><br />
| <math>{\mathrm{d}^2\!g}\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[4pt]<br />
\texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
\\[4pt]<br />
\texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
\\[4pt]<br />
\texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;69 gives a graphical picture of the difference map <math>\mathrm{D}F = (\mathrm{D}f, \mathrm{D}g)\!</math> for the transformation <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math> This represents the same information about <math>\mathrm{D}f~\!</math> and <math>\mathrm{D}g~\!</math> that was given in the corresponding rows of Tables&nbsp;66-i and 66-ii, for ease of reference repeated below.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[8pt]<br />
\mathrm{D}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 69 -- Difference Map (Short Form).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 69.} ~~ \text{Difference Map of}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;70-a shows a way of visualizing the tangent functor map <math>\mathrm{d}F = (\mathrm{d}f, \mathrm{d}g)\!</math> for the transformation <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math> This amounts to the same information about <math>\mathrm{d}f~\!</math> and <math>\mathrm{d}g~\!</math> that was given in Tables&nbsp;66-i and 66-ii, the corresponding rows of which are repeated below.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{d}f<br />
& = & u \!\cdot\! v \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[8pt]<br />
\mathrm{d}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 70-a -- Tangent Functor Diagram.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 70-a.} ~~ \text{Tangent Functor Diagram for}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math><br />
|}<br />
<br />
<br><br />
<br />
Figure 70-b shows another way to picture the action of the tangent functor on the logical transformation <math>F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 70-b -- Tangent Functor Diagram.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 70-b.} ~~ \text{Tangent Functor Ferris Wheel for}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math><br />
|}<br />
<br />
<br><br />
<br />
* '''Note.''' The original Figure&nbsp;70-b lost some of its labeling in a succession of platform metamorphoses over the years, so we have included an ASCII version below to indicate where the missing labels go.<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| /////\ /////\ | | /XXXX\ /XXXX\ | | /\\\\\ /\\\\\ |<br />
| ///////o//////\ | | /XXXXXXoXXXXXX\ | | /\\\\\\o\\\\\\\ |<br />
| //////// \//////\ | | /XXXXXX/ \XXXXXX\ | | /\\\\\\/ \\\\\\\\ |<br />
| o/////// \//////o | | oXXXXXX/ \XXXXXXo | | o\\\\\\/ \\\\\\\o |<br />
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |<br />
| |/du//| |//dv/| | | |XXXXX| |XXXXX| | | |\du\\| |\\dv\| |<br />
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |<br />
| o//////\ ///////o | | oXXXXXX\ /XXXXXXo | | o\\\\\\\ /\\\\\\o |<br />
| \//////\ //////// | | \XXXXXX\ /XXXXXX/ | | \\\\\\\\ /\\\\\\/ |<br />
| \//////o/////// | | \XXXXXXoXXXXXX/ | | \\\\\\\o\\\\\\/ |<br />
| \///// \///// | | \XXXX/ \XXXX/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ (u)(v) o-----------------------o dv' @ (u)(v) =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| / \ /////\ | | /\\\\\ /XXXX\ | | /\\\\\ /\\\\\ |<br />
| / o//////\ | | /\\\\\\oXXXXXX\ | | /\\\\\\o\\\\\\\ |<br />
| / //\//////\ | | /\\\\\\//\XXXXXX\ | | /\\\\\\/ \\\\\\\\ |<br />
| o ////\//////o | | o\\\\\\////\XXXXXXo | | o\\\\\\/ \\\\\\\o |<br />
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |<br />
| | du |/////|//dv/| | | |\\\\\|/////|XXXXX| | | |\du\\| |\\dv\| |<br />
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |<br />
| o \//////////o | | o\\\\\\\////XXXXXXo | | o\\\\\\\ /\\\\\\o |<br />
| \ \///////// | | \\\\\\\\//XXXXXX/ | | \\\\\\\\ /\\\\\\/ |<br />
| \ o/////// | | \\\\\\\oXXXXXX/ | | \\\\\\\o\\\\\\/ |<br />
| \ / \///// | | \\\\\/ \XXXX/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ (u) v o-----------------------o dv' @ (u) v =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| /////\ / \ | | /XXXX\ /\\\\\ | | /\\\\\ /\\\\\ |<br />
| ///////o \ | | /XXXXXXo\\\\\\\ | | /\\\\\\o\\\\\\\ |<br />
| /////////\ \ | | /XXXXXX//\\\\\\\\ | | /\\\\\\/ \\\\\\\\ |<br />
| o//////////\ o | | oXXXXXX////\\\\\\\o | | o\\\\\\/ \\\\\\\o |<br />
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |<br />
| |/du//|/////| dv | | | |XXXXX|/////|\\\\\| | | |\du\\| |\\dv\| |<br />
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |<br />
| o//////\//// o | | oXXXXXX\////\\\\\\o | | o\\\\\\\ /\\\\\\o |<br />
| \//////\// / | | \XXXXXX\//\\\\\\/ | | \\\\\\\\ /\\\\\\/ |<br />
| \//////o / | | \XXXXXXo\\\\\\/ | | \\\\\\\o\\\\\\/ |<br />
| \///// \ / | | \XXXX/ \\\\\/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ u (v) o-----------------------o dv' @ u (v) =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| / \ / \ | | /\\\\\ /\\\\\ | | /\\\\\ /\\\\\ |<br />
| / o \ | | /\\\\\\o\\\\\\\ | | /\\\\\\o\\\\\\\ |<br />
| / / \ \ | | /\\\\\\/ \\\\\\\\ | | /\\\\\\/ \\\\\\\\ |<br />
| o / \ o | | o\\\\\\/ \\\\\\\o | | o\\\\\\/ \\\\\\\o |<br />
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |<br />
| | du | | dv | | | |\\\\\| |\\\\\| | | |\du\\| |\\dv\| |<br />
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |<br />
| o \ / o | | o\\\\\\\ /\\\\\\o | | o\\\\\\\ /\\\\\\o |<br />
| \ \ / / | | \\\\\\\\ /\\\\\\/ | | \\\\\\\\ /\\\\\\/ |<br />
| \ o / | | \\\\\\\o\\\\\\/ | | \\\\\\\o\\\\\\/ |<br />
| \ / \ / | | \\\\\/ \\\\\/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ u v o-----------------------o dv' @ u v =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| U | |\U\\\\\\\\\\\\\\\\\\\\\| |\U\\\\\\\\\\\\\\\\\\\\\|<br />
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|<br />
| /////\ /////\ | |\\\\\/////\\/////\\\\\\| |\\\\\/ \\/ \\\\\\|<br />
| ///////o//////\ | |\\\\///////o//////\\\\\| |\\\\/ o \\\\\|<br />
| /////////\//////\ | |\\\////////X\//////\\\\| |\\\/ /\\ \\\\|<br />
| o//////////\//////o | |\\o///////XXX\//////o\\| |\\o /\\\\ o\\|<br />
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|<br />
| |//u//|/////|//v//| | |\\|//u//|XXXXX|//v//|\\| |\\| u |\\\\\| v |\\|<br />
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|<br />
| o//////\//////////o | |\\o//////\XXX///////o\\| |\\o \\\\/ o\\|<br />
| \//////\///////// | |\\\\//////\X////////\\\| |\\\\ \\/ /\\\|<br />
| \//////o/////// | |\\\\\//////o///////\\\\| |\\\\\ o /\\\\|<br />
| \///// \///// | |\\\\\\/////\\/////\\\\\| |\\\\\\ /\\ /\\\\\|<br />
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|<br />
| | |\\\\\\\\\\\\\\\\\\\\\\\| |\\\\\\\\\\\\\\\\\\\\\\\|<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= u' o-----------------------o v' =<br />
= | U' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/u'//|XXXXX|\\v'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
Figure 70-b. Tangent Functor Ferris Wheel for F<u, v> = <((u)(v)), ((u, v))><br />
</pre><br />
|}<br />
<br />
==Epilogue, Enchoiry, Exodus==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
It is time to explain myself . . . . let us stand up.<br />
| width="4%" | &nbsp;<br />
|- <br />
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 79]<br />
|}<br />
<br />
==Appendices==<br />
<br />
===Appendix 1. Propositional Forms and Differential Expansions===<br />
<br />
====Table A1. Propositional Forms on Two Variables====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A1.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="background:ghostwhite"<br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}</math><br />
| width="25%" | <math>\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{0}\\f_{1}\\f_{2}\\f_{3}\\f_{4}\\f_{5}\\f_{6}\\f_{7}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0000}\\f_{0001}\\f_{0010}\\f_{0011}\\f_{0100}\\f_{0101}\\f_{0110}\\f_{0111}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~0\\0~0~0~1\\0~0~1~0\\0~0~1~1\\0~1~0~0\\0~1~0~1\\0~1~1~0\\0~1~1~1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)~ ~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~ ~(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{,~} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{~~} y \texttt{)}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{false}<br />
\\<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\<br />
y ~\text{without}~ x<br />
\\<br />
\text{not}~ x<br />
\\<br />
x ~\text{without}~ y<br />
\\<br />
\text{not}~ y<br />
\\<br />
x ~\text{not equal to}~ y<br />
\\<br />
\text{not both}~ x ~\text{and}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\<br />
\lnot x \land \lnot y<br />
\\<br />
\lnot x \land y<br />
\\<br />
\lnot x<br />
\\<br />
x \land \lnot y<br />
\\<br />
\lnot y<br />
\\<br />
x \ne y<br />
\\<br />
\lnot x \lor \lnot y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{8}\\f_{9}\\f_{10}\\f_{11}\\f_{12}\\f_{13}\\f_{14}\\f_{15}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{1000}\\f_{1001}\\f_{1010}\\f_{1011}\\f_{1100}\\f_{1101}\\f_{1110}\\f_{1111}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1~0~0~0\\1~0~0~1\\1~0~1~0\\1~0~1~1\\1~1~0~0\\1~1~0~1\\1~1~1~0\\1~1~1~1<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~ ~ ~ ~} y \texttt{~~}<br />
\\<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{~~} x \texttt{~ ~ ~ ~}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{((~))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{and}~ y<br />
\\<br />
x ~\text{equal to}~ y<br />
\\<br />
y<br />
\\<br />
\text{not}~ x ~\text{without}~ y<br />
\\<br />
x<br />
\\<br />
\text{not}~ y ~\text{without}~ x<br />
\\<br />
x ~\text{or}~ y<br />
\\<br />
\text{true}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x \land y<br />
\\<br />
x = y<br />
\\<br />
y<br />
\\<br />
x \Rightarrow y<br />
\\<br />
x<br />
\\<br />
x \Leftarrow y<br />
\\<br />
x \lor y<br />
\\<br />
1<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A2. Propositional Forms on Two Variables====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A2.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="background:ghostwhite"<br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}</math><br />
| width="25%" | <math>\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>f_{0000}\!</math><br />
| <math>0~0~0~0</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0001}\\f_{0010}\\f_{0100}\\f_{1000}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~1\\0~0~1~0\\0~1~0~0\\1~0~0~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\<br />
y ~\text{without}~ x<br />
\\<br />
x ~\text{without}~ y<br />
\\<br />
x ~\text{and}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot x \land \lnot y<br />
\\<br />
\lnot x \land y<br />
\\<br />
x \land \lnot y<br />
\\<br />
x \land y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{3}\\f_{12}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0011}\\f_{1100}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~1~1\\1~1~0~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ x<br />
\\<br />
x<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot x<br />
\\<br />
x<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{6}\\f_{9}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0110}\\f_{1001}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~0\\1~0~0~1<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{not equal to}~ y<br />
\\<br />
x ~\text{equal to}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x \ne y<br />
\\<br />
x = y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{5}\\f_{10}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0101}\\f_{1010}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~0~1\\1~0~1~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ y<br />
\\<br />
y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot y<br />
\\<br />
y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{7}\\f_{11}\\f_{13}\\f_{14}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0111}\\f_{1011}\\f_{1101}\\f_{1110}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~1\\1~0~1~1\\1~1~0~1\\1~1~1~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not both}~ x ~\text{and}~ y<br />
\\<br />
\text{not}~ x ~\text{without}~ y<br />
\\<br />
\text{not}~ y ~\text{without}~ x<br />
\\<br />
x ~\text{or}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot x \lor \lnot y<br />
\\<br />
x \Rightarrow y<br />
\\<br />
x \Leftarrow y<br />
\\<br />
x \lor y<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>f_{1111}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A3. E''f'' Expanded Over Differential Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A3.} ~~ \mathrm{E}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{T}_{11}f\\\mathrm{E}f|_{\mathrm{d}x ~ \mathrm{d}y}\end{matrix}</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{T}_{10}f\\\mathrm{E}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}\end{matrix}</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{T}_{01}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}\end{matrix}</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{T}_{00}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{3}\\f_{12}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{6}\\f_{9}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{5}\\f_{10}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{7}\\f_{11}\\f_{13}\\f_{14}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
|- style="background:ghostwhite"<br />
| style="border-top:1px solid black" colspan="2" | <math>\text{Fixed Point Total}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>4\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>4\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>4\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>16\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A4. D''f'' Expanded Over Differential Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A4.} ~~ \mathrm{D}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\mathrm{D}f|_{\mathrm{d}x ~ \mathrm{d}y}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}~\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
y<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
y<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
x<br />
\\<br />
x<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
x<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
y<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
y<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <br />
<math>\begin{matrix}<br />
x<br />
\\<br />
x<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A5. E''f'' Expanded Over Ordinary Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A5.} ~~ \mathrm{E}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\mathrm{E}f|_{xy}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{E}f|_{x \texttt{(} y \texttt{)}}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{E}f|_{\texttt{(} x \texttt{)} y}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{E}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{3}\\f_{12}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{6}\\f_{9}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{5}\\f_{10}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{7}\\f_{11}\\f_{13}\\f_{14}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A6. D''f'' Expanded Over Ordinary Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A6.} ~~ \mathrm{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\mathrm{D}f|_{xy}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{x \texttt{(} y \texttt{)}}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} x \texttt{)} y}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\texttt{(} x \texttt{)}\\\texttt{~} x \texttt{~}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Appendix 2. Differential Forms===<br />
<br />
The actions of the difference operator <math>\mathrm{D}\!</math> and the tangent operator <math>\mathrm{d}\!</math> on the 16 bivariate propositions are shown in Tables&nbsp;A7 and A8.<br />
<br />
Table A7 expands the differential forms that result over a ''logical basis'':<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
|<br />
<math>\{~ \texttt{(}\mathrm{d}x\texttt{)(}\mathrm{d}y\texttt{)}, ~\mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}, ~\texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.\!</math><br />
|}<br />
<br />
This set consists of the singular propositions in the first order differential variables, indicating mutually exclusive and exhaustive ''cells'' of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the cell-basis, point-basis, or singular differential basis. In this setting it is frequently convenient to use the following abbreviations:<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
|<br />
<math>\partial x ~=~ \mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}\!</math> &nbsp; &nbsp; and &nbsp; &nbsp; <math>\partial y ~=~ \texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y.\!</math><br />
|}<br />
<br />
Table A8 expands the differential forms that result over an ''algebraic basis'':<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
| <math>\{~ 1, ~\mathrm{d}x, ~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.\!</math><br />
|}<br />
<br />
This set consists of the ''positive propositions'' in the first order differential variables, indicating overlapping positive regions of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the ''positive differential basis''.<br />
<br />
====Table A7. Differential Forms Expanded on a Logical Basis====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A7.} ~~ \text{Differential Forms Expanded on a Logical Basis}\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| &nbsp;<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" | <math>\mathrm{D}f~\!</math><br />
| <math>\mathrm{d}f~\!</math><br />
|-<br />
| <math>f_{0}\!</math><br />
| style="border-right:none" | <math>\texttt{(~)}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x) ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\\<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\\<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\\<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\partial x<br />
\\<br />
\partial x<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
\\<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\partial x & + & \partial y<br />
\\<br />
\partial x & + & \partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\partial y<br />
\\<br />
\partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x) ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\\<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\\<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\\<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| style="border-right:none" | <math>\texttt{((~))}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A8. Differential Forms Expanded on an Algebraic Basis====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A8.} ~~ \text{Differential Forms Expanded on an Algebraic Basis}\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| &nbsp;<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" | <math>\mathrm{D}f~\!</math><br />
| <math>\mathrm{d}f~\!</math><br />
|-<br />
| <math>f_{0}\!</math><br />
| style="border-right:none" | <math>\texttt{(~)}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x<br />
\\<br />
\mathrm{d}x<br />
\end{matrix}\!</math><br />
| <math>\begin{matrix}<br />
\mathrm{d}x<br />
\\<br />
\mathrm{d}x<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\\<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\\<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}y<br />
\\<br />
\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y<br />
\\<br />
\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| style="border-right:none" | <math>\texttt{((~))}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A9. Tangent Proposition as Pointwise Linear Approximation====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A9.} ~~ \text{Tangent Proposition}~ \mathrm{d}f = \text{Pointwise Linear Approximation to the Difference Map}~ \mathrm{D}f\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}f =<br />
\\[2pt]<br />
\partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}^2\!f =<br />
\\[2pt]<br />
\partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\mathrm{d}f|_{x \, y}</math><br />
| <math>\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math><br />
|-<br />
| style="border-right:none" | <math>f_0\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{5}\\f_{10}\end{matrix}\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\end{matrix}\!</math><br />
| <math>\begin{matrix}<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>f_{15}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A10. Taylor Series Expansion Df = d''f'' + d<sup>2</sup>''f''====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" |<br />
<math>\text{Table A10.} ~~ \text{Taylor Series Expansion}~ {\mathrm{D}f = \mathrm{d}f + \mathrm{d}^2\!f}\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{D}f<br />
\\<br />
= & \mathrm{d}f & + & \mathrm{d}^2\!f<br />
\\<br />
= & \partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y & + & \partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\mathrm{d}f|_{x \, y}</math><br />
| <math>\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math><br />
|-<br />
| style="border-right:none" | <math>f_0\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>f_{15}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A11. Partial Differentials and Relative Differentials====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A11.} ~~ \text{Partial Differentials and Relative Differentials}\!</math><br />
|- style="background:ghostwhite; height:50px"<br />
| &nbsp;<br />
| <math>f\!</math><br />
| <math>\frac{\partial f}{\partial x}\!</math><br />
| <math>\frac{\partial f}{\partial y}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}f =<br />
\\[2pt]<br />
\partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\left. \frac{\partial x}{\partial y} \right| f\!</math><br />
| <math>\left. \frac{\partial y}{\partial x} \right| f\!</math><br />
|-<br />
| <math>f_0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A12. Detail of Calculation for the Difference Map====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:4px double black; border-left:4px double black; border-right:4px double black; border-top:4px double black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A12.} ~~ \text{Detail of Calculation for}~ {\mathrm{E}f + f = \mathrm{D}f}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:6%" | &nbsp;<br />
| style="width:14%; border-left:1px solid black" | <math>f\!</math><br />
| style="width:20%; border-left:4px double black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\mathrm{d}x ~ \mathrm{d}y}<br />
\\[4pt]<br />
+ & f|_{\mathrm{d}x ~ \mathrm{d}y}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\mathrm{d}x ~ \mathrm{d}y}<br />
\end{array}</math><br />
| style="width:20%; border-left:1px solid black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}<br />
\\[4pt]<br />
+ & f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}<br />
\end{array}</math><br />
| style="width:20%; border-left:1px solid black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
+ & f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}<br />
\end{array}</math><br />
| style="width:20%; border-left:1px solid black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
+ & f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}<br />
\end{array}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{0}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:4px double black; border-left:4px double black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{1}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{)(} y \texttt{)~}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{2}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{)~} y \texttt{~~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{4}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~~} x \texttt{~(} y \texttt{)~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{8}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~~} x \texttt{~~} y \texttt{~~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{3}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{(} x \texttt{)}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{12}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>x\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{6}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{,~} y \texttt{)~}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{9}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} x \texttt{,~} y \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{5}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{(} y \texttt{)}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{10}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>y\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{7}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{~~} y \texttt{)~}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{11}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{~(} y \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{13}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} x \texttt{)~} y \texttt{)~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{14}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} x \texttt{)(} y \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{15}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:4px double black; border-left:4px double black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Appendix 3. Computational Details===<br />
<br />
====Operator Maps for the Logical Conjunction ''f''<sub>8</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{8}~\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{8}<br />
& = && f_{8}(u, v)<br />
\\[4pt]<br />
& = && uv<br />
\\[4pt]<br />
& = && uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & uv \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & uv \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{8}<br />
& = && uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & uv \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & uv \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & uv \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.2-i} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{E}f_{8}<br />
& = & f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{8}(\mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{8}(\mathrm{d}u, \mathrm{d}v)<br />
\\[4pt]<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[20pt]<br />
\mathrm{E}f_{8}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
\\[4pt]<br />
&&&&& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&&&&&& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.2-ii} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{c}}<br />
\mathrm{E}f_{8}<br />
& = & (u + \mathrm{d}u) \cdot (v + \mathrm{d}v)<br />
\\[6pt]<br />
& = & u \cdot v<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}f_{8}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{8}<br />
& = && \mathrm{E}f_{8}<br />
& + & \boldsymbol\varepsilon f_{8}<br />
\\[4pt]<br />
& = && f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{8}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & uv<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{~}<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}f_{8}<br />
& = & \boldsymbol\varepsilon f_{8}<br />
& + & \mathrm{E}f_{8}<br />
\\[6pt]<br />
& = & f_{8}(u, v)<br />
& + & f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[6pt]<br />
& = & uv<br />
& + & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
& = & 0<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}f_{8}<br />
& = & 0<br />
& + & u \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.3-iii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 3)}\!</math><br />
|<br />
<math>\begin{array}{c*{9}{l}}<br />
\mathrm{D}f_{8}<br />
& = & \boldsymbol\varepsilon f_{8} ~+~ \mathrm{E}f_{8}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{8}<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ u \,\cdot\, v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~ u \;\cdot\; v \;\cdot\; \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}f_{8}<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u ~ \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} ~ v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot\, \mathrm{d}u ~ \mathrm{d}v<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = & ~ ~ 0 ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~ ~ u ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ ~ ~ v ~~ \cdot ~ \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
=====Computation of d''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.4} ~~ \text{Computation of}~ \mathrm{d}f_{8}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{8}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.5} ~~ \text{Computation of}~ \mathrm{r}f_{8}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{8} & = & \mathrm{D}f_{8} ~+~ \mathrm{d}f_{8}<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[20pt]<br />
\mathrm{r}f_{8}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Conjunction=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.6} ~~ \text{Computation Summary for}~ f_{8}(u, v) = uv\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{8}<br />
& = & uv \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}f_{8}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{r}f_{8}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Operator Maps for the Logical Equality ''f''<sub>9</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{9}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{9}<br />
& = && f_{9}(u, v)<br />
\\[4pt]<br />
& = && \texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{9}(1, 1)<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{9}(1, 0)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{9}(0, 1)<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{9}(0, 0)<br />
\\[4pt]<br />
& = && u v & + & 0 & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{9}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.2} ~~ \text{Computation of}~ \mathrm{E}f_{9}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{E}f_{9}<br />
& = && f_{9}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! f_{9}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! f_{9}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{ } \mathrm{d}v \texttt{ })<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! f_{9}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! f_{9}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{ } \mathrm{d}v \texttt{ })<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{ (} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{) }<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{ (} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{) }<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{E}f_{9}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{9}<br />
& = && \mathrm{E}f_{9}<br />
& + & \boldsymbol\varepsilon f_{9}<br />
\\[4pt]<br />
& = && f_{9}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{9}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{9}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[20pt]<br />
\mathrm{D}f_{9}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}f_{9}<br />
& = & 0 \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & 1 \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & 1 \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & 0 \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of d''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.4} ~~ \text{Computation of}~ \mathrm{d}f_{9}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.5} ~~ \text{Computation of}~ \mathrm{r}f_{9}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{9} & = & \mathrm{D}f_{9} ~+~ \mathrm{d}f_{9}<br />
\\[20pt]<br />
\mathrm{D}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[20pt]<br />
\mathrm{r}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Equality=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.6} ~~ \text{Computation Summary for}~ f_{9}(u, v) = \texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{9}<br />
& = & uv \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}f_{9}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f_{9}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{9}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}f_{9}<br />
& = & uv \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Operator Maps for the Logical Implication ''f''<sub>11</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{11}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{11}<br />
& = && f_{11}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(} u \texttt{(} v \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{11}(1, 1)<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{11}(1, 0)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{11}(0, 1)<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{11}(0, 0)<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ }<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ }<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{11}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.2} ~~ \text{Computation of}~ \mathrm{E}f_{11}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{E}f_{11}<br />
& = && f_{11}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = &&<br />
\texttt{(}<br />
\\<br />
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\\<br />
&&& \texttt{(}<br />
\\<br />
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\<br />
&&& \texttt{))}<br />
\\[4pt]<br />
& = &&<br />
u v<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)((} \mathrm{d}v \texttt{)))}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u \texttt{((} \mathrm{d}v \texttt{)))}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[4pt]<br />
& = &&<br />
u v<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{E}f_{11}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{11}<br />
& = && \mathrm{E}f_{11}<br />
& + & \boldsymbol\varepsilon f_{11}<br />
\\[4pt]<br />
& = && f_{11}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{11}(u, v)<br />
\\[4pt]<br />
& = &&<br />
\texttt{(} \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\texttt{(} \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{(} v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & u v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & 0<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & 0<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = && u v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{c*{9}{l}}<br />
\mathrm{D}f_{11}<br />
& = & \boldsymbol\varepsilon f_{11} ~+~ \mathrm{E}f_{11}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{11}<br />
& = & u v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}f_{11}<br />
& = &<br />
u v<br />
\cdot<br />
\texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\cdot<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\cdot<br />
\texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\cdot<br />
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = &<br />
u v<br />
\cdot<br />
\texttt{~(} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{~}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\cdot<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\cdot<br />
\texttt{~} \mathrm{d}u ~ \mathrm{d}v \texttt{~}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\cdot<br />
\texttt{~} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of d''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.4} ~~ \text{Computation of}~ \mathrm{d}f_{11}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{11}<br />
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{11}<br />
& = & u v \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.5} ~~ \text{Computation of}~ \mathrm{r}f_{11}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{11} & = & \mathrm{D}f_{11} ~+~ \mathrm{d}f_{11}<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{11}<br />
& = & u v \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u<br />
\\[20pt]<br />
\mathrm{r}f_{11}<br />
& = & u v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Implication=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.6} ~~ \text{Computation Summary for}~ f_{11}(u, v) = \texttt{(} u \texttt{(} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{11}<br />
& = & u v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}f_{11}<br />
& = & u v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f_{11}<br />
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{11}<br />
& = & u v \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u<br />
\\[6pt]<br />
\mathrm{r}f_{11}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Operator Maps for the Logical Disjunction ''f''<sub>14</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{14}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{14}<br />
& = && f_{14}(u, v)<br />
\\[4pt]<br />
& = && \texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{14}(1, 1)<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{14}(1, 0)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{14}(0, 1)<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{14}(0, 0)<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ }<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ }<br />
& + & 0<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{14}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.2} ~~ \text{Computation of}~ \mathrm{E}f_{14}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{E}f_{14}<br />
& = && f_{14}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = &&<br />
\texttt{((}<br />
\\<br />
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\\<br />
&&& \texttt{)(}<br />
\\<br />
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\<br />
&&& \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! f_{14}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! f_{14}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{ } \mathrm{d}v \texttt{ })<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! f_{14}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! f_{14}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{ } \mathrm{d}v \texttt{ })<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{E}f_{14}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{14}<br />
& = && \mathrm{E}f_{14}<br />
& + & \boldsymbol\varepsilon f_{14}<br />
\\[4pt]<br />
& = && f_{14}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{14}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{))((} v \texttt{,} \mathrm{d}v \texttt{)))}<br />
& + & \texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{14}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
\\[4pt]<br />
&& + & 0<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
\\[4pt]<br />
&& + & uv \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
\\[20pt]<br />
\mathrm{D}f_{14}<br />
& = && uv \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}f_{14}<br />
& = & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & 0 \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of d''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.4} ~~ \text{Computation of}~ \mathrm{d}f_{14}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.5} ~~ \text{Computation of}~ \mathrm{r}f_{14}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{14} & = & \mathrm{D}f_{14} ~+~ \mathrm{d}f_{14}<br />
\\[20pt]<br />
\mathrm{D}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{d}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[20pt]<br />
\mathrm{r}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Disjunction=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.6} ~~ \text{Computation Summary for}~ f_{14}(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{14}<br />
& = & uv \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 1<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}f_{14}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f_{14}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{d}f_{14}<br />
& = & uv \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}f_{14}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
===Appendix 4. Source Materials===<br />
<br />
===Appendix 5. Various Definitions of the Tangent Vector===<br />
<br />
==References==<br />
<br />
===Works Cited===<br />
<br />
{| cellpadding=3<br />
| valign=top | [AuM]<br />
| Auslander, L., and MacKenzie, R.E., ''Introduction to Differentiable Manifolds'', McGraw-Hill, 1963. Reprinted, Dover, New York, NY, 1977.<br />
|-<br />
| valign=top | [BiG]<br />
| Bishop, R.L., and Goldberg, S.I., ''Tensor Analysis on Manifolds'', Macmillan, 1968. Reprinted, Dover, New York, NY, 1980.<br />
|-<br />
| valign=top | [Boo]<br />
| Boole, G., ''An Investigation of The Laws of Thought'', Macmillan, 1854. Reprinted, Dover, New York, NY, 1958.<br />
|-<br />
| valign=top | [BoT]<br />
| Bott, R., and Tu, L.W., ''Differential Forms in Algebraic Topology'', Springer-Verlag, New York, NY, 1982.<br />
|-<br />
| valign=top | [dCa]<br />
| do Carmo, M.P., ''Riemannian Geometry''. Originally published in Portuguese, 1st editiom 1979, 2nd edition 1988. Translated by F. Flaherty, Birkhäuser, Boston, MA, 1992.<br />
|-<br />
| valign=top | [Che46]<br />
| Chevalley, C., ''Theory of Lie Groups'', Princeton University Press, Princeton, NJ, 1946.<br />
|-<br />
| valign=top | [Che56]<br />
| Chevalley, C., ''Fundamental Concepts of Algebra'', Academic Press, 1956.<br />
|-<br />
| valign=top | [Cho86]<br />
| Chomsky, N., ''Knowledge of Language : Its Nature, Origin, and Use'', Praeger, New York, NY, 1986.<br />
|-<br />
| valign=top | [Cho93]<br />
| Chomsky, N., ''Language and Thought'', Moyer Bell, Wakefield, RI, 1993.<br />
|-<br />
| valign=top | [DoM]<br />
| Doolin, B.F., and Martin, C.F., ''Introduction to Differential Geometry for Engineers'', Marcel Dekker, New York, NY, 1990.<br />
|-<br />
| valign=top | [Fuji]<br />
| Fujiwara, H., ''Logic Testing and Design for Testability'', MIT Press, Cambridge, MA, 1985.<br />
|-<br />
| valign=top | [Hic]<br />
| Hicks, N.J., ''Notes on Differential Geometry'', Van Nostrand, Princeton, NJ, 1965.<br />
|-<br />
| valign=top | [Hir]<br />
| Hirsch, M.W., ''Differential Topology'', Springer-Verlag, New York, NY, 1976.<br />
|-<br />
| valign=top | [How]<br />
| Howard, W.A., "The Formulae-as-Types Notion of Construction", Notes circulated from 1969. Reprinted in [SeH, 479-490].<br />
|-<br />
| valign=top | [JGH]<br />
| Jones, A., Gray, A., and Hutton, R., ''Manifolds and Mechanics'', Cambridge University Press, Cambridge, UK, 1987.<br />
|-<br />
| valign=top | [KoA]<br />
| Kosinski, A.A., ''Differential Manifolds'', Academic Press, San Diego, CA, 1993.<br />
|-<br />
| valign=top | [Koh]<br />
| Kohavi, Z., ''Switching and Finite Automata Theory'', 2nd edition, McGraw-Hill, New York, NY, 1978.<br />
|-<br />
| valign=top | [LaS]<br />
| Lambek, J., and Scott, P.J., ''Introduction to Higher Order Categorical Logic'', Cambridge University Press, Cambridge, UK, 1986.<br />
|-<br />
| valign=top | [La83]<br />
| Lang, S., ''Real Analysis'', 2nd edition, Addison-Wesley, Reading, MA, 1983.<br />
|-<br />
| valign=top | [La84]<br />
| Lang, S., ''Algebra'', 2nd edition, Addison-Wesley, Menlo Park, CA, 1984.<br />
|-<br />
| valign=top | [La85]<br />
| Lang, S., ''Differential Manifolds'', Springer-Verlag, New York, NY, 1985.<br />
|-<br />
| valign=top | [La93]<br />
| Lang, S., ''Real and Functional Analysis'', 3rd edition, Springer-Verlag, New York, NY, 1993.<br />
|-<br />
| valign=top | [Lie80]<br />
| Lie, S., "Sophus Lie's 1880 Transformation Group Paper", in ''Lie Groups : History, Frontiers, and Applications, Volume 1'', translated by M. Ackerman, comments by R. Hermann, Math Sci Press, Brookline, MA, 1975. Original paper 1880.<br />
|-<br />
| valign=top | [Lie84]<br />
| Lie, S., "Sophus Lie's 1884 Differential Invariant Paper", in ''Lie Groups : History, Frontiers, and Applications, Volume 3'', translated by M. Ackerman, comments by R. Hermann, Math Sci Press, Brookline, MA, 1976. Original paper 1884.<br />
|-<br />
| valign=top | [LoS]<br />
| Loomis, L.H., and Sternberg, S., ''Advanced Calculus'', Addison-Wesley, Reading, MA, 1968.<br />
|-<br />
| valign=top | [Mel]<br />
| Melzak, Z.A., ''Companion to Concrete Mathematics, Volume 2 : Mathematical Ideas, Modeling, and Applications'', John Wiley amd Sons, New York, NY, 1976.<br />
|-<br />
| valign=top | [Men]<br />
| Menabrea, L.F., "Sketch of the Analytical Engine Invented by Charles Babbage" with Notes by the Translator, Ada Augusta (Byron), Countess of Lovelace'', in [M&M, 225–297]. Originally published 1842.<br />
|-<br />
| valign=top | [M&M]<br />
| Morrison, P., and Morrison, E. (eds.), ''Charles Babbage on the Principles and Development of the Calculator, and Other Seminal Writings by Charles Babbage and Others, With an Introduction by the Editors'', Dover, Mineola, NY, 1961.<br />
|-<br />
| valign=top | [P1]<br />
| Peirce, C.S., ''Collected Papers of Charles Sanders Peirce'', vols. 1–8, C. Hartshorne, P. Weiss, and A.W. Burks (eds.), Harvard University Press, Cambridge, MA, 1931–1960. Cited as CP [volume].[paragraph].<br />
|-<br />
| valign=top | [P2]<br />
| Peirce, C.S., "Qualitative Logic", in ''The New Elements of Mathematics, Volume 4'', C. Eisele (ed.), Mouton, The Hague, 1976. Cited as NE [volume], [page].<br />
|-<br />
| valign=top | [Rob]<br />
| Roberts, D.D., ''The Existential Graphs of Charles S. Peirce'', Mouton, The Hague, 1973.<br />
|-<br />
| valign=top | [SeH]<br />
| Seldin, J.P., and Hindley, J.R. (eds.), ''To H.B. Curry : Essays on Combinatory Logic, Lambda Calculus, and Formalism'', Academic Press, London, UK, 1980.<br />
|-<br />
| valign=top | [SpB]<br />
| Spencer-Brown, G., ''Laws of Form'', George Allen and Unwin, London, UK, 1969.<br />
|-<br />
| valign=top | [Sp65]<br />
| Spivak, M., ''Calculus on Manifolds : A Modern Approach to Classical Theorems of Advanced Calculus'', W.A. Benjamin, New York, NY, 1965.<br />
|-<br />
| valign=top | [Sp79]<br />
| Spivak, M., ''A Comprehensive Introduction to Differential Geometry'', vols. 1–2. 1st edition 1970. 2nd edition, Publish or Perish Inc., Houston, TX, 1979.<br />
|-<br />
| valign=top | [Sty]<br />
| Styazhkin, N.I., ''History of Mathematical Logic from Leibniz to Peano'', 1st published in Russian, Nauka, Moscow, 1964. MIT Press, Cambridge, MA, 1969.<br />
|-<br />
| valign=top | [Wie]<br />
| Wiener, N., ''Cybernetics : or Control and Communication in the Animal and the Machine'', 1st edition 1948. 2nd edition, MIT Press, Cambridge, MA, 1961.<br />
|}<br />
<br />
===Works Consulted===<br />
<br />
{| cellpadding=3<br />
| valign=top | [Ami]<br />
| Amit, D.J., ''Modeling Brain Function : The World of Attractor Neural Networks'', Cambridge University Press, Cambridge, UK, 1989.<br />
|-<br />
| valign=top | [Ed87]<br />
| Edelman, G.M., ''Neural Darwinism : The Theory of Neuronal Group Selection'', Basic Books, New York, NY, 1987.<br />
|-<br />
| valign=top | [Ed88]<br />
| Edelman, G.M., ''Topobiology : An Introduction to Molecular Embryology'', Basic Books, New York, NY, 1988.<br />
|-<br />
| valign=top | [Fla]<br />
| Flanders, H., ''Differential Forms with Applications to the Physical Sciences'', Academic Press, 1963. Reprinted, Dover, Mineola, NY, 1989. <br />
|-<br />
| valign=top | [Has]<br />
| Hassoun, M.H. (ed.), ''Associative Neural Memories : Theory and Implementation'', Oxford University Press, New York, NY, 1993.<br />
|-<br />
| valign=top | [KoB]<br />
| Kosko, B., ''Neural Networks and Fuzzy Systems : A Dynamical Systems Approach to Machine Intelligence'', Prentice-Hall, Englewood Cliffs, NJ, 1992.<br />
|-<br />
| valign=top | [MaB]<br />
| Mac Lane, S., and Birkhoff, G., ''Algebra'', 3rd edition, Chelsea, New York, NY, 1993.<br />
|-<br />
| valign=top | [Mac]<br />
| Mac Lane, S., ''Categories for the Working Mathematician'', Springer-Verlag, New York, NY, 1971.<br />
|-<br />
| valign=top | [McC]<br />
| McCulloch, W.S., ''Embodiments of Mind'', MIT Press, Cambridge, MA, 1965.<br />
|-<br />
| valign=top | [Mc1]<br />
| McCulloch, W.S., "A Heterarchy of Values Determined by the Topology of Nervous Nets", Bulletin of Mathematical Biophysics, vol. 7 (1945), pp. 89–93. Reprinted in [McC].<br />
|-<br />
| valign=top | [MiP]<br />
| Minsky, M.L., and Papert, S.A., ''Perceptrons : An Introduction to Computational Geometry'', MIT Press, Cambridge, MA, 1969. 2nd printing 1972. Expanded edition 1988.<br />
|-<br />
| valign=top | [Rum]<br />
| Rumelhart, D.E., Hinton, G.E., and McClelland, J.L., "A General Framework for Parallel Distributed Processing" = Chapter 2 in Rumelhart, McClelland, and the PDP Research Group, ''Parallel Distributed Processing, Explorations in the Microstructure of Cognition, Volume 1 : Foundations'', MIT Press, Cambridge, MA, 1986.<br />
|}<br />
<br />
===Incidental Works===<br />
<br />
{| cellpadding=3<br />
| valign=top | [Dew]<br />
| Dewey, John, ''How We Think'', D.C. Heath, Lexington, MA, 1910. Reprinted, Prometheus Books, Buffalo, NY, 1991.<br />
|-<br />
| valign=top | [Fou]<br />
| Foucault, Michel, ''The Archaeology of Knowledge and The Discourse on Language'', A.M. Sheridan-Smith and Rupert Swyer (trans.), Pantheon, New York, NY, 1972. Originally published as ''L´Archéologie du Savoir et L´ordre du discours'', Editions Gallimard, 1969 & 1971.<br />
|-<br />
| valign=top | [Hom]<br />
| Homer, ''The Odyssey'', with an English translation by A.T. Murray, Loeb Classical Library, Harvard University Press, Cambridge, MA, 1980. First printed 1919.<br />
|-<br />
| valign=top | [Jam]<br />
| James, William, ''Pragmatism : A New Name for Some Old Ways of Thinking'', Longmans, Green, and Company, New York, NY, 1907.<br />
|-<br />
| valign=top | [Ler]<br />
| Leroux, Gaston, ''The Phantom of the Opera'', foreword by P. Haining, Dorset Press, New York, NY, 1988. Originally published in French, 1911.<br />
|-<br />
| valign=top | [Mus]<br />
| Musil, Robert, ''The Man Without Qualities'', 3 volumes, translated with a foreword by Eithne Wilkins and Ernst Kaiser, Pan Books, London, UK, 1979. English edition first published by Secker and Warburg, 1954. Originally published in German, ''Der Mann ohne Eigenschaften'', 1930 & 1932.<br />
|-<br />
| valign=top | [PlaR]<br />
| Plato, ''The Republic'', with an English translation by Paul Shorey, Loeb Classical Library, Harvard University Press, Cambridge, MA, 1980. First printed 1930 & 1935.<br />
|-<br />
| valign=top | [PlaS]<br />
| Plato, ''The Sophist'', Loeb Classical Library, William Heinemann, London, 1921, 1987.<br />
|-<br />
| valign=top | [Qui]<br />
| Quine, W.V., ''Mathematical Logic'', 1st edition, 1940. Revised edition, 1951. Harvard University Press, Cambridge, MA, 1981.<br />
|-<br />
| valign=top | [SaD]<br />
| de Santillana, Giorgio, and von Dechend, Hertha, ''Hamlet's Mill : An Essay on Myth and the Frame of Time'', David R. Godine, Publisher, Boston, MA, 1977. 1st published 1969.<br />
|-<br />
| valign=top | [Sha]<br />
| Shakespeare, William, '' William Shakespeare : The Complete Works'', Compact Edition, S. Wells and G. Taylor (eds.), Oxford University Press, Oxford, UK, 1988.<br />
|-<br />
| valign=top | [Sh1]<br />
| Shakespeare, William, ''A Midsummer Night's Dream'', Washington Square Press, New York, NY, 1958.<br />
|-<br />
| valign=top | [Sh2]<br />
| Shakespeare, William, ''The Tragedy of Hamlet, Prince of Denmark'', In [Sha], pp. 654&ndash;690.<br />
|-<br />
| valign=top | [Sh3]<br />
| Shakespeare, William, ''Measure for Measure'', Washington Square Press, New York, NY, 1965.<br />
|-<br />
| valign=top | [Web]<br />
| ''Webster's Ninth New Collegiate Dictionary'', Merriam-Webster, Springfield, MA, 1983.<br />
|-<br />
| valign=top | [Whi]<br />
| Whitman, Walt, ''Leaves of Grass'', Vintage Books / The Library of America, New York, NY, 1992. Originally published in numerous editions, 1855&ndash;1892.<br />
|-<br />
| valign=top | [Wil]<br />
| Wilhelm, R., and Baynes, C.F. (trans.), ''The I Ching, or Book of Changes'', foreword by C.G. Jung, preface by H. Wilhelm, 3rd edition, Bollingen Series XIX, Princeton University Press, Princeton, NJ, 1967.<br />
|}<br />
<br />
==Document History==<br />
<br />
<pre><br />
Author: Jon Awbrey<br />
Created: 16 Dec 1993<br />
Relayed: 31 Oct 1994<br />
Revised: 03 Jun 2003<br />
Recoded: 03 Jun 2007<br />
</pre><br />
<br />
[[Category:Adaptive Systems]]<br />
[[Category:Artificial Intelligence]]<br />
[[Category:Boolean Algebra]]<br />
[[Category:Boolean Functions]]<br />
[[Category:Charles Sanders Peirce]]<br />
[[Category:Combinatorics]]<br />
[[Category:Computer Science]]<br />
[[Category:Cybernetics]]<br />
[[Category:Differential Logic]]<br />
[[Category:Discrete Systems]]<br />
[[Category:Dynamical Systems]]<br />
[[Category:Formal Languages]]<br />
[[Category:Formal Sciences]]<br />
[[Category:Formal Systems]]<br />
[[Category:Functional Logic]]<br />
[[Category:Graph Theory]]<br />
[[Category:Group Theory]]<br />
[[Category:Inquiry]]<br />
[[Category:Knowledge Representation]]<br />
[[Category:Linguistics]]<br />
[[Category:Logic]]<br />
[[Category:Logical Graphs]]<br />
[[Category:Mathematics]]<br />
[[Category:Mathematical Systems Theory]]<br />
[[Category:Philosophy]]<br />
[[Category:Science]]<br />
[[Category:Semiotics]]<br />
[[Category:Systems Science]]<br />
[[Category:Visualization]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Directory:Jon_Awbrey/Differential_Logic_and_Dynamic_Systems_2.0&diff=469877Directory:Jon Awbrey/Differential Logic and Dynamic Systems 2.02021-01-13T18:24:32Z<p>Jon Awbrey: parse test</p>
<hr />
<div>{{DISPLAYTITLE:Differential Logic and Dynamic Systems 2.0}}<br />
'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''<br />
<br />
{| align="center" cellpadding="10"<br />
| [[Image:Tangent_Functor_Ferris_Wheel.gif]]<br />
|}<br />
<br />
{| style="height:36px; width:100%"<br />
| align="left" | ''Stand and unfold yourself.''<br />
| align="right" | Hamlet: Francsico&mdash;1.1.2<br />
|}<br />
<br />
This article develops a differential extension of propositional calculus and applies it to a context of problems arising in dynamic systems. The work pursued here is coordinated with a parallel application that focuses on neural network systems, but the dependencies are arranged to make the present article the main and the more self-contained work, to serve as a conceptual frame and a technical background for the network project.<br />
<br />
==Review and Transition==<br />
<br />
This note continues a previous discussion on the problem of dealing with change and diversity in logic-based intelligent systems. It is useful to begin by summarizing essential material from previous reports.<br />
<br />
Table 1 outlines a notation for propositional calculus based on two types of logical connectives, both of variable <math>k\!</math>-ary scope.<br />
<br />
* A bracketed list of propositional expressions in the form <math>\texttt{(} e_1, e_2, \ldots, e_{k-1}, e_k \texttt{)}\!</math> indicates that exactly one of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> is false.<br />
<br />
* A concatenation of propositional expressions in the form <math>e_1 ~ e_2 ~ \ldots ~ e_{k-1} ~ e_k</math> indicates that all of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> are true, in other words, that their [[logical conjunction]] is true.<br />
<br />
All other propositional connectives can be obtained in a very efficient style of representation through combinations of these two forms. Strictly speaking, the concatenation form is dispensable in light of the bracketed form, but it is convenient to maintain it as an abbreviation of more complicated bracket expressions.<br />
<br />
This treatment of propositional logic is derived from the work of C.S. Peirce [P1, P2], who gave this approach an extensive development in his graphical systems of predicate, relational, and modal logic [Rob]. More recently, these ideas were revived and supplemented in an alternative interpretation by George Spencer-Brown [SpB]. Both of these authors used other forms of enclosure where I use parentheses, but the structural topologies of expression and the functional varieties of interpretation are fundamentally the same.<br />
<br />
While working with expressions solely in propositional calculus, it is easiest to use plain parentheses for logical connectives. In contexts where parentheses are needed for other purposes &ldquo;teletype&rdquo; parentheses <math>\texttt{(} \ldots \texttt{)}\!</math> or barred parentheses <math>(\!| \ldots |\!)</math> may be used for logical operators.<br />
<br />
The briefest expression for logical truth is the empty word, usually denoted by <math>{}^{\backprime\backprime} \boldsymbol\varepsilon {}^{\prime\prime}\!</math> or <math>{}^{\backprime\backprime} \boldsymbol\lambda {}^{\prime\prime}\!</math> in formal languages, where it forms the identity element for concatenation. To make it visible in this text, it may be denoted by the equivalent expression <math>{}^{\backprime\backprime} \texttt{((~))} {}^{\prime\prime},\!</math> or, especially if operating in an algebraic context, by a simple <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}.\!</math> Also when working in an algebraic mode, the plus sign <math>{}^{\backprime\backprime} + {}^{\prime\prime}\!</math> may be used for [[exclusive disjunction]]. For example, we have the following paraphrases of algebraic expressions by bracket expressions:<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
|<br />
<math>\begin{matrix}<br />
x + y ~=~ \texttt{(} x, y \texttt{)}<br />
\\[6pt]<br />
x + y + z ~=~ \texttt{((} x, y \texttt{)}, z \texttt{)} ~=~ \texttt{(} x, \texttt{(} y, z \texttt{))}<br />
\end{matrix}</math><br />
|}<br />
<br />
It is important to note that the last expressions are not equivalent to the triple bracket <math>\texttt{(} x, y, z \texttt{)}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 1.} ~~ \text{Syntax and Semantics of a Calculus for Propositional Logic}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Expression}~\!</math><br />
| <math>\text{Interpretation}\!</math><br />
| <math>\text{Other Notations}\!</math><br />
|-<br />
| &nbsp;<br />
| <math>\text{True}\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{False}\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>x\!</math><br />
| <math>x\!</math><br />
| <math>x\!</math><br />
|-<br />
| <math>\texttt{(} x \texttt{)}\!</math><br />
| <math>\text{Not}~ x\!</math><br />
|<br />
<math>\begin{matrix}<br />
x'<br />
\\<br />
\tilde{x}<br />
\\<br />
\lnot x<br />
\end{matrix}\!</math><br />
|-<br />
| <math>x~y~z\!</math><br />
| <math>x ~\text{and}~ y ~\text{and}~ z\!</math><br />
| <math>x \land y \land z\!</math><br />
|-<br />
| <math>\texttt{((} x \texttt{)(} y \texttt{)(} z \texttt{))}\!</math><br />
| <math>x ~\text{or}~ y ~\text{or}~ z\!</math><br />
| <math>x \lor y \lor z\!</math><br />
|-<br />
| <math>\texttt{(} x ~ \texttt{(} y \texttt{))}\!</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{implies}~ y<br />
\\<br />
\mathrm{If}~ x ~\text{then}~ y<br />
\end{matrix}</math><br />
| <math>x \Rightarrow y\!</math><br />
|-<br />
| <math>\texttt{(} x \texttt{,} y \texttt{)}\!</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{not equal to}~ y<br />
\\<br />
x ~\text{exclusive or}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x \ne y<br />
\\<br />
x + y<br />
\end{matrix}</math><br />
|-<br />
| <math>\texttt{((} x \texttt{,} y \texttt{))}\!</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{is equal to}~ y<br />
\\<br />
x ~\text{if and only if}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x = y<br />
\\<br />
x \Leftrightarrow y<br />
\end{matrix}</math><br />
|-<br />
| <math>\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Just one of}<br />
\\<br />
x, y, z<br />
\\<br />
\text{is false}.<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x'y~z~ & \lor<br />
\\<br />
x~y'z~ & \lor<br />
\\<br />
x~y~z' &<br />
\end{matrix}</math><br />
|-<br />
| <math>\texttt{((} x \texttt{),(} y \texttt{),(} z \texttt{))}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Just one of}<br />
\\<br />
x, y, z<br />
\\<br />
\text{is true}.<br />
\\<br />
&<br />
\\<br />
\text{Partition all}<br />
\\<br />
\text{into}~ x, y, z.<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x~y'z' & \lor<br />
\\<br />
x'y~z' & \lor<br />
\\<br />
x'y'z~ &<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,} y \texttt{),} z \texttt{)}<br />
\\<br />
&<br />
\\<br />
\texttt{(} x \texttt{,(} y \texttt{,} z \texttt{))}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Oddly many of}<br />
\\<br />
x, y, z<br />
\\<br />
\text{are true}.<br />
\end{matrix}\!</math><br />
|<br />
<p><math>x + y + z\!</math></p><br />
<br><br />
<p><math>\begin{matrix}<br />
x~y~z~ & \lor<br />
\\<br />
x~y'z' & \lor<br />
\\<br />
x'y~z' & \lor<br />
\\<br />
x'y'z~ &<br />
\end{matrix}\!</math></p><br />
|-<br />
| <math>\texttt{(} w \texttt{,(} x \texttt{),(} y \texttt{),(} z \texttt{))}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Partition}~ w<br />
\\<br />
\text{into}~ x, y, z.<br />
\\<br />
&<br />
\\<br />
\text{Genus}~ w ~\text{comprises}<br />
\\<br />
\text{species}~ x, y, z.<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
w'x'y'z' & \lor<br />
\\<br />
w~x~y'z' & \lor<br />
\\<br />
w~x'y~z' & \lor<br />
\\<br />
w~x'y'z~ &<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
'''Note.''' The usage that one often sees, of a plus sign "<math>+\!</math>" to represent inclusive disjunction, and the reference to this operation as ''boolean addition'', is a misnomer on at least two counts. Boole used the plus sign to represent exclusive disjunction (at any rate, an operation of aggregation restricted in its logical interpretation to cases where the represented sets are disjoint (Boole, 32)), as any mathematician with a sensitivity to the ring and field properties of algebra would do:<br />
<br />
<blockquote><br />
The expression <math>x + y\!</math> seems indeed uninterpretable, unless it be assumed that the things represented by <math>x\!</math> and the things represented by <math>y\!</math> are entirely separate; that they embrace no individuals in common. (Boole, 66).<br />
</blockquote><br />
<br />
It was only later that Peirce and Jevons treated inclusive disjunction as a fundamental operation, but these authors, with a respect for the algebraic properties that were already associated with the plus sign, used a variety of other symbols for inclusive disjunction (Sty, 177, 189). It seems to have been Schröder who later reassigned the plus sign to inclusive disjunction (Sty, 208). Additional information, discussion, and references can be found in (Boole) and (Sty, 177&ndash;263). Aside from these historical points, which never really count against a current practice that has gained a life of its own, this usage does have a further disadvantage of cutting or confounding the lines of communication between algebra and logic. For this reason, it will be avoided here.<br />
<br />
==A Functional Conception of Propositional Calculus==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Out of the dimness opposite equals advance . . . .<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Always substance and increase,<br><br />
Always a knit of identity . . . . always distinction . . . .<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;always a breed of life.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]<br />
|}<br />
<br />
In the general case, we start with a set of logical features <math>\{a_1, \ldots, a_n\}</math> that represent properties of objects or propositions about the world. In concrete examples the features <math>\{a_i\!\}</math> commonly appear as capital letters from an ''alphabet'' like <math>\{A, B, C, \ldots\}</math> or as meaningful words from a linguistic ''vocabulary'' of codes. This language can be drawn from any sources, whether natural, technical, or artificial in character and interpretation. In the application to dynamic systems we tend to use the letters <math>\{x_1, \ldots, x_n\}</math> as our coordinate propositions, and to interpret them as denoting properties of a system's ''state'', that is, as propositions about its location in configuration space. Because I have to consider non-deterministic systems from the outset, I often use the word ''state'' in a loose sense, to denote the position or configuration component of a contemplated state vector, whether or not it ever gets a deterministic completion.<br />
<br />
The set of logical features <math>\{a_1, \ldots, a_n\}</math> provides a basis for generating an <math>n\!</math>-dimensional ''[[universe of discourse]]'' that I denote as <math>[a_1, \ldots, a_n].</math> It is useful to consider each universe of discourse as a unified categorical object that incorporates both the set of points <math>\langle a_1, \ldots, a_n \rangle</math> and the set of propositions <math>f : \langle a_1, \ldots, a_n \rangle \to \mathbb{B}</math> that are implicit with the ordinary picture of a venn diagram on <math>n\!</math> features. Thus, we may regard the universe of discourse <math>[a_1, \ldots, a_n]</math> as an ordered pair having the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}),</math> and we may abbreviate this last type designation as <math>\mathbb{B}^n\ +\!\to \mathbb{B},</math> or even more succinctly as <math>[\mathbb{B}^n].</math> (Used this way, the angle brackets <math>\langle\ldots\rangle</math> are referred to as ''generator brackets''.)<br />
<br />
Table 2 exhibits the scheme of notation I use to formalize the domain of propositional calculus, corresponding to the logical content of truth tables and venn diagrams. Although it overworks the square brackets a bit, I also use either one of the equivalent notations <math>[n]\!</math> or <math>\mathbf{n}</math> to denote the data type of a finite set on <math>n\!</math> elements.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 2.} ~~ \text{Propositional Calculus : Basic Notation}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Symbol}\!</math><br />
| <math>\text{Notation}\!</math><br />
| <math>\text{Description}\!</math><br />
| <math>\text{Type}\!</math><br />
|-<br />
| <math>\mathfrak{A}\!</math><br />
| <math>\{ {}^{\backprime\backprime} a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} a_n {}^{\prime\prime} \}\!</math><br />
| <math>\text{Alphabet}\!</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>\mathcal{A}\!</math><br />
| <math>\{ a_1, \ldots, a_n \}\!</math><br />
| <math>\text{Basis}\!</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>A_i\!</math><br />
| <math>\{ \texttt{(} a_i \texttt{)}, a_i \}\!</math><br />
| <math>\text{Dimension}~ i\!</math><br />
| <math>\mathbb{B}\!</math><br />
|-<br />
| <math>A\!</math><br />
|<br />
<math>\begin{matrix}<br />
\langle \mathcal{A} \rangle<br />
\\[2pt]<br />
\langle a_1, \ldots, a_n \rangle<br />
\\[2pt]<br />
\{ (a_1, \ldots, a_n) \}<br />
\\[2pt]<br />
A_1 \times \ldots \times A_n<br />
\\[2pt]<br />
\textstyle \prod_{i=1}^n A_i<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Set of cells},<br />
\\[2pt]<br />
\text{coordinate tuples},<br />
\\[2pt]<br />
\text{points, or vectors}<br />
\\[2pt]<br />
\text{in the universe}<br />
\\[2pt]<br />
\text{of discourse}<br />
\end{matrix}</math><br />
| <math>\mathbb{B}^n\!</math><br />
|-<br />
| <math>A^*\!</math><br />
| <math>(\mathrm{hom} : A \to \mathbb{B})\!</math><br />
| <math>\text{Linear functions}\!</math><br />
| <math>(\mathbb{B}^n)^* \cong \mathbb{B}^n\!</math><br />
|-<br />
| <math>A^\uparrow\!</math><br />
| <math>(A \to \mathbb{B})\!</math><br />
| <math>\text{Boolean functions}\!</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}\!</math><br />
|-<br />
| <math>A^\bullet\!</math><br />
|<br />
<math>\begin{matrix}<br />
[\mathcal{A}]<br />
\\[2pt]<br />
(A, A^\uparrow)<br />
\\[2pt]<br />
(A ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
(A, (A \to \mathbb{B}))<br />
\\[2pt]<br />
[a_1, \ldots, a_n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Universe of discourse}<br />
\\[2pt]<br />
\text{based on the features}<br />
\\[2pt]<br />
\{ a_1, \ldots, a_n \}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))<br />
\\[2pt]<br />
(\mathbb{B}^n ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
[\mathbb{B}^n]<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
===Qualitative Logic and Quantitative Analogy===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
''Logical'', however, is used in a third sense, which is at once more vital and more practical; to denote, namely, the systematic care, negative and positive, taken to safeguard reflection so that it may yield the best results under the given conditions.<br />
| width="4%" | &nbsp;<br />
|- <br />
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 56]<br />
|}<br />
<br />
These concepts and notations may now be explained in greater detail. In order to begin as simply as possible, let us distinguish two levels of analysis and set out initially on the easier path. On the first level of analysis we take spaces like <math>\mathbb{B},</math> <math>\mathbb{B}^n,</math> and <math>(\mathbb{B}^n \to \mathbb{B})</math> at face value and treat them as the primary objects of interest. On the second level of analysis we use these spaces as coordinate charts for talking about points and functions in more fundamental spaces.<br />
<br />
A pair of spaces, of types <math>\mathbb{B}^n</math> and <math>(\mathbb{B}^n \to \mathbb{B}),</math> give typical expression to everything that we commonly associate with the ordinary picture of a venn diagram. The dimension, <math>n,\!</math> counts the number of "circles" or simple closed curves that are inscribed in the universe of discourse, corresponding to its relevant logical features or basic propositions. Elements of type <math>\mathbb{B}^n</math> correspond to what are often called propositional ''interpretations'' in logic, that is, the different assignments of truth values to sentence letters. Relative to a given universe of discourse, these interpretations are visualized as its ''cells'', in other words, the smallest enclosed areas or undivided regions of the venn diagram. The functions <math>f : \mathbb{B}^n \to \mathbb{B}</math> correspond to the different ways of shading the venn diagram to indicate arbitrary propositions, regions, or sets. Regions included under a shading indicate the ''models'', and regions excluded represent the ''non-models'' of a proposition. To recognize and formalize the natural cohesion of these two layers of concepts into a single universe of discourse, we introduce the type notations <math>[\mathbb{B}^n] = \mathbb{B}^n\ +\!\to \mathbb{B}</math> to stand for the pair of types <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})).</math> The resulting "stereotype" serves to frame the universe of discourse as a unified categorical object, and makes it subject to prescribed sets of evaluations and transformations (categorical morphisms or ''arrows'') that affect the universe of discourse as an integrated whole.<br />
<br />
Most of the time we can serve the algebraic, geometric, and logical interests of our study without worrying about their occasional conflicts and incidental divergences. The conventions and definitions already set down will continue to cover most of the algebraic and functional aspects of our discussion, but to handle the logical and qualitative aspects we will need to add a few more. In general, abstract sets may be denoted by gothic, greek, or script capital variants of <math>A, B, C,\!</math> and so on, with their elements being denoted by a corresponding set of subscripted letters in plain lower case, for example, <math>\mathcal{A} = \{a_i\}.</math> Most of the time, a set such as <math>\mathcal{A} = \{a_i\}\!</math> will be employed as the ''alphabet'' of a [[formal language]]. These alphabet letters serve to name the logical features (properties or propositions) that generate a particular universe of discourse. When we want to discuss the particular features of a universe of discourse, beyond the abstract designation of a type like <math>(\mathbb{B}^n\ +\!\to \mathbb{B}),</math> then we may use the following notations. If <math>\mathcal{A} = \{a_1, \ldots, a_n\}</math> is an alphabet of logical features, then <math>A = \langle \mathcal{A} \rangle = \langle a_1, \ldots, a_n \rangle</math> is the set of interpretations, <math>A^\uparrow = (A \to \mathbb{B})</math> is the set of propositions, and <math>A^\bullet = [\mathcal{A}] = [a_1, \ldots, a_n]</math> is the combination of these interpretations and propositions into the universe of discourse that is based on the features <math>\{a_1, \ldots, a_n\}.</math><br />
<br />
As always, especially in concrete examples, these rules may be dropped whenever necessary, reverting to a free assortment of feature labels. However, when we need to talk about the logical aspects of a space that is already named as a vector space, it will be necessary to make special provisions. At any rate, these elaborations can be deferred until actually needed.<br />
<br />
===Philosophy of Notation : Formal Terms and Flexible Types===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Where number is irrelevant, regimented mathematical technique has hitherto tended to be lacking. Thus it is that the progress of natural science has depended so largely upon the discernment of measurable quantity of one sort or another.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]<br />
|}<br />
<br />
For much of our discussion propositions and boolean functions are treated as the same formal objects, or as different interpretations of the same formal calculus. This rule of interpretation has exceptions, though. There is a distinctively logical interest in the use of propositional calculus that is not exhausted by its functional interpretation. It is part of our task in this study to deal with these uniquely logical characteristics as they present themselves both in our subject matter and in our formal calculus. Just to provide a hint of what's at stake: In logic, as opposed to the more imaginative realms of mathematics, we consider it a good thing to always know what we are talking about. Where mathematics encourages tolerance for uninterpreted symbols as intermediate terms, logic exerts a keener effort to interpret directly each oblique carrier of meaning, no matter how slight, and to unfold the complicities of every indirection in the flow of information. Translated into functional terms, this means that we want to maintain a continual, immediate, and persistent sense of both the converse relation <math>f^{-1} \subseteq \mathbb{B} \times \mathbb{B}^n,</math> or what is the same thing, <math>f^{-1} : \mathbb{B} \to \mathcal{P}(\mathbb{B}^n),</math> and the ''fibers'' or inverse images <math>f^{-1}(0)\!</math> and <math>f^{-1}(1),\!</math> associated with each boolean function <math>f : \mathbb{B}^n \to \mathbb{B}</math> that we use. In practical terms, the desired implementation of a propositional interpreter should incorporate our intuitive recognition that the induced partition of the functional domain into level sets <math>f^{-1}(b),\!</math> for <math>b \in \mathbb{B},</math> is part and parcel of understanding the denotative uses of each propositional function <math>f.\!</math><br />
<br />
===Special Classes of Propositions===<br />
<br />
It is important to remember that the coordinate propositions <math>\{a_i\},\!</math> besides being projection maps <math>a_i : \mathbb{B}^n \to \mathbb{B},</math> are propositions on an equal footing with all others, even though employed as a basis in a particular moment. This set of <math>n\!</math> propositions may sometimes be referred to as the ''basic propositions'', the ''coordinate propositions'', or the ''simple propositions'' that found a universe of discourse. Either one of the equivalent notations, <math>\{a_i : \mathbb{B}^n \to \mathbb{B}\}</math> or <math>(\mathbb{B}^n \xrightarrow{i} \mathbb{B}),</math> may be used to indicate the adoption of the propositions <math>a_i\!</math> as a basis for describing a universe of discourse.<br />
<br />
Among the <math>2^{2^n}</math> propositions in <math>[a_1, \ldots, a_n]</math> are several families of <math>2^n\!</math> propositions each that take on special forms with respect to the basis <math>\{ a_1, \ldots, a_n \}.</math> Three of these families are especially prominent in the present context, the ''linear'', the ''positive'', and the ''singular'' propositions. Each family is naturally parameterized by the coordinate <math>n\!</math>-tuples in <math>\mathbb{B}^n</math> and falls into <math>n + 1\!</math> ranks, with a binomial coefficient <math>\tbinom{n}{k}</math> giving the number of propositions that have rank or weight <math>k.\!</math><br />
<br />
<ul><br />
<br />
<li><br />
<p>The ''linear propositions'', <math>\{ \ell : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{\ell} \mathbb{B}),\!</math> may be written as sums:</p><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\sum_{i=1}^n e_i ~=~ e_1 + \ldots + e_n<br />
~\text{where}~<br />
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 0 \end{matrix}\right\}<br />
~\text{for}~ i = 1 ~\text{to}~ n.\!</math><br />
|}<br />
</li><br />
<br />
<li><br />
<p>The ''positive propositions'', <math>\{ p : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{p} \mathbb{B}),\!</math> may be written as products:</p><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\prod_{i=1}^n e_i ~=~ e_1 \cdot \ldots \cdot e_n<br />
~\text{where}~<br />
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 1 \end{matrix}\right\}<br />
~\text{for}~ i = 1 ~\text{to}~ n.\!</math><br />
|}<br />
</li><br />
<br />
<li><br />
<p>The ''singular propositions'', <math>\{ \mathbf{x} : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{s} \mathbb{B}),\!</math> may be written as products:</p><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\prod_{i=1}^n e_i ~=~ e_1 \cdot \ldots \cdot e_n<br />
~\text{where}~<br />
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = \texttt{(} a_i \texttt{)} \end{matrix}\right\}<br />
~\text{for}~ i = 1 ~\text{to}~ n.\!</math><br />
|}<br />
</li><br />
<br />
</ul><br />
<br />
In each case the rank <math>k\!</math> ranges from <math>0\!</math> to <math>n\!</math> and counts the number of positive appearances of the coordinate propositions <math>a_1, \ldots, a_n\!</math> in the resulting expression. For example, for <math>{n = 3},\!</math> the linear proposition of rank <math>0\!</math> is <math>0,\!</math> the positive proposition of rank <math>0\!</math> is <math>1,\!</math> and the singular proposition of rank <math>0\!</math> is <math>\texttt{(} a_1 \texttt{)(} a_2 \texttt{)(} a_3\texttt{)}.\!</math><br />
<br />
The basic propositions <math>a_i : \mathbb{B}^n \to \mathbb{B}</math> are both linear and positive. So these two kinds of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions.<br />
<br />
Linear propositions and positive propositions are generated by taking boolean sums and products, respectively, over selected subsets of basic propositions, so both families of propositions are parameterized by the powerset <math>\mathcal{P}(\mathcal{I}),</math> that is, the set of all subsets <math>J\!</math> of the basic index set <math>\mathcal{I} = \{1, \ldots, n\}.\!</math><br />
<br />
Let us define <math>\mathcal{A}_J</math> as the subset of <math>\mathcal{A}</math> that is given by <math>\{a_i : i \in J\}.\!</math> Then we may comprehend the action of the linear and the positive propositions in the following terms:<br />
<br />
* The linear proposition <math>\ell_J : \mathbb{B}^n \to \mathbb{B}</math> evaluates each cell <math>\mathbf{x}</math> of <math>\mathbb{B}^n</math> by looking at the coefficients of <math>\mathbf{x}</math> with respect to the features that <math>\ell_J</math> "likes", namely those in <math>\mathcal{A}_J,</math> and then adds them up in <math>\mathbb{B}.</math> Thus, <math>\ell_J(\mathbf{x})</math> computes the parity of the number of features that <math>\mathbf{x}</math> has in <math>\mathcal{A}_J,</math> yielding one for odd and zero for even. Expressed in this idiom, <math>\ell_J(\mathbf{x}) = 1</math> says that <math>\mathbf{x}</math> seems ''odd'' (or ''oddly true'') to <math>\mathcal{A}_J,</math> whereas <math>\ell_J(\mathbf{x}) = 0</math> says that <math>\mathbf{x}</math> seems ''even'' (or ''evenly true'') to <math>\mathcal{A}_J,</math> so long as we recall that ''zero times'' is evenly often, too.<br />
<br />
* The positive proposition <math>p_J : \mathbb{B}^n \to \mathbb{B}</math> evaluates each cell <math>\mathbf{x}</math> of <math>\mathbb{B}^n</math> by looking at the coefficients of <math>\mathbf{x}</math> with regard to the features that <math>p_J\!</math> "likes", namely those in <math>\mathcal{A}_J,</math> and then takes their product in <math>\mathbb{B}.</math> Thus, <math>p_J(\mathbf{x})</math> assesses the unanimity of the multitude of features that <math>\mathbf{x}</math> has in <math>\mathcal{A}_J,</math> yielding one for all and aught for else. In these consensual or contractual terms, <math>p_J(\mathbf{x}) = 1</math> means that <math>\mathbf{x}</math> is ''AOK'' or congruent with all of the conditions of <math>\mathcal{A}_J,</math> while <math>p_J(\mathbf{x}) = 0</math> means that <math>\mathbf{x}</math> defaults or dissents from some condition of <math>\mathcal{A}_J.</math><br />
<br />
===Basis Relativity and Type Ambiguity===<br />
<br />
Finally, two things are important to keep in mind with regard to the simplicity, linearity, positivity, and singularity of propositions.<br />
<br />
First, all of these properties are relative to a particular basis. For example, a singular proposition with respect to a basis <math>\mathcal{A}</math> will not remain singular if <math>\mathcal{A}</math> is extended by a number of new and independent features. Even if we stick to the original set of pairwise options <math>\{a_i\} \cup \{ \texttt{(} a_i \texttt{)} \}\!</math> to select a new basis, the sets of linear and positive propositions are determined by the choice of simple propositions, and this determination is tantamount to the conventional choice of a cell as origin.<br />
<br />
Second, the singular propositions <math>\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B},</math> picking out as they do a single cell or a coordinate tuple <math>\mathbf{x}</math> of <math>\mathbb{B}^n,</math> become the carriers or the vehicles of a certain type-ambiguity that vacillates between the dual forms <math>\mathbb{B}^n</math> and <math>(\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B})</math> and infects the whole hierarchy of types built on them. In other words, the terms that signify the interpretations <math>\mathbf{x} : \mathbb{B}^n</math> and the singular propositions <math>\mathbf{x} : \mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B}</math> are fully equivalent in information, and this means that every token of the type <math>\mathbb{B}^n</math> can be reinterpreted as an appearance of the subtype <math>\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B}.</math> And vice versa, the two types can be exchanged with each other everywhere that they turn up. In practical terms, this allows the use of singular propositions as a way of denoting points, forming an alternative to coordinate tuples.<br />
<br />
For example, relative to the universe of discourse <math>[a_1, a_2, a_3]\!</math> the singular proposition <math>a_1 a_2 a_3 : \mathbb{B}^3 \xrightarrow{s} \mathbb{B}</math> could be explicitly retyped as <math>a_1 a_2 a_3 : \mathbb{B}^3</math> to indicate the point <math>(1, 1, 1)\!</math> but in most cases the proper interpretation could be gathered from context. Both notations remain dependent on a particular basis, but the code that is generated under the singular option has the advantage in its self-commenting features, in other words, it constantly reminds us of its basis in the process of denoting points. When the time comes to put a multiplicity of different bases into play, and to search for objects and properties that remain invariant under the transformations between them, this infinitesimal potential advantage may well evolve into an overwhelming practical necessity.<br />
<br />
===The Analogy Between Real and Boolean Types===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Measurement consists in correlating our subject matter with the series of real numbers; and such correlations are desirable because, once they are set up, all the well-worked theory of numerical mathematics lies ready at hand as a tool for our further reasoning.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]<br />
|}<br />
<br />
There are two further reasons why it useful to spend time on a careful treatment of types, and they both have to do with our being able to take full computational advantage of certain dimensions of flexibility in the types that apply to terms. First, the domains of differential geometry and logic programming are connected by analogies between real and boolean types of the same pattern. Second, the types involved in these patterns have important isomorphisms connecting them that apply on both the real and the boolean sides of the picture.<br />
<br />
Amazingly enough, these isomorphisms are themselves schematized by the axioms and theorems of propositional logic. This fact is known as the ''propositions as types'' analogy or the Curry&ndash;Howard isomorphism [How]. In another formulation it says that terms are to types as proofs are to propositions. See [LaS, 42&ndash;46] and [SeH] for a good discussion and further references. To anticipate the bearing of these issues on our immediate topic, Table&nbsp;3 sketches a partial overview of the Real to Boolean analogy that may serve to illustrate the paradigm in question.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 3.} ~~ \text{Analogy Between Real and Boolean Types}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Real Domain} ~ \mathbb{R}\!</math><br />
| <math>\longleftrightarrow\!</math><br />
| <math>\text{Boolean Domain} ~ \mathbb{B}\!</math><br />
|-<br />
| <math>\mathbb{R}^n\!</math><br />
| <math>\text{Basic Space}\!</math><br />
| <math>\mathbb{B}^n\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}\!</math><br />
| <math>\text{Function Space}\!</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R}\!</math><br />
| <math>\text{Tangent Vector}\!</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to ((\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R})\!</math><br />
| <math>\text{Vector Field}\!</math><br />
| <math>\mathbb{B}^n \to ((\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B})\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \times (\mathbb{R}^n \to \mathbb{R})) \to \mathbb{R}\!</math><br />
| '''<font size="4">"</font>'''<br />
| <math>(\mathbb{B}^n \times (\mathbb{B}^n \to \mathbb{B})) \to \mathbb{B}\!</math><br />
|-<br />
| <math>((\mathbb{R}^n \to \mathbb{R}) \times \mathbb{R}^n) \to \mathbb{R}\!</math><br />
| '''<font size="4">"</font>'''<br />
| <math>((\mathbb{B}^n \to \mathbb{B}) \times \mathbb{B}^n) \to \mathbb{B}\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^n \to \mathbb{R})~\!</math><br />
| <math>\text{Derivation}\!</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^n \to \mathbb{B})\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}^m\!</math><br />
| <math>\begin{matrix}\text{Basic}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}^m\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^m \to \mathbb{R})\!</math><br />
| <math>\begin{matrix}\text{Function}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^m \to \mathbb{B})\!</math><br />
|}<br />
<br />
<br><br />
<br />
The Table exhibits a sample of likely parallels between the real and boolean domains. The central column gives a selection of terminology that is borrowed from differential geometry and extended in its meaning to the logical side of the Table. These are the varieties of spaces that come up when we turn to analyzing the dynamics of processes that pursue their courses through the states of an arbitrary space <math>X.\!</math> Moreover, when it becomes necessary to approach situations of overwhelming dynamic complexity in a succession of qualitative reaches, then the methods of logic that are afforded by the boolean domains, with their declarative means of synthesis and deductive modes of analysis, supply a natural battery of tools for the task.<br />
<br />
It is usually expedient to take these spaces two at a time, in dual pairs of the form <math>X\!</math> and <math>(X \to \mathbb{K}).</math> In general, one creates pairs of type schemas by replacing any space <math>X\!</math> with its dual <math>(X \to \mathbb{K}),</math> for example, pairing the type <math>X \to Y</math> with the type <math>(X \to \mathbb{K}) \to (Y \to \mathbb{K}),</math> and <math>X \times Y</math> with <math>(X \to \mathbb{K}) \times (Y \to \mathbb{K}).</math> The word ''dual'' is used here in its broader sense to mean all of the functionals, not just the linear ones. Given any function <math>f : X \to \mathbb{K},</math> the ''converse'' or inverse relation corresponding to <math>f\!</math> is denoted <math>f^{-1},\!</math> and the subsets of <math>X\!</math> that are defined by <math>f^{-1}(k),\!</math> taken over <math>k\!</math> in <math>\mathbb{K},</math> are called the ''fibers'' or the ''level sets'' of the function <math>f.\!</math><br />
<br />
===Theory of Control and Control of Theory===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
You will hardly know who I am or what I mean,<br><br />
But I shall be good health to you nevertheless,<br><br />
And filter and fibre your blood.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]<br />
|}<br />
<br />
In the boolean context a function <math>f : X \to \mathbb{B}</math> is tantamount to a ''proposition'' about elements of <math>X,\!</math> and the elements of <math>X\!</math> constitute the ''interpretations'' of that proposition. The fiber <math>f^{-1}(1)\!</math> comprises the set of ''models'' of <math>f,\!</math> or examples of elements in <math>X\!</math> satisfying the proposition <math>f.\!</math> The fiber <math>f^{-1}(0)\!</math> collects the complementary set of ''anti-models'', or the exceptions to the proposition <math>f\!</math> that exist in <math>X.\!</math> Of course, the space of functions <math>(X \to \mathbb{B})\!</math> is isomorphic to the set of all subsets of <math>X,\!</math> called the ''power set'' of <math>X,\!</math> and often denoted <math>\mathcal{P}(X)</math> or <math>2^X.\!</math><br />
<br />
The operation of replacing <math>X\!</math> by <math>(X \to \mathbb{B})\!</math> in a type schema corresponds to a certain shift of attitude towards the space <math>X,\!</math> in which one passes from a focus on the ostensibly individual elements of <math>X\!</math> to a concern with the states of information and uncertainty that one possesses about objects and situations in <math>X.\!</math> The conceptual obstacles in the path of this transition can be smoothed over by using singular functions <math>(\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B})</math> as stepping stones. First of all, it's an easy step from an element <math>\mathbf{x}</math> of type <math>\mathbb{B}^n</math> to the equivalent information of a singular proposition <math>\mathbf{x} : X \xrightarrow{s} \mathbb{B}, </math> and then only a small jump of generalization remains to reach the type of an arbitrary proposition <math>f : X \to \mathbb{B},</math> perhaps understood to indicate a relaxed constraint on the singularity of points or a neighborhood circumscribing the original <math>\mathbf{x}.</math> This is frequently a useful transformation, communicating between the ''objective'' and the ''intentional'' perspectives, in spite perhaps of the open objection that this distinction is transient in the mean time and ultimately superficial.<br />
<br />
It is hoped that this measure of flexibility, allowing us to stretch a point into a proposition, can be useful in the examination of inquiry driven systems, where the differences between empirical, intentional, and theoretical propositions constitute the discrepancies and the distributions that drive experimental activity. I can give this model of inquiry a cybernetic cast by realizing that theory change and theory evolution, as well as the choice and the evaluation of experiments, are actions that are taken by a system or its agent in response to the differences that are detected between observational contents and theoretical coverage.<br />
<br />
All of the above notwithstanding, there are several points that distinguish these two tasks, namely, the ''theory of control'' and the ''control of theory'', features that are often obscured by too much precipitation in the quickness with which we understand their similarities. In the control of uncertainty through inquiry, some of the actuators that we need to be concerned with are axiom changers and theory modifiers, operators with the power to compile and to revise the theories that generate expectations and predictions, effectors that form and edit our grammars for the languages of observational data, and agencies that rework the proposed model to fit the actual sequences of events and the realized relationships of values that are observed in the environment. Moreover, when steps must be taken to carry out an experimental action, there must be something about the particular shape of our uncertainty that guides us in choosing what directions to explore, and this impression is more than likely influenced by previous accumulations of experience. Thus it must be anticipated that much of what goes into scientific progress, or any sustainable effort toward a goal of knowledge, is necessarily predicated on long term observation and modal expectations, not only on the more local or short term prediction and correction.<br />
<br />
===Propositions as Types and Higher Order Types===<br />
<br />
The types collected in Table&nbsp;3 (repeated below) serve to illustrate the themes of ''higher order propositional expressions'' and the ''propositions as types'' (PAT) analogy.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 3.} ~~ \text{Analogy Between Real and Boolean Types}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Real Domain} ~ \mathbb{R}\!</math><br />
| <math>\longleftrightarrow\!</math><br />
| <math>\text{Boolean Domain} ~ \mathbb{B}\!</math><br />
|-<br />
| <math>\mathbb{R}^n\!</math><br />
| <math>\text{Basic Space}\!</math><br />
| <math>\mathbb{B}^n\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}\!</math><br />
| <math>\text{Function Space}\!</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R}\!</math><br />
| <math>\text{Tangent Vector}\!</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to ((\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R})\!</math><br />
| <math>\text{Vector Field}\!</math><br />
| <math>\mathbb{B}^n \to ((\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B})\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \times (\mathbb{R}^n \to \mathbb{R})) \to \mathbb{R}\!</math><br />
| '''<font size="4">"</font>'''<br />
| <math>(\mathbb{B}^n \times (\mathbb{B}^n \to \mathbb{B})) \to \mathbb{B}\!</math><br />
|-<br />
| <math>((\mathbb{R}^n \to \mathbb{R}) \times \mathbb{R}^n) \to \mathbb{R}\!</math><br />
| '''<font size="4">"</font>'''<br />
| <math>((\mathbb{B}^n \to \mathbb{B}) \times \mathbb{B}^n) \to \mathbb{B}\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^n \to \mathbb{R})~\!</math><br />
| <math>\text{Derivation}\!</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^n \to \mathbb{B})\!</math><br />
|-<br />
| <math>\mathbb{R}^n \to \mathbb{R}^m\!</math><br />
| <math>\begin{matrix}\text{Basic}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>\mathbb{B}^n \to \mathbb{B}^m\!</math><br />
|-<br />
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^m \to \mathbb{R})\!</math><br />
| <math>\begin{matrix}\text{Function}\\[2pt]\text{Transformation}\end{matrix}</math><br />
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^m \to \mathbb{B})\!</math><br />
|}<br />
<br />
<br><br />
<br />
First, observe that the type of a ''tangent vector at a point'', also known as a ''directional derivative'' at that point, has the form <math>(\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K},</math> where <math>\mathbb{K}</math> is the chosen ground field, in the present case either <math>\mathbb{R}</math> or <math>\mathbb{B}.</math> At a point in a space of type <math>\mathbb{K}^n,</math> a directional derivative operator <math>\vartheta\!</math> takes a function on that space, an <math>f\!</math> of type <math>(\mathbb{K}^n \to \mathbb{K}),</math> and maps it to a ground field value of type <math>\mathbb{K}.</math> This value is known as the ''derivative'' of <math>f\!</math> in the direction <math>\vartheta\!</math> [Che46, 76&ndash;77]. In the boolean case <math>\vartheta : (\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math> has the form of a proposition about propositions, in other words, a proposition of the next higher type.<br />
<br />
Next, by way of illustrating the propositions as types idea, consider a proposition of the form <math>X \Rightarrow (Y \Rightarrow Z).</math> One knows from propositional calculus that this is logically equivalent to a proposition of the form <math>(X \land Y) \Rightarrow Z.</math> But this equivalence should remind us of the functional isomorphism that exists between a construction of the type <math>X \to (Y \to Z)</math> and a construction of the type <math>(X \times Y) \to Z.</math> The propositions as types analogy permits us to take a functional type like this and, under the right conditions, replace the functional arrows &ldquo;<math>\to~\!</math>&rdquo; and products &ldquo;<math>\times\!</math>&rdquo; with the respective logical arrows &ldquo;<math>\Rightarrow\!</math>&rdquo; and products &ldquo;<math>\land\!</math>&rdquo;. Accordingly, viewing the result as a proposition, we can employ axioms and theorems of propositional calculus to suggest appropriate isomorphisms among the categorical and functional constructions.<br />
<br />
Finally, examine the middle four rows of Table&nbsp;3. These display a series of isomorphic types that stretch from the categories that are labeled ''Vector Field'' to those that are labeled ''Derivation''. A ''vector field'', also known as an ''infinitesimal transformation'', associates a tangent vector at a point with each point of a space. In symbols, a vector field is a function of the form <math>\textstyle \xi : X \to \bigcup_{x \in X} \xi_x\!</math> that assigns to each point <math>x\!</math> of the space <math>X\!</math> a tangent vector to <math>X\!</math> at that point, namely, the tangent vector <math>\xi_x\!</math> [Che46, 82&ndash;83]. If <math>X\!</math> is of the type <math>\mathbb{K}^n,</math> then <math>\xi\!</math> is of the type <math>\mathbb{K}^n \to ((\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K}).</math> This has the pattern <math>X \to (Y \to Z),</math> with <math>X = \mathbb{K}^n,</math> <math>Y = (\mathbb{K}^n \to \mathbb{K}),</math> and <math>Z = \mathbb{K}.</math><br />
<br />
Applying the propositions as types analogy, one can follow this pattern through a series of metamorphoses from the type of a vector field to the type of a derivation, as traced out in Table&nbsp;4. Observe how the function <math>f : X \to \mathbb{K},</math> associated with the place of <math>Y\!</math> in the pattern, moves through its paces from the second to the first position. In this way, the vector field <math>\xi,\!</math> initially viewed as attaching each tangent vector <math>\xi_x\!</math> to the site <math>x\!</math> where it acts in <math>X,\!</math> now comes to be seen as acting on each scalar potential <math>f : X \to \mathbb{K}</math> like a generalized species of differentiation, producing another function <math>\xi f : X \to \mathbb{K}</math> of the same type.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 4.} ~~ \text{An Equivalence Based on the Propositions as Types Analogy}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Pattern}\!</math><br />
| <math>\text{Construct}\!</math><br />
| <math>\text{Instance}\!</math><br />
|-<br />
| <math>X \to (Y \to Z)\!</math><br />
| <math>\text{Vector Field}\!</math><br />
| <math>\mathbb{K}^n \to ((\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K})\!</math><br />
|-<br />
| <math>(X \times Y) \to Z\!</math><br />
| <math>\Uparrow\!</math><br />
| <math>(\mathbb{K}^n \times (\mathbb{K}^n \to \mathbb{K})) \to \mathbb{K}\!</math><br />
|-<br />
| <math>(Y \times X) \to Z\!</math><br />
| <math>\Downarrow\!</math><br />
| <math>((\mathbb{K}^n \to \mathbb{K}) \times \mathbb{K}^n) \to \mathbb{K}\!</math><br />
|-<br />
| <math>Y \to (X \to Z)\!</math><br />
| <math>\text{Derivation}\!</math><br />
| <math>(\mathbb{K}^n \to \mathbb{K}) \to (\mathbb{K}^n \to \mathbb{K})\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Reality at the Threshold of Logic===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
But no science can rest entirely on measurement, and many scientific investigations are quite out of reach of that device. To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]<br />
|}<br />
<br />
Table 5 accumulates an array of notation that I hope will not be too distracting. Some of it is rarely needed, but has been filled in for the sake of completeness. Its purpose is simple, to give literal expression to the visual intuitions that come with venn diagrams, and to help build a bridge between our qualitative and quantitative outlooks on dynamic systems.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 5.} ~~ \text{A Bridge Over Troubled Waters}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| align="center" | <math>\text{Linear Space}\!</math><br />
| align="center" | <math>\text{Liminal Space}\!</math><br />
| align="center" | <math>\text{Logical Space}\!</math><br />
|-<br />
| <math>\begin{matrix}\mathcal{X} & = & \{ x_1, \ldots, x_n \}\end{matrix}</math><br />
| <math>\begin{matrix}\underline{\mathcal{X}} & = & \{ \underline{x}_1, \ldots, \underline{x}_n \}\end{matrix}</math><br />
| <math>\begin{matrix}\mathcal{A} & = & \{ a_1, \ldots, a_n \}\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X_i & = & \langle x_i \rangle<br />
\\<br />
& \cong & \mathbb{K}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}_i & = & \{ \texttt{(} \underline{x}_i \texttt{)}, \underline{x}_i \}<br />
\\<br />
& \cong & \mathbb{B}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A_i & = & \{ \texttt{(} a_i \texttt{)}, a_i \}<br />
\\<br />
& \cong & \mathbb{B}<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X<br />
\\<br />
= & \langle \mathcal{X} \rangle<br />
\\<br />
= & \langle x_1, \ldots, x_n \rangle<br />
\\<br />
= & X_1 \times \ldots \times X_n<br />
\\<br />
= & \prod_{i=1}^n X_i<br />
\\<br />
\cong & \mathbb{K}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}<br />
\\<br />
= & \langle \underline{\mathcal{X}} \rangle<br />
\\<br />
= & \langle \underline{x}_1, \ldots, \underline{x}_n \rangle<br />
\\<br />
= & \underline{X}_1 \times \ldots \times \underline{X}_n<br />
\\<br />
= & \prod_{i=1}^n \underline{X}_i<br />
\\<br />
\cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A<br />
\\<br />
= & \langle \mathcal{A} \rangle<br />
\\<br />
= & \langle a_1, \ldots, a_n \rangle<br />
\\<br />
= & A_1 \times \ldots \times A_n<br />
\\<br />
= & \prod_{i=1}^n A_i<br />
\\<br />
\cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X^* & = & (\ell : X \to \mathbb{K})<br />
\\<br />
& \cong & \mathbb{K}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}^* & = & (\ell : \underline{X} \to \mathbb{B})<br />
\\<br />
& \cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A^* & = & (\ell : A \to \mathbb{B})<br />
\\<br />
& \cong & \mathbb{B}^n<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X^\uparrow & = & (X \to \mathbb{K})<br />
\\<br />
& \cong & (\mathbb{K}^n \to \mathbb{K})<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}^\uparrow & = & (\underline{X} \to \mathbb{B})<br />
\\<br />
& \cong & (\mathbb{B}^n \to \mathbb{B})<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A^\uparrow & = & (A \to \mathbb{B})<br />
\\<br />
& \cong & (\mathbb{B}^n \to \mathbb{B})<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
X^\bullet<br />
\\<br />
= & [\mathcal{X}]<br />
\\<br />
= & [x_1, \ldots, x_n]<br />
\\<br />
= & (X, X^\uparrow)<br />
\\<br />
= & (X ~+\!\to \mathbb{K})<br />
\\<br />
= & (X, (X \to \mathbb{K}))<br />
\\<br />
\cong & (\mathbb{K}^n, (\mathbb{K}^n \to \mathbb{K}))<br />
\\<br />
= & (\mathbb{K}^n ~+\!\to \mathbb{K})<br />
\\<br />
= & [\mathbb{K}^n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\underline{X}^\bullet<br />
\\<br />
= & [\underline{\mathcal{X}}]<br />
\\<br />
= & [\underline{x}_1, \ldots, \underline{x}_n]<br />
\\<br />
= & (\underline{X}, \underline{X}^\uparrow)<br />
\\<br />
= & (\underline{X} ~+\!\to \mathbb{B})<br />
\\<br />
= & (\underline{X}, (\underline{X} \to \mathbb{B}))<br />
\\<br />
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))<br />
\\<br />
= & (\mathbb{B}^n ~+\!\to \mathbb{B})<br />
\\<br />
= & [\mathbb{B}^n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A^\bullet<br />
\\<br />
= & [\mathcal{A}]<br />
\\<br />
= & [a_1, \ldots, a_n]<br />
\\<br />
= & (A, A^\uparrow)<br />
\\<br />
= & (A ~+\!\to \mathbb{B})<br />
\\<br />
= & (A, (A \to \mathbb{B}))<br />
\\<br />
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))<br />
\\<br />
= & (\mathbb{B}^n ~+\!\to \mathbb{B})<br />
\\<br />
= & [\mathbb{B}^n]<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
The left side of the Table collects mostly standard notation for an <math>n\!</math>-dimensional vector space over a field <math>\mathbb{K}.</math> The right side of the table repeats the first elements of a notation that I sketched above, to be used in further developments of propositional calculus. (I plan to use this notation in the logical analysis of neural network systems.) The middle column of the table is designed as a transitional step from the case of an arbitrary field <math>\mathbb{K},</math> with a special interest in the continuous line <math>\mathbb{R},</math> to the qualitative and discrete situations that are instanced and typified by <math>\mathbb{B}.</math><br />
<br />
I now proceed to explain these concepts in more detail. The most important ideas developed in Table&nbsp;5 are these:<br />
<br />
* The idea of a universe of discourse, which includes both a space of ''points'' and a space of ''maps'' on those points.<br />
<br />
* The idea of passing from a more complex universe to a simpler universe by a process of ''thresholding'' each dimension of variation down to a single bit of information.<br />
<br />
For the sake of concreteness, let us suppose that we start with a continuous <math>n\!</math>-dimensional vector space like <math>X = \langle x_1, \ldots, x_n \rangle \cong \mathbb{R}^n.</math> The coordinate system <math>\mathcal{X} = \{x_i\}</math> is a set of maps <math>x_i : \mathbb{R}^n \to \mathbb{R},</math> also known as the ''coordinate projections''. Given a "dataset" of points <math>\mathbf{x}</math> in <math>\mathbb{R}^n,</math> there are numerous ways of sensibly reducing the data down to one bit for each dimension. One strategy that is general enough for our present purposes is as follows. For each <math>i\!</math> we choose an <math>n\!</math>-ary relation <math>L_i\!</math> on <math>\mathbb{R}^n,</math> that is, a subset of <math>\mathbb{R}^n,</math> and then we define the <math>i^\mathrm{th}\!</math> threshold map, or ''limen'' <math>\underline{x}_i</math> as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\underline{x}_i : \mathbb{R}^n \to \mathbb{B}\ \text{such that:}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
\underline{x}_i(\mathbf{x}) = 1 & \text{if} & \mathbf{x} \in L_i,<br />
\\[4pt]<br />
\underline{x}_i(\mathbf{x}) = 0 & \text{if} & \mathbf{x} \not\in L_i.<br />
\end{matrix}</math><br />
|}<br />
<br />
In other notations that are sometimes used, the operator <math>\chi (\ldots)</math> or the corner brackets <math>\lceil\ldots\rceil</math> can be used to denote a ''characteristic function'', that is, a mapping from statements to their truth values in <math>\mathbb{B}.</math> Finally, it is not uncommon to use the name of the relation itself as a predicate that maps <math>n\!</math>-tuples into truth values. Thus we have the following notational variants of the above definition:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
\underline{x}_i (\mathbf{x}) & = & \chi (\mathbf{x} \in L_i) & = & \lceil \mathbf{x} \in L_i \rceil & = & L_i (\mathbf{x}).<br />
\end{matrix}</math><br />
|}<br />
<br />
Notice that, as defined here, there need be no actual relation between the <math>n\!</math>-dimensional subsets <math>\{L_i\}\!</math> and the coordinate axes corresponding to <math>\{x_i\},\!</math> aside from the circumstance that the two sets have the same cardinality. In concrete cases, though, one usually has some reason for associating these "volumes" with these "lines", for instance, <math>L_i\!</math> is bounded by some hyperplane that intersects the <math>i^\text{th}\!</math> axis at a unique threshold value <math>r_i \in \mathbb{R}.\!</math> Often, the hyperplane is chosen normal to the axis. In recognition of this motive, let us make the following convention. When the set <math>L_i\!</math> has points on the <math>i^\text{th}\!</math> axis, that is, points of the form <math>(0, \ldots, 0, r_i, 0, \ldots, 0)</math> where only the <math>x_i\!</math> coordinate is possibly non-zero, we may pick any one of these coordinate values as a parametric index of the relation. In this case we say that the indexing is ''real'', otherwise the indexing is ''imaginary''. For a knowledge based system <math>X,\!</math> this should serve once again to mark the distinction between ''acquaintance'' and ''opinion''.<br />
<br />
States of knowledge about the location of a system or about the distribution of a population of systems in a state space <math>X = \mathbb{R}^n</math> can now be expressed by taking the set <math>\underline{\mathcal{X}} = \{\underline{x}_i\}</math> as a basis of logical features. In picturesque terms, one may think of the underscore and the subscript as combining to form a subtextual spelling for the <math>i^\text{th}\!</math> threshold map. This can help to remind us that the ''threshold operator'' <math>(\underline{~})_i</math> acts on <math>\mathbf{x}</math> by setting up a kind of a &ldquo;hurdle&rdquo; for it. In this interpretation the coordinate proposition <math>\underline{x}_i</math> asserts that the representative point <math>\mathbf{x}</math> resides ''above'' the <math>i^\mathrm{th}\!</math> threshold.<br />
<br />
Primitive assertions of the form <math>\underline{x}_i (\mathbf{x})</math> may then be negated and joined by means of propositional connectives in the usual ways to provide information about the state <math>\mathbf{x}</math> of a contemplated system or a statistical ensemble of systems. Parentheses <math>\texttt{(} \ldots \texttt{)}\!</math> may be used to indicate logical negation. Eventually one discovers the usefulness of the <math>k\!</math>-ary ''just one false'' operators of the form <math>\texttt{(} a_1 \texttt{,} \ldots \texttt{,} a_k \texttt{)},\!</math> as treated in earlier reports. This much tackle generates a space of points (cells, interpretations), <math>\underline{X} \cong \mathbb{B}^n,</math> and a space of functions (regions, propositions), <math>\underline{X}^\uparrow \cong (\mathbb{B}^n \to \mathbb{B}).</math> Together these form a new universe of discourse <math>\underline{X}^\bullet</math> of the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})),</math> which we may abbreviate as <math>\mathbb{B}^n\ +\!\to \mathbb{B}</math> or most succinctly as <math>[\mathbb{B}^n].</math><br />
<br />
The square brackets have been chosen to recall the rectangular frame of a venn diagram. In thinking about a universe of discourse it is a good idea to keep this picture in mind, graphically illustrating the links among the elementary cells <math>\underline{\mathbf{x}},</math> the defining features <math>\underline{x}_i,</math> and the potential shadings <math>f : \underline{X} \to \mathbb{B}</math> all at the same time, not to mention the arbitrariness of the way we choose to inscribe our distinctions in the medium of a continuous space.<br />
<br />
Finally, let <math>X^*\!</math> denote the space of linear functions, <math>(\ell : X \to \mathbb{K}),</math> which has in the finite case the same dimensionality as <math>X,\!</math> and let the same notation be extended across the Table.<br />
<br />
We have just gone through a lot of work, apparently doing nothing more substantial than spinning a complex spell of notational devices through a labyrinth of baffled spaces and baffling maps. The reason for doing this was to bind together and to constitute the intuitive concept of a universe of discourse into a coherent categorical object, the kind of thing, once grasped, that can be turned over in the mind and considered in all its manifold changes and facets. The effort invested in these preliminary measures is intended to pay off later, when we need to consider the state transformations and the time evolution of neural network systems.<br />
<br />
===Tables of Propositional Forms===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope. It provides explicit techniques for manipulating the most basic ingredients of discourse.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7&ndash;8]<br />
|}<br />
<br />
To prepare for the next phase of discussion, Tables&nbsp;6 and 7 collect and summarize all of the propositional forms on one and two variables. These propositional forms are represented over bases of boolean variables as complete sets of boolean-valued functions. Adjacent to their names and specifications are listed what are roughly the simplest expressions in the ''[[cactus language]]'', the particular syntax for propositional calculus that I use in formal and computational contexts. For the sake of orientation, the English paraphrases and the more common notations are listed in the last two columns. As simple and circumscribed as these low-dimensional universes may appear to be, a careful exploration of their differential extensions will involve us in complexities sufficient to demand our attention for some time to come.<br />
<br />
Propositional forms on one variable correspond to boolean functions <math>f : \mathbb{B}^1 \to \mathbb{B}.</math> In Table&nbsp;6 these functions are listed in a variant form of [[truth table]], one in which the axes of the usual arrangement are rotated through a right angle. Each function <math>f_i\!</math> is indexed by the string of values that it takes on the points of the universe <math>X^\bullet = [x] \cong \mathbb{B}^1.</math> The binary index generated in this way is converted to its decimal equivalent and these are used as conventional names for the <math>f_i,\!</math> as shown in the first column of the Table. In their own right the <math>2^1\!</math> points of the universe <math>X^\bullet</math> are coordinated as a space of type <math>\mathbb{B}^1,</math> this in light of the universe <math>X^\bullet</math> being a functional domain where the coordinate projection <math>x\!</math> takes on its values in <math>\mathbb{B}.</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 6.} ~~ \text{Propositional Forms on One Variable}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}</math><br />
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_0\!</math><br />
| <math>f_{00}\!</math><br />
| <math>0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>f_1\!</math><br />
| <math>f_{01}\!</math><br />
| <math>0~1\!</math><br />
| <math>\texttt{(} x \texttt{)}\!</math><br />
| <math>\text{not}~ x\!</math><br />
| <math>\lnot x\!</math><br />
|-<br />
| <math>f_2\!</math><br />
| <math>f_{10}\!</math><br />
| <math>1~0\!</math><br />
| <math>x\!</math><br />
| <math>x\!</math><br />
| <math>x\!</math><br />
|-<br />
| <math>f_3\!</math><br />
| <math>f_{11}\!</math><br />
| <math>1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
Propositional forms on two variables correspond to boolean functions <math>f : \mathbb{B}^2 \to \mathbb{B}.</math> In Table&nbsp;7 each function <math>f_i\!</math> is indexed by the values that it takes on the points of the universe <math>X^\bullet = [x, y] \cong \mathbb{B}^2.</math> Converting the binary index thus generated to a decimal equivalent, we obtain the functional nicknames that are listed in the first column. The <math>2^2\!</math> points of the universe <math>X^\bullet</math> are coordinated as a space of type <math>\mathbb{B}^2,</math> as indicated under the heading of the Table, where the coordinate projections <math>x\!</math> and <math>y\!</math> run through the various combinations of their values in <math>\mathbb{B}.</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 7-a.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}\!</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}\!</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}\!</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}\!</math><br />
| style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}\!</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}\!</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[4pt]<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\\[4pt]<br />
f_{3}<br />
\\[4pt]<br />
f_{4}<br />
\\[4pt]<br />
f_{5}<br />
\\[4pt]<br />
f_{6}<br />
\\[4pt]<br />
f_{7}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0000}<br />
\\[4pt]<br />
f_{0001}<br />
\\[4pt]<br />
f_{0010}<br />
\\[4pt]<br />
f_{0011}<br />
\\[4pt]<br />
f_{0100}<br />
\\[4pt]<br />
f_{0101}<br />
\\[4pt]<br />
f_{0110}<br />
\\[4pt]<br />
f_{0111}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~0<br />
\\[4pt]<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~0~1~1<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
0~1~0~1<br />
\\[4pt]<br />
0~1~1~0<br />
\\[4pt]<br />
0~1~1~1<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{)} ~ y ~<br />
\\[4pt]<br />
\texttt{(} x \texttt{)}<br />
\\[4pt]<br />
~ x ~ \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{,} ~ y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x ~ y \texttt{)}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{false}<br />
\\[4pt]<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\[4pt]<br />
y ~\text{without}~ x<br />
\\[4pt]<br />
\text{not}~ x<br />
\\[4pt]<br />
x ~\text{without}~ y<br />
\\[4pt]<br />
\text{not}~ y<br />
\\[4pt]<br />
x ~\text{not equal to}~ y<br />
\\[4pt]<br />
\text{not both}~ x ~\text{and}~ y<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
\lnot x \land \lnot y<br />
\\[4pt]<br />
\lnot x \land y<br />
\\[4pt]<br />
\lnot x<br />
\\[4pt]<br />
x \land \lnot y<br />
\\[4pt]<br />
\lnot y<br />
\\[4pt]<br />
x \ne y<br />
\\[4pt]<br />
\lnot x \lor \lnot y<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{8}<br />
\\[4pt]<br />
f_{9}<br />
\\[4pt]<br />
f_{10}<br />
\\[4pt]<br />
f_{11}<br />
\\[4pt]<br />
f_{12}<br />
\\[4pt]<br />
f_{13}<br />
\\[4pt]<br />
f_{14}<br />
\\[4pt]<br />
f_{15}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1000}<br />
\\[4pt]<br />
f_{1001}<br />
\\[4pt]<br />
f_{1010}<br />
\\[4pt]<br />
f_{1011}<br />
\\[4pt]<br />
f_{1100}<br />
\\[4pt]<br />
f_{1101}<br />
\\[4pt]<br />
f_{1110}<br />
\\[4pt]<br />
f_{1111}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
1~0~0~0<br />
\\[4pt]<br />
1~0~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\\[4pt]<br />
1~1~1~1<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x ~ y<br />
\\[4pt]<br />
\texttt{((} x \texttt{,} ~ y \texttt{))}<br />
\\[4pt]<br />
y<br />
\\[4pt]<br />
\texttt{(} x ~ \texttt{(} y \texttt{))}<br />
\\[4pt]<br />
x<br />
\\[4pt]<br />
\texttt{((} x \texttt{)} ~ y \texttt{)}<br />
\\[4pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
\texttt{((~))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x ~\text{and}~ y<br />
\\[4pt]<br />
x ~\text{equal to}~ y<br />
\\[4pt]<br />
y<br />
\\[4pt]<br />
\text{not}~ x ~\text{without}~ y<br />
\\[4pt]<br />
x<br />
\\[4pt]<br />
\text{not}~ y ~\text{without}~ x<br />
\\[4pt]<br />
x ~\text{or}~ y<br />
\\[4pt]<br />
\text{true}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x \land y<br />
\\[4pt]<br />
x = y<br />
\\[4pt]<br />
y<br />
\\[4pt]<br />
x \Rightarrow y<br />
\\[4pt]<br />
x<br />
\\[4pt]<br />
x \Leftarrow y<br />
\\[4pt]<br />
x \lor y<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 7-b.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}</math><br />
| style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}</math><br />
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>f_{0000}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\\[4pt]<br />
f_{4}<br />
\\[4pt]<br />
f_{8}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0001}<br />
\\[4pt]<br />
f_{0010}<br />
\\[4pt]<br />
f_{0100}<br />
\\[4pt]<br />
f_{1000}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
1~0~0~0<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[4pt]<br />
\texttt{(} x \texttt{)} ~ y ~<br />
\\[4pt]<br />
~ x ~ \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
~ x ~~ y ~<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\[4pt]<br />
y ~\text{without}~ x<br />
\\[4pt]<br />
x ~\text{without}~ y<br />
\\[4pt]<br />
x ~\text{and}~ y<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot x \land \lnot y<br />
\\[4pt]<br />
\lnot x \land y<br />
\\[4pt]<br />
x \land \lnot y<br />
\\[4pt]<br />
x \land y<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{3}<br />
\\[4pt]<br />
f_{12}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0011}<br />
\\[4pt]<br />
f_{1100}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\[4pt]<br />
x<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{not}~ x<br />
\\[4pt]<br />
x<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot x<br />
\\[4pt]<br />
x<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[4pt]<br />
f_{9}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0110}<br />
\\[4pt]<br />
f_{1001}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[4pt]<br />
1~0~0~1<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{,} y \texttt{)}<br />
\\[4pt]<br />
\texttt{((} x \texttt{,} y \texttt{))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x ~\text{not equal to}~ y<br />
\\[4pt]<br />
x ~\text{equal to}~ y<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
x \ne y<br />
\\[4pt]<br />
x = y<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{5}<br />
\\[4pt]<br />
f_{10}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0101}<br />
\\[4pt]<br />
f_{1010}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\[4pt]<br />
y<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{not}~ y<br />
\\[4pt]<br />
y<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot y<br />
\\[4pt]<br />
y<br />
\end{matrix}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[4pt]<br />
f_{11}<br />
\\[4pt]<br />
f_{13}<br />
\\[4pt]<br />
f_{14}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0111}<br />
\\[4pt]<br />
f_{1011}<br />
\\[4pt]<br />
f_{1101}<br />
\\[4pt]<br />
f_{1110}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} ~ x ~~ y ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{(} ~ x ~ \texttt{(} y \texttt{))}<br />
\\[4pt]<br />
\texttt{((} x \texttt{)} ~ y ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\text{not both}~ x ~\text{and}~ y<br />
\\[4pt]<br />
\text{not}~ x ~\text{without}~ y<br />
\\[4pt]<br />
\text{not}~ y ~\text{without}~ x<br />
\\[4pt]<br />
x ~\text{or}~ y<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\lnot x \lor \lnot y<br />
\\[4pt]<br />
x \Rightarrow y<br />
\\[4pt]<br />
x \Leftarrow y<br />
\\[4pt]<br />
x \lor y<br />
\end{matrix}\!</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>f_{1111}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
==A Differential Extension of Propositional Calculus==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Fire over water:<br><br />
The image of the condition before transition.<br><br />
Thus the superior man is careful<br><br />
In the differentiation of things,<br><br />
So that each finds its place.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; ''I Ching'', Hexagram 64, [Wil, 249]<br />
|}<br />
<br />
This much preparation is enough to begin introducing my subject, if I excuse myself from giving full arguments for my definitional choices until some later stage. I am trying to develop a ''differential theory of qualitative equations'' that parallels the application of differential geometry to dynamic systems. The idea of a tangent vector is key to this work and a major goal is to find the right logical analogues of tangent spaces, bundles, and functors. The strategy is taken of looking for the simplest versions of these constructions that can be discovered within the realm of propositional calculus, so long as they serve to fill out the general theme.<br />
<br />
===Differential Propositions : Qualitative Analogues of Differential Equations===<br />
<br />
In order to define the differential extension of a universe of discourse <math>[\mathcal{A}],</math> the initial alphabet <math>\mathcal{A}</math> must be extended to include a collection of symbols for ''differential features'', or basic ''changes'' that are capable of occurring in <math>[\mathcal{A}].</math> Intuitively, these symbols may be construed as denoting primitive features of change, qualitative attributes of motion, or propositions about how things or points in the universe may be changing or moving with respect to the features that are noted in the initial alphabet.<br />
<br />
Therefore, let us define the corresponding ''differential alphabet'' or ''tangent alphabet'' as <math>\mathrm{d}\mathcal{A}\!</math> <math>=\!</math> <math>\{\mathrm{d}a_1, \ldots, \mathrm{d}a_n\},</math> in principle just an arbitrary alphabet of symbols, disjoint from the initial alphabet <math>\mathcal{A}\!</math> <math>=\!</math> <math>\{ a_1, \ldots, a_n \},\!</math> that is intended to be interpreted in the way just indicated. It only remains to be understood that the precise interpretation of the symbols in <math>\mathrm{d}\mathcal{A}\!</math> is often conceived to be changeable from point to point of the underlying space <math>A.\!</math> Indeed, for all we know, the state space <math>A\!</math> might well be the state space of a language interpreter, one that is concerned, among other things, with the idiomatic meanings of the dialect generated by <math>\mathcal{A}</math> and <math>\mathrm{d}\mathcal{A}.\!</math><br />
<br />
The ''tangent space'' to <math>A\!</math> at one of its points <math>x,\!</math> sometimes written <math>\mathrm{T}_x(A),</math> takes the form <math>\mathrm{d}A</math> <math>=\!</math> <math>\langle \mathrm{d}\mathcal{A} \rangle</math> <math>=\!</math> <math>\langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle.\!</math> Strictly speaking, the name ''cotangent space'' is probably more correct for this construction, but the fact that we take up spaces and their duals in pairs to form our universes of discourse allows our language to be pliable here.<br />
<br />
Proceeding as we did with the base space <math>A,\!</math> the tangent space <math>\mathrm{d}A</math> at a point of <math>A\!</math> can be analyzed as a product of distinct and independent factors:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\mathrm{d}A ~=~ \prod_{i=1}^n \mathrm{d}A_i ~=~ \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n.\!</math><br />
|}<br />
<br />
Here, <math>\mathrm{d}A_i\!</math> is a set of two differential propositions, <math>\mathrm{d}A_i = \{ (\mathrm{d}a_i), \mathrm{d}a_i \},\!</math> where <math>\texttt{(} \mathrm{d}a_i \texttt{)}\!</math> is a proposition with the logical value of <math>\text{not} ~ \mathrm{d}a_i.\!</math> Each component <math>\mathrm{d}A_i\!</math> has the type <math>\mathbb{B},\!</math> operating under the ordered correspondence <math>\{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \} \cong \{ 0, 1 \}.\!</math> However, clarity is often served by acknowledging this differential usage with a superficially distinct type <math>\mathbb{D},\!</math> whose intension may be indicated as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\mathbb{D} = \{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \} = \{ \text{same}, \text{different} \} = \{ \text{stay}, \text{change} \} = \{ \text{stop}, \text{step} \}.\!</math><br />
|}<br />
<br />
Viewed within a coordinate representation, spaces of type <math>\mathbb{B}^n\!</math> and <math>\mathbb{D}^n\!</math> may appear to be identical sets of binary vectors, but taking a view at this level of abstraction would be like ignoring the qualitative units and the diverse dimensions that distinguish position and momentum, or the different roles of quantity and impulse.<br />
<br />
===An Interlude on the Path===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
There would have been no beginnings: instead, speech would proceed from me, while I stood in its path &ndash; a slender gap &ndash; the point of its possible disappearance.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
A sense of the relation between <math>\mathbb{B}</math> and <math>\mathbb{D}</math> may be obtained by considering the ''path classifier'' (or the ''equivalence class of curves'') approach to tangent vectors. Consider a universe <math>[\mathcal{X}].\!</math> Given the boolean value system, a path in the space <math>X = \langle \mathcal{X} \rangle</math> is a map <math>q : \mathbb{B} \to X.</math> In this context the set of paths <math>(\mathbb{B} \to X)</math> is isomorphic to the cartesian square <math>X^2 = X \times X,</math> or the set of ordered pairs chosen from <math>X.\!</math><br />
<br />
We may analyze <math>X^2 = \{ (u, v) : u, v \in X \}</math> into two parts, specifically, the ordered pairs <math>(u, v)\!</math> that lie on and off the diagonal:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}X^2 & = & \{ (u, v) : u = v \} & \cup & \{ (u, v) : u \ne v \}.\end{matrix}</math><br />
|}<br />
<br />
This partition may also be expressed in the following symbolic form:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}X^2 & \cong & \operatorname{diag} (X) & + & 2 \binom{X}{2}.\end{matrix}</math><br />
|}<br />
<br />
The separate terms of this formula are defined as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}\operatorname{diag} (X) & = & \{ (x, x) : x \in X \}.\end{matrix}\!</math><br />
|}<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}\binom{X}{k} & = & X ~\text{choose}~ k & = & \{ k\text{-sets from}~ X \}.\end{matrix}\!</math><br />
|}<br />
<br />
Thus we have:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\begin{matrix}\binom{X}{2} & = & \{ \{ u, v \} : u, v \in X \}.\end{matrix}</math><br />
|}<br />
<br />
We may now use the features in <math>\mathrm{d}\mathcal{X} = \{ \mathrm{d}x_i \} = \{ \mathrm{d}x_1, \ldots, \mathrm{d}x_n \}</math> to classify the paths of <math>(\mathbb{B} \to X)</math> by way of the pairs in <math>X^2.\!</math> If <math>X \cong \mathbb{B}^n,</math> then a path <math>q\!</math> in <math>X\!</math> has the following form:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
q : (\mathbb{B} \to \mathbb{B}^n) & \cong & \mathbb{B}^n \times \mathbb{B}^n & \cong & \mathbb{B}^{2n} & \cong & (\mathbb{B}^2)^n.<br />
\end{matrix}</math><br />
|}<br />
<br />
Intuitively, we want to map this <math>(\mathbb{B}^2)^n</math> onto <math>\mathbb{D}^n</math> by mapping each component <math>\mathbb{B}^2</math> onto a copy of <math>\mathbb{D}.</math> But in the presenting context <math>{}^{\backprime\backprime} \mathbb{D} {}^{\prime\prime}</math> is just a name associated with, or an incidental quality attributed to, coefficient values in <math>\mathbb{B}</math> when they are attached to features in <math>\mathrm{d}\mathcal{X}.</math><br />
<br />
Taking these intentions into account, define <math>\mathrm{d}x_i : X^2 \to \mathbb{B}</math> in the following manner:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcrcl}<br />
\mathrm{d}x_i(u, v)<br />
& = & \texttt{(} ~ x_i(u) & \texttt{,} & x_i(v) ~ \texttt{)}<br />
\\<br />
& = & x_i(u) & + & x_i(v)<br />
\\<br />
& = & x_i(v) & - & x_i(u).<br />
\end{array}</math><br />
|}<br />
<br />
In the above transcription, the operator bracket of the form <math>\texttt{(} \ldots \texttt{,} \ldots \texttt{)}\!</math> is a ''cactus lobe'', in general signifying that just one of the arguments listed is false. In the case of two arguments this is the same thing as saying that the arguments are not equal. The plus sign signifies boolean addition, in the sense of addition in <math>\mathrm{GF}(2),</math> and thus means the same thing in this context as the minus sign, in the sense of adding the additive inverse.<br />
<br />
The above definition of <math>\mathrm{d}x_i : X^2 \to \mathbb{B}</math> is equivalent to defining <math>\mathrm{d}x_i : (\mathbb{B} \to X) \to \mathbb{B}\!</math> in the following way:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcrcl}<br />
\mathrm{d}x_i (q)<br />
& = & \texttt{(} ~ x_i(q_0) & \texttt{,} & x_i(q_1) ~ \texttt{)}<br />
\\<br />
& = & x_i(q_0) & + & x_i(q_1)<br />
\\<br />
& = & x_i(q_1) & - & x_i(q_0).<br />
\end{array}</math><br />
|}<br />
<br />
In this definition <math>q_b = q(b),\!</math> for each <math>b\!</math> in <math>\mathbb{B}.</math> Thus, the proposition <math>\mathrm{d}x_i</math> is true of the path <math>q = (u, v)\!</math> exactly if the terms of <math>q,\!</math> the endpoints <math>u\!</math> and <math>v,\!</math> lie on different sides of the question <math>x_i.\!</math><br />
<br />
The language of features in <math>\langle \mathrm{d}\mathcal{X} \rangle,</math> indeed the whole calculus of propositions in <math>[\mathrm{d}\mathcal{X}],</math> may now be used to classify paths and sets of paths. In other words, the paths can be taken as models of the propositions <math>g : \mathrm{d}X \to \mathbb{B}.</math> For example, the paths corresponding to <math>\mathrm{diag}(X)</math> fall under the description <math>\texttt{(} \mathrm{d}x_1 \texttt{)} \cdots \texttt{(} \mathrm{d}x_n \texttt{)},\!</math> which says that nothing changes against the backdrop of the coordinate frame <math>\{ x_1, \ldots, x_n \}.\!</math><br />
<br />
Finally, a few words of explanation may be in order. If this concept of a path appears to be described in a roundabout fashion, it is because I am trying to avoid using any assumption of vector space properties for the space <math>X\!</math> that contains its range. In many ways the treatment is still unsatisfactory, but improvements will have to wait for the introduction of substitution operators acting on singular propositions.<br />
<br />
===The Extended Universe of Discourse===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
At the moment of speaking, I would like to have perceived a nameless voice, long preceding me, leaving me merely to enmesh myself in it, taking up its cadence, and to lodge myself, when no one was looking, in its interstices as if it had paused an instant, in suspense, to beckon to me.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
Next we define the ''extended alphabet'' or ''bundled alphabet'' <math>\mathrm{E}\mathcal{A}</math> as follows:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lclcl}<br />
\mathrm{E}\mathcal{A}<br />
& = & \mathcal{A} \cup \mathrm{d}\mathcal{A}<br />
& = & \{ a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}.<br />
\end{array}</math><br />
|}<br />
<br />
This supplies enough material to construct the ''differential extension'' <math>\mathrm{E}A,</math> or the ''tangent bundle'' over the initial space <math>A,\!</math> in the following fashion:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcl}<br />
\mathrm{E}A<br />
& = & \langle \mathrm{E}\mathcal{A} \rangle<br />
\\[4pt]<br />
& = & \langle \mathcal{A} \cup \mathrm{d}\mathcal{A} \rangle<br />
\\[4pt]<br />
& = & \langle a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle,<br />
\end{array}</math><br />
|}<br />
<br />
and also:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcl}<br />
\mathrm{E}A<br />
& = & A \times \mathrm{d}A<br />
\\[4pt]<br />
& = & A_1 \times \ldots \times A_n \times \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n.<br />
\end{array}</math><br />
|}<br />
<br />
This gives <math>\mathrm{E}A</math> the type <math>\mathbb{B}^n \times \mathbb{D}^n.</math><br />
<br />
Finally, the tangent universe <math>\mathrm{E}A^\bullet = [\mathrm{E}\mathcal{A}]\!</math> is constituted from the totality of points and maps, or interpretations and propositions, that are based on the extended set of features <math>\mathrm{E}\mathcal{A},</math> and this fact is summed up in the following notation:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lclcl}<br />
\mathrm{E}A^\bullet<br />
& = & [\mathrm{E}\mathcal{A}]<br />
& = & [a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n].<br />
\end{array}</math><br />
|}<br />
<br />
This gives the tangent universe <math>\mathrm{E}A^\bullet\!</math> the type:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lcl}<br />
(\mathbb{B}^n \times \mathbb{D}^n\ +\!\to \mathbb{B})<br />
& = & (\mathbb{B}^n \times \mathbb{D}^n, (\mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B})).<br />
\end{array}</math><br />
|}<br />
<br />
A proposition in the tangent universe <math>[\mathrm{E}\mathcal{A}]</math> is called a ''differential proposition'' and forms the analogue of a system of differential equations, constraints, or relations in ordinary calculus.<br />
<br />
With these constructions, the differential extension <math>\mathrm{E}A</math> and the space of differential propositions <math>(\mathrm{E}A \to \mathbb{B}),\!</math> we have arrived, in main outline, at one of the major subgoals of this study. Table&nbsp;8 summarizes the concepts that have been introduced for working with differentially extended universes of discourse.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 8.} ~~ \text{Differential Extension : Basic Notation}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Symbol}\!</math><br />
| <math>\text{Notation}\!</math><br />
| <math>\text{Description}\!</math><br />
| <math>\text{Type}\!</math><br />
|-<br />
| <math>\mathrm{d}\mathfrak{A}\!</math><br />
| <math>\{ {}^{\backprime\backprime} \mathrm{d}a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} \mathrm{d}a_n {}^{\prime\prime} \}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Alphabet of}<br />
\\[2pt]<br />
\text{differential symbols}<br />
\end{matrix}</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>\mathrm{d}\mathcal{A}\!</math><br />
| <math>\{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Basis of}<br />
\\[2pt]<br />
\text{differential features}<br />
\end{matrix}</math><br />
| <math>[n] = \mathbf{n}\!</math><br />
|-<br />
| <math>\mathrm{d}A_i\!</math><br />
| <math>\{ \texttt{(} \mathrm{d}a_i \texttt{)}, \mathrm{d}a_i \}\!</math><br />
| <math>\text{Differential dimension}~ i\!</math><br />
| <math>\mathbb{D}\!</math><br />
|-<br />
| <math>\mathrm{d}A\!</math><br />
|<br />
<math>\begin{matrix}<br />
\langle \mathrm{d}\mathcal{A} \rangle<br />
\\[2pt]<br />
\langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle<br />
\\[2pt]<br />
\{ (\mathrm{d}a_1, \ldots, \mathrm{d}a_n) \}<br />
\\[2pt]<br />
\mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n<br />
\\[2pt]<br />
\textstyle \prod_i \mathrm{d}A_i<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Tangent space at a point:}<br />
\\[2pt]<br />
\text{Set of changes, motions,}<br />
\\[2pt]<br />
\text{steps, tangent vectors}<br />
\\[2pt]<br />
\text{at a point}<br />
\end{matrix}</math><br />
| <math>\mathbb{D}^n\!</math><br />
|-<br />
| <math>\mathrm{d}A^*\!</math><br />
| <math>(\mathrm{hom} : \mathrm{d}A \to \mathbb{B})\!</math><br />
| <math>\text{Linear functions on}~ \mathrm{d}A\!</math><br />
| <math>(\mathbb{D}^n)^* \cong \mathbb{D}^n\!</math><br />
|-<br />
| <math>\mathrm{d}A^\uparrow\!</math><br />
| <math>(\mathrm{d}A \to \mathbb{B})\!</math><br />
| <math>\text{Boolean functions on}~ \mathrm{d}A\!</math><br />
| <math>\mathbb{D}^n \to \mathbb{B}\!</math><br />
|-<br />
| <math>\mathrm{d}A^\bullet\!</math><br />
|<br />
<math>\begin{matrix}<br />
[\mathrm{d}\mathcal{A}]<br />
\\[2pt]<br />
(\mathrm{d}A, \mathrm{d}A^\uparrow)<br />
\\[2pt]<br />
(\mathrm{d}A ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
(\mathrm{d}A, (\mathrm{d}A \to \mathbb{B}))<br />
\\[2pt]<br />
[\mathrm{d}a_1, \ldots, \mathrm{d}a_n]<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{Tangent universe at a point of}~ A^\bullet,<br />
\\[2pt]<br />
\text{based on the tangent features}<br />
\\[2pt]<br />
\{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
(\mathbb{D}^n, (\mathbb{D}^n \to \mathbb{B}))<br />
\\[2pt]<br />
(\mathbb{D}^n ~+\!\to \mathbb{B})<br />
\\[2pt]<br />
[\mathbb{D}^n]<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
The adjectives ''differential'' or ''tangent'' are systematically attached to every construct based on the differential alphabet <math>\mathrm{d}\mathfrak{A},</math> taken by itself. Strictly speaking, we probably ought to call <math>\mathrm{d}\mathcal{A}</math> the set of ''cotangent'' features derived from <math>\mathcal{A},</math> but the only time this distinction really seems to matter is when we need to distinguish the tangent vectors as maps of type <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math> from cotangent vectors as elements of type <math>\mathbb{D}^n.</math> In like fashion, having defined <math>\mathrm{E}\mathcal{A} = \mathcal{A} \cup \mathrm{d}\mathcal{A},</math> we can systematically attach the adjective ''extended'' or the substantive ''bundle'' to all of the constructs associated with this full complement of <math>{2n}\!</math> features.<br />
<br />
It eventually becomes necessary to extend the initial alphabet even further, to allow for the discussion of higher order differential expressions. Table&nbsp;9 provides a suggestion of how these further extensions can be carried out.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 9.} ~~ \text{Higher Order Differential Features}\!</math><br />
|<br />
<p><math>\begin{array}{lllll}<br />
\mathrm{d}^0 \mathcal{A} & = & \{ a_1, \ldots, a_n \} & = & \mathcal{A}<br />
\\<br />
\mathrm{d}^1 \mathcal{A} & = & \{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \} & = & \mathrm{d}\mathcal{A}<br />
\end{array}</math></p><br />
<br />
<p><math>\begin{array}{lll}<br />
\mathrm{d}^k \mathcal{A} & = & \{ \mathrm{d}^k a_1, \ldots, \mathrm{d}^k a_n \}<br />
\\<br />
\mathrm{d}^* \mathcal{A} & = & \{ \mathrm{d}^0 \mathcal{A}, \ldots, \mathrm{d}^k \mathcal{A}, \ldots \}<br />
\end{array}</math></p><br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}^0 \mathcal{A} & = & \mathrm{d}^0 \mathcal{A}<br />
\\<br />
\mathrm{E}^1 \mathcal{A} & = & \mathrm{d}^0 \mathcal{A} ~\cup~ \mathrm{d}^1 \mathcal{A}<br />
\\<br />
\mathrm{E}^k \mathcal{A} & = & \mathrm{d}^0 \mathcal{A} ~\cup~ \ldots ~\cup~ \mathrm{d}^k \mathcal{A}<br />
\\<br />
\mathrm{E}^\infty \mathcal{A} & = & \bigcup~ \mathrm{d}^* \mathcal{A}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
===Intentional Propositions===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Do you guess I have some intricate purpose?<br><br />
Well I have . . . . for the April rain has, and the mica on<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the side of a rock has.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 45]<br />
|}<br />
<br />
In order to analyze the behavior of a system at successive moments in time, while staying within the limitations of propositional logic, it is necessary to create independent alphabets of logical features for each moment of time that we contemplate using in our discussion. These moments have reference to typical instances and relative intervals, not actual or absolute times. For example, to discuss ''velocities'' (first order rates of change) we need to consider points of time in pairs. There are a number of natural ways of doing this. Given an initial alphabet, we could use its symbols as a lexical basis to generate successive alphabets of compound symbols, say, with temporal markers appended as suffixes.<br />
<br />
As a standard way of dealing with these situations, the following scheme of notation suggests a way of extending any alphabet of logical features through as many temporal moments as a particular order of analysis may demand. The lexical operators <math>\mathrm{p}^k</math> and <math>\mathrm{Q}^k</math> are convenient in many contexts where the accumulation of prime symbols and union symbols would otherwise be cumbersome.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 10.} ~~ \text{A Realm of Intentional Features}\!</math><br />
|<br />
<p><math>\begin{array}{lllll}<br />
\mathrm{p}^0 \mathcal{A} & = & \{ a_1, \ldots, a_n \} & = & \mathcal{A}<br />
\\<br />
\mathrm{p}^1 \mathcal{A} & = & \{ a_1^\prime, \ldots, a_n^\prime \} & = & \mathcal{A}^\prime<br />
\\<br />
\mathrm{p}^2 \mathcal{A} & = & \{ a_1^{\prime\prime}, \ldots, a_n^{\prime\prime} \} & = & \mathcal{A}^{\prime\prime}<br />
\\<br />
\cdots & & \cdots &<br />
\end{array}</math></p><br />
<br />
<p><math>\begin{array}{lll}<br />
\mathrm{p}^k \mathcal{A} & = & \{ \mathrm{p}^k a_1, \ldots, \mathrm{p}^k a_n \}<br />
\end{array}</math></p><br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{Q}^0 \mathcal{A} & = & \mathcal{A}<br />
\\<br />
\mathrm{Q}^1 \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}'<br />
\\<br />
\mathrm{Q}^2 \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}' \cup \mathcal{A}''<br />
\\<br />
\cdots & & \cdots<br />
\\<br />
\mathrm{Q}^k \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}' \cup \ldots \cup \mathrm{p}^k \mathcal{A}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
The resulting augmentations of our logical basis determine a series of discursive universes that may be called the ''intentional extension'' of propositional calculus. This extension follows a pattern analogous to the differential extension, which was developed in terms of the operators <math>\mathrm{d}^k</math> and <math>\mathrm{E}^k,</math> and there is a natural relation between these two extensions that bears further examination. In contexts displaying this pattern, where a sequence of domains stretches from an anchoring domain <math>X\!</math> through an indefinite number of higher reaches, a particular collection of domains based on <math>X\!</math> will be referred to as a ''realm'' of <math>X,\!</math> and when the succession exhibits a temporal aspect, as a ''reign'' of <math>X.\!</math><br />
<br />
For the purposes of this discussion, an ''intentional proposition'' is defined as a proposition in the universe of discourse <math>\mathrm{Q}X^\bullet = [\mathrm{Q}\mathcal{X}],</math> in other words, a map <math>q : \mathrm{Q}X \to \mathbb{B}.</math> The sense of this definition may be seen if we consider the following facts. First, the equivalence <math>\mathrm{Q}X = X \times X'</math> motivates the following chain of isomorphisms between spaces:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lllcl}<br />
(\mathrm{Q}X \to \mathbb{B})<br />
& \cong & (X & \times & ~X' \to \mathbb{B})<br />
\\[4pt]<br />
& \cong & (X & \to & (X' \to \mathbb{B}))<br />
\\[4pt]<br />
& \cong & (X' & \to & (X~ \to \mathbb{B})).<br />
\end{array}</math><br />
|}<br />
<br />
Viewed in this light, an intentional proposition <math>q\!</math> may be rephrased as a map <math>q : X \times X' \to \mathbb{B},</math> which judges the juxtaposition of states in <math>X\!</math> from one moment to the next. Alternatively, <math>q\!</math> may be parsed in two stages in two different ways, as <math>q : X \to (X' \to \mathbb{B})</math> and as <math>q : X' \to (X \to \mathbb{B}),</math> which associate to each point of <math>X\!</math> or <math>X'\!</math> a proposition about states in <math>X'\!</math> or <math>X,\!</math> respectively. In this way, an intentional proposition embodies a type of value system, in effect, a proposal that places a value on a collection of ends-in-view, or a project that evaluates a set of goals as regarded from each point of view in the state space of a system.<br />
<br />
In sum, the intentional proposition <math>q\!</math> indicates a method for the systematic selection of local goals. As a general form of description, a map of the type <math>q : \mathrm{Q}^i X \to \mathbb{B}\!</math> may be referred to as an "<math>i^\text{th}</math> order intentional proposition". Naturally, when we speak of intentional propositions without qualification, we usually mean first order intentions.<br />
<br />
Many different realms of discourse have the same structure as the extensions that have been indicated here. From a strictly logical point of view, each new layer of terms is composed of independent logical variables that are no different in kind from those that go before, and each further course of logical atoms is treated like so many individual, but otherwise indifferent bricks by the prototype computer program that I use as a propositional interpreter. Thus, the names that I use to single out the differential and the intentional extensions, and the lexical paradigms that I follow to construct them, are meant to suggest the interpretations that I have in mind, but they can only hint at the extra meanings that human communicators may pack into their terms and inflections.<br />
<br />
As applied here, the word ''intentional'' is drawn from common use and may have little bearing on its technical use in other, more properly philosophical, contexts. I am merely using the complex of intentional concepts &mdash; aims, ends, goals, objectives, purposes, and so on &mdash; metaphorically to flesh out and vividly to represent any situation where one needs to contemplate a system in multiple aspects of state and destination, that is, its being in certain states and at the same time acting as if headed through certain states. If confusion arises, more neutral words like ''conative'', ''contingent'', ''discretionary'', ''experimental'', ''kinetic'', ''progressive'', ''tentative'', or ''trial'' would probably serve as well.<br />
<br />
===Life on Easy Street===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Failing to fetch me at first keep encouraged,<br><br />
Missing me one place search another,<br><br />
I stop some where waiting for you<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]<br />
|}<br />
<br />
The finite character of the extended universe <math>[\mathrm{E}\mathcal{A}]</math> makes the problem of solving differential propositions relatively straightforward, at least, in principle. The solution set of the differential proposition <math>q : \mathrm{E}A \to \mathbb{B}</math> is the set of models <math>q^{-1}(1)\!</math> in <math>\mathrm{E}A.</math> Finding all the models of <math>q,\!</math> the extended interpretations in <math>\mathrm{E}A</math> that satisfy <math>q,\!</math> can be carried out by a finite search. Being in possession of complete algorithms for propositional calculus modeling, theorem checking, or theorem proving makes the analytic task fairly simple in principle, though the question of efficiency in the face of arbitrary complexity may always remain another matter entirely. While the fact that propositional satisfiability is NP-complete may be discouraging for the prospects of a single efficient algorithm that covers the whole space <math>[\mathrm{E}\mathcal{A}]</math> with equal facility, there appears to be much room for improvement in classifying special forms and in developing algorithms that are tailored to their practical processing.<br />
<br />
In view of these constraints and contingencies, our focus shifts to the tasks of approximation and interpretation that support intuition, especially in dealing with the natural kinds of differential propositions that arise in applications, and in the effort to understand, in succinct and adaptive forms, their dynamic implications. In the absence of direct insights, these tasks are partially carried out by forging analogies with the familiar situations and customary routines of ordinary calculus. But the indirect approach, going by way of specious analogy and intuitive habit, forces us to remain on guard against the circumstance that occurs when the word ''forging'' takes on its shadier nuance, indicting the constant risk of a counterfeit in the proportion.<br />
<br />
==Back to the Beginning : Exemplary Universes==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
I would have preferred to be enveloped in words, borne way beyond all possible beginnings.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
To anchor our understanding of differential logic, let us look at how the various concepts apply in the simplest possible concrete cases, where the initial dimension is only 1 or 2. In spite of the obvious simplicity of these cases, it is possible to observe how central difficulties of the subject begin to arise already at this stage.<br />
<br />
===A One-Dimensional Universe===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
There was never any more inception than there is now,<br><br />
Nor any more youth or age than there is now;<br><br />
And will never be any more perfection than there is now,<br><br />
Nor any more heaven or hell than there is now.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]<br />
|}<br />
<br />
Let <math>\mathcal{X} = \{ x_1 \} = \{ A \}</math> be an alphabet that represents one boolean variable or a single logical feature. In this example the capital letter <math>{}^{\backprime\backprime} A {}^{\prime\prime}\!</math> is used usual informally, to name a feature and not a space, in departure from our formerly stated formal conventions. At any rate, the basis element <math>A = x_1\!</math> may be interpreted as a simple proposition or a coordinate projection <math>A = x_1 : \mathbb{B} \xrightarrow{i} \mathbb{B}.</math> The space <math>X = \langle A \rangle = \{ \texttt{(} A \texttt{)}, A \}</math> of points (cells, vectors, interpretations) has cardinality <math>2^n = 2^1 = 2\!</math> and is isomorphic to <math>\mathbb{B} = \{ 0, 1 \}.</math> Moreover, <math>X\!</math> may be identified with the set of singular propositions <math>\{ x : \mathbb{B} \xrightarrow{s} \mathbb{B} \}.</math> The space of linear propositions <math>X^* = \{ \mathrm{hom} : \mathbb{B} \xrightarrow{\ell} \mathbb{B} \} = \{ 0, A \}</math> is algebraically dual to <math>X\!</math> and also has cardinality <math>2.\!</math> Here, <math>{}^{\backprime\backprime} 0 {}^{\prime\prime}\!</math> is interpreted as denoting the constant function <math>0 : \mathbb{B} \to \mathbb{B},</math> amounting to the linear proposition of rank <math>0,\!</math> while <math>A\!</math> is the linear proposition of rank <math>1.\!</math> Last but not least we have the positive propositions <math>\{ \mathrm{pos} : \mathbb{B} \xrightarrow{p} \mathbb{B} \} = \{ A, 1 \},\!</math> of rank <math>1\!</math> and <math>0,\!</math> respectively, where <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}\!</math> is understood as denoting the constant function <math>1 : \mathbb{B} \to \mathbb{B}.</math> In sum, there are <math>2^{2^n} = 2^{2^1} = 4</math> propositions altogether in the universe of discourse, comprising the set <math>X^\uparrow = \{ f : X \to \mathbb{B} \} = \{ 0, \texttt{(} A \texttt{)}, A, 1 \} \cong (\mathbb{B} \to \mathbb{B}).</math><br />
<br />
The first order differential extension of <math>\mathcal{X}</math> is <math>\mathrm{E}\mathcal{X} = \{ x_1, \mathrm{d}x_1 \} = \{ A, \mathrm{d}A \}.</math> If the feature <math>A\!</math> is understood as applying to some object or state, then the feature <math>\mathrm{d}A</math> may be interpreted as an attribute of the same object or state that says that it is changing ''significantly'' with respect to the property <math>A,\!</math> or that it has an ''escape velocity'' with respect to the state <math>A.\!</math> In practice, differential features acquire their logical meaning through a class of ''temporal inference rules''.<br />
<br />
For example, relative to a frame of observation that is left implicit for now, one is permitted to make the following sorts of inference: From the fact that <math>A\!</math> and <math>\mathrm{d}A</math> are true at a given moment one may infer that <math>\texttt{(} A \texttt{)}\!</math> will be true in the next moment of observation. Altogether in the present instance, there is the fourfold scheme of inference that is shown below:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:50%"<br />
|<br />
<math>\begin{matrix}<br />
\text{From} & \texttt{(} A \texttt{)}<br />
& \text{and} & \texttt{(} \mathrm{d}A \texttt{)}<br />
& \text{infer} & \texttt{(} A \texttt{)} & \text{next.}<br />
\\[8pt]<br />
\text{From} & \texttt{(} A \texttt{)}<br />
& \text{and} & \mathrm{d}A<br />
& \text{infer} & A & \text{next.}<br />
\\[8pt]<br />
\text{From} & A<br />
& \text{and} & \texttt{(} \mathrm{d}A \texttt{)}<br />
& \text{infer} & A & \text{next.}<br />
\\[8pt]<br />
\text{From} & A<br />
& \text{and} & \mathrm{d}A<br />
& \text{infer} & \texttt{(} A \texttt{)} & \text{next.}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
It might be thought that an independent time variable needs to be brought in at this point, but it is an insight of fundamental importance that the idea of process is logically prior to the notion of time. A time variable is a reference to a ''clock'' &mdash; a canonical, conventional process that is accepted or established as a standard of measurement, but in essence no different than any other process. This raises the question of how different subsystems in a more global process can be brought into comparison, and what it means for one process to serve the function of a local standard for others. But these inquiries only wrap up puzzles in further riddles, and are obviously too involved to be handled at our current level of approximation.<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
The clock indicates the moment . . . . but what does<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;eternity indicate?<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 79]<br />
|}<br />
<br />
Observe that the secular inference rules, used by themselves, involve a loss of information, since nothing in them can tell us whether the momenta <math>\{ \texttt{(} \mathrm{d}A \texttt{)}, \mathrm{d}A \}\!</math> are changed or unchanged in the next instance. In order to know this, one would have to determine <math>\mathrm{d}^2 A,\!</math> and so on, pursuing an infinite regress. Ultimately, in order to rest with a finitely determinate system, it is necessary to make an infinite assumption, for example, that <math>\mathrm{d}^k A = 0\!</math> for all <math>k\!</math> greater than some fixed value <math>M.\!</math> Another way to escape the regress is through the provision of a dynamic law, in typical form making higher order differentials dependent on lower degrees and estates.<br />
<br />
===Example 1. A Square Rigging===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
Urge and urge and urge,<br><br />
Always the procreant urge of the world.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 28]<br />
|}<br />
<br />
By way of example, suppose that we are given the initial condition <math>A = \mathrm{d}A\!</math> and the law <math>\mathrm{d}^2 A = \texttt{(} A \texttt{)}.\!</math> Since the equation <math>A = \mathrm{d}A\!</math> is logically equivalent to the disjunction <math>A ~ \mathrm{d}A ~\text{or}~ \texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)},\!</math> we may infer two possible trajectories, as displayed in Table&nbsp;11. In either case the state <math>A ~ \texttt{(} \mathrm{d}A \texttt{)(} \mathrm{d}^2 A \texttt{)}\!</math> is a stable attractor or a terminal condition for both starting points.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 11.} ~~ \text{A Pair of Commodious Trajectories}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| <math>\text{Time}\!</math><br />
| <math>\text{Trajectory 1}\!</math><br />
| <math>\text{Trajectory 2}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
2<br />
\\[4pt]<br />
3<br />
\\[4pt]<br />
4<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
A & \mathrm{d}A & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
\texttt{(} A \texttt{)} & \mathrm{d}A & \mathrm{d}^2 A<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
{}^{\shortparallel} & {}^{\shortparallel} & {}^{\shortparallel}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{)} & \texttt{(} \mathrm{d}A \texttt{)} & \mathrm{d}^2 A<br />
\\[4pt]<br />
\texttt{(} A \texttt{)} & \mathrm{d}A & \mathrm{d}^2 A<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}<br />
\\[4pt]<br />
{}^{\shortparallel} & {}^{\shortparallel} & {}^{\shortparallel}<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Because the initial space <math>X = \langle A \rangle\!</math> is one-dimensional, we can easily fit the second order extension <math>\mathrm{E}^2 X = \langle A, \mathrm{d}A, \mathrm{d}^2 A \rangle\!</math> within the compass of a single venn diagram, charting the couple of converging trajectories as shown in Figure&nbsp;12.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 12 -- The Anchor.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 12.} ~~ \text{The Anchor}\!</math><br />
|}<br />
<br />
If we eliminate from view the regions of <math>\mathrm{E}^2 X\!</math> that are ruled out by the dynamic law <math>\mathrm{d}^2 A = \texttt{(} A \texttt{)},\!</math> then what remains is the quotient structure that is shown in Figure&nbsp;13. This picture makes it easy to see that the dynamically allowable portion of the universe is partitioned between the properties <math>A\!</math> and <math>\mathrm{d}^2 A\!.</math> As it happens, this fact might have been expressed &ldquo;right off the bat&rdquo; by an equivalent formulation of the differential law, one that uses the exclusive disjunction to state the law as <math>\texttt{(} A \texttt{,} \mathrm{d}^2 A \texttt{)}\!.</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 13 -- The Tiller.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 13.} ~~ \text{The Tiller}\!</math><br />
|}<br />
<br />
What we have achieved in this example is to give a differential description of a simple dynamic process. In effect, we did this by embedding a directed graph, which can be taken to represent the state transitions of a finite automaton, in a dynamically allotted quotient structure that is created from a boolean lattice or an <math>n\!</math>-cube by nullifying all of the regions that the dynamics outlaws. With growth in the dimensions of our contemplated universes, it becomes essential, both for human comprehension and for computer implementation, that the dynamic structures of interest to us be represented not actually, by acquaintance, but virtually, by description. In our present study, we are using the language of propositional calculus to express the relevant descriptions, and to comprehend the structure that is implicit in the subsets of a <math>n\!</math>-cube without necessarily being forced to actualize all of its points.<br />
<br />
One of the reasons for engaging in this kind of extremely reduced, but explicitly controlled case study is to throw light on the general study of languages, formal and natural, in their full array of syntactic, semantic, and pragmatic aspects. Propositional calculus is one of the last points of departure where we can view these three aspects interacting in a non-trivial way without being immediately and totally overwhelmed by the complexity they generate. Often this complexity causes investigators of formal and natural languages to adopt the strategy of focusing on a single aspect and to abandon all hope of understanding the whole, whether it's the still living natural language or the dynamics of inquiry that lies crystallized in formal logic.<br />
<br />
From the perspective that I find most useful here, a language is a syntactic system that is designed or evolved in part to express a set of descriptions. When the explicit symbols of a language have extensions in its object world that are actually infinite, or when the implicit categories and generative devices of a linguistic theory have extensions in its subject matter that are potentially infinite, then the finite characters of terms, statements, arguments, grammars, logics, and rhetorics force an excess of intension to reside in all these symbols and functions, across the spectrum from the object language to the metalinguistic uses. In the aphorism from W. von Humboldt that Chomsky often cites, for example, in [Cho86, 30] and [Cho93, 49], language requires &ldquo;the infinite use of finite means&rdquo;. This is necessarily true when the extensions are infinite, when the referential symbols and grammatical categories of a language possess infinite sets of models and instances. But it also voices a practical truth when the extensions, though finite at every stage, tend to grow at exponential rates.<br />
<br />
This consequence of dealing with extensions that are &ldquo;practically infinite&rdquo; becomes crucial when one tries to build neural network systems that learn, since the learning competence of any intelligent system is limited to the objects and domains that it is able to represent. If we want to design systems that operate intelligently with the full deck of propositions dealt by intact universes of discourse, then we must supply them with succinct representations and efficient transformations in this domain. Furthermore, in the project of constructing inquiry driven systems, we find ourselves forced to contemplate the level of generality that is embodied in propositions, because the dynamic evolution of these systems is driven by the measurable discrepancies that occur among their expectations, intentions, and observations, and because each of these subsystems or components of knowledge constitutes a propositional modality that can take on the fully generic character of an empirical summary or an axiomatic theory.<br />
<br />
A compression scheme by any other name is a symbolic representation, and this is what the differential extension of propositional calculus, through all of its many universes of discourse, is intended to supply. Why is this particular program of mental calisthenics worth carrying out in general? By providing a uniform logical medium for describing dynamic systems we can make the task of understanding complex systems much easier, both in looking for invariant representations of individual cases and in finding points of comparison among diverse structures that would otherwise appear as isolated systems. All of this goes to facilitate the search for compact knowledge and to adapt what is learned from individual cases to the general realm.<br />
<br />
===Back to the Feature===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
I guess it must be the flag of my disposition, out of hopeful<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;green stuff woven.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 31]<br />
|}<br />
<br />
Let us assume that the sense intended for differential features is well enough established in the intuition, for now, that we may continue with outlining the structure of the differential extension <math>[\mathrm{E}\mathcal{X}] = [A, \mathrm{d}A].\!</math> Over the extended alphabet <math>\mathrm{E}\mathcal{X} = \{ x_1, \mathrm{d}x_1 \} = \{ A, \mathrm{d}A \}\!</math> of cardinality <math>2^n = 2\!</math> we generate the set of points <math>\mathrm{E}X\!</math> of cardinality <math>2^{2n} = 4\!</math> that bears the following chain of equivalent descriptions:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}X & = & \langle A, \mathrm{d}A \rangle<br />
\\[4pt]<br />
& = & \{ \texttt{(} A \texttt{)}, A \} ~\times~ \{ \texttt{(} \mathrm{d}A \texttt{)}, \mathrm{d}A \}<br />
\\[4pt]<br />
& = &<br />
\{<br />
\texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)},~<br />
\texttt{(} A \texttt{)} \mathrm{d}A,~<br />
A \texttt{(} \mathrm{d}A \texttt{)},~<br />
A ~ \mathrm{d}A<br />
\}.<br />
\end{array}</math><br />
|}<br />
<br />
The space <math>\mathrm{E}X\!</math> may be assigned the mnemonic type <math>\mathbb{B} \times \mathbb{D},\!</math> which is really no different than <math>\mathbb{B} \times \mathbb{B} = \mathbb{B}^2.\!</math> An individual element of <math>\mathrm{E}X\!</math> may be regarded as a ''disposition at a point'' or a ''situated direction'', in effect, a singular mode of change occurring at a single point in the universe of discourse. In applications, the modality of this change can be interpreted in various ways, for example, as an expectation, an intention, or an observation with respect to the behavior of a system.<br />
<br />
To complete the construction of the extended universe of discourse <math>\mathrm{E}X^\bullet = [x_1, \mathrm{d}x_1] = [A, \mathrm{d}A]\!</math> one must add the set of differential propositions <math>\mathrm{E}X^\uparrow = \{ g : \mathrm{E}X \to \mathbb{B} \} \cong (\mathbb{B} \times \mathbb{D} \to \mathbb{B})\!</math> to the set of dispositions in <math>\mathrm{E}X.\!</math> There are <math>2^{2^{2n}} = 16\!</math> propositions in <math>\mathrm{E}X^\uparrow,\!</math> as detailed in Table&nbsp;14.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 14.} ~~ \text{Differential Propositions}\!</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>A\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>\mathrm{d}A\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>g_{0}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{1}<br />
\\[4pt]<br />
g_{2}<br />
\\[4pt]<br />
g_{4}<br />
\\[4pt]<br />
g_{8}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
1~0~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
\texttt{(} A \texttt{)} ~ \mathrm{d}A ~<br />
\\[4pt]<br />
~ A ~ \texttt{(} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
~ A ~~ \mathrm{d}A ~<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{neither}~ A ~\text{nor}~ \mathrm{d}A<br />
\\[4pt]<br />
\mathrm{d}A ~\text{and not}~ A<br />
\\[4pt]<br />
A ~\text{and not}~ \mathrm{d}A<br />
\\[4pt]<br />
A ~\text{and}~ \mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot A \land \lnot \mathrm{d}A<br />
\\[4pt]<br />
\lnot A \land \mathrm{d}A<br />
\\[4pt]<br />
A \land \lnot \mathrm{d}A<br />
\\[4pt]<br />
A \land \mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
g_{3}<br />
\\[4pt]<br />
g_{12}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{)}<br />
\\[4pt]<br />
A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ A<br />
\\[4pt]<br />
A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot A<br />
\\[4pt]<br />
A<br />
\end{matrix}\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{6}<br />
\\[4pt]<br />
g_{9}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[4pt]<br />
1~0~0~1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} A \texttt{,} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
\texttt{((} A \texttt{,} \mathrm{d}A \texttt{))}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
A ~\text{not equal to}~ \mathrm{d}A<br />
\\[4pt]<br />
A ~\text{equal to}~ \mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
A \ne \mathrm{d}A<br />
\\[4pt]<br />
A = \mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{5}<br />
\\[4pt]<br />
g_{10}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}A \texttt{)}<br />
\\[4pt]<br />
\mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ \mathrm{d}A<br />
\\[4pt]<br />
\mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot \mathrm{d}A<br />
\\[4pt]<br />
\mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
| &nbsp;<br />
|<br />
<math>\begin{matrix}<br />
g_{7}<br />
\\[4pt]<br />
g_{11}<br />
\\[4pt]<br />
g_{13}<br />
\\[4pt]<br />
g_{14}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} ~ A ~~ \mathrm{d}A ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{(} ~ A ~ \texttt{(} \mathrm{d}A \texttt{))}<br />
\\[4pt]<br />
\texttt{((} A \texttt{)} ~ \mathrm{d}A ~ \texttt{)}<br />
\\[4pt]<br />
\texttt{((} A \texttt{)(} \mathrm{d}A \texttt{))}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not both}~ A ~\text{and}~ \mathrm{d}A<br />
\\[4pt]<br />
\text{not}~ A ~\text{without}~ \mathrm{d}A<br />
\\[4pt]<br />
\text{not}~ \mathrm{d}A ~\text{without}~ A<br />
\\[4pt]<br />
A ~\text{or}~ \mathrm{d}A<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot A \lor \lnot \mathrm{d}A<br />
\\[4pt]<br />
A \Rightarrow \mathrm{d}A<br />
\\[4pt]<br />
A \Leftarrow \mathrm{d}A<br />
\\[4pt]<br />
A \lor \mathrm{d}A<br />
\end{matrix}\!</math><br />
|-<br />
| <math>f_{3}\!</math><br />
| <math>g_{15}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
Aside from changing the names of variables and shuffling the order of rows, this Table follows the format that was used previously for boolean functions of two variables. The rows are grouped to reflect natural similarity classes among the propositions. In a future discussion, these classes will be given additional explanation and motivation as the orbits of a certain transformation group acting on the set of 16 propositions. Notice that four of the propositions, in their logical expressions, resemble those given in the table for <math>X^\uparrow.\!</math> Thus the first set of propositions <math>\{ f_i \}\!</math> is automatically embedded in the present set <math>\{ g_j \}\!</math> and the corresponding inclusions are indicated at the far left margin of the Table.<br />
<br />
===Tacit Extensions===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
I would really like to have slipped imperceptibly into this lecture, as into all the others I shall be delivering, perhaps over the years ahead.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]<br />
|}<br />
<br />
Strictly speaking, however, there is a subtle distinction in type between the function <math>f_i : X \to \mathbb{B}</math> and the corresponding function <math>g_j : \mathrm{E}X \to \mathbb{B},</math> even though they share the same logical expression. Naturally, we want to maintain the logical equivalence of expressions that represent the same proposition while appreciating the full diversity of that proposition's functional and typical representatives. Both perspectives, and all the levels of abstraction extending through them, have their reasons, as will develop in time.<br />
<br />
Because this special circumstance points up an important general theme, it is a good idea to discuss it more carefully. Whenever there arises a situation like this, where one alphabet <math>\mathcal{X}</math> is a subset of another alphabet <math>\mathcal{Y},</math> then we say that any proposition <math>f : \langle \mathcal{X} \rangle \to \mathbb{B}</math> has a ''tacit extension'' to a proposition <math>\boldsymbol\varepsilon f : \langle \mathcal{Y} \rangle \to \mathbb{B},\!</math> and that the space <math>(\langle \mathcal{X} \rangle \to \mathbb{B})</math> has an ''automatic embedding'' within the space <math>(\langle \mathcal{Y} \rangle \to \mathbb{B}).</math> The extension is defined in such a way that <math>\boldsymbol\varepsilon f\!</math> puts the same constraint on the variables of <math>\mathcal{X}</math> that are contained in <math>\mathcal{Y}</math> as the proposition <math>f\!</math> initially did, while it puts no constraint on the variables of <math>\mathcal{Y}</math> outside of <math>\mathcal{X},</math> in effect, conjoining the two constraints.<br />
<br />
If the variables in question are indexed as <math>\mathcal{X} = \{ x_1, \ldots, x_n \}</math> and <math>\mathcal{Y} = \{ x_1, \ldots, x_n, \ldots, x_{n+k} \},</math> then the definition of the tacit extension from <math>\mathcal{X}</math> to <math>\mathcal{Y}</math> may be expressed in the form of an equation:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\boldsymbol\varepsilon f(x_1, \ldots, x_n, \ldots, x_{n+k}) ~=~ f(x_1, \ldots, x_n).\!</math><br />
|}<br />
<br />
On formal occasions, such as the present context of definition, the tacit extension from <math>\mathcal{X}</math> to <math>\mathcal{Y}</math> is explicitly symbolized by the operator <math>\boldsymbol\varepsilon : (\langle \mathcal{X} \rangle \to \mathbb{B}) \to (\langle \mathcal{Y} \rangle \to \mathbb{B}),</math> where the appropriate alphabets <math>\mathcal{X}</math> and <math>\mathcal{Y}</math> are understood from context, but normally one may leave the "<math>\boldsymbol\varepsilon\!</math>" silent.<br />
<br />
Let's explore what this means for the present Example. Here, <math>\mathcal{X} = \{ A \}</math> and <math>\mathcal{Y} = \mathrm{E}\mathcal{X} = \{ A, \mathrm{d}A \}.</math> For each of the propositions <math>f_i\!</math> over <math>X\!,</math> specifically, those whose expression <math>e_i\!</math> lies in the collection <math>\{ 0, \texttt{(} A \texttt{)}, A, 1 \},\!</math> the tacit extension <math>\boldsymbol\varepsilon f\!</math> of <math>f\!</math> to <math>\mathrm{E}X</math> can be phrased as a logical conjunction of two factors, <math>f_i = e_i \cdot \tau ~ ,\!</math> where <math>\tau\!</math> is a logical tautology that uses all the variables of <math>\mathcal{Y} - \mathcal{X}.</math> Working in these terms, the tacit extensions <math>\boldsymbol\varepsilon f\!</math> of <math>f\!</math> to <math>\mathrm{E}X</math> may be explicated as shown in Table&nbsp;15.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:60%"<br />
|+ style="height:30px" | <math>\text{Table 15.} ~~ \text{Tacit Extension of}~ [A] ~\text{to}~ [A, \mathrm{d}A]\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0<br />
& = & 0 & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))} & = & & 0<br />
\\[8pt]<br />
\texttt{(} A \texttt{)}<br />
& = & \texttt{(} A \texttt{)} & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))}<br />
& = & \texttt{(} A \texttt{)} \, \mathrm{d}A ~ & + & \texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)}<br />
\\[8pt]<br />
A<br />
& = & ~A~ & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))}<br />
& = & ~A~ ~\mathrm{d}A~ & + & ~A~ \texttt{(} \mathrm{d}A \texttt{)}<br />
\\[8pt]<br />
1<br />
& = & 1 & \cdot & \texttt{(} \mathrm{d}A \texttt{,(} \mathrm{d}A \texttt{))} & = & & 1<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
In its effect on the singular propositions over <math>X,\!</math> this analysis has an interesting interpretation. The tacit extension takes us from thinking about a particular state, like <math>A\!</math> or <math>\texttt{(} A \texttt{)},\!</math> to considering the collection of outcomes, the outgoing changes or the singular dispositions, that spring from that state.<br />
<br />
===Example 2. Drives and Their Vicissitudes===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
I open my scuttle at night and see the far-sprinkled systems,<br><br />
And all I see, multiplied as high as I can cipher, edge but<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the rim of the farther systems.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 81]<br />
|}<br />
<br />
Before we leave the one-feature case let's look at a more substantial example, one that illustrates a general class of curves that can be charted through the extended feature spaces and that provides an opportunity to discuss a number of important themes concerning their structure and dynamics.<br />
<br />
Again, let <math>\mathcal{X} = \{ x_1 \} = \{ A \}.\!</math> In the discussion that follows we will consider a class of trajectories having the property that <math>\mathrm{d}^k A = 0\!</math> for all <math>k\!</math> greater than some fixed <math>m\!</math> and we may indulge in the use of some picturesque terms that describe salient classes of such curves. Given the finite order condition, there is a highest order non-zero difference <math>\mathrm{d}^m A\!</math> exhibited at each point in the course of any determinate trajectory that one may wish to consider. With respect to any point of the corresponding orbit or curve let us call this highest order differential feature <math>\mathrm{d}^m A\!</math> the ''drive'' at that point. Curves of constant drive <math>\mathrm{d}^m A\!</math> are then referred to as ''<math>m^\text{th}\!</math>-gear curves''.<br />
<br />
* '''Scholium.''' The fact that a difference calculus can be developed for boolean functions is well known [Fuji], [Koh, &sect; 8-4] and was probably familiar to Boole, who was an expert in difference equations before he turned to logic. And of course there is the strange but true story of how the Turin machines of the 1840s prefigured the Turing machines of the 1940s [Men, 225-297]. At the very outset of general purpose, mechanized computing we find that the motive power driving the Analytical Engine of Babbage, the kernel of an idea behind all of his wheels, was exactly his notion that difference operations, suitably trained, can serve as universal joints for any conceivable computation [M&M], [Mel, ch. 4].<br />
<br />
Given this language, the Example we take up here can be described as the family of <math>4^\text{th}\!</math>-gear curves through <math>\mathrm{E}^4 X\!</math> <math>=\!</math> <math>\langle A, ~\mathrm{d}A, ~\mathrm{d}^2\!A, ~\mathrm{d}^3\!A, ~\mathrm{d}^4\!A \rangle.</math> These are the trajectories generated subject to the dynamic law <math>\mathrm{d}^4 A = 1,\!</math> where it is understood in such a statement that all higher order differences are equal to <math>0.\!</math> Since <math>\mathrm{d}^4 A\!</math> and all higher <math>\mathrm{d}^k A\!</math> are fixed, the temporal or transitional conditions (initial, mediate, terminal &mdash; transient or stable states) vary only with respect to their projections as points of <math>\mathrm{E}^3 X = \langle A, ~\mathrm{d}A, ~\mathrm{d}^2\!A, ~\mathrm{d}^3\!A \rangle.</math> Thus, there is just enough space in a planar venn diagram to plot all of these orbits and to show how they partition the points of <math>\mathrm{E}^3 X.\!</math> It turns out that there are exactly two possible orbits, of eight points each, as illustrated in Figure&nbsp;16.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 16 -- A Couple of Fourth Gear Orbits.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 16.} ~~ \text{A Couple of Fourth Gear Orbits}\!</math><br />
|}<br />
<br />
With a little thought it is possible to devise an indexing scheme for the general run of dynamic states that allows for comparing universes of discourse that weigh in on different scales of observation. With this end in sight, let us index the states <math>q \in \mathrm{E}^m X\!</math> with the dyadic rationals (or the binary fractions) in the half-open interval <math>[0, 2).\!</math> Formally and canonically, a state <math>q_r\!</math> is indexed by a fraction <math>r = \tfrac{s}{t}\!</math> whose denominator is the power of two <math>t = 2^m\!</math> and whose numerator is a binary numeral formed from the coefficients of state in a manner to be described next. The ''differential coefficients'' of the state <math>q\!</math> are just the values <math>\mathrm{d}^k\!A(q)</math> for <math>k = 0 ~\text{to}~ m,\!</math> where <math>\mathrm{d}^0\!A</math> is defined as being identical to <math>A.\!</math> To form the binary index <math>d_0.d_1 \ldots d_m\!</math> of the state <math>q\!</math> the coefficient <math>\mathrm{d}^k\!A(q)</math> is read off as the binary digit <math>d_k\!</math> associated with the place value <math>2^{-k}.\!</math> Expressed by way of algebraic formulas, the rational index <math>r\!</math> of the state <math>q\!</math> can be given by the following equivalent formulations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
r(q)<br />
& = &<br />
\displaystyle\sum_k d_k \cdot 2^{-k}<br />
& = &<br />
\displaystyle\sum_k \text{d}^k A(q) \cdot 2^{-k}<br />
\\[8pt]<br />
=<br />
\\[8pt]<br />
\displaystyle\frac{s(q)}{t}<br />
& = &<br />
\displaystyle\frac{\sum_k d_k \cdot 2^{(m-k)}}{2^m}<br />
& = &<br />
\displaystyle\frac{\sum_k \text{d}^k A(q) \cdot 2^{(m-k)}}{2^m}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Applied to the example of <math>4^\text{th}\!</math>-gear curves, this scheme results in the data of Tables&nbsp;17-a and 17-b, which exhibit one period for each orbit. The states in each orbit are listed as ordered pairs <math>(p_i, q_j),\!</math> where <math>p_i\!</math> may be read as a temporal parameter that indicates the present time of the state and where <math>j\!</math> is the decimal equivalent of the binary numeral <math>s.\!</math> Informally and more casually, the Tables exhibit the states <math>q_s\!</math> as subscripted with the numerators of their rational indices, taking for granted the constant denominators of <math>2^m\! = 2^4 = 16.\!</math> In this set-up the temporal successions of states can be reckoned as given by a kind of ''parallel round-up rule''. That is, if <math>(d_k, d_{k+1})\!</math> is any pair of adjacent digits in the state index <math>r,\!</math> then the value of <math>d_k\!</math> in the next state is <math>{d_k}' = d_k + d_{k+1}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:55%"<br />
|+ style="height:30px" | <math>\text{Table 17-a.} ~~ \text{A Couple of Orbits in Fourth Gear : Orbit 1}\!</math><br />
|- style="background:ghostwhite"<br />
| <math>\text{Time}\!</math><br />
| <math>\text{State}\!</math><br />
| <math>A\!</math><br />
| <math>\mathrm{d}A\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| <math>p_i\!</math><br />
| <math>q_j\!</math><br />
| <math>\mathrm{d}^0\!A</math><br />
| <math>\mathrm{d}^1\!A</math><br />
| <math>\mathrm{d}^2\!A</math><br />
| <math>\mathrm{d}^3\!A</math><br />
| <math>\mathrm{d}^4\!A</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
p_0<br />
\\[4pt]<br />
p_1<br />
\\[4pt]<br />
p_2<br />
\\[4pt]<br />
p_3<br />
\\[4pt]<br />
p_4<br />
\\[4pt]<br />
p_5<br />
\\[4pt]<br />
p_6<br />
\\[4pt]<br />
p_7<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
q_{01}<br />
\\[4pt]<br />
q_{03}<br />
\\[4pt]<br />
q_{05}<br />
\\[4pt]<br />
q_{15}<br />
\\[4pt]<br />
q_{17}<br />
\\[4pt]<br />
q_{19}<br />
\\[4pt]<br />
q_{21}<br />
\\[4pt]<br />
q_{31}<br />
\end{matrix}\!</math><br />
| colspan="5" |<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="text-align:center; width:100%"<br />
|<br />
<math>\begin{matrix}<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
1.<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:55%"<br />
|+ style="height:30px" | <math>\text{Table 17-b.} ~~ \text{A Couple of Orbits in Fourth Gear : Orbit 2}\!</math><br />
|- style="background:ghostwhite"<br />
| <math>\text{Time}\!</math><br />
| <math>\text{State}\!</math><br />
| <math>A\!</math><br />
| <math>\mathrm{d}A\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| <math>p_i\!</math><br />
| <math>q_j\!</math><br />
| <math>\mathrm{d}^0\!A</math><br />
| <math>\mathrm{d}^1\!A</math><br />
| <math>\mathrm{d}^2\!A</math><br />
| <math>\mathrm{d}^3\!A</math><br />
| <math>\mathrm{d}^4\!A</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
p_0<br />
\\[4pt]<br />
p_1<br />
\\[4pt]<br />
p_2<br />
\\[4pt]<br />
p_3<br />
\\[4pt]<br />
p_4<br />
\\[4pt]<br />
p_5<br />
\\[4pt]<br />
p_6<br />
\\[4pt]<br />
p_7<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
q_{25}<br />
\\[4pt]<br />
q_{11}<br />
\\[4pt]<br />
q_{29}<br />
\\[4pt]<br />
q_{07}<br />
\\[4pt]<br />
q_{09}<br />
\\[4pt]<br />
q_{27}<br />
\\[4pt]<br />
q_{13}<br />
\\[4pt]<br />
q_{23}<br />
\end{matrix}\!</math><br />
| colspan="5" |<br />
{| align="center" border="0" cellpadding="6" cellspacing="0" style="text-align:center; width:100%"<br />
|<br />
<math>\begin{matrix}<br />
1.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\\[4pt]<br />
0.<br />
\\[4pt]<br />
1.<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
==Transformations of Discourse==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
It is understandable that an engineer should be completely absorbed in his speciality, instead of pouring himself out into the freedom and vastness of the world of thought, even though his machines are being sent off to the ends of the earth; for he no more needs to be capable of applying to his own personal soul what is daring and new in the soul of his subject than a machine is in fact capable of applying to itself the differential calculus on which it is based. The same thing cannot, however, be said about mathematics; for here we have the new method of thought, pure intellect, the very well-spring of the times, the ''fons et origo'' of an unfathomable transformation.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 39]<br />
|}<br />
<br />
In this section we take up the general study of logical transformations, or maps that relate one universe of discourse to another. In many ways, and especially as applied to the subject of intelligent dynamic systems, my argument develops the antithesis of the statement just quoted. Along the way, if incidental to my ends, I hope this essay can pose a fittingly irenic epitaph to the frankly ironic epigraph inscribed at its head.<br />
<br />
My goal in this section is to answer a single question: What is a propositional tangent functor? In other words, my aim is to develop a clear conception of what manner of thing would pass in the logical realm for a genuine analogue of the tangent functor, an object conceived to generalize as far as possible in the abstract terms of category theory the ordinary notions of functional differentiation and the all too familiar operations of taking derivatives.<br />
<br />
As a first step I discuss the kinds of transformations that we already know as ''extensions'' and ''projections'', and I use these special cases to illustrate several different styles of logical and visual representation that will figure heavily in the sequel.<br />
<br />
===Foreshadowing Transformations : Extensions and Projections of Discourse===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
And, despite the care which she took to look behind her at every moment, she failed to see a shadow which followed her like her own shadow, which stopped when she stopped, which started again when she did, and which made no more noise than a well-conducted shadow should.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 126]<br />
|}<br />
<br />
Many times in our discussion we have occasion to place one universe of discourse in the context of a larger universe of discourse. An embedding of the general type <math>[\mathcal{X}] \to [\mathcal{Y}]\!</math> is implied any time that we make use of one alphabet <math>[\mathcal{X}]\!</math> that happens to be included in another alphabet <math>[\mathcal{Y}].\!</math> When we are discussing differential issues we usually have in mind that the extended alphabet <math>[\mathcal{Y}]\!</math> has a special construction or a specific lexical relation with respect to the initial alphabet <math>[\mathcal{X}],\!</math> one that is marked by characteristic types of accents, indices, or inflected forms.<br />
<br />
====Extension from 1 to 2 Dimensions====<br />
<br />
Figure 18-a lays out the ''angular form'' of venn diagram for universes of 1 and 2 dimensions, indicating the embedding map of type <math>\mathbb{B}^1 \to \mathbb{B}^2\!</math> and detailing the coordinates that are associated with individual cells. Because all points, cells, or logical interpretations are represented as connected geometric areas, we can say that these pictures provide us with an ''areal view'' of each universe of discourse.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-a -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-a.} ~~ \text{Extension from 1 to 2 Dimensions : Areal}\!</math><br />
|}<br />
<br />
Figure 18-b shows the differential extension from <math>X^\bullet = [x]\!</math> to <math>\mathrm{E}X^\bullet = [x, \mathrm{d}x]\!</math> in a ''bundle of boxes'' form of venn diagram. As awkward as it may seem at first, this type of picture is often the most natural and the most easily available representation when we want to conceptualize the localized information or momentary knowledge of an intelligent dynamic system. It gives a ready picture of a ''proposition at a point'', in the present instance, of a proposition about changing states which is itself associated with a particular dynamic state of a system. It is easy to see how this application might be extended to conceive of more general types of instantaneous knowledge that are possessed by a system.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-b -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-b.} ~~ \text{Extension from 1 to 2 Dimensions : Bundle}\!</math><br />
|}<br />
<br />
Figure 18-c shows the same extension in a ''compact'' style of venn diagram, where the differential features at each position are represented by arrows extending from that position that cross or do not cross, as the case may be, the corresponding feature boundaries.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-c -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-c.} ~~ \text{Extension from 1 to 2 Dimensions : Compact}\!</math><br />
|}<br />
<br />
Figure 18-d compresses the picture of the differential extension even further, yielding a directed graph or ''digraph'' form of representation. (Notice that my definition of a digraph allows for loops or ''slings'' at individual points, in addition to arcs or ''arrows'' between the points.)<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 18-d -- Extension from 1 to 2 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 18-d.} ~~ \text{Extension from 1 to 2 Dimensions : Digraph}\!</math><br />
|}<br />
<br />
====Extension from 2 to 4 Dimensions====<br />
<br />
Figure 19-a lays out the ''areal view'' or the ''angular form'' of venn diagram for universes of 2 and 4 dimensions, indicating the embedding map of type <math>\mathbb{B}^2 \to \mathbb{B}^4.\!</math> In many ways these pictures are the best kind there is, giving full canvass to an ideal vista. Their style allows the clearest, the fairest, and the plainest view that we can form of a universe of discourse, affording equal representation to all dispositions and maintaining a balance with respect to ordinary and differential features. If only we could extend this view! Unluckily, an obvious difficulty beclouds this prospect, and that is how precipitately we run into the limits of our plane and visual intuitions. Even within the scope of the spare few dimensions that we have scanned up to this point subtle discrepancies have crept in already. The circumstances that bind us and the frameworks that block us, the flat distortion of the planar projection and the inevitable ineffability that precludes us from wrapping its rhomb figure into rings around a torus, all of these factors disguise the underlying but true connectivity of the universe of discourse.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-a -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-a.} ~~ \text{Extension from 2 to 4 Dimensions : Areal}\!</math><br />
|}<br />
<br />
Figure 19-b shows the differential extension from <math>U^\bullet = [u, v]\!</math> to <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v]\!</math> in the ''bundle of boxes'' form of venn diagram.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-b -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-b.} ~~ \text{Extension from 2 to 4 Dimensions : Bundle}\!</math><br />
|}<br />
<br />
As dimensions increase, this factorization of the extended universe along the lines that are marked out by the bundle picture begins to look more and more like a practical necessity. But whenever we use a propositional model to address a real situation in the context of nature we need to remain aware that this articulation into factors, affecting our description, may be wholly artificial in nature and cleave to nothing, no joint in nature, nor any juncture in time to be in or out of joint.<br />
<br />
Figure 19-c illustrates the extension from 2 to 4 dimensions in the ''compact'' style of venn diagram. Here, just the changes with respect to the center cell are shown.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-c -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-c.} ~~ \text{Extension from 2 to 4 Dimensions : Compact}\!</math><br />
|}<br />
<br />
Figure 19-d gives the ''digraph'' form of representation for the differential extension <math>U^\bullet \to \mathrm{E}U^\bullet,\!</math> where the 4 nodes marked with a circle <math>{}^{\bigcirc}\!</math> are the cells <math>uv,\, u \texttt{(} v \texttt{)},\, \texttt{(} u \texttt{)} v,\, \texttt{(} u \texttt{)(} v \texttt{)},\!</math> respectively, and where a 2-headed arc counts as 2 arcs of the differential digraph.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 19-d -- Extension from 2 to 4 Dimensions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 19-d.} ~~ \text{Extension from 2 to 4 Dimensions : Digraph}\!</math><br />
|}<br />
<br />
===Thematization of Functions : And a Declaration of Independence for Variables===<br />
<br />
{| width="100%"<br />
| align="left" |<br />
''And as imagination bodies forth''<br><br />
''The forms of things unknown, the poet's pen''<br><br />
''Turns them to shapes, and gives to airy nothing''<br><br />
''A local habitation and a name.''<br />
| align="right" valign="bottom" | A Midsummer Night's Dream, 5.1.18<br />
|}<br />
<br />
In the representation of propositions as functions it is possible to notice different degrees of explicitness in the way their functional character is symbolized. To indicate what I mean by this, the next series of Figures illustrates a set of graphic conventions that will be put to frequent use in the remainder of this discussion, both to mark the relevant distinctions and to help us convert between related expressions at different levels of explicitness in their functionality.<br />
<br />
====Thematization : Venn Diagrams====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
The known universe has one complete lover and that is the greatest poet. He consumes an eternal passion and is indifferent which chance happens and which possible contingency of fortune or misfortune and persuades daily and hourly his delicious pay.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 11&ndash;12]<br />
|}<br />
<br />
Figure 20-i traces the first couple of steps in this order of ''thematic'' progression, that will gradually run the gamut through a complete series of degrees of functional explicitness in the expression of logical propositions. The first venn diagram represents a situation where the function is indicated by a shaded figure and a logical expression. At this stage one may be thinking of the proposition only as expressed by a formula in a particular language and its content only as a subset of the universe of discourse, as when considering the proposition <math>u\!\cdot\!v</math> in the universe <math>[u, v].\!</math> The second venn diagram depicts a situation in which two significant steps have been taken. First, one has taken the trouble to give the proposition <math>u\!\cdot\!v</math> a distinctive functional name <math>{}^{\backprime\backprime} J {}^{\prime\prime}.\!</math> Second, one has come to think explicitly about the target domain that contains the functional values of <math>J,\!</math> as when writing <math>J : \langle u, v \rangle \to \mathbb{B}.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 20-i -- Thematization of Conjunction (Stage 1).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 20-i.} ~~ \text{Thematization of Conjunction (Stage 1)}\!</math><br />
|}<br />
<br />
In Figure 20-ii the proposition <math>J\!</math> is viewed explicitly as a transformation from one universe of discourse to another.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 20-ii -- Thematization of Conjunction (Stage 2).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 20-ii.} ~~ \text{Thematization of Conjunction (Stage 2)}\!</math><br />
|}<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
o-------------------------------o o-------------------------------o<br />
| | | |<br />
| o-----o o-----o | | o-----o o-----o |<br />
| / \ / \ | | / \ / \ |<br />
| / o \ | | / o \ |<br />
| / /`\ \ | | / /`\ \ |<br />
| o o```o o | | o o```o o |<br />
| | u |```| v | | | | u |```| v | |<br />
| o o```o o | | o o```o o |<br />
| \ \`/ / | | \ \`/ / |<br />
| \ o / | | \ o / |<br />
| \ / \ / | | \ / \ / |<br />
| o-----o o-----o | | o-----o o-----o |<br />
| | | |<br />
o-------------------------------o o-------------------------------o<br />
\ / \ /<br />
\ / \ /<br />
\ / \ J /<br />
\ / \ /<br />
\ / \ /<br />
o----------\---------/----------o o----------\---------/----------o<br />
| \ / | | \ / |<br />
| \ / | | \ / |<br />
| o-----@-----o | | o-----@-----o |<br />
| /`````````````\ | | /`````````````\ |<br />
| /```````````````\ | | /```````````````\ |<br />
| /`````````````````\ | | /`````````````````\ |<br />
| o```````````````````o | | o```````````````````o |<br />
| |```````````````````| | | |```````````````````| |<br />
| |```````` J ````````| | | |```````` x ````````| |<br />
| |```````````````````| | | |```````````````````| |<br />
| o```````````````````o | | o```````````````````o |<br />
| \`````````````````/ | | \`````````````````/ |<br />
| \```````````````/ | | \```````````````/ |<br />
| \`````````````/ | | \`````````````/ |<br />
| o-----------o | | o-----------o |<br />
| | | |<br />
| | | |<br />
o-------------------------------o o-------------------------------o<br />
J = u v x = J<u, v><br />
<br />
Figure 20-ii. Thematization of Conjunction (Stage 2)<br />
</pre><br />
|}<br />
<br />
In the first venn diagram the name that is assigned to a composite proposition, function, or region in the source universe is delegated to a simple feature in the target universe. This can result in a single character or term exceeding the responsibilities it can carry off well. Allowing the name of a function <math>J : \langle u, v \rangle \to \mathbb{B}\!</math> to serve as the name of its dependent variable <math>J : \mathbb{B}\!</math> does not mean that one has to confuse a function with any of its values, but it does put one at risk for a number of obvious problems, and we should not be surprised, on numerous and limiting occasions, when quibbling arises from the attempts of a too original syntax to serve these two masters.<br />
<br />
The second venn diagram circumvents these difficulties by introducing a new variable name for each basic feature of the target universe, as when writing <math>J : \langle u, v \rangle \to \langle x \rangle,\!</math> and thereby assigns a concrete type <math>\langle x \rangle</math> to the abstract codomain <math>\mathbb{B}.\!</math> To make this induction of variables more formal one can append subscripts, as in <math>x_J,\!</math> to indicate the origin or derivation of the new characters. Or we may use a lexical modifier to convert function names into variable names, for example, associating the function name <math>J\!</math> with the variable name <math>\check{J}.\!</math> Thus we may think of <math>x = x_J = \check{J}\!</math> as the ''cache variable'' corresponding to the function <math>J\!</math> or the symbol <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> considered as a contingent variable.<br />
<br />
In Figure 20-iii we arrive at a stage where the functional equations <math>J = u\!\cdot\!v</math> and <math>x = u\!\cdot\!v</math> are regarded as propositions in their own right, reigning in and ruling over the 3-feature universes of discourse <math>[u, v, J]~\!</math> and <math>[u, v, x],\!</math> respectively. Subject to the cautions already noted, the function name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> can be reinterpreted as the name of a feature <math>\check{J}</math> and the equation <math>J = u\!\cdot\!v</math> can be read as the logical equivalence <math>\texttt{((} J, u ~ v \texttt{))}.\!</math> To give it a generic name let us call this newly expressed, collateral proposition the ''thematization'' or the ''thematic extension'' of the original proposition <math>J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 20-iii -- Thematization of Conjunction (Stage 3).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 20-iii.} ~~ \text{Thematization of Conjunction (Stage 3)}\!</math><br />
|}<br />
<br />
The first venn diagram represents the thematization of the conjunction <math>J\!</math> with shading in the appropriate regions of the universe <math>[u, v, J].\!</math> Also, it illustrates a quick way of constructing a thematic extension. First, draw a line, in practice or the imagination, that bisects every cell of the original universe, placing half of each cell under the aegis of the thematized proposition and the other half under its antithesis. Next, on the scene where the theme applies leave the shade wherever it lies, and off the stage, where it plays otherwise, stagger the pattern in a harlequin guise.<br />
<br />
In the final venn diagram of this sequence the thematic progression comes full circle and completes one round of its development. The ambiguities that were occasioned by the changing role of the name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> are resolved by introducing a new variable name <math>{}^{\backprime\backprime} x {}^{\prime\prime}</math> to take the place of <math>\check{J},\!</math> and the region that represents this fresh featured <math>x\!</math> is circumscribed in a more conventional symmetry of form and placement. Just as we once gave the name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> to the proposition <math>u\!\cdot\!v,</math> we now give the name <math>{}^{\backprime\backprime} \iota {}^{\prime\prime}</math> to its thematization <math>\texttt{((} x, u ~ v \texttt{))}.\!</math> Already, again, at this culminating stage of reflection, we begin to think of the newly named proposition as a distinctive individual, a particular function <math>\iota : \langle u, v, x \rangle \to \mathbb{B}.\!</math><br />
<br />
From now on, the terms ''thematic extension'' and ''thematization'' will be used to describe both the process and degree of explication that progresses through this series of pictures, both the operation of increasingly explicit symbolization and the dimension of variation that is swept out by it. To speak of this change in general, that takes us in our current example from <math>J\!</math> to <math>\iota,\!</math> we introduce a class of operators symbolized by the Greek letter <math>\theta,\!</math> writing <math>\iota = \theta J\!</math> in the present instance. The operator <math>\theta,\!</math> in the present situation bearing the type <math>\theta : [u, v] \to [u, v, x],\!</math> provides us with a convenient way of recapitulating and summarizing the complete cycle of thematic developments.<br />
<br />
Figure 21 shows how the thematic extension operator <math>\theta\!</math> acts on two further examples, the disjunction <math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math> and the equality <math>\texttt{((} u, v \texttt{))}.\!</math> Referring to the disjunction as <math>f(u, v)\!</math> and the equality as <math>f(u, v),\!</math> we may express the thematic extensions as <math>\varphi = \theta f\!</math> and <math>\gamma = \theta g.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 21 -- Thematization of Disjunction and Equality.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 21.} ~~ \text{Thematization of Disjunction and Equality}\!</math><br />
|}<br />
<br />
====Thematization : Truth Tables====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
That which distorts honest shapes or which creates unearthly beings or places or contingencies is a nuisance and a revolt.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 19]<br />
|}<br />
<br />
Tables 22 through 25 outline a method for computing the thematic extensions of propositions in terms of their coordinate values.<br />
<br />
A preliminary step, as illustrated in Table&nbsp;22, is to write out the truth table representations of the propositional forms whose thematic extensions one wants to compute, in the present instance, the functions <math>f(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math> and <math>g(u, v) = \texttt{((} u, v \texttt{))}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:50%"<br />
|+ style="height:30px" | <math>\text{Table 22.} ~~ \text{Disjunction}~ f ~\text{and Equality}~ g\!</math><br />
|- style="height:35px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black" | <math>f\!</math><br />
| <math>g\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Next, each propositional form is individually represented in the fashion shown in Tables&nbsp;23-i and 23-ii, using <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} g {}^{\prime\prime}\!</math> as function names and creating new variables <math>x\!</math> and <math>y\!</math> to hold the associated functional values. This pair of Tables outlines the first stage in the transition from the <math>2\!</math>-dimensional universes of <math>f\!</math> and <math>g\!</math> to the <math>3\!</math>-dimensional universes of <math>\theta f\!</math> and <math>\theta g.\!</math> The top halves of the Tables replicate the truth table patterns for <math>f\!</math> and <math>g\!</math> in the form <math>f : [u, v] \to [x]\!</math> and <math>g : [u, v] \to [y].\!</math> The bottom halves of the tables print the negatives of these pictures, as it were, and paste the truth tables for <math>\texttt{(} f \texttt{)}\!</math> and <math>\texttt{(} g \texttt{)}\!</math> under the copies for <math>f\!</math> and <math>g.\!</math> At this stage, the columns for <math>\theta f\!</math> and <math>\theta g\!</math> are appended almost as afterthoughts, amounting to indicator functions for the sets of ordered triples that make up the functions <math>f\!</math> and <math>g.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 23-i and 23-ii.} ~~ \text{Thematics of Disjunction and Equality (1)}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 23-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black" | <math>f\!</math><br />
| <math>x\!</math><br />
| <math>\varphi\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |<br />
<math>\begin{matrix}\to\\\to\\\to\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" | &nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 23-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black" | <math>g\!</math><br />
| <math>y\!</math><br />
| <math>\gamma\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |<br />
<math>\begin{matrix}\to\\\to\\\to\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" | &nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
All the data are now in place to give the truth tables for <math>\theta f\!</math> and <math>\theta g.\!</math> All that remains to be done is to permute the rows and change the roles of <math>x\!</math> and <math>y\!</math> from dependent to independent variables. In Tables&nbsp;24-i and 24-ii the rows are arranged in such a way as to put the 3-tuples <math>(u, v, x)\!</math> and <math>(u, v, y)\!</math> in binary numerical order, suitable for viewing as the arguments of the maps <math>\theta f = \varphi : [u, v, x] \to \mathbb{B}\!</math> and <math>\theta g = \gamma : [u, v, y] \to \mathbb{B}.\!</math> Moreover, the structure of the tables is altered slightly, allowing the now vestigial functions <math>\theta f\!</math> and <math>\theta g\!</math> to be passed over without further attention and shifting the heavy vertical bars a notch to the right. In effect, this clinches the fact that the thematic variables <math>x := \check{f}\!</math> and <math>y := \check{g}\!</math> are now treated as independent variables.<br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 24-i and 24-ii.} ~~ \text{Thematics of Disjunction and Equality (2)}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 24-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>f\!</math><br />
| <math>x\!</math><br />
| style="border-left:1px solid black" | <math>\varphi\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <font size="+2">&nbsp;<br></font><math>\begin{matrix}\to\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 24-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>g\!</math><br />
| <math>y\!</math><br />
| style="border-left:1px solid black" | <math>\gamma\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | &nbsp;<br><math>\begin{matrix}\to\\\to\end{matrix}\!</math><br>&nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
An optional reshuffling of the rows brings additional features of the thematic extensions to light. Leaving the columns in place for the sake of comparison, Tables&nbsp;25-i and 25-ii sort the rows in a different order, in effect treating <math>x\!</math> and <math>y\!</math> as the primary variables in their respective 3-tuples. Regarding the thematic extensions in the form <math>\varphi : [x, u, v] \to \mathbb{B}\!</math> and <math>\gamma : [y, u, v] \to \mathbb{B}\!</math> makes it easier to see in this tabular setting a property that was graphically obvious in the venn diagrams above. Specifically, when the thematic variable <math>\check{F}\!</math> is true then <math>\theta F\!</math> exhibits the pattern of the original <math>F,\!</math> and when <math>\check{F}\!</math> is false then <math>\theta F\!</math> exhibits the pattern of its negation <math>\texttt{(} F \texttt{)}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 25-i and 25-ii.} ~~ \text{Thematics of Disjunction and Equality (3)}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 25-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>f\!</math><br />
| <math>x\!</math><br />
| style="border-left:1px solid black" | <math>\varphi\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>{\to}\!</math><br><font size="+2">&nbsp;<br>&nbsp;<br>&nbsp;<br></font><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <font size="+2">&nbsp;<br></font><math>\begin{matrix}\to\\\to\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 25-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>g\!</math><br />
| <math>y\!</math><br />
| style="border-left:1px solid black" | <math>\gamma\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | &nbsp;<br><math>\begin{matrix}\to\\\to\end{matrix}\!</math><br>&nbsp;<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
Finally, Tables&nbsp;26-i and 26-ii compare the tacit extensions <math>\boldsymbol\varepsilon : [u, v] \to [u, v, x]\!</math> and <math>\boldsymbol\varepsilon : [u, v] \to [u, v, y]\!</math> with the thematic extensions of the same types, as applied to the propositions <math>f\!</math> and <math>g,\!</math> respectively.<br />
<br />
<br><br />
<br />
{| align="center" border="0" style="width:90%"<br />
|+ style="height:25px" | <math>\text{Tables 26-i and 26-ii.} ~~ \text{Tacit Extension and Thematization}\!</math><br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 26-i.} ~~ \text{Disjunction}~ f\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>x\!</math><br />
| style="border-left:1px solid black" | <math>\boldsymbol\varepsilon f\!</math><br />
| <math>\theta f\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
|}<br />
| width="50%" |<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:25px" | <math>\text{Table 26-ii.} ~~ \text{Equality}~ g\!</math><br />
|- style="height:25px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| <math>y\!</math><br />
| style="border-left:1px solid black" | <math>\boldsymbol\varepsilon g\!</math><br />
| <math>\theta g\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
|}<br />
<br />
<br><br />
<br />
Table 27 summarizes the thematic extensions of all propositions on two variables. Column&nbsp;4 lists the equations of form <math>\texttt{((} \check{f_i}, f_i (u, v) \texttt{))}\!</math> and Column&nbsp;5 simplifies these equations into the form of algebraic expressions. As always, <math>{}^{\backprime\backprime} + {}^{\prime\prime}\!</math> refers to exclusive disjunction and each <math>{}^{\backprime\backprime} \check{f} {}^{\prime\prime}\!</math> appearing in the last two Columns refers to the corresponding variable name <math>{}^{\backprime\backprime} \check{f_i} {}^{\prime\prime}.~\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 27.} ~~ \text{Thematization of Bivariate Propositions}\!</math><br />
|- style="height:30px; background:ghostwhite"<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>{f}\!</math><br />
| <math>\theta f\!</math><br />
| <math>\theta f\!</math><br />
|- style="background:ghostwhite"<br />
| align="right" | <math>u\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|- style="background:ghostwhite"<br />
| align="right" | <math>v\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| align="left" | <math>\texttt{((} \check{f} \texttt{,~(~)~))}\!</math><br />
| align="left" | <math>\check{f} + 1\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[4pt]<br />
f_{2}<br />
\\[4pt]<br />
f_{4}<br />
\\[4pt]<br />
f_{8}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[4pt]<br />
0~0~1~0<br />
\\[4pt]<br />
0~1~0~0<br />
\\[4pt]<br />
1~0~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} u \texttt{)~} v \texttt{~}<br />
\\[4pt]<br />
\texttt{~} u \texttt{~(} v \texttt{)}<br />
\\[4pt]<br />
\texttt{~} u \texttt{~~} v \texttt{~}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(u)(v)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~(u)~v~~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~u~(v)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~u~~v~~))}<br />
\end{array}</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + u + v + uv<br />
\\[4pt]<br />
\check{f} + v + uv + 1<br />
\\[4pt]<br />
\check{f} + u + uv + 1<br />
\\[4pt]<br />
\check{f} + uv + 1<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{3}<br />
\\[4pt]<br />
f_{12}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~1~1<br />
\\[4pt]<br />
1~1~0~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{)}<br />
\\[4pt]<br />
\texttt{~} u \texttt{~}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(u)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~u~~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + u<br />
\\[4pt]<br />
\check{f} + u + 1<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[4pt]<br />
f_{9}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[4pt]<br />
1~0~0~1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{,} v \texttt{)}<br />
\\[4pt]<br />
\texttt{((} u \texttt{,} v \texttt{))}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~~(} u \texttt{,} v \texttt{)~~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~((} u \texttt{,} v \texttt{))~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + u + v + 1<br />
\\[4pt]<br />
\check{f} + u + v<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{5}<br />
\\[4pt]<br />
f_{10}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~0~1<br />
\\[4pt]<br />
1~0~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} v \texttt{)}<br />
\\[4pt]<br />
\texttt{~} v \texttt{~}<br />
\end{matrix}</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(} v \texttt{)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~~} v \texttt{~~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + v<br />
\\[4pt]<br />
\check{f} + v + 1<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[4pt]<br />
f_{11}<br />
\\[4pt]<br />
f_{13}<br />
\\[4pt]<br />
f_{14}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[4pt]<br />
1~0~1~1<br />
\\[4pt]<br />
1~1~0~1<br />
\\[4pt]<br />
1~1~1~0<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(~} u \texttt{~~} v \texttt{~)}<br />
\\[4pt]<br />
\texttt{(~} u \texttt{~(} v \texttt{))}<br />
\\[4pt]<br />
\texttt{((} u \texttt{)~} v \texttt{~)}<br />
\\[4pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\end{matrix}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\texttt{((} \check{f} \texttt{,~(~} u \texttt{~~} v \texttt{~)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~(~} u \texttt{~(} v \texttt{))~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~((} u \texttt{)~} v \texttt{~)~))}<br />
\\[4pt]<br />
\texttt{((} \check{f} \texttt{,~((} u \texttt{)(} v \texttt{))~))}<br />
\end{array}\!</math><br />
| align="left" |<br />
<math>\begin{array}{l}<br />
\check{f} + uv<br />
\\[4pt]<br />
\check{f} + u + uv<br />
\\[4pt]<br />
\check{f} + v + uv<br />
\\[4pt]<br />
\check{f} + u + v + uv + 1<br />
\end{array}\!</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| align="left" | <math>\texttt{((} \check{f} \texttt{,~((~))~))}\!</math><br />
| align="left" | <math>\check{f}\!</math><br />
|}<br />
<br />
<br><br />
<br />
In order to show what all of the thematic extensions from two dimensions to three dimensions look like in terms of coordinates, Tables&nbsp;28 and 29 present ordinary truth tables for the functions <math>f_i : \mathbb{B}^2 \to \mathbb{B}\!</math> and for the corresponding thematizations <math>\theta f_i = \varphi_i : \mathbb{B}^3 \to \mathbb{B}.\!</math><br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 28.} ~~ \text{Propositions on Two Variables}\!</math><br />
|- style="height:35px; background:ghostwhite"<br />
| style="width:5%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:5%; border-bottom:1px solid black; border-left:1px solid black" | <math>f_{0}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{1}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{2}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{3}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{4}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{5}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{6}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{7}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{8}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{9}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{10}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{11}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{12}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{13}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{14}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>f_{15}\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 29.} ~~ \text{Thematic Extensions of Bivariate Propositions}\!</math><br />
|- style="height:35px; background:ghostwhite"<br />
| style="width:5%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\check{f}\!</math><br />
| style="width:5%; border-bottom:1px solid black; border-left:1px solid black" | <math>\varphi_{0}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{1}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{2}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{3}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{4}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{5}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{6}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{7}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{8}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{9}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{10}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{11}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{12}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{13}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{14}\!</math><br />
| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{15}\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
|-<br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
| &nbsp;<br />
| <math>1\!</math><br />
|-<br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|-<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>0\!</math><br />
| style="border-left:1px solid black" | <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
|-<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| style="border-left:1px solid black" | &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| &nbsp;<br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Propositional Transformations===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
If only the word &lsquo;artificial&rsquo; were associated with the idea of ''art'', or expert skill gained through voluntary apprenticeship (instead of suggesting the factitious and unreal), we might say that ''logical'' refers to artificial thought.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 56&ndash;57]<br />
|}<br />
<br />
In this section we develop a comprehensive set of concepts for dealing with transformations between universes of discourse. In this most general setting the source and target universes of a transformation are allowed to be different, but may be the same. When we apply these concepts to dynamic systems we focus on the important special case of transformations that map a universe into itself, regarding them as the state transitions of a discrete dynamical process and placing them among the myriad ways that a universe of discourse might change, and by that change turn into itself.<br />
<br />
====Alias and Alibi Transformations====<br />
<br />
There are customarily two modes of understanding a transformation, at least, when we try to interpret its relevance to something in reality. A transformation always refers to a changing prospect, to say it in a unified but equivocal way, but this can be taken to mean either a subjective change in the interpreting observer's point of view or an objective change in the systematic subject of discussion. In practice these variant uses of the transformation concept are distinguished in the following terms:<br />
<br />
# A ''perspectival'' or ''alias'' transformation refers to a shift in perspective or a change in language that takes place in the observer's frame of reference.<br />
# A ''transitional'' or ''alibi'' transformation refers to a change of position or an alteration of state that occurs in the object system as it falls under study.<br />
<br />
(For a recent discussion of the alias vs. alibi issue, as it relates to linear transformations in vector spaces and to other issues of an algebraic nature, see [MaB, 256, 582-4].)<br />
<br />
Naturally, when we are concerned with the dynamical properties of a system, the transitional aspect of transformation is the factor that comes to the fore, and this involves us in contemplating all of the ways of changing a universe into itself while remaining under the rule of established dynamical laws. In the prospective application to dynamic systems, and to neural networks viewed in this light, our interest lies chiefly with the transformations of a state space into itself that constitute the state transitions of a discrete dynamic process. Nevertheless, many important properties of these transformations, and some constructions that we need to see most clearly, are independent of the transitional interpretation and are likely to be confounded with irrelevant features if presented first and only in that association.<br />
<br />
In addition, and in partial contrast, intelligent systems are exactly that species of dynamic agents that have the capacity to have a point of view, and we cannot do justice to their peculiar properties without examining their ability to form and transform their own frames of reference in exposure to the elements of their own experience. In this setting, the perspectival aspect of transformation is the facet that shines most brightly, perhaps too often leaving us fascinated with mere glimmerings of its actual potential. It needs to be emphasized that nothing of the ordinary sort needs be moved in carrying out a transformation under the alias interpretation, that it may only involve a change in the forms of address, an amendment of the terms which are customed to approach and fashioned to describe the very same things in the very same world. But again, working within a discipline of realistic computation, we know how formidably complex and resource-consuming such transformations of perspective can be to implement in practice, much less to endow in the self-governed form of a nascently intelligent dynamical system.<br />
<br />
====Transformations of General Type====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
''Es ist passiert'', &ldquo;it just sort of happened&rdquo;, people said there when other people in other places thought heaven knows what had occurred. It was a peculiar phrase, not known in this sense to the Germans and with no equivalent in other languages, the very breath of it transforming facts and the bludgeonings of fate into something light as eiderdown, as thought itself.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 34]<br />
|}<br />
<br />
Consider the situation illustrated in Figure&nbsp;30, where the alphabets <math>\mathcal{U} = \{ u, v \}\!</math> and <math>\mathcal{X} = \{ x, y, z \}\!</math> are used to label basic features in two different logical universes, <math>U^\bullet = [u, v]\!</math> and <math>X^\bullet = [x, y, z].\!</math><br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
o-------------------------------------------------------o<br />
| U |<br />
| |<br />
| o-----------o o-----------o |<br />
| / \ / \ |<br />
| / o \ |<br />
| / / \ \ |<br />
| / / \ \ |<br />
| o o o o |<br />
| | | | | |<br />
| | u | | v | |<br />
| | | | | |<br />
| o o o o |<br />
| \ \ / / |<br />
| \ \ / / |<br />
| \ o / |<br />
| \ / \ / |<br />
| o-----------o o-----------o |<br />
| |<br />
| |<br />
o---------------------------o---------------------------o<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
/ \ / \ / \<br />
o-------------------------o o-------------------------o o-------------------------o<br />
| U | | U | | U |<br />
| o---o o---o | | o---o o---o | | o---o o---o |<br />
| / \ / \ | | / \ / \ | | / \ / \ |<br />
| / o \ | | / o \ | | / o \ |<br />
| / / \ \ | | / / \ \ | | / / \ \ |<br />
| o o o o | | o o o o | | o o o o |<br />
| | u | | v | | | | u | | v | | | | u | | v | |<br />
| o o o o | | o o o o | | o o o o |<br />
| \ \ / / | | \ \ / / | | \ \ / / |<br />
| \ o / | | \ o / | | \ o / |<br />
| \ / \ / | | \ / \ / | | \ / \ / |<br />
| o---o o---o | | o---o o---o | | o---o o---o |<br />
| | | | | |<br />
o-------------------------o o-------------------------o o-------------------------o<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ g | \ f / | h /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ | \ / | /<br />
\ o----------|-----------\-----/-----------|----------o /<br />
\ | X | \ / | | /<br />
\ | | \ / | | /<br />
\ | | o-----o-----o | | /<br />
\| | / \ | |/<br />
\ | / \ | /<br />
|\ | / \ | /|<br />
| \ | / \ | / |<br />
| \ | / \ | / |<br />
| \ | o x o | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \ | | | | / |<br />
| \| | | |/ |<br />
| o--o--------o o--------o--o |<br />
| / \ \ / / \ |<br />
| / \ \ / / \ |<br />
| / \ o / \ |<br />
| / \ / \ / \ |<br />
| / \ / \ / \ |<br />
| o o--o-----o--o o |<br />
| | | | | |<br />
| | | | | |<br />
| | | | | |<br />
| | y | | z | |<br />
| | | | | |<br />
| | | | | |<br />
| o o o o |<br />
| \ \ / / |<br />
| \ \ / / |<br />
| \ o / |<br />
| \ / \ / |<br />
| \ / \ / |<br />
| o-----------o o-----------o |<br />
| |<br />
| |<br />
o---------------------------------------------------o<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ p , q /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
\ /<br />
o<br />
<br />
Figure 30. Generic Frame of a Logical Transformation<br />
</pre><br />
|}<br />
<br />
Enter the picture, as we usually do, in the middle of things, with features like <math>x, y , z\!</math> that present themselves to be simple enough in their own right and that form a satisfactory, if temporary foundation to provide a basis for discussion. In this universe and on these terms we find expression for various propositions and questions of principal interest to ourselves, as indicated by the maps <math>p, q : X \to \mathbb{B}.\!</math> Then we discover that the simple features <math>\{ x, y, z \}\!</math> are really more complex than we thought at first, and it becomes useful to regard them as functions <math>\{ f, g, h \}\!</math> of other features <math>\{ u, v \}\!</math> that we place in a preface to our original discourse, or suppose as topics of a preliminary universe of discourse <math>U^\bullet = [u, v].\!</math> It may happen that these late-blooming but pre-ambling features are found to lie closer, in a sense that may be our job to determine, to the central nature of the situation of interest, in which case they earn our regard as being more fundamental, but these functions and features are only required to supply a critical stance on the universe of discourse or an alternate perspective on the nature of things in order to be preserved as useful.<br />
<br />
A particular transformation <math>F : [u, v] \to [x, y, z]\!</math> may be expressed by a system of equations, as shown below. Here, <math>F\!</math> is defined by its component maps <math>F = (F_1, F_2, F_3) = (f, g, h),\!</math> where each component map in <math>\{ f, g, h \}\!</math> is a proposition of type <math>\mathbb{B}^n \to \mathbb{B}^1.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:50%"<br />
|<br />
<math>\begin{matrix}<br />
x & = & f(u, v)<br />
\\[10pt]<br />
y & = & g(u, v)<br />
\\[10pt]<br />
z & = & h(u, v)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Regarded as a logical statement, this system of equations expresses a relation between a collection of freely chosen propositions <math>\{ f, g, h \}\!</math> in one universe of discourse and the special collection of simple propositions <math>\{ x, y, z \}\!</math> on which is founded another universe of discourse. Growing familiarity with a particular transformation of discourse, and the desire to achieve a ready understanding of its implications, requires that we be able to convert this information about generals and simples into information about all the main subtypes of propositions, including the linear and singular propositions.<br />
<br />
===Analytic Expansions : Operators and Functors===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Consider what effects that might ''conceivably'' have practical bearings you ''conceive'' the objects of your ''conception'' to have. Then, your ''conception'' of those effects is the whole of your ''conception'' of the object.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; C.S. Peirce, &ldquo;The Maxim of Pragmatism&rdquo;, CP 5.438<br />
|}<br />
<br />
Given the barest idea of a logical transformation, as suggested by the sketch in Figure&nbsp;30, and having conceptualized the universe of discourse, with all of its points and propositions, as a beginning object of discussion, we are ready to enter the next phase of our investigation.<br />
<br />
====Operators on Propositions and Transformations====<br />
<br />
The next step is naturally inclined toward objects of the next higher order, namely, with operators that take in argument lists of logical transformations and that give back specified types of logical transformations as their results. For our present aims, we do not need to consider the most general class of such operators, nor any one of them for its own sake. Rather, we are interested in the special sorts of operators that arise in the study and analysis of logical transformations. Figuratively speaking, these operators serve as instruments for the live tomography (and hopefully not the vivisection) of the forms of change under view. Beyond that, they open up ways to implement the changes of view that we need to grasp all the variations on a transformational theme, or to appreciate enough of its significant features to &ldquo;get the drift&rdquo; of the change occurring, to form a passing acquaintance or a synthetic comprehension of its general character and disposition.<br />
<br />
The simplest type of operator is one that takes a single transformation as an argument and returns a single transformation as a result, and most of the operators explicitly considered in our discussion will be of this kind. Figure&nbsp;31 illustrates the typical situation.<br />
<br />
{| align="center" border="0" cellpadding="20"<br />
|<br />
<pre><br />
o---------------------------------------o<br />
| |<br />
| |<br />
| U% F X% |<br />
| o------------------>o |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| !W! | | !W! |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| v v |<br />
| o------------------>o |<br />
| !W!U% !W!F !W!X% |<br />
| |<br />
| |<br />
o---------------------------------------o<br />
Figure 31. Operator Diagram (1)<br />
</pre><br />
|}<br />
<br />
In this Figure <math>{}^{\backprime\backprime} \mathsf{W} {}^{\prime\prime}\!</math> stands for a generic operator <math>\mathsf{W},\!</math> in this case one that takes a logical transformation <math>F\!</math> of type <math>(U^\bullet \to X^\bullet)\!</math> into a logical transformation <math>\mathsf{W}F\!</math> of type <math>(\mathsf{W}U^\bullet \to \mathsf{W}X^\bullet).\!</math> Thus, the operator <math>\mathsf{W}\!</math> must be viewed as making assignments for both families of objects we have previously considered, that is, for universes of discourse like <math>{U^\bullet}\!</math> and <math>{X^\bullet}\!</math> and for logical transformations like <math>F.\!</math><br />
<br />
'''Note.''' Strictly speaking, an operator like <math>\mathsf{W}\!</math> works between two whole categories of universes and transformations, which we call the ''source'' and the ''target'' categories of <math>\mathsf{W}.\!</math> Given this setting, <math>\mathsf{W}\!</math> specifies for each universe <math>U^\bullet\!</math> in its source category a definite universe <math>\mathsf{W}U^\bullet\!</math> in its target category, and to each transformation <math>F\!</math> in its source category it assigns a unique transformation <math>\mathsf{W}F\!</math> in its target category. Naturally, this only works if <math>\mathsf{W}\!</math> takes the source <math>U^\bullet</math> and the target <math>X^\bullet</math> of the map <math>F\!</math> over to the source <math>\mathsf{W}U^\bullet\!</math> and the target <math>\mathsf{W}X^\bullet\!</math> of the map <math>\mathsf{W}F.\!</math> With luck or care enough, we can avoid ever having to put anything like that in words again, letting diagrams do the work. In the situations of present concern we are usually focused on a single transformation <math>F,\!</math> and thus we can take it for granted that the assignment of universes under <math>\mathsf{W}\!</math> is defined appropriately at the source and target ends of <math>F.\!</math> It is not always the case, though, that we need to use the particular names (like <math>{}^{\backprime\backprime} \mathsf{W}U^\bullet {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \mathsf{W}X^\bullet {}^{\prime\prime}\!</math>) that <math>\mathsf{W}\!</math> assigns by default to its operative image universes. In most contexts we will usually have a prior acquaintance with these universes under other names and it is necessary only that we can tell from the information associated with an operator <math>\mathsf{W}\!</math> what universes they are.<br />
<br />
In Figure&nbsp;31 the maps <math>F\!</math> and <math>\mathsf{W}F\!</math> are displayed horizontally, the way one normally orients functional arrows in a written text, and <math>\mathsf{W}\!</math> rolls the map <math>F\!</math> downward into the images that are associated with <math>\mathsf{W}F.\!</math> In Figure&nbsp;32 the same information is redrawn so that the maps <math>F\!</math> and <math>\mathsf{W}F\!</math> flow down the page, and <math>\mathsf{W}\!</math> unfurls the map <math>F\!</math> rightward into domains that are the eminent purview of <math>\mathsf{W}F.\!</math><br />
<br />
{| align="center" border="0" cellpadding="20"<br />
|<br />
<pre><br />
o---------------------------------------o<br />
| |<br />
| |<br />
| U% !W! !W!U% |<br />
| o------------------>o |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| F | | !W!F |<br />
| | | |<br />
| | | |<br />
| | | |<br />
| v v |<br />
| o------------------>o |<br />
| X% !W! !W!X% |<br />
| |<br />
| |<br />
o---------------------------------------o<br />
Figure 32. Operator Diagram (2)<br />
</pre><br />
|}<br />
<br />
The latter arrangement, as exhibited in Figure&nbsp;32, is more congruent with the thinking about operators that we shall do in the rest of this discussion, since all logical transformations from here on out will be pictured vertically, after the fashion of Figure&nbsp;30.<br />
<br />
====Differential Analysis of Propositions and Transformations====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" | The resultant metaphysical problem now is this: ''Does the man go round the squirrel or not?''<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 43]<br />
|}<br />
<br />
The approach to the differential analysis of logical propositions and transformations of discourse to be pursued here is carried out in terms of particular operators <math>\mathsf{W}\!</math> that act on propositions <math>F\!</math> or on transformations <math>F\!</math> to yield the corresponding operator maps <math>\mathsf{W}F.\!</math> The operator results then become the subject of a series of further stages of analysis, which take them apart into their propositional components, rendering them as a set of purely logical constituents. After this is done, all the parts are then re-integrated to reconstruct the original object in the light of a more complete understanding, at least in ways that enable one to appreciate certain aspects of it with fresh insight.<br />
<br />
* '''Remark on Strategy.''' At this point we run into a set of conceptual difficulties that force us to make a strategic choice in how we proceed. Part of the problem can be remedied by extending our discussion of tacit extensions to the transformational context. But the troubles that remain are much more obstinate and lead us to try two different types of solution. The approach that we develop first makes use of a variant type of extension operator, the ''trope extension'', to be defined below. This method is more conservative and requires less preparation, but has features which make it seem unsatisfactory in the long run. A more radical approach, but one with a better hope of long term success, makes use of the notion of ''contingency spaces''. These are an even more generous type of extended universe than the kind we currently use, but are defined subject to certain internal constraints. The extra work needed to set up this method forces us to put it off to a later stage. However, as a compromise, and to prepare the ground for the next pass, we call attention to the various conceptual difficulties as they arise along the way and try to give an honest estimate of how well our first approach deals with them.<br />
<br />
We now describe in general terms the particular operators that are instrumental to this form of analysis. The main series of operators all have the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathsf{W}<br />
& : &<br />
( U^\bullet \to X^\bullet )<br />
& \to &<br />
( \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet )<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
If we assume that the source universe <math>U^\bullet</math> and the target universe <math>X^\bullet</math> have finite dimensions <math>n\!</math> and <math>k,\!</math> respectively, then each operator <math>\mathsf{W}\!</math> is encompassed by the same abstract type:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathsf{W}<br />
& : &<br />
( [\mathbb{B}^n] \to [\mathbb{B}^k] )<br />
& \to &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k \times \mathbb{D}^k] )<br />
\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Since the range features of the operator result <math>\mathsf{W}F : [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k \times \mathbb{D}^k]</math> can be sorted by their ordinary versus differential qualities and the component maps can be examined independently, the complete operator <math>\mathsf{W}\!</math> can be separated accordingly into two components, in the form <math>\mathsf{W} = (\boldsymbol\varepsilon, \mathrm{W}).\!</math> Given a fixed context of source and target universes, <math>\boldsymbol\varepsilon\!</math> is always the same type of operator, a multiple component version of the tacit extension operators that were described earlier. In this context <math>\boldsymbol\varepsilon\!</math> has the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{array}{lccccc}<br />
\text{Concrete type}<br />
& \boldsymbol\varepsilon<br />
& : &<br />
( U^\bullet \to X^\bullet )<br />
& \to &<br />
( \mathrm{E}U^\bullet \to X^\bullet )<br />
\\[10pt]<br />
\text{Abstract type}<br />
& \boldsymbol\varepsilon<br />
& : &<br />
( [\mathbb{B}^n] \to [\mathbb{B}^k] )<br />
& \to &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k] )<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
On the other hand, the operator <math>\mathrm{W}\!</math> is specific to each <math>\mathsf{W}.\!</math> In this context <math>\mathrm{W}\!</math> always has the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{array}{lccccc}<br />
\text{Concrete type}<br />
& W<br />
& : &<br />
( U^\bullet \to X^\bullet )<br />
& \to &<br />
( \mathrm{E}U^\bullet \to \mathrm{d}X^\bullet )<br />
\\[10pt]<br />
\text{Abstract type}<br />
& W<br />
& : &<br />
( [\mathbb{B}^n] \to [\mathbb{B}^k] )<br />
& \to &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{D}^k] )<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
In the types just assigned to <math>\boldsymbol\varepsilon\!</math> and <math>\mathrm{W}\!</math> and by implication to their results <math>\boldsymbol\varepsilon F\!</math> and <math>\mathrm{W}F,\!</math> we have listed the most restrictive ranges defined for them rather than the more expansive target spaces that subsume these ranges. When there is need to recognize both, we may use type indications like the following:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\boldsymbol\varepsilon F<br />
& : &<br />
( \mathrm{E}U^\bullet \to X^\bullet \subseteq \mathrm{E}X^\bullet )<br />
& \cong &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k] \subseteq [\mathbb{B}^k \times \mathbb{D}^k] )<br />
\\[10pt]<br />
WF<br />
& : &<br />
( \mathrm{E}U^\bullet \to \mathrm{d}X^\bullet \subseteq \mathrm{E}X^\bullet )<br />
& \cong &<br />
( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{D}^k] \subseteq [\mathbb{B}^k \times \mathbb{D}^k] )<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Hopefully, though, a general appreciation of these subsumptions will prevent us from having to make such declarations more often than absolutely necessary.<br />
<br />
In giving names to these operators we try to preserve as much of the traditional nomenclature and as many of the classical associations as possible. The chief difficulty in doing this is occasioned by the distinction between the &ldquo;sans&nbsp;serif&rdquo; operators <math>\mathsf{W}\!</math> and their &ldquo;serified&rdquo; components <math>\mathrm{W},\!</math> which forces us to find two distinct but parallel sets of terminology. Here is a plan to that purpose. First, the component operators <math>\mathrm{W}\!</math> are named by analogy with the corresponding operators in the classical difference calculus. Next, the complete operators <math>\mathsf{W} = (\boldsymbol\varepsilon, \mathrm{W})</math> are assigned titles according to their roles in a geometric or trigonometric allegory, if only to ensure that the tangent functor, that belongs to this family and whose exposition we are still working toward, comes out fit with its customary name. Finally, the operator results <math>\mathsf{W}F\!</math> and <math>\mathrm{W}F\!</math> can be fixed in our frame of reference by tethering the operative adjective for <math>\mathsf{W}\!</math> or <math>\mathrm{W}\!</math> to the anchoring epithet &ldquo;map&rdquo;, in conformity with an already standard practice.<br />
<br />
=====The Secant Operator : '''E'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Mr. Peirce, after pointing out that our beliefs are really rules for action, said that, to develop a thought's meaning, we need only determine what conduct it is fitted to produce: that conduct is for us its sole significance.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 46]<br />
|}<br />
<br />
Figures&nbsp;33-i and 33-ii depict two stages in the form of analysis that will be applied to transformations throughout the remainder of this study. From now on our interest is staked on an operator denoted <math>{}^{\backprime\backprime} \mathsf{E} {}^{\prime\prime},\!</math> which receives the principal investment of analytic attention, and on the constituent parts of <math>\mathsf{E},\!</math> which derive their shares of significance as developed by the analysis. In the sequel, we refer to <math>\mathsf{E}\!</math> as the ''secant operator'', taking it for granted that a context has been chosen that defines its type. The secant operator has the component description <math>\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}),\!</math> and its active ingredient <math>\mathrm{E}\!</math> is known as the ''enlargement operator''. (Here, we name <math>\mathrm{E}\!</math> after the literal ancestor of the shift operator in the calculus of finite differences, defined so that <math>\mathrm{E}f(x) = f(x+1)\!</math> for any suitable function <math>f,\!</math> though of course the logical analogue that we take up here must have a rather different definition.)<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
U% $E$ $E$U% $E$U% $E$U%<br />
o------------------>o============o============o<br />
| | | |<br />
| | | |<br />
| | | |<br />
| | | |<br />
F | | $E$F = | $d$^0.F + | $r$^0.F<br />
| | | |<br />
| | | |<br />
| | | |<br />
v v v v<br />
o------------------>o============o============o<br />
X% $E$ $E$X% $E$X% $E$X%<br />
<br />
Figure 33-i. Analytic Diagram (1)<br />
</pre><br />
|}<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
U% $E$ $E$U% $E$U% $E$U% $E$U%<br />
o------------------>o============o============o============o<br />
| | | | |<br />
| | | | |<br />
| | | | |<br />
| | | | |<br />
F | | $E$F = | $d$^0.F + | $d$^1.F + | $r$^1.F<br />
| | | | |<br />
| | | | |<br />
| | | | |<br />
v v v v v<br />
o------------------>o============o============o============o<br />
X% $E$ $E$X% $E$X% $E$X% $E$X%<br />
<br />
Figure 33-ii. Analytic Diagram (2)<br />
</pre><br />
|}<br />
<br />
In its action on universes <math>\mathsf{E}\!</math> yields the same result as <math>\mathrm{E},\!</math> a fact that can be expressed in equational form by writing <math>\mathsf{E}U^\bullet = \mathrm{E}U^\bullet\!</math> for any universe <math>U^\bullet.\!</math> Notice that the extended universes across the top and bottom of the diagram are indicated to be strictly identical, rather than requiring a corresponding decomposition for them. In a certain sense, the functional parts of <math>\mathsf{E}F\!</math> are partitioned into separate contexts that have to be re-integrated again, but the best image to use is that of making transparent copies of each universe and then overlapping their functional contents once more at the conclusion of the analysis, as suggested by the graphic conventions that are used at the top of Figure&nbsp;30.<br />
<br />
Acting on a transformation <math>F\!</math> from universe <math>U^\bullet\!</math> to universe <math>X^\bullet,\!</math> the operator <math>\mathsf{E}\!</math> determines a transformation <math>\mathsf{E}F\!</math> from <math>\mathsf{E}U^\bullet\!</math> to <math>\mathsf{E}X^\bullet.\!</math> The map <math>\mathsf{E}F\!</math> forms the main body of evidence to be investigated in performing a differential analysis of <math>F.\!</math> Because we shall frequently be focusing on small pieces of this map for considerable lengths of time, and consequently lose sight of the &ldquo;big picture&rdquo;, it is critically important to emphasize that the map <math>\mathsf{E}F\!</math> is a transformation that determines a relation from one extended universe into another. This means that we should not be satisfied with our understanding of a transformation <math>F\!</math> until we can lay out the full &ldquo;parts diagram&rdquo; of <math>\mathsf{E}F\!</math> along the lines of the generic frame in Figure&nbsp;30.<br />
<br />
Working within the confines of propositional calculus, it is possible to give an elementary definition of <math>\mathsf{E}F\!</math> by means of a system of propositional equations, as we now describe.<br />
<br />
Given a transformation<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>F = (F_1, \ldots, F_k) : \mathbb{B}^n \to \mathbb{B}^k\!</math><br />
|}<br />
<br />
of concrete type<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>F : [u_1, \ldots, u_n] \to [x_1, \ldots, x_k],\!</math><br />
|}<br />
<br />
the transformation<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\mathsf{E}F = (F_1, \ldots, F_k, \mathrm{E}F_1, \ldots, \mathrm{E}F_k) : \mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B}^k \times \mathbb{D}^k\!</math><br />
|}<br />
<br />
of concrete type<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\mathsf{E}F : [u_1, \dots, u_n, \mathrm{d}u_1, \dots, \mathrm{d}u_n] \to [x_1, \ldots, x_k, \mathrm{d}x_1, \ldots, \mathrm{d}x_k]\!</math><br />
|}<br />
<br />
is defined by means of the following system of logical equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\\[16pt]<br />
\mathrm{d}x_1<br />
& = & \mathrm{E}F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1 + \mathrm{d}u_1, \ldots, u_n + \mathrm{d}u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
\mathrm{d}x_k<br />
& = & \mathrm{E}F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1 + \mathrm{d}u_1, \ldots, u_n + \mathrm{d}u_n)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
It is important to note that this system of equations can be read as a conjunction of equational propositions, in effect, as a single proposition in the universe of discourse generated by all the named variables. Specifically, this is the universe of discourse over <math>2(n+k)\!</math> variables denoted by:<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
\mathrm{E}[\mathcal{U} \cup \mathcal{X}]<br />
& = &<br />
[u_1, \ldots, u_n, ~ x_1, \ldots, x_k, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n, ~ \mathrm{d}x_1, \ldots, \mathrm{d}x_k].<br />
\end{matrix}</math><br />
|}<br />
<br />
In this light, it should be clear that the system of equations defining <math>\mathsf{E}F\!</math> embodies, in a higher rank and differentially extended version, an analogy with the process of thematization that we treated earlier for propositions of type <math>F : \mathbb{B}^n \to \mathbb{B}.\!</math><br />
<br />
The entire collection of constraints that is represented in the above system of equations may be abbreviated by writing <math>\mathsf{E}F = (\boldsymbol\varepsilon F, \mathrm{E}F),\!</math> for any map <math>F.\!</math> This is tantamount to regarding <math>\mathsf{E}\!</math> as a complex operator, <math>\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}),\!</math> with a form of application that distributes each component of the operator to work on each component of the operand, as follows:<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
|<br />
<math>\begin{matrix}<br />
\mathsf{E}F<br />
& = &<br />
(\boldsymbol\varepsilon, \mathrm{E})F<br />
& = &<br />
(\boldsymbol\varepsilon F, \mathrm{E}F)<br />
& = &<br />
(\boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k, ~ \mathrm{E}F_1, \ldots, \mathrm{E}F_k).<br />
\end{matrix}</math><br />
|}<br />
<br />
Quite a lot of &ldquo;thematic infrastructure&rdquo; or interpretive information is being swept under the rug in the use of such abbreviations. When confusion arises about the meaning of such constructions, one always has recourse to the defining system of equations, in its totality a purely propositional expression. This means that the parenthesized argument lists, that were used in this context to build an image of multi-component transformations, should not be expected to determine a well-defined product in themselves but only to serve as reminders of the prior thematic decisions (choices of variable names, etc.) that have to be made in order to determine one. Accordingly, the argument list notation can be regarded as a kind of ''thematic frame'', an interpretive storage device that preserves the proper associations of concrete logical features between the extended universes at the source and target of <math>\mathsf{E}F.\!</math><br />
<br />
The generic notations <math>\mathsf{d}^0\!F, \mathsf{d}^1\!F, \ldots, \mathsf{d}^m\!F\!</math> in Figure&nbsp;33 refer to the increasing orders of differentials that are extracted in the course of analyzing <math>F.\!</math> When the analysis is halted at a partial stage of development, notations like <math>\mathsf{r}^0\!F, \mathsf{r}^1\!F, \ldots, \mathsf{r}^m\!F\!</math> may be used to summarize the contributions to <math>\mathsf{E}F\!</math> that remain to be analyzed. The Figure illustrates a convention that makes <math>\mathsf{r}^m\!F,\!</math> in effect, the sum of all differentials of order strictly greater than <math>m.\!</math><br />
<br />
We next discuss the operators that figure into this form of analysis, describing their effects on transformations. In simplified or specialized contexts these operators tend to take on a variety of different names and notations, some of whose number we introduce along the way.<br />
<br />
=====The Radius Operator : '''e'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
And the tangible fact at the root of all our thought-distinctions, however subtle, is that there is no one of them so fine as to consist in anything but a possible difference of practice.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 46]<br />
|}<br />
<br />
The operator identified as <math>\mathrm{d}^0\!</math> in the analytic diagram (Figure&nbsp;33) has the sole purpose of creating a proxy for <math>F\!</math> in the appropriately extended context. Construed in terms of its broadest components, <math>\mathrm{d}^0\!</math> is equivalent to the doubly tacit extension operator <math>(\boldsymbol\varepsilon, \boldsymbol\varepsilon),\!</math> in recognition of which let us redub it as <math>{}^{\backprime\backprime} \mathsf{e} {}^{\prime\prime}.\!</math> Pursuing a geometric analogy, we may refer to <math>\mathsf{e} =(\boldsymbol\varepsilon, \boldsymbol\varepsilon) = \mathrm{d}^0\!</math> as the ''radius operator''. The operation intended by all of these forms is defined by the following equation:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathsf{e}F<br />
& = &<br />
(\boldsymbol\varepsilon, \boldsymbol\varepsilon)F<br />
\\[4pt]<br />
& = &<br />
(\boldsymbol\varepsilon F, ~ \boldsymbol\varepsilon F)<br />
\\[4pt]<br />
& = &<br />
(\boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k, ~ \boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k).<br />
\end{array}</math><br />
|}<br />
<br />
which is tantamount to the system of equations below.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\\[16pt]<br />
\mathrm{d}x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
\mathrm{d}x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
=====The Phantom of the Operators : '''&eta;'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
I was wondering what the reason could be, when I myself raised my head and everything within me seemed drawn towards the Unseen, ''which was playing the most perfect music''!<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 81]<br />
|}<br />
<br />
We now describe an operator whose persistent but elusive action behind the scenes, whose slightly twisted and ambivalent character, and whose fugitive disposition, caught somewhere in flight between the arrantly negative and the positive but errant intent, has cost us some painstaking trouble to detect. In the end we shall place it among the other extensions and projections, as a shade among shadows, of muted tones and motley hue, that adumbrates its own thematic frame and paradoxically lights the way toward a whole new spectrum of values.<br />
<br />
Given a transformation <math>F : [u_1, \ldots, u_n] \to [x_1, \dots, x_k],\!</math> we often have call to consider a family of related transformations, all having the form:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>F^\dagger : [u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n] \to [\mathrm{d}x_1, \dots, \mathrm{d}x_k].\!</math><br />
|}<br />
<br />
The operator <math>\eta</math> is introduced to deal with the simplest one of these maps:<br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
| <math>\eta F : [u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n] \to [\mathrm{d}x_1, \ldots \mathrm{d}x_k],\!</math><br />
|}<br />
<br />
which is defined by the following equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x_1<br />
& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_1 (u_1, \ldots, u_n)<br />
\\[4pt]<br />
\cdots && \cdots && \cdots<br />
\\[4pt]<br />
\mathrm{d}x_k<br />
& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n)<br />
& = & F_k (u_1, \ldots, u_n)<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
In effect, the operator <math>\eta\!</math> is nothing but the stand-alone version of a procedure that is otherwise invoked subordinate to the work of the radius operator <math>\mathsf{e}.\!</math> Operating independently, <math>\eta\!</math> achieves precisely the same results that the second <math>\boldsymbol\varepsilon\!</math> in <math>(\boldsymbol\varepsilon, \boldsymbol\varepsilon)\!</math> accomplishes by working within the context of its ordered pair thematic frame. From this point on, because the use of <math>\boldsymbol\varepsilon\!</math> and <math>\eta\!</math> in this setting combines the aims of both the tacit and the thematic extensions, and because <math>\eta\!</math> reflects in regard to <math>\boldsymbol\varepsilon\!</math> little more than the application of a differential twist, a mere turn of phrase, we refer to <math>\eta\!</math> as the ''trope extension'' operator.<br />
<br />
=====The Chord Operator : '''D'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
What difference would it practically make to any one if this notion rather than that notion were true? If no practical difference whatever can be traced, then the alternatives mean practically the same thing, and all dispute is idle.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 45]<br />
|}<br />
<br />
Next we discuss an operator that is always immanent in this form of analysis, and remains implicitly present in the entire proceeding. It may appear once as a record: a relic or revenant that reprises the reminders of an earlier stage of development. Or it may appear always as a resource: a reserve or redoubt that caches in advance an echo of what remains to be played out, cleared up, and requited in full at a future stage. And all of this remains true whether or not we recall the key at any time, and whether or not the subtending theme is recited explicitly at any stage of play.<br />
<br />
This is the operator that is referred to as <math>\mathsf{r}^0\!</math> in the initial stage of analysis (Figure&nbsp;33-i) and that is expanded as <math>\mathsf{d}^1 + \mathsf{r}^1\!</math> in the subsequent step (Figure&nbsp;33-ii). In congruence, but not quite harmony with our allusions of analogy that are not quite geometry, we call this the ''chord operator'' and denote it <math>\mathsf{D}.\!</math> In the more casual terms that are here introduced, <math>\mathsf{D}</math> is defined as the remainder of <math>\mathsf{E}\!</math> and <math>\mathsf{e}\!</math> and it assigns a due measure to each undertone of accord or discord that is struck between the note of enterprise <math>\mathsf{E}\!</math> and the bar of exigency <math>\mathsf{e}.\!</math><br />
<br />
The tension between these counterposed notions, in balance transient but regular in stridence, may be refracted along familiar lines, though never by any such fraction resolved. In this style we write <math>\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}),\!</math> calling <math>\mathrm{D}\!</math> the ''difference operator'' and noting that it plays a role in this realm of mutable and diverse discourse that is analogous to the part taken by the discrete difference operator in the ordinary difference calculus. Finally, we should note that the chord <math>\mathsf{D}\!</math> is not one that need be lost at any stage of development. At the <math>m^\text{th}\!</math> stage of play it can always be reconstituted in the following form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathsf{D}<br />
& = & \mathsf{E} - \mathsf{e}<br />
\\[6pt]<br />
& = & \mathsf{r}^0<br />
\\[6pt]<br />
& = & \mathsf{d}^1 + \mathsf{r}^1<br />
\\[6pt]<br />
& = & \mathsf{d}^1 + \ldots + \mathsf{d}^m + \mathsf{r}^m<br />
\\[6pt]<br />
& = & \displaystyle \sum_{i=1}^m \mathsf{d}^i + \mathsf{r}^m<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====The Tangent Operator : '''T'''=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
They take part in scenes of whose significance they have no inkling. They are merely tangent to curves of history the beginnings and ends and forms of which pass wholly beyond their ken. So we are tangent to the wider life of things.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; William James, ''Pragmatism'', [Jam, 300]<br />
|}<br />
<br />
The operator tagged as <math>\mathsf{d}^1\!</math> in the analytic diagram (Figure&nbsp;33) is called the ''tangent operator'' and is usually denoted in this text as <math>\mathsf{d}\!</math> or <math>\mathsf{T}.\!</math> Because it has the properties required to qualify as a functor, namely, preserving the identity element of the composition operation and the articulated form of every composition of transformations, it also earns the title of a ''tangent functor''. According to the custom adopted here, we dissect it as <math>\mathsf{T} = \mathsf{d} = (\boldsymbol\varepsilon, \mathrm{d}),\!</math> where <math>\mathrm{d}\!</math> is the operator that yields the first order differential <math>\mathrm{d}F\!</math> when applied to a transformation <math>F,\!</math> and whose name is legion.<br />
<br />
Figure&nbsp;34 illustrates a stage of analysis where we ignore everything but the tangent functor <math>\mathsf{T}\!</math> and attend to it chiefly as it bears on the first order differential <math>\mathrm{d}F\!</math> in the analytic expansion of <math>F.\!</math> In this situation we often refer to the extended universes <math>\mathrm{E}U^\bullet\!</math> and <math>\mathrm{E}X^\bullet\!</math> under the equivalent designations <math>\mathsf{T}U^\bullet\!</math> and <math>\mathsf{T}X^\bullet,\!</math> respectively. The purpose of the tangent functor <math>\mathsf{T}\!</math> is to extract the tangent map <math>\mathsf{T}F\!</math> at each point of <math>U^\bullet,\!</math> and the tangent map <math>\mathsf{T}F = (\boldsymbol\varepsilon, \mathrm{d})F\!</math> tells us not only what the transformation <math>F\!</math> is doing at each point of the universe <math>U^\bullet\!</math> but also what <math>F\!</math> is doing to states in the neighborhood of that point, approximately, linearly, and relatively speaking.<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
U% $T$ $T$U% $T$U%<br />
o------------------>o============o<br />
| | |<br />
| | |<br />
| | |<br />
| | |<br />
F | | $T$F = | <!e!, d> F<br />
| | |<br />
| | |<br />
| | |<br />
v v v<br />
o------------------>o============o<br />
X% $T$ $T$X% $T$X%<br />
<br />
Figure 34. Tangent Functor Diagram<br />
</pre><br />
|}<br />
<br />
* '''NB.''' There is one aspect of the preceding construction that remains especially problematic. Why did we define the operators <math>\mathrm{W}\!</math> in <math>\{ \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}\!</math> so that the ranges of their resulting maps all fall within the realms of differential quality, even fabricating a variant of the tacit extension operator to have that character? Clearly, not all of the operator maps <math>\mathrm{W}F\!</math> have equally good reasons for placing their values in differential stocks. The reason for it appears to be that, without doing this, we cannot justify the comparison and combination of their functional values in the various analytic steps. By default, only those values in the same functional component can be brought into algebraic modes of interaction. Up till now the only mechanism provided for their broader association has been a purely logical one, their common placement in a target universe of discourse, but the task of converting this logical circumstance into algebraic forms of application has not yet been taken up.<br />
<br />
===Transformations of Type '''B'''<sup>2</sup> &rarr; '''B'''<sup>1</sup>===<br />
<br />
To study the effects of these analytic operators in the simplest possible setting, let us revert to a still more primitive case. Consider the singular proposition <math>J(u, v)= u\!\cdot\!v,\!</math> regarded either as the functional product of the maps <math>u\!</math> and <math>v\!</math> or as the logical conjunction of the features <math>u\!</math> and <math>v,\!</math> a map whose fiber of truth <math>J^{-1}(1)\!</math> picks out the single cell of that logical description in the universe of discourse <math>U^\bullet.\!</math> Thus <math>J,\!</math> or <math>u\!\cdot\!v,\!</math> may be treated as another name for the point whose coordinates are <math>(1, 1)\!</math> in <math>U^\bullet.\!</math><br />
<br />
====Analytic Expansion of Conjunction====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
<p>In her sufferings she read a great deal and discovered that she had lost something, the possession of which she had previously not been much aware of: a&nbsp;soul.</p><br />
<br />
<p>What is that? It is easily defined negatively: it is simply what curls up and hides when there is any mention of algebraic series.</p><br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 118]<br />
|}<br />
<br />
Figure&nbsp;35 pictures the form of conjunction <math>J : \mathbb{B}^2 \to \mathbb{B}\!</math> as a transformation from the <math>2\!</math>-dimensional universe <math>[u, v]\!</math> to the <math>1\!</math>-dimensional universe <math>[x].\!</math> This is a subtle but significant change of viewpoint on the proposition, attaching an arbitrary but concrete quality to its functional value. Using the language introduced earlier, we can express this change by saying that the proposition <math>J : \langle u, v \rangle \to \mathbb{B}\!</math> is being recast into the thematized role of a transformation <math>J : [u, v] \to [x],\!</math> where the new variable <math>x\!</math> takes the part of a thematic variable <math>\check{J}.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 35 -- A Conjunction Viewed as a Transformation.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 35.} ~~ \text{Conjunction as Transformation}\!</math><br />
|}<br />
<br />
=====Tacit Extension of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
I teach straying from me, yet who can stray from me?<br><br />
I follow you whoever you are from the present hour;<br><br />
My words itch at your ears till you understand them.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 83]<br />
|}<br />
<br />
Earlier we defined the tacit extension operators <math>\boldsymbol\varepsilon : X^\bullet \to Y^\bullet\!</math> as maps embedding each proposition of a given universe <math>X^\bullet~\!</math> in a more generously given universe <math>Y^\bullet \supset X^\bullet.\!</math> Of immediate interest are the tacit extensions <math>\boldsymbol\varepsilon : U^\bullet \to \mathrm{E}U^\bullet,\!</math> that locate each proposition of <math>U^\bullet\!</math> in the enlarged context of <math>\mathrm{E}U^\bullet.\!</math> In its application to the propositional conjunction <math>J = u\!\cdot\!v</math> in <math>[u, v],\!</math> the tacit extension operator <math>\boldsymbol\varepsilon\!</math> yields the proposition <math>\boldsymbol\varepsilon J\!</math> in <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v].\!</math> The extended proposition <math>\boldsymbol\varepsilon J\!</math> may be computed according to the scheme in Table&nbsp;36, in effect doing nothing more that conjoining a tautology of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> to <math>J\!</math> in <math>U^\bullet.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 36.} ~~ \text{Computation of}~ \boldsymbol\varepsilon J\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\boldsymbol\varepsilon J & = & J {}_{^\langle} u, v {}_{^\rangle}<br />
\\[4pt]<br />
& = & u \cdot v<br />
\\[4pt]<br />
& = & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{ } \mathrm{d}v \texttt{ }<br />
& + & u \cdot v \cdot \texttt{ } \mathrm{d}u \texttt{ } \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \cdot v \cdot \texttt{ } \mathrm{d}u \texttt{ } \cdot \texttt{ } \mathrm{d}v \texttt{ }<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{array}{*{4}{l}}<br />
\boldsymbol\varepsilon J<br />
& = && u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & u \cdot v \cdot \texttt{~} \mathrm{d}u \texttt{~} \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & u \cdot v \cdot \texttt{~} \mathrm{d}u \texttt{~} \cdot \texttt{~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
The lower portion of the Table contains the dispositional features of <math>\boldsymbol\varepsilon J\!</math> arranged in such a way that the variety of ordinary features spreads across the rows and the variety of differential features runs through the columns. This organization serves to facilitate pattern matching in the remainder of our computations. Again, the tacit extension is usually so trivial a concern that we do not always bother to make an explicit note of it, taking it for granted that any function <math>F\!</math> being employed in a differential context is equivalent to <math>\boldsymbol\varepsilon F\!</math> for a suitable <math>\boldsymbol\varepsilon.\!</math><br />
<br />
Figures&nbsp;37-a through 37-d present several pictures of the proposition <math>J\!</math> and its tacit extension <math>\boldsymbol\varepsilon J.\!</math> Notice in these Figures how <math>\boldsymbol\varepsilon J\!</math> in <math>\mathrm{E}U^\bullet\!</math> visibly extends <math>J\!</math> in <math>U^\bullet\!</math> by annexing to the indicated cells of <math>J\!</math> all the arcs that exit from or flow out of them. In effect, this extension attaches to these cells all the dispositions that spring from them, in other words, it attributes to these cells all the conceivable changes that are their issue.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-a -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-a.} ~~ \text{Tacit Extension of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-b -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-b.} ~~ \text{Tacit Extension of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-c -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-c.} ~~ \text{Tacit Extension of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 37-d -- Tacit Extension of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 37-d.} ~~ \text{Tacit Extension of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
The computational scheme shown in Table&nbsp;36 treated <math>J\!</math> as a proposition in <math>U^\bullet\!</math> and formed <math>\boldsymbol\varepsilon J\!</math> as a proposition in <math>\mathrm{E}U^\bullet.\!</math> When <math>J\!</math> is regarded as a mapping <math>J : U^\bullet \to X^\bullet\!</math> then <math>\boldsymbol\varepsilon J\!</math> must be obtained as a mapping <math>\boldsymbol\varepsilon J : \mathrm{E}U^\bullet \to X^\bullet.\!</math> By default, the tacit extension of the map <math>J : [u, v] \to [x]\!</math> is naturally taken to be a particular map,<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\boldsymbol\varepsilon J : [u, v, \mathrm{d}u, \mathrm{d}v] \to [x] \subseteq [x, \mathrm{d}x],\!</math><br />
|}<br />
<br />
namely, the one that looks like <math>J\!</math> when painted in the frame of the extended source universe and that takes the same thematic variable in the extended target universe as the one that <math>J\!</math> already takes.<br />
<br />
But the choice of a particular thematic variable, for example <math>x\!</math> for <math>\check{J},\!</math> is a shade more arbitrary than the choice of original variable names <math>\{ u, v \},\!</math> so the map we are calling the ''trope extension'',<br />
<br />
{| align="center" cellpadding="6" width="90%"<br />
| <math>\eta J : [u, v, \mathrm{d}u, \mathrm{d}v] \to [\mathrm{d}x] \subseteq [x, \mathrm{d}x],\!</math><br />
|}<br />
<br />
since it looks just the same as <math>\boldsymbol\varepsilon J\!</math> in the way its fibers paint the source domain, belongs just as fully to the family of tacit extensions, generically considered.<br />
<br />
These considerations have the practical consequence that all of our computations and illustrations of <math>\boldsymbol\varepsilon J\!</math> perform the double duty of capturing <math>\eta J\!</math> as well. In other words, we are saved the work of carrying out calculations and drawing figures for the trope extension <math>\eta J,\!</math> because it would be identical to the work already done for <math>\boldsymbol\varepsilon J.\!</math> Since the computations given for <math>\boldsymbol\varepsilon J\!</math> are expressed solely in terms of the variables <math>\{ u, v, \mathrm{d}u, \mathrm{d}v \},\!</math> they work equally well for finding <math>\eta J.\!</math> Further, since each of the above Figures shows only how the level sets of <math>\boldsymbol\varepsilon J\!</math> partition the extended source universe <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v],\!</math> all of them serve equally well as portraits of <math>\eta J.\!</math><br />
<br />
=====Enlargement Map of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
No one could have established the existence of any details that might not just as well have existed in earlier times too; but all the relations between things had shifted slightly. Ideas that had once been of lean account grew fat.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 62]<br />
|}<br />
<br />
The enlargement map <math>\mathrm{E}J\!</math> is computed from the proposition <math>J\!</math> by making a particular class of formal substitutions for its variables, in this case <math>u + \mathrm{d}u\!</math> for <math>u\!</math> and <math>v + \mathrm{d}v\!</math> for <math>v,\!</math> and afterwards expanding the result in whatever way is found convenient.<br />
<br />
Table&nbsp;38 shows a typical scheme of computation, following a systematic method of exploiting boolean expansions over selected variables and ultimately developing <math>\mathrm{E}J\!</math> over the cells of <math>[u, v].\!</math> The critical step of this procedure uses the facts that <math>\texttt{(} 0, x \texttt{)} = 0 + x = x\!</math> and <math>\texttt{(} 1, x \texttt{)} = 1 + x = \texttt{(} x \texttt{)}\!</math> for any boolean variable <math>x.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 38.} ~~ \text{Computation of}~ \mathrm{E}J ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{E}J & = & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}<br />
\\[4pt]<br />
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
& = & \texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot J_{(1 + \mathrm{d}u, 1 + \mathrm{d}v)}<br />
& + & \texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot J_{(1 + \mathrm{d}u, \mathrm{d}v)}<br />
& + & \texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot J_{(\mathrm{d}u, 1 + \mathrm{d}v)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot J_{(\mathrm{d}u, \mathrm{d}v)}<br />
\\[4pt]<br />
& = &<br />
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot J_{(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})}<br />
& + &<br />
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot J_{(\texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v)}<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot J_{(\mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)})}<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot J_{(\mathrm{d}u, \mathrm{d}v)}<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{E}J<br />
& = &<br />
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&& + &<br />
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{ } \mathrm{d}v \texttt{ }<br />
\\[4pt]<br />
&&&&& + &<br />
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot \texttt{ } \mathrm{d}u \texttt{ } \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&&&&&& + &<br />
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \texttt{ } \mathrm{d}u \texttt{ }~\texttt{ } \mathrm{d}v \texttt{ }<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;39 exhibits another method that happens to work quickly in this particular case, using distributive laws to multiply things out in an algebraic manner, arranging the notations of feature and fluxion according to a scale of simple character and degree. Proceeding this way leads through an intermediate step which, in chiming the changes of ordinary calculus, should take on a familiar ring. Consequential properties of exclusive disjunction then carry us on to the concluding line.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 39.} ~~ \text{Computation of}~ \mathrm{E}J ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{c}}<br />
\mathrm{E}J<br />
& = & (u + \mathrm{d}u) \cdot (v + \mathrm{d}v)<br />
\\[6pt]<br />
& = & u \cdot v<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = &<br />
\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{ } \mathrm{d}v \texttt{ }<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot \texttt{ } \mathrm{d}u \texttt{ } \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \texttt{ } \mathrm{d}u \texttt{ }~\texttt{ } \mathrm{d}v \texttt{ }<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;40-a through 40-d present several views of the enlarged proposition <math>\mathrm{E}J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-a -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-a.} ~~ \text{Enlargement of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-b -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-b.} ~~ \text{Enlargement of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-c -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-c.} ~~ \text{Enlargement of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 40-d -- Enlargement of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 40-d.} ~~ \text{Enlargement of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
An intuitive reading of the proposition <math>\mathrm{E}J\!</math> becomes available at this point. Recall that propositions in the extended universe <math>\mathrm{E}U^\bullet\!</math> express the ''dispositions'' of a system and the constraints that are placed on them. In other words, a differential proposition in <math>\mathrm{E}U^\bullet\!</math> can be read as referring to various changes that a system might undergo in and from its various states. In particular, we can understand <math>\mathrm{E}J\!</math> as a statement that tells us what changes need to be made with regard to each state in the universe of discourse in order to reach the ''truth'' of <math>J,\!</math> that is, the region of the universe where <math>J\!</math> is true. This interpretation is visibly clear in the Figures above and appeals to the imagination in a satisfying way but it has the added benefit of giving fresh meaning to the original name of the shift operator <math>\mathrm{E}.\!</math> Namely, <math>\mathrm{E}J\!</math> can be read as a proposition that ''enlarges'' on the meaning of <math>J,\!</math> in the sense of explaining its practical bearings and clarifying what it means in terms of actions and effects &mdash; the available options for differential action and the consequential effects that result from each choice.<br />
<br />
Read this way, the enlargement <math>\mathrm{E}J\!</math> has strong ties to the normal use of <math>J,\!</math> no matter whether it is understood as a proposition or a function, namely, to act as a figurative device for indicating the models of <math>J,\!</math> in effect, pointing to the interpretive elements in its fiber of truth <math>J^{-1}(1).\!</math> It is this kind of &ldquo;use&rdquo; that is often contrasted with the &ldquo;mention&rdquo; of a proposition, and thereby hangs a tale.<br />
<br />
=====Digression : Reflection on Use and Mention=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Reflection is turning a topic over in various aspects and in various lights so that nothing significant about it shall be overlooked &mdash; almost as one might turn a stone over to see what its hidden side is like or what is covered by it.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 57]<br />
|}<br />
<br />
The contrast drawn in logic between the ''use'' and the ''mention'' of a proposition corresponds to the difference that we observe in functional terms between using <math>{}^{\backprime\backprime} J \, {}^{\prime\prime}\!</math> to indicate the region <math>J^{-1}(1)\!</math> and using <math>{}^{\backprime\backprime} J \, {}^{\prime\prime}\!</math> to indicate the function <math>J.\!</math> You may think that one of these uses ought to be proscribed, and logicians are quick to prescribe against their confusion. But there seems to be no likelihood in practice that their interactions can be avoided. If the name <math>{}^{\backprime\backprime} J \, {}^{\prime\prime}\!</math> is used as a sign of the function <math>J,\!</math> and if the function <math>J\!</math> has its use in signifying something else, as would constantly be the case when some future theory of signs has given a functional meaning to every sign whatsoever, then is not <math>J,\!</math> by transitivity a sign of the thing itself? There are, of course, two answers to this question. Not every act of signifying or referring need be transitive. Not every warrant or guarantee or certificate is automatically transferable, indeed, not many. Not every feature of a feature is a feature of the featuree. Otherwise, if a buffalo is white, and white is a color, then a buffalo would ''be'' a color.<br />
<br />
The logical or pragmatic distinction between use and mention is cogent and necessary, and so is the analogous functional distinction between determining a value and determining what determines that value, but so are the normal techniques that we use to make these distinctions apply flexibly in practice. The way that the hue and cry about use and mention is raised in logical discussions, you might be led to think that this single dimension of choices embraces the only kinds of use worth mentioning and the only kinds of mention worth using. It will constitute the expeditionary and taxonomic tasks of that future theory of signs to explore and to classify the many other constellations and dimensions of use and mention that are yet to be opened up by the generative potential of full-fledged sign relations.<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
The well-known capacity that thoughts have &mdash; as doctors have discovered &mdash; for dissolving and dispersing those hard lumps of deep, ingrowing, morbidly entangled conflict that arise out of gloomy regions of the self probably rests on nothing other than their social and worldly nature, which links the individual being with other people and things; but unfortunately what gives them their power of healing seems to be the same as what diminishes the quality of personal experience in them.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 130]<br />
|}<br />
<br />
=====Difference Map of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
&ldquo;It doesn't matter what one does,&rdquo; the Man Without Qualities said to himself, shrugging his shoulders. &ldquo;In a tangle of forces like this it doesn't make a scrap of difference.&rdquo; He turned away like a man who has learned renunciation, almost indeed like a sick man who shrinks from any intensity of contact. And then, striding through his adjacent dressing-room, he passed a punching-ball that hung there; he gave it a blow far swifter and harder than is usual in moods of resignation or states of weakness.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 8]<br />
|}<br />
<br />
With the tacit extension map <math>\boldsymbol\varepsilon J\!</math> and the enlargement map <math>\mathrm{E}J\!</math> well in place, the difference map <math>\mathrm{D}J\!</math> can be computed along the lines displayed in Table&nbsp;41, ending up with an expansion of <math>\mathrm{D}J\!</math> over the cells of <math>[u, v].\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 41.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & \mathrm{E}J<br />
& + & \boldsymbol\varepsilon J<br />
\\[6pt]<br />
& = & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}<br />
& + & J_{(u, v)}<br />
\\[6pt]<br />
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \cdot v<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = &<br />
u \cdot v \cdot \qquad 0<br />
\\[6pt]<br />
& + &<br />
u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v<br />
& + &<br />
u \cdot \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v<br />
\\[6pt]<br />
& + &<br />
u \cdot v \cdot \texttt{~} \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
&&& + &<br />
\texttt{(} u \texttt{)} \cdot v \cdot \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
& + &<br />
u \cdot v \cdot \texttt{~} \mathrm{d}u \;\cdot\; \mathrm{d}v \texttt{~}<br />
&&&&& + &<br />
\texttt{(} u \texttt{)} \cdot \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = &<br />
u \cdot v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
u \cdot \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v<br />
& + &<br />
\texttt{(} u \texttt{)} \cdot v \cdot \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)} \cdot \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Alternatively, the difference map <math>\mathrm{D}J\!</math> can be expanded over the cells of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> to arrive at the formulation shown in Table&nbsp;42. The same development would be obtained from the previous Table by collecting terms in an alternate manner, along the rows rather than the columns in the middle portion of the Table.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 42.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & \boldsymbol\varepsilon J<br />
& + & \mathrm{E}J<br />
\\[6pt]<br />
& = & J_{(u, v)}<br />
& + & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}<br />
\\[6pt]<br />
& = & u \cdot v<br />
& + & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)} \cdot \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
& = & 0<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}J<br />
& = & 0<br />
& + & u \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Even more simply, the same result is reached by matching up the propositional coefficients of <math>\boldsymbol\varepsilon J</math> and <math>\mathrm{E}J\!</math> along the cells of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> and adding the pairs under boolean addition, that is, &ldquo;mod 2&rdquo;, where 1&nbsp;+&nbsp;1&nbsp;=&nbsp;0, as shown in Table&nbsp;43.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 43.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 3)}\!</math><br />
|<br />
<math>\begin{array}{*{5}{l}}<br />
\mathrm{D}J & = & \boldsymbol\varepsilon J & + & \mathrm{E}J<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\boldsymbol\varepsilon J<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ u \,\cdot\, v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~ u \;\cdot\; v \;\cdot\; \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u ~ \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} ~ v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot\, \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & ~~ 0 ~~ \,\cdot\, ~ \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~~ u ~ \,\cdot\, ~~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ ~ v ~~ \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
The difference map <math>\mathrm{D}J\!</math> can also be given a ''dispositional'' interpretation. First, recall that <math>\boldsymbol\varepsilon J\!</math> exhibits the dispositions to change from anywhere in <math>J\!</math> to anywhere at all in the universe of discourse and <math>\mathrm{E}J\!</math> exhibits the dispositions to change from anywhere in the universe to anywhere in <math>J.\!</math> Next, observe that each of these classes of dispositions may be divided in accordance with the case of <math>J\!</math> versus <math>\texttt{(} J \texttt{)}\!</math> that applies to their points of departure and destination, as shown below. Then, since the dispositions corresponding to <math>\boldsymbol\varepsilon J</math> and <math>\mathrm{E}J\!</math> have in common the dispositions to preserve <math>J,\!</math> their symmetric difference <math>\texttt{(} \boldsymbol\varepsilon J, \mathrm{E}J \texttt{)}\!</math> is made up of all the remaining dispositions, which are in fact disposed to cross the boundary of <math>J\!</math> in one direction or the other. In other words, we may conclude that <math>\mathrm{D}J\!</math> expresses the collective disposition to make a definite change with respect to <math>J,\!</math> no matter what value it holds in the current state of affairs.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
\boldsymbol\varepsilon J<br />
& = & \{ \text{Dispositions from}~ J ~\text{to}~ J \}<br />
& + & \{ \text{Dispositions from}~ J ~\text{to}~ \texttt{(} J \texttt{)} \}<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = & \{ \text{Dispositions from}~ J ~\text{to}~ J \}<br />
& + & \{ \text{Dispositions from}~ \texttt{(} J \texttt{)} ~\text{to}~ J \}<br />
\\[6pt]<br />
\mathrm{D}J<br />
& = & \{ \text{Dispositions from}~ J ~\text{to}~ \texttt{(} J \texttt{)} \}<br />
& + & \{ \text{Dispositions from}~ \texttt{(} J \texttt{)} ~\text{to}~ J \}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;44-a through 44-d illustrate the difference proposition <math>\mathrm{D}J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-a -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-a.} ~~ \text{Difference Map of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-b -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-b.} ~~ \text{Difference Map of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-c -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-c.} ~~ \text{Difference Map of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 44-d -- Difference Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 44-d.} ~~ \text{Difference Map of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
=====Differential of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
By deploying discourse throughout a calendar, and by giving a date to each of its elements, one does not obtain a definitive hierarchy of precessions and originalities; this hierarchy is never more than relative to the systems of discourse that it sets out to evaluate.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Michel Foucault, ''The Archaeology of Knowledge'', [Fou, 143]<br />
|}<br />
<br />
Finally, at long last, the differential proposition <math>\mathrm{d}J\!</math> can be gleaned from the difference proposition <math>\mathrm{D}J\!</math> by ranging over the cells of <math>[u, v]\!</math> and picking out the linear proposition of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> that is &ldquo;closest&rdquo; to the portion of <math>\mathrm{D}J\!</math> that touches on each point. The idea of distance that would give this definition unequivocal sense has been referred to in cautionary quotes, the kind we use to distance ourselves from taking a final position. There are obvious notions of approximation that suggest themselves, but finding one that can be justified as ultimately correct is not as straightforward as it seems.<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
He had drifted into the very heart of the world. From him to the distant beloved was as far as to the next tree.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 144]<br />
|}<br />
<br />
Let us venture a guess as to where these developments might be heading. From the present vantage point it appears that the ultimate answer to the quandary of distances and the question of a fitting measure may be that, rather than having the constitution of an analytic series depend on our familiar notions of approach, proximity, and approximation, it will be found preferable, and perhaps unavoidable, to turn the tables and let the orders of approximation be defined in terms of our favored and operative notions of formal analysis. Only the aftermath of this conversion, if it does converge, could be hoped to prove whether this hortatory form of analysis and the cohort idea of an analytic form &mdash; the limitary concept of a self-corrective process and the coefficient concept of a completable product &mdash; are truly (in practical reality) the more inceptive and persistent of principles and really (for all practical purposes) the more effective and regulative of ideas.<br />
<br />
Awaiting that determination, I proceed with what seems like the obvious course, and compute <math>\mathrm{d}J\!</math> according to the pattern in Table&nbsp;45.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 45.} ~~ \text{Computation of}~ \mathrm{d}J\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}J<br />
& = &<br />
u\!\cdot\!v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
u \, \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \, \mathrm{d}v<br />
& + &<br />
\texttt{(} u \texttt{)} \, v \cdot \mathrm{d}u \, \texttt{(} \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \!\cdot\! \mathrm{d}v \texttt{~}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}J<br />
& = &<br />
u\!\cdot\!v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + &<br />
u \, \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + &<br />
\texttt{(} u \texttt{)} \, v \cdot \mathrm{d}u<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;46-a through 46-d illustrate the proposition <math>{\mathrm{d}J},\!</math> rounded out in our usual array of prospects. This proposition of <math>\mathrm{E}U^\bullet\!</math> is what we refer to as the (first order) differential of <math>J,\!</math> and normally regard as ''the'' differential proposition corresponding to <math>J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-a -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-a.} ~~ \text{Differential of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-b -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-b.} ~~ \text{Differential of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-c -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-c.} ~~ \text{Differential of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 46-d -- Differential of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 46-d.} ~~ \text{Differential of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
=====Remainder of Conjunction=====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
<p>I bequeath myself to the dirt to grow from the grass I love,<br><br />
If you want me again look for me under your bootsoles.</p><br />
<br />
<p>You will hardly know who I am or what I mean,<br><br />
But I shall be good health to you nevertheless,<br><br />
And filter and fibre your blood.</p><br />
<br />
<p>Failing to fetch me at first keep encouraged,<br><br />
Missing me one place search another,<br><br />
I stop some where waiting for you</p><br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]<br />
|}<br />
<br />
<br><br />
<br />
Let us recapitulate the story so far. We have in effect been carrying out a decomposition of the enlarged proposition <math>\mathrm{E}J\!</math> in a series of stages. First, we considered the equation <math>\mathrm{E}J = \boldsymbol\varepsilon J + \mathrm{D}J,\!</math> which was involved in the definition of <math>\mathrm{D}J\!</math> as the difference <math>\mathrm{E}J - \boldsymbol\varepsilon J.\!</math> Next, we contemplated the equation <math>\mathrm{D}J = \mathrm{d}J + \mathrm{r}J,\!</math> which expresses <math>\mathrm{D}J\!</math> in terms of two components, the differential <math>\mathrm{d}J\!</math> that was just extracted and the residual component <math>\mathrm{r}J = \mathrm{D}J - \mathrm{d}J.~\!</math> This remaining proposition <math>\mathrm{r}J\!</math> can be computed as shown in Table&nbsp;47.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 47.} ~~ \text{Computation of}~ \mathrm{r}J\!</math><br />
|<br />
<math>\begin{array}{*{5}{l}}<br />
\mathrm{r}J & = & \mathrm{D}J & + & \mathrm{d}J<br />
\end{array}\!</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}J<br />
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}J<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|-<br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{r}J ~<br />
& = & u \!\cdot\! v \cdot ~ \mathrm{d}u \cdot \mathrm{d}v ~ ~ ~ ~ ~<br />
& + & u \texttt{(} v \texttt{)} \cdot \, \mathrm{d}u \cdot \mathrm{d}v \,<br />
& + & \texttt{(} u \texttt{)} v \cdot \, \mathrm{d}u \cdot \mathrm{d}v \,<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \, \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
As it happens, the remainder <math>\mathrm{r}J\!</math> falls under the description of a second order differential <math>\mathrm{r}J = \mathrm{d}^2 J.\!</math> This means that the expansion of <math>\mathrm{E}J\!</math> in the form:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|<br />
<math>\begin{array}{*{7}{l}}<br />
\mathrm{E}J<br />
& = & \boldsymbol\varepsilon J<br />
& + & \mathrm{D}J<br />
\\[6pt]<br />
& = & \boldsymbol\varepsilon J<br />
& + & \mathrm{d}J<br />
& + & \mathrm{r}J<br />
\\[6pt]<br />
& = & \mathrm{d}^0 J<br />
& + & \mathrm{d}^1 J<br />
& + & \mathrm{d}^2 J<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
which is nothing other than the propositional analogue of a Taylor series, is a decomposition that terminates in a finite number of steps.<br />
<br />
Figures&nbsp;48-a through 48-d illustrate the proposition <math>\mathrm{r}J = \mathrm{d}^2 J,\!</math> which forms the remainder map of <math>J\!</math> and also, in this instance, the second order differential of <math>J.\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-a -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-a.} ~~ \text{Remainder of}~ J ~\text{(Areal)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-b -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-b.} ~~ \text{Remainder of}~ J ~\text{(Bundle)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-c -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-c.} ~~ \text{Remainder of}~ J ~\text{(Compact)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 48-d -- Remainder of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 48-d.} ~~ \text{Remainder of}~ J ~\text{(Digraph)}\!</math><br />
|}<br />
<br />
=====Summary of Conjunction=====<br />
<br />
To establish a convenient reference point for further discussion, Table&nbsp;49 summarizes the operator actions that have been computed for the form of conjunction, as exemplified by the proposition <math>J.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 49.} ~~ \text{Computation Summary for}~ J~\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon J<br />
& = & u \!\cdot\! v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}J<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}J<br />
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}J<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{r}J<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Analytic Series : Coordinate Method====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
And if he is told that something ''is'' the way it is, then he thinks: Well, it could probably just as easily be some other way. So the sense of possibility might be defined outright as the capacity to think how everything could &ldquo;just as easily&rdquo; be, and to attach no more importance to what is than to what is not.<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Robert Musil, ''The Man Without Qualities'', [Mus, 12]<br />
|}<br />
<br />
Table&nbsp;50 exhibits a truth table method for computing the analytic series (or the differential expansion) of a proposition in terms of coordinates.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:70%"<br />
|+ style="height:30px" | <math>\text{Table 50.} ~~ \text{Computation of an Analytic Series in Terms of Coordinates}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:8%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:8%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:8%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}u\!</math><br />
| style="width:8%; border-bottom:1px solid black" | <math>\mathrm{d}v\!</math><br />
| style="width:8%; border-bottom:1px solid black; border-left:1px solid black" | <math>u'\!</math><br />
| style="width:8%; border-bottom:1px solid black" | <math>v'\!</math><br />
| style="width:10%; border-bottom:1px solid black; border-left:4px double black" | <math>\boldsymbol\varepsilon J\!</math><br />
| style="width:10%; border-bottom:1px solid black" | <math>\mathrm{E}J\!</math><br />
| style="width:12%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{D}J\!</math><br />
| style="width:10%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}J\!</math><br />
| style="width:10%; border-bottom:1px solid black" | <math>\mathrm{d}^2\!J\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
The first six columns of the Table, taken as a whole, represent the variables of a construct called the ''contingent universe'' <math>[u, v, \mathrm{d}u, \mathrm{d}v, u', v'],\!</math> or the bundle of ''contingency spaces'' <math>[\mathrm{d}u, \mathrm{d}v, u', v']\!</math> over the universe <math>[u, v].\!</math> Their placement to the left of the double bar indicates that all of them amount to independent variables, but there is a co-dependency among them, as described by the following equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
u' & = & u + \mathrm{d}u & = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\\[8pt]<br />
v' & = & v + \mathrm{d}v & = & \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
These relations correspond to the formal substitutions that are made in defining <math>\mathrm{E}J\!</math> and <math>\mathrm{D}J.\!</math> For now, the whole rigamarole of contingency spaces can be regarded as a technical device for achieving the effect of these substitutions, adapted to a setting where functional compositions and other symbolic manipulations are difficult to contemplate and execute.<br />
<br />
The five columns to the right of the double bar in Table&nbsp;50 contain the values of the dependent variables <math>\{ \boldsymbol\varepsilon J, ~\mathrm{E}J, ~\mathrm{D}J, ~\mathrm{d}J, ~\mathrm{d}^2\!J \}.\!</math> These are normally interpreted as values of functions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B}\!</math> or as values of propositions in the extended universe <math>[u, v, \mathrm{d}u, \mathrm{d}v]\!</math> but the dependencies prevailing in the contingent universe make it possible to regard these same final values as arising via functions on alternative lists of arguments, for example, the set <math>\{ u, v, u', v' \}.\!</math><br />
<br />
The column for <math>\boldsymbol\varepsilon J\!</math> is computed as <math>J(u, v) = uv\!</math> and together with the columns for <math>u\!</math> and <math>v\!</math> illustrates how we &ldquo;share structure&rdquo; in the Table by listing only the first entries of each constant block.<br />
<br />
The column for <math>\mathrm{E}J\!</math> is computed by means of the following chain of identities, where the contingent variables <math>u'\!</math> and <math>v'\!</math> are defined as <math>u' = u + \mathrm{d}u\!</math> and <math>v' = v + \mathrm{d}v.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"<br />
|<br />
<math>\begin{matrix}<br />
\mathrm{E}J(u, v, \mathrm{d}u, \mathrm{d}v)<br />
& = &<br />
J(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& = &<br />
J(u', v')<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
This makes it easy to determine <math>\mathrm{E}J\!</math> by inspection, computing the conjunction <math>J(u', v') = u'v'\!</math> from the columns headed <math>u'\!</math> and <math>v'.\!</math> Since each of these forms expresses the same proposition <math>\mathrm{E}J\!</math> in <math>\mathrm{E}U^\bullet,\!</math> the dependence on <math>\mathrm{d}u\!</math> and <math>\mathrm{d}v\!</math> is still present but merely left implicit in the final variant <math>J(u', v').\!</math><br />
<br />
* '''Note.''' On occasion, it is tempting to use the further notation <math>J'(u, v) = J(u', v'),\!</math> especially to suggest a transformation that acts on whole propositions, for example, taking the proposition <math>J\!</math> into the proposition <math>J' = \mathrm{E}J.\!</math> The prime <math>( {}^{\prime} )\!</math> then signifies an action that is mediated by a field of choices, namely, the values that are picked out for the contingent variables in sweeping through the initial universe. But this heaps an unwieldy lot of construed intentions on a rather slight character and puts too high a premium on the constant correctness of its interpretation. In practice, therefore, it is best to avoid this usage.<br />
<br />
Given the values of <math>\boldsymbol\varepsilon J\!</math> and <math>\mathrm{E}J,\!</math> the columns for the remaining functions can be filled in quickly. The difference map is computed according to the relation <math>\mathrm{D}J = \boldsymbol\varepsilon J + \mathrm{E}J.\!</math> The first order differential <math>\mathrm{d}J\!</math> is found by looking in each block of constant argument pairs <math>u, v\!</math> and choosing the linear function of <math>\mathrm{d}u, \mathrm{d}v\!</math> that best approximates <math>\mathrm{D}J\!</math> in that block. Finally, the remainder is computed as <math>\mathrm{r}J = \mathrm{D}J + \mathrm{d}J,\!</math> in this case yielding the second order differential <math>\mathrm{d}^2\!J.\!</math><br />
<br />
====Analytic Series : Recap====<br />
<br />
Let us now summarize the results of Table&nbsp;50 by writing down for each column and for each block of constant argument pairs <math>u, v\!</math> a reasonably canonical symbolic expression for the function of <math>\mathrm{d}u, \mathrm{d}v\!</math> that appears there. The synopsis formed in this way is presented in Table&nbsp;51. As one has a right to expect, it confirms the results that were obtained previously by operating solely in terms of the formal calculus.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:70%"<br />
|+ style="height:30px" | <math>\text{Table 51.} ~~ \text{Computation of an Analytic Series in Symbolic Terms}\!</math><br />
|- style="height:35px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black" | <math>J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{E}J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{D}J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{d}J\!</math><br />
| style="border-left:1px solid black" | <math>\mathrm{d}^2\!J\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}u \!\;\cdot\;\! \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}u \!\;\cdot\;\! \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
\mathrm{d}u<br />
\\[4pt]<br />
\mathrm{d}v<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Figures&nbsp;52 and 53 provide a quick overview of the analysis performed so far, giving the successive decompositions of <math>\mathrm{E}J = J + \mathrm{D}J\!</math> and <math>\mathrm{D}J = \mathrm{d}J + \mathrm{r}J\!</math> in two different styles of diagram.<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 52 -- Decomposition of EJ.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 52.} ~~ \text{Decomposition of}~ \mathrm{E}J\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 53 -- Decomposition of DJ.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 53.} ~~ \text{Decomposition of}~ \mathrm{D}J\!</math><br />
|}<br />
<br />
====Terminological Interlude====<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
Lastly, my attention was especially attracted, not so much to the scene, as to the mirrors that produced it. These mirrors were broken in parts. Yes, they were marked and scratched; they had been &ldquo;starred&rdquo;, in spite of their solidity &hellip;<br />
| width="4%" | &nbsp;<br />
|-<br />
| align="right" colspan="3" | &mdash; Gaston Leroux, ''The Phantom of the Opera'', [Ler, 230]<br />
|}<br />
<br />
At this point several issues of terminology have accrued enough substance to intrude on our discussion. The remarks of this Subsection are intended to accomplish two goals. First, we call attention to significant aspects of the previous series of Figures, translating into literal terms what they depict in iconic forms, and we re-stress the most important structural elements they indicate. Next, we prepare the way for taking on more complex examples of transformations, those whose target universes have more than one dimension.<br />
<br />
In talking about the actions of operators it is important to keep in mind the distinctions between the operators per&nbsp;se, their operands, and their results. Furthermore, in working with composite forms of operators <math>\mathrm{W} = (\mathrm{W}_1, \ldots, \mathrm{W}_n),\!</math> transformations <math>\mathrm{F} = (\mathrm{F}_1, \ldots, \mathrm{F}_n),\!</math> and target domains <math>X^\bullet = [x_1, \ldots, x_n],\!</math> we need to preserve a clear distinction between the compound entity of each given type and any one of its separate components. It is curious, given the usefulness of the concepts ''operator'' and ''operand'', that we seem to lack a generic term, formed on the same root, for the corresponding result of an operation. Following the obvious paradigm would lead to words like ''opus'', ''opera'', and ''operant'', but these words are too affected with clang associations to work well at present, though they might be adapted in time. One current usage gets around this problem by using the substantive ''map'' as a systematic epithet to express the result of each operator's action. We will follow this practice as far as possible, for example, using the phrase ''tangent map'' to denote the end product of the tangent functor acting on its operand map.<br />
<br />
* '''Scholium.''' See [JGH, 6-9] for a good account of tangent functors and tangent maps in ordinary analysis and for examples of their use in mechanics. This work as a whole is a model of clarity in applying functorial principles to problems in physical dynamics.<br />
<br />
Whenever we focus on isolated propositions, on single components of composite operators, or on the portions of transformations that have <math>1\!</math>-dimensional ranges, we are free to shift between the native form of a proposition <math>J : U \to \mathbb{B}\!</math> and the thematized form of a mapping <math>J : U^\bullet \to [x]\!</math> without much trouble. In these cases we are able to tolerate a higher degree of ambiguity about the precise nature of an operator's input and output domains than we otherwise might. For example, in the preceding treatment of the example <math>J,\!</math> and for each operator <math>\mathrm{W}\!</math> in the set <math>\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \},\!</math> both the operand <math>J\!</math> and the result <math>\mathrm{W}J\!</math> could be viewed in either one of two ways. On one hand we may treat them as propositions <math>J : U \to \mathbb{B}\!</math> and <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B},\!</math> ignoring the distinction between the range <math>[x] \cong \mathbb{B}\!</math> of <math>\boldsymbol\varepsilon J\!</math> and the range <math>[\mathrm{d}x] \cong \mathbb{D}\!</math> of the other types of <math>\mathrm{W}J.\!</math> This is what we usually do when we content ourselves with simply coloring in regions of venn diagrams. On the other hand we may view these entities as maps <math>J : U^\bullet \to [x] = X^\bullet\!</math> and <math>\boldsymbol\varepsilon J : \mathrm{E}U^\bullet \to [x] \subseteq \mathrm{E}X^\bullet\!</math> or <math>\mathrm{W}J : \mathrm{E}U^\bullet \to [\mathrm{d}x] \subseteq \mathrm{E}X^\bullet,\!</math> in which case the qualitative characters of the output features are not ignored.<br />
<br />
At the beginning of this Section we recast the natural form of a proposition <math>J : U \to \mathbb{B}\!</math> into the thematic role of a transformation <math>J : U^\bullet \to [x],\!</math> where <math>x\!</math> was a variable recruited to express the newly independent <math>\check{J}.\!</math> However, in our computations and representations of operator actions we immediately lapsed back to viewing the results as native elements of the extended universe <math>\mathrm{E}U^\bullet,\!</math> in other words, as propositions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B},\!</math> where <math>\mathrm{W}\!</math> ranged over the set <math>\{ \boldsymbol\varepsilon, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}.\!</math> That is as it should be. We have worked hard to devise a language that gives us these advantages &mdash; the flexibility to exchange terms and types of equal information value and the capacity to reflect as quickly and as wittingly as a controlled reflex on the fibers of our propositions, independently of whether they express amusements, beliefs, or conjectures.<br />
<br />
As we take on target spaces of increasing dimension, however, these types of confusions (and confusions of types) become less and less permissible. For this reason, Tables&nbsp;54 and 55 present a rather detailed summary of the notation and the terminology we are using, as applied to the case <math>J = uv.\!</math> The rationale of these Tables is not so much to train more elephant guns on this poor drosophila of a concrete example but to invest our paradigm with enough solidity to bear the weight of abstraction to come.<br />
<br />
Table&nbsp;54 provides basic notation and descriptive information for the objects and operators used in this Example, giving the generic type (or broadest defined type) for each entity. Here, the sans&nbsp;serif operators <math>\mathsf{W} \in \{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{d}, \mathsf{r} \}\!</math> and their components <math>\mathrm{W} \in \{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}\!</math> both have the same broad type <math>\mathsf{W}, \mathrm{W} : (U^\bullet \to X^\bullet) \to (\mathrm{E}U^\bullet \to \mathrm{E}X^\bullet),\!</math> as appropriate to operators that map transformations <math>J : U^\bullet \to X^\bullet\!</math> to extended transformations <math>\mathsf{W}J, \mathrm{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 54.} ~~ \text{Cast of Characters : Expansive Subtypes of Objects and Operators}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| align="center" | <math>\text{Symbol}\!</math><br />
| align="center" | <math>\text{Notation}\!</math><br />
| align="center" | <math>\text{Description}\!</math><br />
| align="center" | <math>\text{Type}\!</math><br />
|-<br />
| align="center" | <math>U^\bullet\!</math><br />
| <math>= [u, v]\!</math><br />
| <math>\text{Source universe}\!</math><br />
| <math>[\mathbb{B}^2]\!</math><br />
|-<br />
| align="center" | <math>X^\bullet~\!</math><br />
| <math>= [x]\!</math><br />
| <math>\text{Target universe}\!</math><br />
| <math>[\mathbb{B}^1]~\!</math><br />
|-<br />
| align="center" | <math>\mathrm{E}U^\bullet\!</math><br />
| <math>= [u, v, \mathrm{d}u, \mathrm{d}v]\!</math><br />
| <math>\text{Extended source universe}\!</math><br />
| <math>[\mathbb{B}^2 \!\times\! \mathbb{D}^2]</math><br />
|-<br />
| align="center" | <math>\mathrm{E}X^\bullet\!</math><br />
| <math>= [x, \mathrm{d}x]~\!</math><br />
| <math>\text{Extended target universe}\!</math><br />
| <math>[\mathbb{B}^1 \!\times\! \mathbb{D}^1]</math><br />
|-<br />
| align="center" | <math>J\!</math><br />
| <math>J : U \!\to\! \mathbb{B}\!</math><br />
| <math>\text{Proposition}\!</math><br />
| <math>(\mathbb{B}^2 \!\to\! \mathbb{B}) \in [\mathbb{B}^2]\!</math><br />
|-<br />
| align="center" | <math>J\!</math><br />
| <math>J : U^\bullet \!\to\! X^\bullet\!</math><br />
| <math>\text{Transformation or Map}\!</math><br />
| <math>[\mathbb{B}^2] \!\to\! [\mathbb{B}^1]\!</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\boldsymbol\varepsilon<br />
\\<br />
\eta<br />
\\<br />
\mathrm{E}<br />
\\<br />
\mathrm{D}<br />
\\<br />
\mathrm{d}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{W} : U^\bullet \!\to\! \mathrm{E}U^\bullet,<br />
\\<br />
\mathrm{W} : X^\bullet \!\to\! \mathrm{E}X^\bullet,<br />
\\<br />
\mathrm{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\\<br />
\text{for each}~ \mathrm{W} ~\text{in the set:}<br />
\\<br />
\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{ll}<br />
\text{Tacit extension operator} & \boldsymbol\varepsilon<br />
\\<br />
\text{Trope extension operator} & \eta<br />
\\<br />
\text{Enlargement operator} & \mathrm{E}<br />
\\<br />
\text{Difference operator} & \mathrm{D}<br />
\\<br />
\text{Differential operator} & \mathrm{d}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^2] \!\to\! [\mathbb{B}^2 \!\times\! \mathbb{D}^2]},<br />
\\<br />
{[\mathbb{B}^1] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]},<br />
\\\\<br />
([\mathbb{B}^2] \!\to\! [\mathbb{B}^1]) \!\to\!<br />
\\<br />
([\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1])<br />
\end{array}</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\mathsf{e}<br />
\\<br />
\mathsf{E}<br />
\\<br />
\mathsf{D}<br />
\\<br />
\mathsf{T}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{W} : U^\bullet \!\to\! \mathsf{T}U^\bullet = \mathrm{E}U^\bullet,<br />
\\<br />
\mathsf{W} : X^\bullet \!\to\! \mathsf{T}X^\bullet = \mathrm{E}X^\bullet,<br />
\\<br />
\mathsf{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathsf{T}U^\bullet \!\to\! \mathsf{T}X^\bullet)<br />
\\<br />
\text{for each}~ \mathsf{W} ~\text{in the set:}<br />
\\<br />
\{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{T} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{lll}<br />
\text{Radius operator} & \mathsf{e} & = (\boldsymbol\varepsilon, \eta)<br />
\\<br />
\text{Secant operator} & \mathsf{E} & = (\boldsymbol\varepsilon, \mathrm{E})<br />
\\<br />
\text{Chord operator} & \mathsf{D} & = (\boldsymbol\varepsilon, \mathrm{D})<br />
\\<br />
\text{Tangent functor} & \mathsf{T} & = (\boldsymbol\varepsilon, \mathrm{d})<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^2] \!\to\! [\mathbb{B}^2 \!\times\! \mathbb{D}^2]},<br />
\\<br />
{[\mathbb{B}^1] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]},<br />
\\\\<br />
([\mathbb{B}^2] \!\to\! [\mathbb{B}^1]) \!\to\!<br />
\\<br />
([\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1])<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;55 supplies a more detailed outline of terminology for operators and their results. Here, we list the restrictive subtype (or narrowest defined subtype) that applies to each entity and we indicate across the span of the Table the whole spectrum of alternative types that color the interpretation of each symbol. For example, all the component operator maps <math>\mathrm{W}J\!</math> have <math>1\!</math>-dimensional ranges, either <math>\mathbb{B}^1\!</math> or <math>\mathbb{D}^1,\!</math> and so they can be viewed either as propositions <math>\mathrm{W}J : \mathrm{E}U \to \mathbb{B}\!</math> or as logical transformations <math>\mathrm{W}J : \mathrm{E}U^\bullet \to X^\bullet.\!</math> As a rule, the plan of the Table allows us to name each entry by detaching the underlined adjective at the left of its row and prefixing it to the generic noun at the top of its column. In one case, however, it is customary to depart from this scheme. Because the phrase ''differential proposition'', applied to the result <math>\mathrm{d}J : \mathrm{E}U \to \mathbb{D},\!</math> does not distinguish it from the general run of differential propositions <math>\mathrm{G}: \mathrm{E}U \to \mathbb{B},\!</math> it is usual to single out <math>\mathrm{d}J\!</math> as the ''tangent proposition'' of <math>J.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 55.} ~~ \text{Synopsis of Terminology : Restrictive and Alternative Subtypes}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| &nbsp;<br />
| align="center" | <math>\text{Operator}\!</math><br />
| align="center" | <math>\text{Proposition}\!</math><br />
| align="center" | <math>\text{Map}\!</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tacit}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\boldsymbol\varepsilon : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\boldsymbol\varepsilon : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{B} \\<br />
\boldsymbol\varepsilon J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{B}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x] \\<br />
\boldsymbol\varepsilon J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Trope}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\eta : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\eta : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\eta J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\eta J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Enlargement}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{E} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{E}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{E}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Difference}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{D} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{D}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{D}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Differential}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{d} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{d} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{d}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}\!</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Remainder}}\\\text{operator}\end{matrix}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{r} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{r} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{r}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x] \\<br />
\mathrm{r}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Radius}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e} = (\boldsymbol\varepsilon, \eta) \\<br />
\mathsf{e} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{e}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Secant}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}) \\<br />
\mathsf{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{E}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Chord}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}) \\<br />
\mathsf{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{D}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tangent}}\\\text{functor}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T} = (\boldsymbol\varepsilon, \mathrm{d}) \\<br />
\mathsf{T} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}J : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}J : \mathbb{B}^2 \!\times\! \mathbb{D}^2 \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T}J : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, \mathrm{d}x] \\<br />
\mathsf{T}J : [\mathbb{B}^2 \!\times\! \mathbb{D}^2] \!\to\! [\mathbb{B}^1 \!\times\! \mathbb{D}^1]<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====End of Perfunctory Chatter : Time to Roll the Clip!====<br />
<br />
Two steps remain to finish the analysis of <math>J\!</math> that we began so long ago. First, we need to paste our accumulated heap of flat pictures into the frames of transformations, filling out the shapes of the operator maps <math>\mathsf{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet.~\!</math> This scheme is executed in two styles, using the ''areal views'' in Figures&nbsp;56-a and the ''box views'' in Figures&nbsp;56-b. Finally, in Figures&nbsp;57-1 to 57-4 we put all the pieces together to construct the full operator diagrams for <math>\mathsf{W} : J \to \mathsf{W}J.\!</math> There is a considerable amount of redundancy among the following three series of Figures but that will hopefully provide a fuller picture of the operations under review, enabling these snapshots to serve as successive frames in the animation of logic they are meant to become.<br />
<br />
=====Operator Maps : Areal Views=====<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a1 -- Radius Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a1.} ~~ \text{Radius Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a2 -- Secant Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a2.} ~~ \text{Secant Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a3 -- Chord Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a3.} ~~ \text{Chord Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-a4 -- Tangent Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-a4.} ~~ \text{Tangent Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
=====Operator Maps : Box Views=====<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b1 -- Radius Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b1.} ~~ \text{Radius Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b2 -- Secant Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b2.} ~~ \text{Secant Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b3 -- Chord Map of J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b3.} ~~ \text{Chord Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 56-b4 -- Tangent Map of J ISW.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 56-b4.} ~~ \text{Tangent Map of the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
=====Operator Diagrams for the Conjunction J = uv=====<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-1 -- Radius Operator Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-1.} ~~ \text{Radius Operator Diagram for the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-2 -- Secant Operator Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-2.} ~~ \text{Secant Operator Diagram for the Conjunction}~ J = uv~\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-3 -- Chord Operator Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-3.} ~~ \text{Chord Operator Diagram for the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 57-4 -- Tangent Functor Diagram for J.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 57-4.} ~~ \text{Tangent Functor Diagram for the Conjunction}~ J = uv\!</math><br />
|}<br />
<br />
===Taking Aim at Higher Dimensional Targets===<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="40%" | &nbsp;<br />
| width="60%" |<br />
The past and present wilt . . . . I have filled them and<br><br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;emptied them,<br><br />
And proceed to fill my next fold of the future.<br />
|-<br />
| &nbsp;<br />
| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 87]<br />
|}<br />
<br />
In the next Section we consider a transformation <math>F\!</math> of concrete type <math>F : [u, v] \to [x, y]\!</math> and abstract type <math>F : [\mathbb{B}^2] \to [\mathbb{B}^2].\!</math> From the standpoint of propositional calculus we naturally approach the task of understanding such a transformation by parsing it into component maps with <math>1\!</math>-dimensional ranges, as follows:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{ccccccl}<br />
F & = & (F_1, F_2) & = & (f, g) & : & [u, v] \to [x, y],<br />
\\[6pt]<br />
&& F_1 & = & f & : & [u, v] \to [x],<br />
\\[6pt]<br />
&& F_2 & = & g & : & [u, v] \to [y].<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Then we tackle the separate components, now viewed as propositions <math>F_i : U \to \mathbb{B},\!</math> one at a time. At the completion of this analytic phase, we return to the task of synthesizing these partial and transient impressions into an agile form of integrity, a solidly coordinated and deeply integrated comprehension of the ongoing transformation. (Very often, of course, in tangling with refractory cases, we never get as far as the beginning again.)<br />
<br />
Let us now refer to the dimension of the target space or codomain as the ''toll'' (or ''tole'') of a transformation, as distinguished from the dimension of the range or image that is customarily called the ''rank''. When we keep to transformations with a toll of <math>1,\!</math> as <math>J : [u, v] \to [x],\!</math> we tend to get lazy about distinguishing a logical transformation from its component propositions. However, if we deal with transformations of a higher toll, this form of indolence can no longer be tolerated.<br />
<br />
Well, perhaps we can carry it a little further. After all, the operator result <math>\mathrm{W}J : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet\!</math> is a map of toll <math>2,\!</math> and cannot be unfolded in one piece as a proposition. But when a map has rank <math>1,\!</math> like <math>\boldsymbol\varepsilon J : \mathrm{E}U \to X \subseteq \mathrm{E}X\!</math> or <math>\mathrm{d}J : \mathrm{E}U \to \mathrm{d}X \subseteq \mathrm{E}X,\!</math> we naturally choose to concentrate on the <math>1\!</math>-dimensional range of the operator result <math>\mathrm{W}J,\!</math> ignoring the final difference in quality between the spaces <math>X\!</math> and <math>\mathrm{d}X,\!</math> and view <math>\mathrm{W}J\!</math> as a proposition about <math>\mathrm{E}U.\!</math><br />
<br />
In this way, an initial ambivalence about the role of the operand <math>J\!</math> conveys a double duty to the result <math>\mathrm{W}J.\!</math> The pivot that is formed by our focus of attention is essential to the linkage that transfers this double moment, as the whole process takes its bearing and wheels around the precise measure of a narrow bead that we can draw on the range of <math>\mathrm{W}J.\!</math> This is the escapement that it takes to get away with what may otherwise seem to be a simple duplicity, and this is the tolerance that is needed to counterbalance a certain arrogance of equivocation, by all of which machinations we make ourselves free to indicate the operator results <math>\mathrm{W}J\!</math> as propositions or as transformations, indifferently.<br />
<br />
But that's it, and no further. Neglect of these distinctions in range and target universes of higher dimensions is bound to cause a hopeless confusion. To guard against these adverse prospects, Tables&nbsp;58 and 59 lay the groundwork for discussing a typical map <math>F : [\mathbb{B}^2] \to [\mathbb{B}^2],\!</math> and begin to pave the way to some extent for discussing any transformation of the form <math>F : [\mathbb{B}^n] \to [\mathbb{B}^k].\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 58.} ~~ \text{Cast of Characters : Expansive Subtypes of Objects and Operators}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| align="center" | <math>\text{Symbol}\!</math><br />
| align="center" | <math>\text{Notation}\!</math><br />
| align="center" | <math>\text{Description}\!</math><br />
| align="center" | <math>\text{Type}\!</math><br />
|-<br />
| align="center" | <math>U^\bullet\!</math><br />
| <math>= [u, v]\!</math><br />
| <math>\text{Source universe}\!</math><br />
| <math>[\mathbb{B}^n]\!</math><br />
|-<br />
| align="center" | <math>X^\bullet~\!</math><br />
| <math>\begin{array}{l}<br />
= [x, y] \\<br />
= [f, g]<br />
\end{array}</math><br />
| <math>\text{Target universe}\!</math><br />
| <math>[\mathbb{B}^k]\!</math><br />
|-<br />
| align="center" | <math>\mathrm{E}U^\bullet\!</math><br />
| <math>= [u, v, \mathrm{d}u, \mathrm{d}v]\!</math><br />
| <math>\text{Extended source universe}\!</math><br />
| <math>[\mathbb{B}^n \!\times\! \mathbb{D}^n]\!</math><br />
|-<br />
| align="center" | <math>\mathrm{E}X^\bullet\!</math><br />
| <math>\begin{array}{l}<br />
= [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
= [f, g, \mathrm{d}f, \mathrm{d}g]<br />
\end{array}</math><br />
| <math>\text{Extended target universe}\!</math><br />
| <math>[\mathbb{B}^k \!\times\! \mathbb{D}^k]\!</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
f \\ g<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{ll}<br />
f : U \!\to\! [x] \cong \mathbb{B} \\<br />
g : U \!\to\! [y] \cong \mathbb{B}<br />
\end{array}</math><br />
| <math>\text{Proposition}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathbb{B}^n \!\to\! \mathbb{B} \\<br />
\in (\mathbb{B}^n, \mathbb{B}^n \!\to\! \mathbb{B}) = [\mathbb{B}^n]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>F\!</math><br />
| <math>F = (f, g) : U^\bullet \!\to\! X^\bullet\!</math><br />
| <math>\text{Transformation of Map}\!</math><br />
| <math>[\mathbb{B}^n] \!\to\! [\mathbb{B}^k]</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\boldsymbol\varepsilon<br />
\\<br />
\eta<br />
\\<br />
\mathrm{E}<br />
\\<br />
\mathrm{D}<br />
\\<br />
\mathrm{d}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{W} : U^\bullet \!\to\! \mathrm{E}U^\bullet,<br />
\\<br />
\mathrm{W} : X^\bullet \!\to\! \mathrm{E}X^\bullet,<br />
\\<br />
\mathrm{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\\<br />
\text{for each}~ \mathrm{W} ~\text{in the set:}<br />
\\<br />
\{ \boldsymbol\varepsilon, \eta, \mathrm{E}, \mathrm{D}, \mathrm{d} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{ll}<br />
\text{Tacit extension operator} & \boldsymbol\varepsilon<br />
\\<br />
\text{Trope extension operator} & \eta<br />
\\<br />
\text{Enlargement operator} & \mathrm{E}<br />
\\<br />
\text{Difference operator} & \mathrm{D}<br />
\\<br />
\text{Differential operator} & \mathrm{d}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^n] \!\to\! [\mathbb{B}^n \!\times\! \mathbb{D}^n]},<br />
\\<br />
{[\mathbb{B}^k] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]},<br />
\\\\<br />
([\mathbb{B}^n] \!\to\! [\mathbb{B}^k]) \!\to\!<br />
\\<br />
([\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k])<br />
\end{array}</math><br />
|-<br />
| align="center" |<br />
<math>\begin{matrix}<br />
\mathsf{e}<br />
\\<br />
\mathsf{E}<br />
\\<br />
\mathsf{D}<br />
\\<br />
\mathsf{T}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{W} : U^\bullet \!\to\! \mathsf{T}U^\bullet = \mathrm{E}U^\bullet,<br />
\\<br />
\mathsf{W} : X^\bullet \!\to\! \mathsf{T}X^\bullet = \mathrm{E}X^\bullet,<br />
\\<br />
\mathsf{W} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathsf{T}U^\bullet \!\to\! \mathsf{T}X^\bullet)<br />
\\<br />
\text{for each}~ \mathsf{W} ~\text{in the set:}<br />
\\<br />
\{ \mathsf{e}, \mathsf{E}, \mathsf{D}, \mathsf{T} \}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{lll}<br />
\text{Radius operator} & \mathsf{e} & = (\boldsymbol\varepsilon, \eta)<br />
\\<br />
\text{Secant operator} & \mathsf{E} & = (\boldsymbol\varepsilon, \mathrm{E})<br />
\\<br />
\text{Chord operator} & \mathsf{D} & = (\boldsymbol\varepsilon, \mathrm{D})<br />
\\<br />
\text{Tangent functor} & \mathsf{T} & = (\boldsymbol\varepsilon, \mathrm{d})<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
{[\mathbb{B}^n] \!\to\! [\mathbb{B}^n \!\times\! \mathbb{D}^n]},<br />
\\<br />
{[\mathbb{B}^k] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]},<br />
\\\\<br />
([\mathbb{B}^n] \!\to\! [\mathbb{B}^k]) \!\to\!<br />
\\<br />
([\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k])<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 59.} ~~ \text{Synopsis of Terminology : Restrictive and Alternative Subtypes}~\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| &nbsp;<br />
| align="center" | <math>\begin{matrix}\text{Operator}\\\text{or}\\\text{Operand}\end{matrix}</math><br />
| align="center" | <math>\begin{matrix}\text{Proposition}\\\text{or}\\\text{Component}\end{matrix}</math><br />
| align="center" | <math>\begin{matrix}\text{Transformation}\\\text{or}\\\text{Map}\end{matrix}</math><br />
|-<br />
| align="center" | <math>\underline{\text{Operand}}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
F = (F_1, F_2) \\<br />
F = (f, g) : U \!\to\! X<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
F_i : \langle u, v \rangle \!\to\! \mathbb{B} \\<br />
F_i : \mathbb{B}^n \!\to\! \mathbb{B}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
F : [u, v] \!\to\! [x, y] \\<br />
F : [\mathbb{B}^n] \!\to\! [\mathbb{B}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tacit}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\boldsymbol\varepsilon : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\boldsymbol\varepsilon : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{B} \\<br />
\boldsymbol\varepsilon F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{B}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\boldsymbol\varepsilon F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y] \\<br />
\boldsymbol\varepsilon F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Trope}}\\\text{extension}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\eta : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\eta : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\eta F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\eta F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\eta F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Enlargement}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{E} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{E}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{E}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{E}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Difference}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{D} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{D}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{D}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{D}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Differential}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{d} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{d} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{d}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}\!</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Remainder}}\\\text{operator}\end{matrix}\!</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r} : U^\bullet \!\to\! \mathrm{E}U^\bullet,~<br />
\mathrm{r} : X^\bullet \!\to\! \mathrm{E}X^\bullet \\<br />
\mathrm{r} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{d}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{r}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{r}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [\mathrm{d}x, \mathrm{d}y] \\<br />
\mathrm{r}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Radius}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e} = (\boldsymbol\varepsilon, \eta) \\<br />
\mathsf{e} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{e}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{e}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Secant}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}) \\<br />
\mathsf{E} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{E}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{E}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Chord}}\\\text{operator}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}) \\<br />
\mathsf{D} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
| &nbsp;<br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{D}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{D}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|-<br />
| align="center" | <math>\begin{matrix}\underline{\text{Tangent}}\\\text{functor}\end{matrix}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T} = (\boldsymbol\varepsilon, \mathrm{d}) \\<br />
\mathsf{T} : (U^\bullet \!\to\! X^\bullet) \!\to\! (\mathrm{E}U^\bullet \!\to\! \mathrm{E}X^\bullet)<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathrm{d}F_i : \langle u, v, \mathrm{d}u, \mathrm{d}v \rangle \!\to\! \mathbb{D} \\<br />
\mathrm{d}F_i : \mathbb{B}^n \!\times\! \mathbb{D}^n \!\to\! \mathbb{D}<br />
\end{array}</math><br />
|<br />
<math>\begin{array}{l}<br />
\mathsf{T}F : [u, v, \mathrm{d}u, \mathrm{d}v] \!\to\! [x, y, \mathrm{d}x, \mathrm{d}y] \\<br />
\mathsf{T}F : [\mathbb{B}^n \!\times\! \mathbb{D}^n] \!\to\! [\mathbb{B}^k \!\times\! \mathbb{D}^k]<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
===Transformations of Type '''B'''<sup>2</sup> &rarr; '''B'''<sup>2</sup>===<br />
<br />
To take up a slightly more complex example, but one that remains simple enough to pursue through a complete series of developments, consider the transformation from <math>U^\bullet = [u, v]\!</math> to <math>X^\bullet = [x, y]\!</math> that is defined by the following system of equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
x<br />
& = & f(u, v)<br />
& = & \texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[8pt]<br />
y<br />
& = & g(u, v)<br />
& = & \texttt{((} u \texttt{,} v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
The component notation <math>F = (F_1, F_2) = (f, g) : U^\bullet \to X^\bullet\!</math> allows us to give a name and a type to this transformation and permits defining it by the compact description that follows:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
(x, y)<br />
& = & F(u, v)<br />
& = & (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Logical Transformations====<br />
<br />
The information that defines the logical transformation <math>F\!</math> can be represented in the form of a truth table, as shown in Table&nbsp;60. To cut down on subscripts in this example we continue to use plain letter equivalents for all components of spaces and maps.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:60%"<br />
|+ style="height:30px" | <math>\text{Table 60.} ~~ \text{A Propositional Transformation}\!</math><br />
|- style="height:40px; background:ghostwhite; width:100%"<br />
| style="width:25%" | <math>u\!</math><br />
| style="width:25%" | <math>v\!</math><br />
| style="width:25%; border-left:1px solid black" | <math>f\!</math><br />
| style="width:25%" | <math>g\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="border-top:1px solid black" | <math>u\!</math><br />
| style="border-top:1px solid black" | <math>v\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;61 shows how we might paint a picture of the transformation <math>F\!</math> in the manner of Figure&nbsp;30.<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 61 -- Propositional Transformation.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 61.} ~~ \text{A Propositional Transformation}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;62 extracts the gist of Figure&nbsp;61, exhibiting a style of diagram that is adequate for most purposes.<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 62 -- Propositional Transformation (Short Form).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 62.} ~~ \text{A Propositional Transformation (Short Form)}\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Local Transformations====<br />
<br />
Figure&nbsp;63 gives a more complete picture of the transformation <math>F,\!</math> showing how the points of <math>U^\bullet\!</math> are transformed into points of <math>X^\bullet.\!</math> The bold lines crossing from one universe to the other trace the action that <math>F\!</math> induces on points, in other words, they show how the transformation acts as a mapping from points to points and chart its effects on the elements that are variously called cells, points, positions, or singular propositions.<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 63 -- Transformation of Positions.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 63.} ~~ \text{A Transformation of Positions}\!</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;64 shows how the action of <math>F\!</math> on cells or points can be computed in terms of coordinates.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 64.} ~~ \text{A Transformation of Positions}\!</math><br />
|- style="height:40px; background:ghostwhite; width:100%"<br />
| style="width:8%" | <math>u\!</math><br />
| style="width:8%" | <math>v\!</math><br />
| style="width:12%; border-left:1px solid black" | <math>x\!</math><br />
| style="width:12%" | <math>y\!</math><br />
| style="width:10%; border-left:1px solid black" | <math>x~y\!</math><br />
| style="width:10%" | <math>x \texttt{(} y \texttt{)}\!</math><br />
| style="width:10%" | <math>\texttt{(} x \texttt{)} y\!</math><br />
| style="width:10%" | <math>\texttt{(} x \texttt{)(} y \texttt{)}\!</math><br />
| style="width:20%; border-left:1px solid black" | <math>X^\bullet = [x, y]\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\uparrow<br />
\\[4pt]<br />
F =<br />
\\[4pt]<br />
(f, g)<br />
\\[4pt]<br />
\uparrow<br />
\end{matrix}</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="border-top:1px solid black" | <math>u\!</math><br />
| style="border-top:1px solid black" | <math>v\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
| style="border-top:1px solid black" | <math>\texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>u~v\!</math><br />
| style="border-top:1px solid black" | <math>\texttt{(} u \texttt{,} v \texttt{)}\!</math><br />
| style="border-top:1px solid black" | <math>\texttt{(} u \texttt{)(} v \texttt{)}\!</math><br />
| style="border-top:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>U^\bullet = [u, v]\!</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;65 extends this scheme from single cells to arbitrary regions, showing how we might compute the action of a logical transformation on arbitrary propositions in the universe of discourse. The effect of a point-transformation on arbitrary propositions, or any other structures erected on points, is referred to as the ''induced action'' of the transformation on the structures in question.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 65-a.} ~~ \text{An Induced Transformation on Propositions}\!</math><br />
|- style="height:50px; background:ghostwhite"<br />
| style="width:20%" | <math>X^\bullet~\!</math><br />
| style="width:20%; border-right:none" | <math>\longleftarrow\!</math><br />
| style="width:20%; border-left:none; border-right:none" | <math>F = (f, g)\!</math><br />
| style="width:20%; border-left:none" | <math>\longleftarrow\!</math><br />
| style="width:20%" | <math>U^\bullet~\!</math><br />
|- style="background:ghostwhite"<br />
| rowspan="2" | <math>f_i (x, y)\!</math><br />
| align="right" | <math>\begin{matrix}u = \\ v =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~0~0\\1~0~1~0\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= u \\ = v\end{matrix}</math><br />
| rowspan="2" style="width:20%" | <math>f_j (u, v)\!</math><br />
|- style="background:ghostwhite"<br />
| align="right" | <math>\begin{matrix}x = \\ y =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~1~0\\1~0~0~1\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= f(u, v) \\ = g(u, v)\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{2}<br />
\\[2pt]<br />
f_{3}<br />
\\[2pt]<br />
f_{4}<br />
\\[2pt]<br />
f_{5}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{7}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)~ ~}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{~ ~(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{,~} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{~~} y \texttt{)}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~0<br />
\\[2pt]<br />
0~0~0~0<br />
\\[2pt]<br />
0~0~0~1<br />
\\[2pt]<br />
0~0~0~1<br />
\\[2pt]<br />
0~1~1~0<br />
\\[2pt]<br />
0~1~1~0<br />
\\[2pt]<br />
0~1~1~1<br />
\\[2pt]<br />
0~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{,~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{,~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{~~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{~~} v \texttt{)}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[2pt]<br />
f_{0}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{7}<br />
\\[2pt]<br />
f_{7}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{8}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{10}<br />
\\[2pt]<br />
f_{11}<br />
\\[2pt]<br />
f_{12}<br />
\\[2pt]<br />
f_{13}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{15}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[2pt]<br />
\texttt{~ ~ ~ ~} y \texttt{~~}<br />
\\[2pt]<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[2pt]<br />
\texttt{~~} x \texttt{~ ~ ~ ~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
1~0~0~0<br />
\\[2pt]<br />
1~0~0~0<br />
\\[2pt]<br />
1~0~0~1<br />
\\[2pt]<br />
1~0~0~1<br />
\\[2pt]<br />
1~1~1~0<br />
\\[2pt]<br />
1~1~1~0<br />
\\[2pt]<br />
1~1~1~1<br />
\\[2pt]<br />
1~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~~} u \texttt{~~} v \texttt{~~}<br />
\\[2pt]<br />
\texttt{~~} u \texttt{~~} v \texttt{~~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{8}<br />
\\[2pt]<br />
f_{8}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{15}<br />
\\[2pt]<br />
f_{15}<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table 65-b.} ~~ \text{An Induced Transformation on Propositions}\!</math><br />
|- style="height:50px; background:ghostwhite"<br />
| style="width:20%" | <math>X^\bullet~\!</math><br />
| style="width:20%; border-right:none" | <math>\longleftarrow\!</math><br />
| style="width:20%; border-left:none; border-right:none" | <math>F = (f, g)\!</math><br />
| style="width:20%; border-left:none" | <math>\longleftarrow\!</math><br />
| style="width:20%" | <math>U^\bullet~\!</math><br />
|- style="background:ghostwhite"<br />
| rowspan="2" | <math>f_i (x, y)\!</math><br />
| align="right" | <math>\begin{matrix}u = \\ v =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~0~0\\1~0~1~0\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= u \\ = v\end{matrix}</math><br />
| rowspan="2" style="width:20%" | <math>f_j (u, v)\!</math><br />
|- style="background:ghostwhite"<br />
| align="right" | <math>\begin{matrix}x = \\ y =\end{matrix}</math><br />
| <math>\begin{matrix}1~1~1~0\\1~0~0~1\end{matrix}</math><br />
| align="left" | <math>\begin{matrix}= f(u, v) \\ = g(u, v)\end{matrix}</math><br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>0~0~0~0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>f_{0}\!</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[2pt]<br />
f_{2}<br />
\\[2pt]<br />
f_{4}<br />
\\[2pt]<br />
f_{8}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~0<br />
\\[2pt]<br />
0~0~0~1<br />
\\[2pt]<br />
0~1~1~0<br />
\\[2pt]<br />
1~0~0~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[2pt]<br />
\texttt{(} u \texttt{,~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{~} u \texttt{~~} v \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{0}<br />
\\[2pt]<br />
f_{1}<br />
\\[2pt]<br />
f_{6}<br />
\\[2pt]<br />
f_{8}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{3}<br />
\\[2pt]<br />
f_{12}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\[2pt]<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~0~0~1<br />
\\[2pt]<br />
1~1~1~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} u \texttt{)(} v \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{1}<br />
\\[2pt]<br />
f_{14}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[2pt]<br />
f_{9}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[2pt]<br />
1~0~0~0<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} u \texttt{~~} v \texttt{)}<br />
\\[2pt]<br />
\texttt{~} u \texttt{~~} v \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[2pt]<br />
f_{8}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{5}<br />
\\[2pt]<br />
f_{10}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\[2pt]<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~0<br />
\\[2pt]<br />
1~0~0~1<br />
\end{matrix}~\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} u \texttt{,~} v \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{6}<br />
\\[2pt]<br />
f_{9}<br />
\end{matrix}</math><br />
|-<br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[2pt]<br />
f_{11}<br />
\\[2pt]<br />
f_{13}<br />
\\[2pt]<br />
f_{14}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
0~1~1~1<br />
\\[2pt]<br />
1~0~0~1<br />
\\[2pt]<br />
1~1~1~0<br />
\\[2pt]<br />
1~1~1~1<br />
\end{matrix}</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
\texttt{~(} u \texttt{~~} v \texttt{)~}<br />
\\[2pt]<br />
\texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[2pt]<br />
\texttt{((~))}<br />
\end{matrix}\!</math><br />
| valign="bottom" |<br />
<math>\begin{matrix}<br />
f_{7}<br />
\\[2pt]<br />
f_{9}<br />
\\[2pt]<br />
f_{14}<br />
\\[2pt]<br />
f_{15}<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>f_{15}\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Difference Operators and Tangent Functors====<br />
<br />
Given the alphabets <math>\mathcal{U} = \{ u, v \}\!</math> and <math>\mathcal{X} = \{ x, y \},\!</math> along with the corresponding universes of discourse <math>U^\bullet, X^\bullet \cong [\mathbb{B}^2],\!</math> how many logical transformations of the general form <math>G = (G_1, G_2) : U^\bullet \to X^\bullet\!</math> are there? Since <math>G_1\!</math> and <math>G_2\!</math> can be any propositions of the type <math>\mathbb{B}^2 \to \mathbb{B},\!</math> there are <math>2^4 = 16\!</math> choices for each of the maps <math>G_1\!</math> and <math>G_2\!</math> and thus there are <math>2^4 \cdot 2^4 = 2^8 = 256\!</math> different mappings altogether of the form <math>G : U^\bullet \to X^\bullet.\!</math> The set of functions of a given type is denoted by placing its type indicator in parentheses, in the present instance writing <math>(U^\bullet \to X^\bullet) = \{ G : U^\bullet \to X^\bullet \},\!</math> and so the cardinality of the ''function space'' <math>(U^\bullet \to X^\bullet)\!</math> is summed up by writing <math>|(U^\bullet \to X^\bullet)| = |(\mathbb{B}^2 \to \mathbb{B}^2)| = 4^4 = 256.\!</math><br />
<br />
Given a transformation <math>G = (G_1, G_2) : U^\bullet \to X^\bullet\!</math> of this type, we proceed to define a pair of further transformations, related to <math>G,\!</math> that operate between the extended universes, <math>\mathrm{E}U^\bullet\!</math> and <math>\mathrm{E}X^\bullet,\!</math> of its source and target domains.<br />
<br />
First, the ''enlargement map'' (or ''secant transformation'') <math>\mathrm{E}G = (\mathrm{E}G_1, \mathrm{E}G_2) : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet\!</math> is defined by the following set of component equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}G_i<br />
& = & G_i (u + \mathrm{d}u, v + \mathrm{d}v)<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Next, the ''difference map'' (or ''chordal transformation'') <math>\mathrm{D}G = (\mathrm{D}G_1, \mathrm{D}G_2) : \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet~\!</math> is defined in component-wise fashion as the boolean sum of the initial proposition <math>G_i\!</math> and the enlarged proposition <math>\mathrm{E}G_i,\!</math> for <math>i = 1, 2,\!</math> according to the following set of equations:<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
\mathrm{D}G_i<br />
& = & G_i (u, v)<br />
& + & \mathrm{E}G_i (u, v, \mathrm{d}u, \mathrm{d}v)<br />
\\[8pt]<br />
& = & G_i (u, v)<br />
& + & G_i (u + \mathrm{d}u, v + \mathrm{d}v)<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Maintaining a strict analogy with ordinary difference calculus would perhaps have us write <math>\mathrm{D}G_i = \mathrm{E}G_i - G_i,\!</math> but the sum and difference operations are the same thing in boolean arithmetic. It is more often natural in the logical context to consider an initial proposition <math>q,\!</math> then to compute the enlargement <math>\mathrm{E}q,\!</math> and finally to determine the difference <math>\mathrm{D}q = q + \mathrm{E}q,\!</math> so we let the variant order of terms reflect this sequence of considerations.<br />
<br />
Viewed in this light the difference operator <math>\mathrm{D}\!</math> is imagined to be a function of very wide scope and polymorphic application, one that is able to realize the association between each transformation <math>G\!</math> and its difference map <math>\mathrm{D}G,\!</math> for example, taking the function space <math>(U^\bullet \to X^\bullet)\!</math> into <math>(\mathrm{E}U^\bullet \to \mathrm{E}X^\bullet).\!</math> When we consider the variety of interpretations permitted to propositions over the contexts in which we put them to use, it should be clear that an operator of this scope is not at all a trivial matter to define in general and that it may take some trouble to work out. For the moment we content ourselves with returning to particular cases.<br />
<br />
Acting on the logical transformation <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;),\!</math> the operators <math>\mathrm{E}\!</math> and <math>\mathrm{D}\!</math> yield the enlarged map <math>\mathrm{E}F = (\mathrm{E}f, \mathrm{E}g)\!</math> and the difference map <math>\mathrm{D}F = (\mathrm{D}f, \mathrm{D}g),\!</math> respectively, whose components are given as follows.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lll}<br />
\mathrm{E}f<br />
& = & \texttt{((} u + \mathrm{d}u \texttt{)(} v + \mathrm{d}v \texttt{))}<br />
\\[8pt]<br />
\mathrm{E}g<br />
& = & \texttt{((} u + \mathrm{d}u \texttt{,~} v + \mathrm{d}v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"<br />
|<br />
<math>\begin{array}{lllll}<br />
\mathrm{D}f<br />
& = & \texttt{((} u \texttt{)(} v \texttt{))}<br />
& + & \texttt{((} u + \mathrm{d}u \texttt{)(} v + \mathrm{d}v \texttt{))}<br />
\\[8pt]<br />
\mathrm{D}g<br />
& = & \texttt{((} u \texttt{,~} v \texttt{))}<br />
& + & \texttt{((} u + \mathrm{d}u \texttt{,~} v + \mathrm{d}v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
But these initial formulas are purely definitional, and help us little in understanding either the purpose of the operators or the meaning of their results. Working symbolically, let us apply the same method to the separate components <math>f\!</math> and <math>g\!</math> that we earlier used on <math>J.\!</math> This work is recorded in Appendix&nbsp;3 and a summary of the results is presented in Tables&nbsp;66-i and 66-ii.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 66-i.} ~~ \text{Computation Summary for}~ f(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f<br />
& = & u \!\cdot\! v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 1<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}f<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \cdot \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{d}f<br />
& = & u \!\cdot\! v \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}f<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 66-ii.} ~~ \text{Computation Summary for}~ g(u, v) = \texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon g<br />
& = & u \!\cdot\! v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}g<br />
& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}g<br />
& = & u \!\cdot\! v \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
Table&nbsp;67 shows how to compute the analytic series for <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math> in terms of coordinates, and Table&nbsp;68 recaps these results in symbolic terms, agreeing with earlier derivations.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 67.} ~~ \text{Computation of an Analytic Series in Terms of Coordinates}\!</math><br />
|- style="height:40px; background:ghostwhite"<br />
| style="width:6%; border-bottom:1px solid black" | <math>u\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>v\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>\mathrm{d}u\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>\mathrm{d}v\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>u'\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>v'\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:4px double black" | <math>f\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>g\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{E}f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{E}g}\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{D}f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{D}g}\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{d}f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{d}g}\!</math><br />
| style="width:6%; border-bottom:1px solid black; border-left:1px solid black" | <math>{\mathrm{d}^2\!f}\!</math><br />
| style="width:6%; border-bottom:1px solid black" | <math>{\mathrm{d}^2\!g}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>0\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|-<br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black" | <math>1\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\1\end{matrix}\!</math><br />
| style="vertical-align:top; border-top:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table 68.} ~~ \text{Computation of an Analytic Series in Symbolic Terms}\!</math><br />
|- style="height:40px; background:ghostwhite; width:100%"<br />
| <math>u\!</math><br />
| <math>v\!</math><br />
| style="border-left:1px solid black" | <math>f\!</math><br />
| <math>g\!</math><br />
| style="border-left:1px solid black" | <math>{\mathrm{D}f}\!</math><br />
| <math>{\mathrm{D}g}\!</math><br />
| style="border-left:1px solid black" | <math>{\mathrm{d}f}\!</math><br />
| <math>{\mathrm{d}g}\!</math><br />
| style="border-left:1px solid black" | <math>{\mathrm{d}^2\!f}\!</math><br />
| <math>{\mathrm{d}^2\!g}\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
1<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[4pt]<br />
\texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
\\[4pt]<br />
\texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
\\[4pt]<br />
\texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
\texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\\[4pt]<br />
\mathrm{d}u \cdot \mathrm{d}v<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\\[4pt]<br />
0<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;69 gives a graphical picture of the difference map <math>\mathrm{D}F = (\mathrm{D}f, \mathrm{D}g)\!</math> for the transformation <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math> This represents the same information about <math>\mathrm{D}f~\!</math> and <math>\mathrm{D}g~\!</math> that was given in the corresponding rows of Tables&nbsp;66-i and 66-ii, for ease of reference repeated below.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f<br />
& = & u \!\cdot\! v \cdot \mathrm{d}u \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[8pt]<br />
\mathrm{D}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 69 -- Difference Map (Short Form).gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 69.} ~~ \text{Difference Map of}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math><br />
|}<br />
<br />
<br><br />
<br />
Figure&nbsp;70-a shows a way of visualizing the tangent functor map <math>\mathrm{d}F = (\mathrm{d}f, \mathrm{d}g)\!</math> for the transformation <math>F = (f, g) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math> This amounts to the same information about <math>\mathrm{d}f~\!</math> and <math>\mathrm{d}g~\!</math> that was given in Tables&nbsp;66-i and 66-ii, the corresponding rows of which are repeated below.<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{d}f<br />
& = & u \!\cdot\! v \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[8pt]<br />
\mathrm{d}g<br />
& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 70-a -- Tangent Functor Diagram.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 70-a.} ~~ \text{Tangent Functor Diagram for}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math><br />
|}<br />
<br />
<br><br />
<br />
Figure 70-b shows another way to picture the action of the tangent functor on the logical transformation <math>F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;).\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Diff Log Dyn Sys -- Figure 70-b -- Tangent Functor Diagram.gif|center]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 70-b.} ~~ \text{Tangent Functor Ferris Wheel for}~ F(u, v) = (\; \texttt{((} u \texttt{)(} v \texttt{))} \;,\; \texttt{((} u \texttt{,} v \texttt{))} \;)</math><br />
|}<br />
<br />
<br><br />
<br />
* '''Note.''' The original Figure&nbsp;70-b lost some of its labeling in a succession of platform metamorphoses over the years, so we have included an ASCII version below to indicate where the missing labels go.<br />
<br />
{| align="center" border="0" cellpadding="10"<br />
|<br />
<pre><br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| /////\ /////\ | | /XXXX\ /XXXX\ | | /\\\\\ /\\\\\ |<br />
| ///////o//////\ | | /XXXXXXoXXXXXX\ | | /\\\\\\o\\\\\\\ |<br />
| //////// \//////\ | | /XXXXXX/ \XXXXXX\ | | /\\\\\\/ \\\\\\\\ |<br />
| o/////// \//////o | | oXXXXXX/ \XXXXXXo | | o\\\\\\/ \\\\\\\o |<br />
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |<br />
| |/du//| |//dv/| | | |XXXXX| |XXXXX| | | |\du\\| |\\dv\| |<br />
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |<br />
| o//////\ ///////o | | oXXXXXX\ /XXXXXXo | | o\\\\\\\ /\\\\\\o |<br />
| \//////\ //////// | | \XXXXXX\ /XXXXXX/ | | \\\\\\\\ /\\\\\\/ |<br />
| \//////o/////// | | \XXXXXXoXXXXXX/ | | \\\\\\\o\\\\\\/ |<br />
| \///// \///// | | \XXXX/ \XXXX/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ (u)(v) o-----------------------o dv' @ (u)(v) =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| / \ /////\ | | /\\\\\ /XXXX\ | | /\\\\\ /\\\\\ |<br />
| / o//////\ | | /\\\\\\oXXXXXX\ | | /\\\\\\o\\\\\\\ |<br />
| / //\//////\ | | /\\\\\\//\XXXXXX\ | | /\\\\\\/ \\\\\\\\ |<br />
| o ////\//////o | | o\\\\\\////\XXXXXXo | | o\\\\\\/ \\\\\\\o |<br />
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |<br />
| | du |/////|//dv/| | | |\\\\\|/////|XXXXX| | | |\du\\| |\\dv\| |<br />
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |<br />
| o \//////////o | | o\\\\\\\////XXXXXXo | | o\\\\\\\ /\\\\\\o |<br />
| \ \///////// | | \\\\\\\\//XXXXXX/ | | \\\\\\\\ /\\\\\\/ |<br />
| \ o/////// | | \\\\\\\oXXXXXX/ | | \\\\\\\o\\\\\\/ |<br />
| \ / \///// | | \\\\\/ \XXXX/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ (u) v o-----------------------o dv' @ (u) v =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| /////\ / \ | | /XXXX\ /\\\\\ | | /\\\\\ /\\\\\ |<br />
| ///////o \ | | /XXXXXXo\\\\\\\ | | /\\\\\\o\\\\\\\ |<br />
| /////////\ \ | | /XXXXXX//\\\\\\\\ | | /\\\\\\/ \\\\\\\\ |<br />
| o//////////\ o | | oXXXXXX////\\\\\\\o | | o\\\\\\/ \\\\\\\o |<br />
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |<br />
| |/du//|/////| dv | | | |XXXXX|/////|\\\\\| | | |\du\\| |\\dv\| |<br />
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |<br />
| o//////\//// o | | oXXXXXX\////\\\\\\o | | o\\\\\\\ /\\\\\\o |<br />
| \//////\// / | | \XXXXXX\//\\\\\\/ | | \\\\\\\\ /\\\\\\/ |<br />
| \//////o / | | \XXXXXXo\\\\\\/ | | \\\\\\\o\\\\\\/ |<br />
| \///// \ / | | \XXXX/ \\\\\/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ u (v) o-----------------------o dv' @ u (v) =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| dU | | dU | | dU |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| / \ / \ | | /\\\\\ /\\\\\ | | /\\\\\ /\\\\\ |<br />
| / o \ | | /\\\\\\o\\\\\\\ | | /\\\\\\o\\\\\\\ |<br />
| / / \ \ | | /\\\\\\/ \\\\\\\\ | | /\\\\\\/ \\\\\\\\ |<br />
| o / \ o | | o\\\\\\/ \\\\\\\o | | o\\\\\\/ \\\\\\\o |<br />
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |<br />
| | du | | dv | | | |\\\\\| |\\\\\| | | |\du\\| |\\dv\| |<br />
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |<br />
| o \ / o | | o\\\\\\\ /\\\\\\o | | o\\\\\\\ /\\\\\\o |<br />
| \ \ / / | | \\\\\\\\ /\\\\\\/ | | \\\\\\\\ /\\\\\\/ |<br />
| \ o / | | \\\\\\\o\\\\\\/ | | \\\\\\\o\\\\\\/ |<br />
| \ / \ / | | \\\\\/ \\\\\/ | | \\\\\/ \\\\\/ |<br />
| o--o o--o | | o--o o--o | | o--o o--o |<br />
| | | | | |<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= du' @ u v o-----------------------o dv' @ u v =<br />
= | dU' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
o-----------------------o o-----------------------o o-----------------------o<br />
| U | |\U\\\\\\\\\\\\\\\\\\\\\| |\U\\\\\\\\\\\\\\\\\\\\\|<br />
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|<br />
| /////\ /////\ | |\\\\\/////\\/////\\\\\\| |\\\\\/ \\/ \\\\\\|<br />
| ///////o//////\ | |\\\\///////o//////\\\\\| |\\\\/ o \\\\\|<br />
| /////////\//////\ | |\\\////////X\//////\\\\| |\\\/ /\\ \\\\|<br />
| o//////////\//////o | |\\o///////XXX\//////o\\| |\\o /\\\\ o\\|<br />
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|<br />
| |//u//|/////|//v//| | |\\|//u//|XXXXX|//v//|\\| |\\| u |\\\\\| v |\\|<br />
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|<br />
| o//////\//////////o | |\\o//////\XXX///////o\\| |\\o \\\\/ o\\|<br />
| \//////\///////// | |\\\\//////\X////////\\\| |\\\\ \\/ /\\\|<br />
| \//////o/////// | |\\\\\//////o///////\\\\| |\\\\\ o /\\\\|<br />
| \///// \///// | |\\\\\\/////\\/////\\\\\| |\\\\\\ /\\ /\\\\\|<br />
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|<br />
| | |\\\\\\\\\\\\\\\\\\\\\\\| |\\\\\\\\\\\\\\\\\\\\\\\|<br />
o-----------------------o o-----------------------o o-----------------------o<br />
= u' o-----------------------o v' =<br />
= | U' | =<br />
= | o--o o--o | =<br />
= | /////\ /\\\\\ | =<br />
= | ///////o\\\\\\\ | =<br />
= | ////////X\\\\\\\\ | =<br />
= | o///////XXX\\\\\\\o | =<br />
= | |/////oXXXXXo\\\\\| | =<br />
= = = = = = = = = = =|/u'//|XXXXX|\\v'\|= = = = = = = = = = =<br />
| |/////oXXXXXo\\\\\| |<br />
| o//////\XXX/\\\\\\o |<br />
| \//////\X/\\\\\\/ |<br />
| \//////o\\\\\\/ |<br />
| \///// \\\\\/ |<br />
| o--o o--o |<br />
| |<br />
o-----------------------o<br />
<br />
Figure 70-b. Tangent Functor Ferris Wheel for F<u, v> = <((u)(v)), ((u, v))><br />
</pre><br />
|}<br />
<br />
==Epilogue, Enchoiry, Exodus==<br />
<br />
{| width="100%" cellpadding="0" cellspacing="0"<br />
| width="4%" | &nbsp;<br />
| width="92%" |<br />
It is time to explain myself . . . . let us stand up.<br />
| width="4%" | &nbsp;<br />
|- <br />
| align="right" colspan="3" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 79]<br />
|}<br />
<br />
==Appendices==<br />
<br />
===Appendix 1. Propositional Forms and Differential Expansions===<br />
<br />
====Table A1. Propositional Forms on Two Variables====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A1.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="background:ghostwhite"<br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}</math><br />
| width="25%" | <math>\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{0}\\f_{1}\\f_{2}\\f_{3}\\f_{4}\\f_{5}\\f_{6}\\f_{7}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0000}\\f_{0001}\\f_{0010}\\f_{0011}\\f_{0100}\\f_{0101}\\f_{0110}\\f_{0111}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~0\\0~0~0~1\\0~0~1~0\\0~0~1~1\\0~1~0~0\\0~1~0~1\\0~1~1~0\\0~1~1~1<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(~)}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)~ ~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~ ~(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{,~} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{~~} y \texttt{)}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{false}<br />
\\<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\<br />
y ~\text{without}~ x<br />
\\<br />
\text{not}~ x<br />
\\<br />
x ~\text{without}~ y<br />
\\<br />
\text{not}~ y<br />
\\<br />
x ~\text{not equal to}~ y<br />
\\<br />
\text{not both}~ x ~\text{and}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0<br />
\\<br />
\lnot x \land \lnot y<br />
\\<br />
\lnot x \land y<br />
\\<br />
\lnot x<br />
\\<br />
x \land \lnot y<br />
\\<br />
\lnot y<br />
\\<br />
x \ne y<br />
\\<br />
\lnot x \lor \lnot y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{8}\\f_{9}\\f_{10}\\f_{11}\\f_{12}\\f_{13}\\f_{14}\\f_{15}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{1000}\\f_{1001}\\f_{1010}\\f_{1011}\\f_{1100}\\f_{1101}\\f_{1110}\\f_{1111}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
1~0~0~0\\1~0~0~1\\1~0~1~0\\1~0~1~1\\1~1~0~0\\1~1~0~1\\1~1~1~0\\1~1~1~1<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~ ~ ~ ~} y \texttt{~~}<br />
\\<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{~~} x \texttt{~ ~ ~ ~}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{((~))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{and}~ y<br />
\\<br />
x ~\text{equal to}~ y<br />
\\<br />
y<br />
\\<br />
\text{not}~ x ~\text{without}~ y<br />
\\<br />
x<br />
\\<br />
\text{not}~ y ~\text{without}~ x<br />
\\<br />
x ~\text{or}~ y<br />
\\<br />
\text{true}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x \land y<br />
\\<br />
x = y<br />
\\<br />
y<br />
\\<br />
x \Rightarrow y<br />
\\<br />
x<br />
\\<br />
x \Leftarrow y<br />
\\<br />
x \lor y<br />
\\<br />
1<br />
\end{matrix}</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A2. Propositional Forms on Two Variables====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A2.} ~~ \text{Propositional Forms on Two Variables}\!</math><br />
|- style="background:ghostwhite"<br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_1\\\text{Decimal}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_2\\\text{Binary}\\\text{Index}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_3\\\text{Truth}\\\text{Table}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_4\\\text{Cactus}\\\text{Language}\end{matrix}</math><br />
| width="25%" | <math>\begin{matrix}\mathcal{L}_5\\\text{English}\\\text{Paraphrase}\end{matrix}</math><br />
| width="15%" | <math>\begin{matrix}\mathcal{L}_6\\\text{Conventional}\\\text{Formula}\end{matrix}</math><br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>x\colon\!</math><br />
| <math>1~1~0~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|- style="background:ghostwhite"<br />
| &nbsp;<br />
| align="right" | <math>y\colon\!</math><br />
| <math>1~0~1~0\!</math><br />
| &nbsp; || &nbsp; || &nbsp;<br />
|-<br />
| <math>f_{0}\!</math><br />
| <math>f_{0000}\!</math><br />
| <math>0~0~0~0</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>\text{false}\!</math><br />
| <math>0\!</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0001}\\f_{0010}\\f_{0100}\\f_{1000}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~0~1\\0~0~1~0\\0~1~0~0\\1~0~0~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{neither}~ x ~\text{nor}~ y<br />
\\<br />
y ~\text{without}~ x<br />
\\<br />
x ~\text{without}~ y<br />
\\<br />
x ~\text{and}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot x \land \lnot y<br />
\\<br />
\lnot x \land y<br />
\\<br />
x \land \lnot y<br />
\\<br />
x \land y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{3}\\f_{12}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0011}\\f_{1100}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~0~1~1\\1~1~0~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ x<br />
\\<br />
x<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot x<br />
\\<br />
x<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{6}\\f_{9}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0110}\\f_{1001}<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~0\\1~0~0~1<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x ~\text{not equal to}~ y<br />
\\<br />
x ~\text{equal to}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
x \ne y<br />
\\<br />
x = y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{5}\\f_{10}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0101}\\f_{1010}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~0~1\\1~0~1~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not}~ y<br />
\\<br />
y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot y<br />
\\<br />
y<br />
\end{matrix}</math><br />
|-<br />
|<br />
<math>\begin{matrix}<br />
f_{7}\\f_{11}\\f_{13}\\f_{14}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
f_{0111}\\f_{1011}\\f_{1101}\\f_{1110}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0~1~1~1\\1~0~1~1\\1~1~0~1\\1~1~1~0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\<br />
\texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\text{not both}~ x ~\text{and}~ y<br />
\\<br />
\text{not}~ x ~\text{without}~ y<br />
\\<br />
\text{not}~ y ~\text{without}~ x<br />
\\<br />
x ~\text{or}~ y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\lnot x \lor \lnot y<br />
\\<br />
x \Rightarrow y<br />
\\<br />
x \Leftarrow y<br />
\\<br />
x \lor y<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>f_{1111}\!</math><br />
| <math>1~1~1~1\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>\text{true}\!</math><br />
| <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A3. E''f'' Expanded Over Differential Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A3.} ~~ \mathrm{E}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{T}_{11}f\\\mathrm{E}f|_{\mathrm{d}x ~ \mathrm{d}y}\end{matrix}</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{T}_{10}f\\\mathrm{E}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}\end{matrix}</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{T}_{01}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}\end{matrix}</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{T}_{00}f\\\mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{3}\\f_{12}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{6}\\f_{9}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{5}\\f_{10}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{7}\\f_{11}\\f_{13}\\f_{14}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
|- style="background:ghostwhite"<br />
| style="border-top:1px solid black" colspan="2" | <math>\text{Fixed Point Total}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>4\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>4\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>4\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>16\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A4. D''f'' Expanded Over Differential Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A4.} ~~ \mathrm{D}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\mathrm{D}f|_{\mathrm{d}x ~ \mathrm{d}y}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}~\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
y<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
y<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
x<br />
\\<br />
x<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
x<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}1\\1\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
y<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
y<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <br />
<math>\begin{matrix}<br />
x<br />
\\<br />
x<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}0\\0\\0\\0\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A5. E''f'' Expanded Over Ordinary Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A5.} ~~ \mathrm{E}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\mathrm{E}f|_{xy}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{E}f|_{x \texttt{(} y \texttt{)}}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{E}f|_{\texttt{(} x \texttt{)} y}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{E}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{1}\\f_{2}\\f_{4}\\f_{8}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{3}\\f_{12}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{6}\\f_{9}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{5}\\f_{10}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" |<br />
<math>\begin{matrix}<br />
f_{7}\\f_{11}\\f_{13}\\f_{14}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{(~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~)}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A6. D''f'' Expanded Over Ordinary Features====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A6.} ~~ \mathrm{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:10%; border-bottom:1px solid black" | &nbsp;<br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" | <math>f\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:4px double black" |<br />
<math>\mathrm{D}f|_{xy}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{x \texttt{(} y \texttt{)}}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} x \texttt{)} y}\!</math><br />
| style="width:18%; border-bottom:1px solid black; border-left:1px solid black" |<br />
<math>\mathrm{D}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{0}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\texttt{(} x \texttt{)}\\\texttt{~} x \texttt{~}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{,~} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
\texttt{~} \mathrm{d}x \texttt{~~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{~} \mathrm{d}x \texttt{~(} \mathrm{d}y \texttt{)}<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)~} \mathrm{d}y \texttt{~}<br />
\\<br />
\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{15}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" | <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Appendix 2. Differential Forms===<br />
<br />
The actions of the difference operator <math>\mathrm{D}\!</math> and the tangent operator <math>\mathrm{d}\!</math> on the 16 bivariate propositions are shown in Tables&nbsp;A7 and A8.<br />
<br />
Table A7 expands the differential forms that result over a ''logical basis'':<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
|<br />
<math>\{~ \texttt{(}\mathrm{d}x\texttt{)(}\mathrm{d}y\texttt{)}, ~\mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}, ~\texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.\!</math><br />
|}<br />
<br />
This set consists of the singular propositions in the first order differential variables, indicating mutually exclusive and exhaustive ''cells'' of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the cell-basis, point-basis, or singular differential basis. In this setting it is frequently convenient to use the following abbreviations:<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
|<br />
<math>\partial x ~=~ \mathrm{d}x~\texttt{(}\mathrm{d}y\texttt{)}\!</math> &nbsp; &nbsp; and &nbsp; &nbsp; <math>\partial y ~=~ \texttt{(}\mathrm{d}x\texttt{)}~\mathrm{d}y.\!</math><br />
|}<br />
<br />
Table A8 expands the differential forms that result over an ''algebraic basis'':<br />
<br />
{| align="center" cellpadding="6" style="text-align:center"<br />
| <math>\{~ 1, ~\mathrm{d}x, ~\mathrm{d}y, ~\mathrm{d}x~\mathrm{d}y ~\}.\!</math><br />
|}<br />
<br />
This set consists of the ''positive propositions'' in the first order differential variables, indicating overlapping positive regions of the tangent universe of discourse. Accordingly, this set of differential propositions may also be referred to as the ''positive differential basis''.<br />
<br />
====Table A7. Differential Forms Expanded on a Logical Basis====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A7.} ~~ \text{Differential Forms Expanded on a Logical Basis}\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| &nbsp;<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" | <math>\mathrm{D}f~\!</math><br />
| <math>\mathrm{d}f~\!</math><br />
|-<br />
| <math>f_{0}\!</math><br />
| style="border-right:none" | <math>\texttt{(~)}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x) ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\\<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\\<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\\<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\partial x<br />
\\<br />
\partial x<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
\\<br />
\mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)} & + & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\partial x & + & \partial y<br />
\\<br />
\partial x & + & \partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y & + & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\partial y<br />
\\<br />
\partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
x & \texttt{(} \mathrm{d}x) ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
y & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{(} x \texttt{,~} y \texttt{)} & \mathrm{d}x ~ \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} & \mathrm{d}x ~ \texttt{(} \mathrm{d}y \texttt{)}<br />
& + &<br />
\texttt{(} x \texttt{)} & \texttt{(} \mathrm{d}x \texttt{)} ~ \mathrm{d}y<br />
& + &<br />
\texttt{((} x \texttt{,~} y \texttt{))} & \mathrm{d}x ~ \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\\<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{~} x \texttt{~} ~\partial y<br />
\\<br />
\texttt{~} y \texttt{~} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\\<br />
\texttt{(} y \texttt{)} ~\partial x<br />
& + &<br />
\texttt{(} x \texttt{)} ~\partial y<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| style="border-right:none" | <math>\texttt{((~))}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A8. Differential Forms Expanded on an Algebraic Basis====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A8.} ~~ \text{Differential Forms Expanded on an Algebraic Basis}\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| &nbsp;<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" | <math>\mathrm{D}f~\!</math><br />
| <math>\mathrm{d}f~\!</math><br />
|-<br />
| <math>f_{0}\!</math><br />
| style="border-right:none" | <math>\texttt{(~)}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x<br />
\\<br />
\mathrm{d}x<br />
\end{matrix}\!</math><br />
| <math>\begin{matrix}<br />
\mathrm{d}x<br />
\\<br />
\mathrm{d}x<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\\<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\\<br />
\mathrm{d}x & + & \mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}y<br />
\\<br />
\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y<br />
\\<br />
\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-right:none" |<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y & + & \mathrm{d}x~\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{~} x \texttt{~}~\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)}~\mathrm{d}x & + & \texttt{(} x \texttt{)}~\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| style="border-right:none" | <math>\texttt{((~))}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A9. Tangent Proposition as Pointwise Linear Approximation====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A9.} ~~ \text{Tangent Proposition}~ \mathrm{d}f = \text{Pointwise Linear Approximation to the Difference Map}~ \mathrm{D}f\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{d}f =<br />
\\[2pt]<br />
\partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}^2\!f =<br />
\\[2pt]<br />
\partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\mathrm{d}f|_{x \, y}</math><br />
| <math>\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math><br />
|-<br />
| style="border-right:none" | <math>f_0\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{5}\\f_{10}\end{matrix}\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}\!</math><br />
|-<br />
| style="border-right:none" |<br />
<math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\end{matrix}\!</math><br />
| <math>\begin{matrix}<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>f_{15}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A10. Taylor Series Expansion Df = d''f'' + d<sup>2</sup>''f''====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" |<br />
<math>\text{Table A10.} ~~ \text{Taylor Series Expansion}~ {\mathrm{D}f = \mathrm{d}f + \mathrm{d}^2\!f}\!</math><br />
|- style="background:ghostwhite; height:40px"<br />
| style="border-right:none" | <math>f\!</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\mathrm{D}f<br />
\\<br />
= & \mathrm{d}f & + & \mathrm{d}^2\!f<br />
\\<br />
= & \partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y & + & \partial_{xy} f \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\mathrm{d}f|_{x \, y}</math><br />
| <math>\mathrm{d}f|_{x \, \texttt{(} y \texttt{)}}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)} \, y}</math><br />
| <math>\mathrm{d}f|_{\texttt{(} x \texttt{)(} y \texttt{)}}</math><br />
|-<br />
| style="border-right:none" | <math>f_0\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\\mathrm{d}x<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 0 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
| style="border-left:4px double black" |<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{~} x \texttt{~} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + &<br />
\texttt{(} x \texttt{)} \cdot \mathrm{d}y & + &<br />
\texttt{~} 1 \texttt{~} \cdot \mathrm{d}x\;\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y\\\mathrm{d}x\\0<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y\\0\\\mathrm{d}x<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}x\\0\\\mathrm{d}x + \mathrm{d}y\\\mathrm{d}y<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
0\\\mathrm{d}x\\\mathrm{d}y\\\mathrm{d}x + \mathrm{d}y<br />
\end{matrix}</math><br />
|-<br />
| style="border-right:none" | <math>f_{15}\!</math><br />
| style="border-left:4px double black" | <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A11. Partial Differentials and Relative Differentials====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="6" cellspacing="0" style="text-align:center; width:90%"<br />
|+ style="height:30px" | <math>\text{Table A11.} ~~ \text{Partial Differentials and Relative Differentials}\!</math><br />
|- style="background:ghostwhite; height:50px"<br />
| &nbsp;<br />
| <math>f\!</math><br />
| <math>\frac{\partial f}{\partial x}\!</math><br />
| <math>\frac{\partial f}{\partial y}\!</math><br />
|<br />
<math>\begin{matrix}<br />
\mathrm{d}f =<br />
\\[2pt]<br />
\partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\left. \frac{\partial x}{\partial y} \right| f\!</math><br />
| <math>\left. \frac{\partial y}{\partial x} \right| f\!</math><br />
|-<br />
| <math>f_0\!</math><br />
| <math>\texttt{(~)}\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|-<br />
| <math>\begin{matrix}f_{1}\\f_{2}\\f_{4}\\f_{8}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)(} y \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)~} y \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~(} y \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{3}\\f_{12}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x\\\mathrm{d}x\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{6}\\f_{9}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\<br />
\texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}x & + & \mathrm{d}y\\\mathrm{d}x & + & \mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{5}\\f_{10}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\end{matrix}</math><br />
| <math>\begin{matrix}0\\0\end{matrix}</math><br />
| <math>\begin{matrix}1\\1\end{matrix}</math><br />
| <math>\begin{matrix}\mathrm{d}y\\\mathrm{d}y\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>\begin{matrix}f_{7}\\f_{11}\\f_{13}\\f_{14}\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{(~} x \texttt{~~} y \texttt{~)}<br />
\\<br />
\texttt{(~} x \texttt{~(} y \texttt{))}<br />
\\<br />
\texttt{((} x \texttt{)~} y \texttt{~)}<br />
\\<br />
\texttt{((} x \texttt{)(} y \texttt{))}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\\<br />
\texttt{~} y \texttt{~}<br />
\\<br />
\texttt{(} y \texttt{)}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{~} x \texttt{~}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\\<br />
\texttt{(} x \texttt{)}<br />
\end{matrix}</math><br />
|<br />
<math>\begin{matrix}<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{~} x \texttt{~} \cdot \mathrm{d}y<br />
\\<br />
\texttt{~} y \texttt{~} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\\<br />
\texttt{(} y \texttt{)} \cdot \mathrm{d}x & + & \texttt{(} x \texttt{)} \cdot \mathrm{d}y<br />
\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
| <math>\begin{matrix}\cdots\\\cdots\\\cdots\\\cdots\end{matrix}</math><br />
|-<br />
| <math>f_{15}\!</math><br />
| <math>\texttt{((~))}\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
| <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
====Table A12. Detail of Calculation for the Difference Map====<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:4px double black; border-left:4px double black; border-right:4px double black; border-top:4px double black; text-align:center; width:80%"<br />
|+ style="height:30px" | <math>\text{Table A12.} ~~ \text{Detail of Calculation for}~ {\mathrm{E}f + f = \mathrm{D}f}\!</math><br />
|- style="background:ghostwhite"<br />
| style="width:6%" | &nbsp;<br />
| style="width:14%; border-left:1px solid black" | <math>f\!</math><br />
| style="width:20%; border-left:4px double black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\mathrm{d}x ~ \mathrm{d}y}<br />
\\[4pt]<br />
+ & f|_{\mathrm{d}x ~ \mathrm{d}y}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\mathrm{d}x ~ \mathrm{d}y}<br />
\end{array}</math><br />
| style="width:20%; border-left:1px solid black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}<br />
\\[4pt]<br />
+ & f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)} \mathrm{d}y}<br />
\end{array}</math><br />
| style="width:20%; border-left:1px solid black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
+ & f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\mathrm{d}x \texttt{(} \mathrm{d}y \texttt{)}}<br />
\end{array}</math><br />
| style="width:20%; border-left:1px solid black" |<br />
<math>\begin{array}{cr}<br />
~ & \mathrm{E}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
+ & f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}<br />
\\[4pt]<br />
= & \mathrm{D}f|_{\texttt{(} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{)}}<br />
\end{array}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{0}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0\!</math><br />
| style="border-top:4px double black; border-left:4px double black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>0 ~+~ 0 ~=~ 0\!</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{1}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{)(} y \texttt{)~}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{2}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{)~} y \texttt{~~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{4}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~~} x \texttt{~(} y \texttt{)~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{8}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~~} x \texttt{~~} y \texttt{~~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{)~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~(} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
+ & \texttt{~~} x \texttt{~~} y \texttt{~~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{3}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{(} x \texttt{)}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{12}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>x\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} x \texttt{)}<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & x<br />
\\[4pt]<br />
+ & x<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{6}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{,~} y \texttt{)~}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{9}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} x \texttt{,~} y \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{5}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{(} y \texttt{)}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{10}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>y\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{(} y \texttt{)}<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 1<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & y<br />
\\[4pt]<br />
+ & y<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{7}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{~~} y \texttt{)~}\!</math><br />
| style="border-top:4px double black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:4px double black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{11}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{~(} x \texttt{~(} y \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~~} x \texttt{~~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{13}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} x \texttt{)~} y \texttt{)~}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{,~} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~~} y \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:1px solid black" | <math>f_{14}\!</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\texttt{((} x \texttt{)(} y \texttt{))}\!</math><br />
| style="border-top:1px solid black; border-left:4px double black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{((} x \texttt{,~} y \texttt{))}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{~(} x \texttt{~(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~~} ~ \texttt{~(} y \texttt{)~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)~} y \texttt{)~}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & \texttt{~(} x \texttt{)~} ~ \texttt{~~}<br />
\end{matrix}</math><br />
| style="border-top:1px solid black; border-left:1px solid black" |<br />
<math>\begin{matrix}<br />
~ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
+ & \texttt{((} x \texttt{)(} y \texttt{))}<br />
\\[4pt]<br />
= & 0<br />
\end{matrix}</math><br />
|-<br />
| style="border-top:4px double black" | <math>f_{15}\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1\!</math><br />
| style="border-top:4px double black; border-left:4px double black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
| style="border-top:4px double black; border-left:1px solid black" | <math>1 ~+~ 1 ~=~ 0\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Appendix 3. Computational Details===<br />
<br />
====Operator Maps for the Logical Conjunction ''f''<sub>8</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{8}~\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{8}<br />
& = && f_{8}(u, v)<br />
\\[4pt]<br />
& = && uv<br />
\\[4pt]<br />
& = && uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & uv \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & uv \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{8}<br />
& = && uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & uv \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & uv \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & uv \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.2-i} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{E}f_{8}<br />
& = & f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{8}(\mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{8}(\mathrm{d}u, \mathrm{d}v)<br />
\\[4pt]<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[20pt]<br />
\mathrm{E}f_{8}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
\\[4pt]<br />
&&&&& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&&&&&&& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.2-ii} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{c}}<br />
\mathrm{E}f_{8}<br />
& = & (u + \mathrm{d}u) \cdot (v + \mathrm{d}v)<br />
\\[6pt]<br />
& = & u \cdot v<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u \cdot \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}f_{8}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{8}<br />
& = && \mathrm{E}f_{8}<br />
& + & \boldsymbol\varepsilon f_{8}<br />
\\[4pt]<br />
& = && f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{8}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & uv<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{~}<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}f_{8}<br />
& = & \boldsymbol\varepsilon f_{8}<br />
& + & \mathrm{E}f_{8}<br />
\\[6pt]<br />
& = & f_{8}(u, v)<br />
& + & f_{8}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[6pt]<br />
& = & uv<br />
& + & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
& = & 0<br />
& + & u \cdot \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u<br />
& + & \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}f_{8}<br />
& = & 0<br />
& + & u \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.3-iii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 3)}\!</math><br />
|<br />
<math>\begin{array}{c*{9}{l}}<br />
\mathrm{D}f_{8}<br />
& = & \boldsymbol\varepsilon f_{8} ~+~ \mathrm{E}f_{8}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{8}<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ u \,\cdot\, v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~ u \;\cdot\; v \;\cdot\; \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{E}f_{8}<br />
& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & u ~ \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} ~ v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot\, \mathrm{d}u ~ \mathrm{d}v<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = & ~ ~ 0 ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & ~ ~ u ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & ~ ~ ~ v ~~ \cdot ~ \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}\!</math><br />
|}<br />
<br />
=====Computation of d''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.4} ~~ \text{Computation of}~ \mathrm{d}f_{8}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{8}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>8</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.5} ~~ \text{Computation of}~ \mathrm{r}f_{8}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{8} & = & \mathrm{D}f_{8} ~+~ \mathrm{d}f_{8}<br />
\\[20pt]<br />
\mathrm{D}f_{8}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[20pt]<br />
\mathrm{r}f_{8}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Conjunction=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F8.6} ~~ \text{Computation Summary for}~ f_{8}(u, v) = uv\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{8}<br />
& = & uv \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{D}f_{8}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\\[6pt]<br />
\mathrm{d}f_{8}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{r}f_{8}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Operator Maps for the Logical Equality ''f''<sub>9</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{9}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{9}<br />
& = && f_{9}(u, v)<br />
\\[4pt]<br />
& = && \texttt{((} u \texttt{,~} v \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{9}(1, 1)<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{9}(1, 0)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{9}(0, 1)<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{9}(0, 0)<br />
\\[4pt]<br />
& = && u v & + & 0 & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{9}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.2} ~~ \text{Computation of}~ \mathrm{E}f_{9}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{E}f_{9}<br />
& = && f_{9}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! f_{9}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! f_{9}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{ } \mathrm{d}v \texttt{ })<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! f_{9}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! f_{9}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{ } \mathrm{d}v \texttt{ })<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{ (} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{) }<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{ (} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{) }<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{E}f_{9}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{9}<br />
& = && \mathrm{E}f_{9}<br />
& + & \boldsymbol\varepsilon f_{9}<br />
\\[4pt]<br />
& = && f_{9}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{9}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))}<br />
& + & \texttt{((} u \texttt{,} v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{9}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[20pt]<br />
\mathrm{D}f_{9}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}f_{9}<br />
& = & 0 \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & 1 \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & 1 \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & 0 \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of d''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.4} ~~ \text{Computation of}~ \mathrm{d}f_{9}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>9</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.5} ~~ \text{Computation of}~ \mathrm{r}f_{9}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{9} & = & \mathrm{D}f_{9} ~+~ \mathrm{d}f_{9}<br />
\\[20pt]<br />
\mathrm{D}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[20pt]<br />
\mathrm{r}f_{9}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Equality=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F9.6} ~~ \text{Computation Summary for}~ f_{9}(u, v) = \texttt{((} u \texttt{,} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{9}<br />
& = & uv \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}f_{9}<br />
& = & uv \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f_{9}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{9}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}f_{9}<br />
& = & uv \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Operator Maps for the Logical Implication ''f''<sub>11</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{11}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{11}<br />
& = && f_{11}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(} u \texttt{(} v \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{11}(1, 1)<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{11}(1, 0)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{11}(0, 1)<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{11}(0, 0)<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ }<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ }<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{11}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}\!</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.2} ~~ \text{Computation of}~ \mathrm{E}f_{11}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{E}f_{11}<br />
& = && f_{11}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = &&<br />
\texttt{(}<br />
\\<br />
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\\<br />
&&& \texttt{(}<br />
\\<br />
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\<br />
&&& \texttt{))}<br />
\\[4pt]<br />
& = &&<br />
u v<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)((} \mathrm{d}v \texttt{)))}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u \texttt{((} \mathrm{d}v \texttt{)))}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[4pt]<br />
& = &&<br />
u v<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\!\cdot\!<br />
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{E}f_{11}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{11}<br />
& = && \mathrm{E}f_{11}<br />
& + & \boldsymbol\varepsilon f_{11}<br />
\\[4pt]<br />
& = && f_{11}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{11}(u, v)<br />
\\[4pt]<br />
& = &&<br />
\texttt{(} \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\texttt{(} \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{(} v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & u v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & 0<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & 0<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = && u v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{c*{9}{l}}<br />
\mathrm{D}f_{11}<br />
& = & \boldsymbol\varepsilon f_{11} ~+~ \mathrm{E}f_{11}<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{11}<br />
& = & u v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}f_{11}<br />
& = &<br />
u v<br />
\cdot<br />
\texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\cdot<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\cdot<br />
\texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\cdot<br />
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = &<br />
u v<br />
\cdot<br />
\texttt{~(} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{~}<br />
& + &<br />
u \texttt{(} v \texttt{)}<br />
\cdot<br />
\texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + &<br />
\texttt{(} u \texttt{)} v<br />
\cdot<br />
\texttt{~} \mathrm{d}u ~ \mathrm{d}v \texttt{~}<br />
& + &<br />
\texttt{(} u \texttt{)(} v \texttt{)}<br />
\cdot<br />
\texttt{~} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of d''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.4} ~~ \text{Computation of}~ \mathrm{d}f_{11}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{11}<br />
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{11}<br />
& = & u v \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>11</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.5} ~~ \text{Computation of}~ \mathrm{r}f_{11}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{11} & = & \mathrm{D}f_{11} ~+~ \mathrm{d}f_{11}<br />
\\[20pt]<br />
\mathrm{D}f_{11}<br />
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{11}<br />
& = & u v \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u<br />
\\[20pt]<br />
\mathrm{r}f_{11}<br />
& = & u v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Implication=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F11.6} ~~ \text{Computation Summary for}~ f_{11}(u, v) = \texttt{(} u \texttt{(} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{11}<br />
& = & u v \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 0<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1<br />
\\[6pt]<br />
\mathrm{E}f_{11}<br />
& = & u v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f_{11}<br />
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{d}f_{11}<br />
& = & u v \cdot \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u<br />
\\[6pt]<br />
\mathrm{r}f_{11}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
====Operator Maps for the Logical Disjunction ''f''<sub>14</sub>(u, v)====<br />
<br />
=====Computation of &epsilon;''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{14}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\boldsymbol\varepsilon f_{14}<br />
& = && f_{14}(u, v)<br />
\\[4pt]<br />
& = && \texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{14}(1, 1)<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{14}(1, 0)<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{14}(0, 1)<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{14}(0, 0)<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ }<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ }<br />
& + & 0<br />
\\[20pt]<br />
\boldsymbol\varepsilon f_{14}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of E''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.2} ~~ \text{Computation of}~ \mathrm{E}f_{14}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{E}f_{14}<br />
& = && f_{14}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
\\[4pt]<br />
& = &&<br />
\texttt{((}<br />
\\<br />
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}<br />
\\<br />
&&& \texttt{)(}<br />
\\<br />
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}<br />
\\<br />
&&& \texttt{))}<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! f_{14}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! f_{14}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{ } \mathrm{d}v \texttt{ })<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! f_{14}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{(} \mathrm{d}v \texttt{)})<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! f_{14}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{ } \mathrm{d}v \texttt{ })<br />
\\[4pt]<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[20pt]<br />
\mathrm{E}f_{14}<br />
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~}<br />
\\[4pt]<br />
&& + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)}<br />
\\[4pt]<br />
&& + & 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of D''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 1)}\!</math><br />
|<br />
<math>\begin{array}{*{10}{l}}<br />
\mathrm{D}f_{14}<br />
& = && \mathrm{E}f_{14}<br />
& + & \boldsymbol\varepsilon f_{14}<br />
\\[4pt]<br />
& = && f_{14}(u + \mathrm{d}u, v + \mathrm{d}v)<br />
& + & f_{14}(u, v)<br />
\\[4pt]<br />
& = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{))((} v \texttt{,} \mathrm{d}v \texttt{)))}<br />
& + & \texttt{((} u \texttt{)(} v \texttt{))}<br />
\\[20pt]<br />
\mathrm{D}f_{14}<br />
& = && 0<br />
& + & 0<br />
& + & 0<br />
& + & 0<br />
\\[4pt]<br />
&& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}<br />
\\[4pt]<br />
&& + & 0<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}<br />
\\[4pt]<br />
&& + & uv \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & 0<br />
& + & 0<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}<br />
\\[20pt]<br />
\mathrm{D}f_{14}<br />
& = && uv \!\cdot\! \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 2)}\!</math><br />
|<br />
<math>\begin{array}{*{9}{l}}<br />
\mathrm{D}f_{14}<br />
& = & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & 0 \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of d''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.4} ~~ \text{Computation of}~ \mathrm{d}f_{14}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{D}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\Downarrow<br />
\\[6pt]<br />
\mathrm{d}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation of r''f''<sub>14</sub>=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.5} ~~ \text{Computation of}~ \mathrm{r}f_{14}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\mathrm{r}f_{14} & = & \mathrm{D}f_{14} ~+~ \mathrm{d}f_{14}<br />
\\[20pt]<br />
\mathrm{D}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{d}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[20pt]<br />
\mathrm{r}f_{14}<br />
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
=====Computation Summary for Disjunction=====<br />
<br />
<br><br />
<br />
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"<br />
|+ style="height:30px" | <math>\text{Table F14.6} ~~ \text{Computation Summary for}~ f_{14}(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math><br />
|<br />
<math>\begin{array}{c*{8}{l}}<br />
\boldsymbol\varepsilon f_{14}<br />
& = & uv \cdot 1<br />
& + & u \texttt{(} v \texttt{)} \cdot 1<br />
& + & \texttt{(} u \texttt{)} v \cdot 1<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0<br />
\\[6pt]<br />
\mathrm{E}f_{14}<br />
& = & uv \cdot \texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}<br />
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{D}f_{14}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}<br />
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}<br />
\\[6pt]<br />
\mathrm{d}f_{14}<br />
& = & uv \cdot 0<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}<br />
\\[6pt]<br />
\mathrm{r}f_{14}<br />
& = & uv \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v<br />
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v<br />
\end{array}</math><br />
|}<br />
<br />
<br><br />
<br />
===Appendix 4. Source Materials===<br />
<br />
===Appendix 5. Various Definitions of the Tangent Vector===<br />
<br />
==References==<br />
<br />
===Works Cited===<br />
<br />
{| cellpadding=3<br />
| valign=top | [AuM]<br />
| Auslander, L., and MacKenzie, R.E., ''Introduction to Differentiable Manifolds'', McGraw-Hill, 1963. Reprinted, Dover, New York, NY, 1977.<br />
|-<br />
| valign=top | [BiG]<br />
| Bishop, R.L., and Goldberg, S.I., ''Tensor Analysis on Manifolds'', Macmillan, 1968. Reprinted, Dover, New York, NY, 1980.<br />
|-<br />
| valign=top | [Boo]<br />
| Boole, G., ''An Investigation of The Laws of Thought'', Macmillan, 1854. Reprinted, Dover, New York, NY, 1958.<br />
|-<br />
| valign=top | [BoT]<br />
| Bott, R., and Tu, L.W., ''Differential Forms in Algebraic Topology'', Springer-Verlag, New York, NY, 1982.<br />
|-<br />
| valign=top | [dCa]<br />
| do Carmo, M.P., ''Riemannian Geometry''. Originally published in Portuguese, 1st editiom 1979, 2nd edition 1988. Translated by F. Flaherty, Birkhäuser, Boston, MA, 1992.<br />
|-<br />
| valign=top | [Che46]<br />
| Chevalley, C., ''Theory of Lie Groups'', Princeton University Press, Princeton, NJ, 1946.<br />
|-<br />
| valign=top | [Che56]<br />
| Chevalley, C., ''Fundamental Concepts of Algebra'', Academic Press, 1956.<br />
|-<br />
| valign=top | [Cho86]<br />
| Chomsky, N., ''Knowledge of Language : Its Nature, Origin, and Use'', Praeger, New York, NY, 1986.<br />
|-<br />
| valign=top | [Cho93]<br />
| Chomsky, N., ''Language and Thought'', Moyer Bell, Wakefield, RI, 1993.<br />
|-<br />
| valign=top | [DoM]<br />
| Doolin, B.F., and Martin, C.F., ''Introduction to Differential Geometry for Engineers'', Marcel Dekker, New York, NY, 1990.<br />
|-<br />
| valign=top | [Fuji]<br />
| Fujiwara, H., ''Logic Testing and Design for Testability'', MIT Press, Cambridge, MA, 1985.<br />
|-<br />
| valign=top | [Hic]<br />
| Hicks, N.J., ''Notes on Differential Geometry'', Van Nostrand, Princeton, NJ, 1965.<br />
|-<br />
| valign=top | [Hir]<br />
| Hirsch, M.W., ''Differential Topology'', Springer-Verlag, New York, NY, 1976.<br />
|-<br />
| valign=top | [How]<br />
| Howard, W.A., "The Formulae-as-Types Notion of Construction", Notes circulated from 1969. Reprinted in [SeH, 479-490].<br />
|-<br />
| valign=top | [JGH]<br />
| Jones, A., Gray, A., and Hutton, R., ''Manifolds and Mechanics'', Cambridge University Press, Cambridge, UK, 1987.<br />
|-<br />
| valign=top | [KoA]<br />
| Kosinski, A.A., ''Differential Manifolds'', Academic Press, San Diego, CA, 1993.<br />
|-<br />
| valign=top | [Koh]<br />
| Kohavi, Z., ''Switching and Finite Automata Theory'', 2nd edition, McGraw-Hill, New York, NY, 1978.<br />
|-<br />
| valign=top | [LaS]<br />
| Lambek, J., and Scott, P.J., ''Introduction to Higher Order Categorical Logic'', Cambridge University Press, Cambridge, UK, 1986.<br />
|-<br />
| valign=top | [La83]<br />
| Lang, S., ''Real Analysis'', 2nd edition, Addison-Wesley, Reading, MA, 1983.<br />
|-<br />
| valign=top | [La84]<br />
| Lang, S., ''Algebra'', 2nd edition, Addison-Wesley, Menlo Park, CA, 1984.<br />
|-<br />
| valign=top | [La85]<br />
| Lang, S., ''Differential Manifolds'', Springer-Verlag, New York, NY, 1985.<br />
|-<br />
| valign=top | [La93]<br />
| Lang, S., ''Real and Functional Analysis'', 3rd edition, Springer-Verlag, New York, NY, 1993.<br />
|-<br />
| valign=top | [Lie80]<br />
| Lie, S., "Sophus Lie's 1880 Transformation Group Paper", in ''Lie Groups : History, Frontiers, and Applications, Volume 1'', translated by M. Ackerman, comments by R. Hermann, Math Sci Press, Brookline, MA, 1975. Original paper 1880.<br />
|-<br />
| valign=top | [Lie84]<br />
| Lie, S., "Sophus Lie's 1884 Differential Invariant Paper", in ''Lie Groups : History, Frontiers, and Applications, Volume 3'', translated by M. Ackerman, comments by R. Hermann, Math Sci Press, Brookline, MA, 1976. Original paper 1884.<br />
|-<br />
| valign=top | [LoS]<br />
| Loomis, L.H., and Sternberg, S., ''Advanced Calculus'', Addison-Wesley, Reading, MA, 1968.<br />
|-<br />
| valign=top | [Mel]<br />
| Melzak, Z.A., ''Companion to Concrete Mathematics, Volume 2 : Mathematical Ideas, Modeling, and Applications'', John Wiley amd Sons, New York, NY, 1976.<br />
|-<br />
| valign=top | [Men]<br />
| Menabrea, L.F., "Sketch of the Analytical Engine Invented by Charles Babbage" with Notes by the Translator, Ada Augusta (Byron), Countess of Lovelace'', in [M&M, 225–297]. Originally published 1842.<br />
|-<br />
| valign=top | [M&M]<br />
| Morrison, P., and Morrison, E. (eds.), ''Charles Babbage on the Principles and Development of the Calculator, and Other Seminal Writings by Charles Babbage and Others, With an Introduction by the Editors'', Dover, Mineola, NY, 1961.<br />
|-<br />
| valign=top | [P1]<br />
| Peirce, C.S., ''Collected Papers of Charles Sanders Peirce'', vols. 1–8, C. Hartshorne, P. Weiss, and A.W. Burks (eds.), Harvard University Press, Cambridge, MA, 1931–1960. Cited as CP [volume].[paragraph].<br />
|-<br />
| valign=top | [P2]<br />
| Peirce, C.S., "Qualitative Logic", in ''The New Elements of Mathematics, Volume 4'', C. Eisele (ed.), Mouton, The Hague, 1976. Cited as NE [volume], [page].<br />
|-<br />
| valign=top | [Rob]<br />
| Roberts, D.D., ''The Existential Graphs of Charles S. Peirce'', Mouton, The Hague, 1973.<br />
|-<br />
| valign=top | [SeH]<br />
| Seldin, J.P., and Hindley, J.R. (eds.), ''To H.B. Curry : Essays on Combinatory Logic, Lambda Calculus, and Formalism'', Academic Press, London, UK, 1980.<br />
|-<br />
| valign=top | [SpB]<br />
| Spencer-Brown, G., ''Laws of Form'', George Allen and Unwin, London, UK, 1969.<br />
|-<br />
| valign=top | [Sp65]<br />
| Spivak, M., ''Calculus on Manifolds : A Modern Approach to Classical Theorems of Advanced Calculus'', W.A. Benjamin, New York, NY, 1965.<br />
|-<br />
| valign=top | [Sp79]<br />
| Spivak, M., ''A Comprehensive Introduction to Differential Geometry'', vols. 1–2. 1st edition 1970. 2nd edition, Publish or Perish Inc., Houston, TX, 1979.<br />
|-<br />
| valign=top | [Sty]<br />
| Styazhkin, N.I., ''History of Mathematical Logic from Leibniz to Peano'', 1st published in Russian, Nauka, Moscow, 1964. MIT Press, Cambridge, MA, 1969.<br />
|-<br />
| valign=top | [Wie]<br />
| Wiener, N., ''Cybernetics : or Control and Communication in the Animal and the Machine'', 1st edition 1948. 2nd edition, MIT Press, Cambridge, MA, 1961.<br />
|}<br />
<br />
===Works Consulted===<br />
<br />
{| cellpadding=3<br />
| valign=top | [Ami]<br />
| Amit, D.J., ''Modeling Brain Function : The World of Attractor Neural Networks'', Cambridge University Press, Cambridge, UK, 1989.<br />
|-<br />
| valign=top | [Ed87]<br />
| Edelman, G.M., ''Neural Darwinism : The Theory of Neuronal Group Selection'', Basic Books, New York, NY, 1987.<br />
|-<br />
| valign=top | [Ed88]<br />
| Edelman, G.M., ''Topobiology : An Introduction to Molecular Embryology'', Basic Books, New York, NY, 1988.<br />
|-<br />
| valign=top | [Fla]<br />
| Flanders, H., ''Differential Forms with Applications to the Physical Sciences'', Academic Press, 1963. Reprinted, Dover, Mineola, NY, 1989. <br />
|-<br />
| valign=top | [Has]<br />
| Hassoun, M.H. (ed.), ''Associative Neural Memories : Theory and Implementation'', Oxford University Press, New York, NY, 1993.<br />
|-<br />
| valign=top | [KoB]<br />
| Kosko, B., ''Neural Networks and Fuzzy Systems : A Dynamical Systems Approach to Machine Intelligence'', Prentice-Hall, Englewood Cliffs, NJ, 1992.<br />
|-<br />
| valign=top | [MaB]<br />
| Mac Lane, S., and Birkhoff, G., ''Algebra'', 3rd edition, Chelsea, New York, NY, 1993.<br />
|-<br />
| valign=top | [Mac]<br />
| Mac Lane, S., ''Categories for the Working Mathematician'', Springer-Verlag, New York, NY, 1971.<br />
|-<br />
| valign=top | [McC]<br />
| McCulloch, W.S., ''Embodiments of Mind'', MIT Press, Cambridge, MA, 1965.<br />
|-<br />
| valign=top | [Mc1]<br />
| McCulloch, W.S., "A Heterarchy of Values Determined by the Topology of Nervous Nets", Bulletin of Mathematical Biophysics, vol. 7 (1945), pp. 89–93. Reprinted in [McC].<br />
|-<br />
| valign=top | [MiP]<br />
| Minsky, M.L., and Papert, S.A., ''Perceptrons : An Introduction to Computational Geometry'', MIT Press, Cambridge, MA, 1969. 2nd printing 1972. Expanded edition 1988.<br />
|-<br />
| valign=top | [Rum]<br />
| Rumelhart, D.E., Hinton, G.E., and McClelland, J.L., "A General Framework for Parallel Distributed Processing" = Chapter 2 in Rumelhart, McClelland, and the PDP Research Group, ''Parallel Distributed Processing, Explorations in the Microstructure of Cognition, Volume 1 : Foundations'', MIT Press, Cambridge, MA, 1986.<br />
|}<br />
<br />
===Incidental Works===<br />
<br />
{| cellpadding=3<br />
| valign=top | [Dew]<br />
| Dewey, John, ''How We Think'', D.C. Heath, Lexington, MA, 1910. Reprinted, Prometheus Books, Buffalo, NY, 1991.<br />
|-<br />
| valign=top | [Fou]<br />
| Foucault, Michel, ''The Archaeology of Knowledge and The Discourse on Language'', A.M. Sheridan-Smith and Rupert Swyer (trans.), Pantheon, New York, NY, 1972. Originally published as ''L´Archéologie du Savoir et L´ordre du discours'', Editions Gallimard, 1969 & 1971.<br />
|-<br />
| valign=top | [Hom]<br />
| Homer, ''The Odyssey'', with an English translation by A.T. Murray, Loeb Classical Library, Harvard University Press, Cambridge, MA, 1980. First printed 1919.<br />
|-<br />
| valign=top | [Jam]<br />
| James, William, ''Pragmatism : A New Name for Some Old Ways of Thinking'', Longmans, Green, and Company, New York, NY, 1907.<br />
|-<br />
| valign=top | [Ler]<br />
| Leroux, Gaston, ''The Phantom of the Opera'', foreword by P. Haining, Dorset Press, New York, NY, 1988. Originally published in French, 1911.<br />
|-<br />
| valign=top | [Mus]<br />
| Musil, Robert, ''The Man Without Qualities'', 3 volumes, translated with a foreword by Eithne Wilkins and Ernst Kaiser, Pan Books, London, UK, 1979. English edition first published by Secker and Warburg, 1954. Originally published in German, ''Der Mann ohne Eigenschaften'', 1930 & 1932.<br />
|-<br />
| valign=top | [PlaR]<br />
| Plato, ''The Republic'', with an English translation by Paul Shorey, Loeb Classical Library, Harvard University Press, Cambridge, MA, 1980. First printed 1930 & 1935.<br />
|-<br />
| valign=top | [PlaS]<br />
| Plato, ''The Sophist'', Loeb Classical Library, William Heinemann, London, 1921, 1987.<br />
|-<br />
| valign=top | [Qui]<br />
| Quine, W.V., ''Mathematical Logic'', 1st edition, 1940. Revised edition, 1951. Harvard University Press, Cambridge, MA, 1981.<br />
|-<br />
| valign=top | [SaD]<br />
| de Santillana, Giorgio, and von Dechend, Hertha, ''Hamlet's Mill : An Essay on Myth and the Frame of Time'', David R. Godine, Publisher, Boston, MA, 1977. 1st published 1969.<br />
|-<br />
| valign=top | [Sha]<br />
| Shakespeare, William, '' William Shakespeare : The Complete Works'', Compact Edition, S. Wells and G. Taylor (eds.), Oxford University Press, Oxford, UK, 1988.<br />
|-<br />
| valign=top | [Sh1]<br />
| Shakespeare, William, ''A Midsummer Night's Dream'', Washington Square Press, New York, NY, 1958.<br />
|-<br />
| valign=top | [Sh2]<br />
| Shakespeare, William, ''The Tragedy of Hamlet, Prince of Denmark'', In [Sha], pp. 654&ndash;690.<br />
|-<br />
| valign=top | [Sh3]<br />
| Shakespeare, William, ''Measure for Measure'', Washington Square Press, New York, NY, 1965.<br />
|-<br />
| valign=top | [Web]<br />
| ''Webster's Ninth New Collegiate Dictionary'', Merriam-Webster, Springfield, MA, 1983.<br />
|-<br />
| valign=top | [Whi]<br />
| Whitman, Walt, ''Leaves of Grass'', Vintage Books / The Library of America, New York, NY, 1992. Originally published in numerous editions, 1855&ndash;1892.<br />
|-<br />
| valign=top | [Wil]<br />
| Wilhelm, R., and Baynes, C.F. (trans.), ''The I Ching, or Book of Changes'', foreword by C.G. Jung, preface by H. Wilhelm, 3rd edition, Bollingen Series XIX, Princeton University Press, Princeton, NJ, 1967.<br />
|}<br />
<br />
==Document History==<br />
<br />
<pre><br />
Author: Jon Awbrey<br />
Created: 16 Dec 1993<br />
Relayed: 31 Oct 1994<br />
Revised: 03 Jun 2003<br />
Recoded: 03 Jun 2007<br />
</pre><br />
<br />
[[Category:Adaptive Systems]]<br />
[[Category:Artificial Intelligence]]<br />
[[Category:Boolean Algebra]]<br />
[[Category:Boolean Functions]]<br />
[[Category:Charles Sanders Peirce]]<br />
[[Category:Combinatorics]]<br />
[[Category:Computer Science]]<br />
[[Category:Cybernetics]]<br />
[[Category:Differential Logic]]<br />
[[Category:Discrete Systems]]<br />
[[Category:Dynamical Systems]]<br />
[[Category:Formal Languages]]<br />
[[Category:Formal Sciences]]<br />
[[Category:Formal Systems]]<br />
[[Category:Functional Logic]]<br />
[[Category:Graph Theory]]<br />
[[Category:Group Theory]]<br />
[[Category:Inquiry]]<br />
[[Category:Knowledge Representation]]<br />
[[Category:Linguistics]]<br />
[[Category:Logic]]<br />
[[Category:Logical Graphs]]<br />
[[Category:Mathematics]]<br />
[[Category:Mathematical Systems Theory]]<br />
[[Category:Philosophy]]<br />
[[Category:Science]]<br />
[[Category:Semiotics]]<br />
[[Category:Systems Science]]<br />
[[Category:Visualization]]<br />
<br />
test</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Directory:Jon_Awbrey/Essays/Inquiry_Driven_Systems_:_Fields_Of_Inquiry&diff=469868Directory:Jon Awbrey/Essays/Inquiry Driven Systems : Fields Of Inquiry2020-12-26T15:54:19Z<p>Jon Awbrey: parse test</p>
<hr />
<div>{{DISPLAYTITLE:Inquiry Driven Systems : Fields Of Inquiry}}<br />
'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''<br />
<br />
{| align="center" width="90%"<br />
| align="left" |<br />
The field denotes<br><br />
this body, and wise men<br><br />
call one who knows it<br><br />
the field-knower.<br />
| align="right" valign="bottom" | &mdash; ''Bhagavad Gita'', 13.1<br />
|}<br />
<br />
==Introduction : Review and Transition==<br />
<br />
{| align="center" width="90%"<br />
| align="left" |<br />
Know me as the field-knower<br><br />
in all fields &mdash; what I deem<br><br />
to be knowledge is knowledge<br><br />
of the field and its knower.<br />
| align="right" valign="bottom" | &mdash; ''Bhagavad Gita,'' 13.2<br />
|}<br />
<br />
In this exposition the character of a logical expansion or analytic form is seen to correspond to a particular perspective on a universe of discourse, a specialized way of viewing its propositions and organizing their interpretations into comprehensible forms.<br />
<br />
In this essay we study the use of analytic forms both in the ordinary expansions and the differential analysis of propositions. The process of logical expansion is formalized in greater detail and, to compensate for the extra level of abstraction, the resulting analytic forms are motivated on intuitive lines as variant perspectives or alternate ways of viewing the situations represented in propositions. By the end of this ascent we hope the reader finds it pleasing to contemplate the whole panorama of differential operations and group actions on a proposition as just so many facets of its differential enlargement which arise from its projection onto complementary perspectives.<br />
<br />
===Proposition Fields===<br />
<br />
{| align="center" width="90%"<br />
| align="left" |<br />
Hear from me in summary<br><br />
what the field is<br><br />
in its character and changes,<br><br />
and of the field-knower's power.<br />
| align="right" valign="bottom" | &mdash; ''Bhagavad Gita,'' 13.3<br />
|}<br />
<br />
This section introduces the class of mathematical objects known as proposition fields. These are structures which arise in a natural way from the analysis of information systems, especially when it comes to systems with &ldquo;reflective&rdquo; intelligence, those which maintain and process components of information about their own states. A system of this kind has its dynamics best understood only if we interpret some of its dimensions of variation as reflecting information about other components of state, and further, only if we assume that the system itself &ldquo;interprets&rdquo; these variables as being informative in just this way.<br />
<br />
A notion of intelligence has just been made to depend on a notion of interpretation. Both issues are to be side-stepped here, but we leave this note as a pointer to future work. To make these ideas explicit, we ought to say what it means for a system to be an interpreter of some qualities of its state as signs for other qualities of its state. But understanding how this is possible is tantamount to a major objective of this whole study, and can only be developed over the course of our investigation. It may not seem like good mathematical practice to let the clearing up of definitions fall to the end, but that is how it sometimes has to be.<br />
<br />
A ''proposition field'' is a mapping from points to propositions, that is, it is a function having the general form <math>U \to (V \to \mathbb{B}).\!</math> We may visualize the proposition field as &ldquo;tagging&rdquo; each point <math>x\!</math> in one universe of discourse <math>U\!</math> with a determinate proposition <math>f_x : V \to \mathbb{B}~\!</math> in another universe of discourse <math>V.\!</math><br />
<br />
Under the general type of a proposition field there are numerous special cases which frequently arise. Common species of proposition fields are listed in Table&nbsp;1, which displays next to the informal name of each an indication how its type is specialized.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" style="border-spacing:0; text-align:center; width:40%"<br />
|+ style="height:30px" | <math>\text{Table 1. Varieties of Proposition Fields}\!</math><br />
<br />
|- style="height:30px; background:ghostwhite"<br />
| style="border-top:0.75pt solid #000000; border-bottom:2.25pt double #000000; border-left:0.75pt solid #000000; border-right:none" | <math>\text{Name}\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:2.25pt double #000000; border-left:0.75pt solid #000000; border-right:0.75pt solid #000000" | <math>\text{Type}\!</math><br />
<br />
|-<br />
| style="border-top:2.25pt double #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\text{Field}\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:0.75pt solid #000000;" | <math>U \to (V \to \mathbb{B})\!</math><br />
<br />
|-<br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\text{Fold}\!</math><br />
| style="border:0.75pt solid #000000;" | <math>U \to (U \to \mathbb{B})\!</math><br />
<br />
|-<br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\text{File}\!</math><br />
| style="border:0.75pt solid #000000;" | <math>U \to (U^\prime \to \mathbb{B})\!</math><br />
<br />
|-<br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\text{Flow}\!</math><br />
| style="border:0.75pt solid #000000;" | <math>U \to (\mathrm{d}U \to \mathbb{B})\!</math><br />
<br />
|-<br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\text{Fray}\!</math><br />
| style="border:0.75pt solid #000000;" | <math>\mathrm{d}U \to (U \to \mathbb{B})\!</math><br />
|}<br />
<br />
<br><br />
<br />
In the midst of a particular application we often find ourselves taking a definite proposition <math>F\!</math> of a certain complexity and deriving from it a proposition field <math>\boldsymbol\iota F.\!</math> The proposition field <math>\boldsymbol\iota F\!</math> may be viewed as setting forth one of the many ways of comprehending <math>F\!</math> as an organized collection of simpler propositions, namely, by associating a unique proposition on a simpler space with each point of another simpler space.<br />
<br />
The proposition <math>F,\!</math> in turn, often arises as a tacit extension <math>\boldsymbol\varepsilon f\!</math> or a differential enlargement <math>\mathrm{E}f\!</math> of a prior proposition <math>f.\!</math> Table&nbsp;2 exhibits typical settings in which proposition fields are developed and applied, taking into consideration the kinds of fields noted above. In each case, composing the whole sequence of operations found in a given row of the Table, we define a transformation <math>T,\!</math> parameterized by the subscripted argument of the last evaluation step, which takes us from propositions in one universe of discourse to propositions in that same or other related universe.<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" style="border-spacing:0; text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 2. Development and Application of Proposition Fields}\!</math><br />
<br />
|- style="height:30px; background:ghostwhite"<br />
| style="border-top:0.75pt solid #000000; border-bottom:2.25pt double #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\text{Origin}~ f\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:2.25pt double #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\to~\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:2.25pt double #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\text{Extension}~ F~\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:2.25pt double #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\to~\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:2.25pt double #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\text{Factorization}~ \Gamma\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:2.25pt double #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\to~\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:2.25pt double #000000; border-left:0.75pt solid #000000; border-right:0.75pt solid #000000;" | <math>\text{Evaluation}~ g\!</math><br />
<br />
|-<br />
| style="border-top:2.25pt double #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f : U \to \mathbb{B}\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\boldsymbol\varepsilon\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\boldsymbol\varepsilon f : U \times V \to \mathbb{B}\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\boldsymbol\iota_1\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\Phi f : U \to (V \to \mathbb{B})\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\boldsymbol\varphi_u\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:0.75pt solid #000000;" | <math>T_u f : V \to \mathbb{B}\!</math><br />
<br />
|-<br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f : U \to \mathbb{B}\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\boldsymbol\varepsilon\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\boldsymbol\varepsilon f : U \times U \to \mathbb{B}\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\boldsymbol\iota_1\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\Phi f : U \to (U \to \mathbb{B})\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\boldsymbol\varphi_u\!</math><br />
| style="border:0.75pt solid #000000;" | <math>T_u f : U \to \mathbb{B}\!</math><br />
<br />
|-<br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f : U \to \mathbb{B}\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\boldsymbol\varepsilon\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\boldsymbol\varepsilon f : U \times U^\prime \to \mathbb{B}\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\boldsymbol\iota_1\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\Phi f : U \to (U^\prime \to \mathbb{B})\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\boldsymbol\varphi_u\!</math><br />
| style="border:0.75pt solid #000000;" | <math>T_u f : U^\prime \to \mathbb{B}\!</math><br />
<br />
|-<br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f : U \to \mathbb{B}\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\mathrm{E}\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\mathrm{E}f : U \times \mathrm{d}U \to \mathbb{B}\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\boldsymbol\iota_1\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\Phi f : U \to (\mathrm{d}U \to \mathbb{B})\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\boldsymbol\varphi_u\!</math><br />
| style="border:0.75pt solid #000000;" | <math>T_u f : \mathrm{d}U \to \mathbb{B}\!</math><br />
<br />
|-<br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f : U \to \mathbb{B}\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\mathrm{E}\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\mathrm{E}f : U \times \mathrm{d}U \to \mathbb{B}\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\boldsymbol\iota_2\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\Psi f : \mathrm{d}U \to (U \to \mathbb{B})\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\boldsymbol\varphi_v\!</math><br />
| style="border:0.75pt solid #000000;" | <math>T_v f : U \to \mathbb{B}\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Analytic Expansions===<br />
<br />
{| align="center" width="90%"<br />
| align="left" |<br />
Ancient seers have sung of this<br><br />
in many ways, with varied meters<br><br />
and with aphorisms on the infinite spirit<br><br />
laced with logical arguments.<br />
| align="right" valign="bottom" | &mdash; ''Bhagavad Gita,'' 13.4<br />
|}<br />
<br />
In &ldquo;Tools and Views&rdquo; we considered the analytic expansions of propositions with respect to various degrees of interpretation of their variables. The significance of these expansions for logical understanding is that they factor the process of interpretation between two stages, in application giving two moments to the clarification of a problematic proposition. In type, an expansion corresponds to an isomorphism,<br />
<br />
{| align="center" cellpadding="8"<br />
|<br />
<math>\begin{matrix}<br />
(\mathbb{B}^n \to \mathbb{B}) & \cong &<br />
(\mathbb{B}^j \times \mathbb{B}^{n-j} \to \mathbb{B})) & \cong &<br />
(\mathbb{B}^j \to (\mathbb{B}^{n-j} \to \mathbb{B})),<br />
\end{matrix}</math><br />
|}<br />
<br />
in which propositions <math>F : \mathbb{B}^n \to \mathbb{B}\!</math> are reconsidered as &ldquo;proposition fields&rdquo; <math>\boldsymbol\iota F : \mathbb{B}^j \to (\mathbb{B}^k \to \mathbb{B}),\!</math> where <math>n = j + k.\!</math> This gives rise to &ldquo;n choose j&rdquo; renditions of the isomorphism <math>\boldsymbol\iota\!</math> for each rendering of the original alphabet into a <math>j\!</math>-set and a <math>k\!</math>-set.<br />
<br />
In the conception of an individual analytic form the set of variables is divided into two parts and the sense of each proposition is apportioned accordingly. The initial set of variables is used to sweep out a range of partial interpretations and thus to set the context of analysis. The remaining set is left uninterpreted and kept embodied in propositional coefficients which are associated with the &ldquo;loci&rdquo; (the points or topics) of this conceptual framework.<br />
<br />
In appearance, the expression of the analytic form is made up of coefficient propositions on a subset of the variables taking their places next to partial interpretations over the complementary set. This may seem like a purely superficial change, since the analytic expansion preserves logical equivalence with the original proposition, but the good of a particular form for a particular situation lies precisely in the facility of its appearance being unaffected by its logical substance. Its choice and use are matters of pragmatic relevance and suitability to a purpose. This means that the worth of a particular form for a problematic situation can only be judged after the fact of its being applied, by the contribution it makes toward clarifying an opportune constellation of propositions, those which comprehend the situation and define the problem.<br />
<br />
If we need to formalize the process, each isomorphism <math>\boldsymbol\iota\!</math> is implicitly indexed by a subset <math>U\!</math> of the original alphabet <math>X,\!</math> but most often we consider only a single expansion at a time and can safely let context determine the sense.<br />
<br />
Let <math>X = \{ x_1, \ldots, x_m \}\!</math> be our principal set of logical variables, and let <math>X\!</math> be divided arbitrarily into a pair of subsets, which we may assume without loss of generality to form an initial segment <math>U = \{ x_1, \ldots, x_j \}\!</math> and a final segment <math>V = \{ x_{j+1}, \ldots, x_m \},\!</math> of cardinalities <math>j\!</math> and <math>k,\!</math> respectively, where <math>m = j + k.\!</math> Corresponding to the choice of <math>U\!</math> (which determines the complementary set <math>V\!</math>) there are a pair of isomorphisms,<br />
<br />
{| align="center" cellpadding="8"<br />
|<br />
<math>\begin{matrix}<br />
\boldsymbol\iota_U : (X \to \mathbb{B}) & = & (U \times V \to \mathbb{B}) & \to & (U \to (V \to \mathbb{B})),<br />
\\<br />
\boldsymbol\iota_V : (X \to \mathbb{B}) & = & (U \times V \to \mathbb{B}) & \to & (V \to (U \to \mathbb{B})),<br />
\end{matrix}</math><br />
|}<br />
<br />
whose employments we describe by saying that the proposition <math>F\!</math> can be factored into the proposition fields <math>\boldsymbol\iota_U F\!</math> and <math>\boldsymbol\iota_V F,\!</math> respectively. These factorizations are commonly expressed by means of the associated analytic forms, which determine a pair of logical expansions for each proposition <math>F\!</math> in the universe <math>X^\bullet,\!</math><br />
<br />
{| align="center" cellpadding="8"<br />
|<br />
<math>\begin{matrix}<br />
F(x) & = & F(u, v) & = & \sum_u \boldsymbol\varphi_u F(v) \cdot u,<br />
\\<br />
F(x) & = & F(u, v) & = & \sum_v \boldsymbol\varphi_v F(u) \cdot v.<br />
\end{matrix}</math><br />
|}<br />
<br />
===Elementary Examples===<br />
<br />
{| align="center" width="90%"<br />
| align="left" |<br />
The field contains the great elements,<br><br />
individuality, understanding,<br><br />
unmanifest nature, the eleven senses,<br><br />
and the five sense realms.<br />
| align="right" valign="bottom" | &mdash; ''Bhagavad Gita,'' 13.5<br />
|}<br />
<br />
A few examples of analytic expansions are appropriate here. The following selection exhibits the concepts and notation we need from previous discussions, indicates how we intend to adapt these materials to the present purpose, and provides a set of building blocks for future constructions.<br />
<br />
To begin with a minimal example, the conjunction <math>J(x, y) = x \cdot y\!</math> yields the partial expansion with respect to <math>x,\!</math><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lrrrr}<br />
J(x, y)<br />
& = & J(1, y) \cdot x<br />
& + & J(0, y) \cdot (x),<br />
\\<br />
& = & \boldsymbol\varphi_x J(y) \cdot x<br />
& + & \boldsymbol\varphi_{(x)} J(y) \cdot (x),<br />
\\<br />
& = & y \cdot x<br />
& + & 0 \cdot (x),<br />
\end{array}</math><br />
|}<br />
<br />
the partial expansion with respect to <math>y,\!</math><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lrrrr}<br />
J(x, y)<br />
& = & J(x, 1) \cdot y<br />
& + & J(x, 0) \cdot (y),<br />
\\<br />
& = & \boldsymbol\varphi_y J(x) \cdot y<br />
& + & \boldsymbol\varphi_{(y)} J(x) \cdot (y),<br />
\\<br />
& = & x \cdot y<br />
& + & 0 \cdot (y),<br />
\end{array}</math><br />
|}<br />
<br />
and the complete expansion with respect to <math>x\!</math> and <math>y,\!</math><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{l*{8}{r}}<br />
J(x, y)<br />
& = & J(1, 1) \cdot xy<br />
& + & J(1, 0) \cdot x(y)<br />
& + & J(0, 1) \cdot (x)y<br />
& + & J(0, 0) \cdot (x)(y),<br />
\\<br />
& = & \boldsymbol\varphi_{xy} J \cdot xy<br />
& + & \boldsymbol\varphi_{x(y)} J \cdot x(y)<br />
& + & \boldsymbol\varphi_{(x)y} J \cdot (x)y<br />
& + & \boldsymbol\varphi_{(x)(y)} J \cdot (x)(y),<br />
\\<br />
& = & 1 \cdot xy<br />
& + & 0 \cdot x(y)<br />
& + & 0 \cdot (x)y<br />
& + & 0 \cdot (x)(y).<br />
\end{array}</math><br />
|}<br />
<br />
Notice how the results of the coefficient extraction (partial interpretation or partial evaluation) operators <math>\boldsymbol\varphi_\alpha,\!</math> where the index <math>\alpha\!</math> refers to a singular proposition of the relevant category of discourse, could almost be defined in terms of the analytic form, namely, as the propositions occupying the designated places of the expansion. This becomes a real possibility if we have an independent way of developing the analytic form in a constructive or computational setting.<br />
<br />
Finally, the reason we are taking such great pains to define all entities in terms of operators on names of expressions is because we wish to anticipate those circumstances of computation when we fail to have the full expressions available all the time, for instance, to evaluate by performing substitutions on.<br />
<br />
Figure&nbsp;3 uses the &ldquo;bundle of boxes&rdquo; style of venn diagram to illustrate the partial expansions of the conjunction <math>J(x, y).\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Fields Of Inquiry 1.jpg|center|400px]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 3. Factorizations of Conjunction}\!</math><br />
|}<br />
<br />
As a slightly more complex example, consider the boolean function determined by the equality or biconditional <math>I(x, y) = ((x, y)).\!</math> This affords the partial expansion with respect to <math>x,\!</math><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lrrrr}<br />
I(x, y)<br />
& = & y \cdot x<br />
& + & (y) \cdot (x),<br />
\end{array}</math><br />
|}<br />
<br />
the partial expansion with respect to <math>y,\!</math><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lrrrr}<br />
I(x, y)<br />
& = & x \cdot y<br />
& + & (x) \cdot (y),<br />
\end{array}</math><br />
|}<br />
<br />
and the complete expansion with respect to <math>x\!</math> and <math>y,\!</math><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{l*{8}{r}}<br />
I(x, y)<br />
& = & 1 \cdot xy<br />
& + & 0 \cdot x(y)<br />
& + & 0 \cdot (x)y<br />
& + & 1 \cdot (x)(y).<br />
\end{array}</math><br />
|}<br />
<br />
Figure&nbsp;4 illustrates the partial expansions of the equality <math>I(x, y).\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Fields Of Inquiry 2.jpg|center|400px]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 4. Factorizations of Equality}\!</math><br />
|}<br />
<br />
For our last example, the implication or conditional <math>K(x, y) = (x(y))~\!</math> results in the partial expansion with respect to <math>x,\!</math><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lrrrr}<br />
K(x, y)<br />
& = & y \cdot x<br />
& + & 1 \cdot (x),<br />
\end{array}</math><br />
|}<br />
<br />
the partial expansion with respect to <math>y,\!</math><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{lrrrr}<br />
K(x, y)<br />
& = & 1 \cdot y<br />
& + & (x) \cdot (y),<br />
\end{array}</math><br />
|}<br />
<br />
and the complete expansion with respect to <math>x\!</math> and <math>y,\!</math><br />
<br />
{| align="center" cellpadding="8" width="90%"<br />
|<br />
<math>\begin{array}{l*{8}{r}}<br />
K(x, y)<br />
& = & 1 \cdot xy<br />
& + & 0 \cdot x(y)<br />
& + & 1 \cdot (x)y<br />
& + & 1 \cdot (x)(y).<br />
\end{array}</math><br />
|}<br />
<br />
Figure&nbsp;5 illustrates the partial expansions of the implication <math>K(x, y).\!</math><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Fields Of Inquiry 3.ISW.jpg|center|400px]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 5. Factorizations of Implication}\!</math><br />
|}<br />
<br />
===Differential Enlargements===<br />
<br />
{| align="center" width="90%"<br />
| align="left" |<br />
Longing, hatred, happiness, suffering,<br><br />
bodily form, consciousness, resolve,<br><br />
thus is this field with its changes<br><br />
defined in summary.<br />
| align="right" valign="bottom" | &mdash; ''Bhagavad Gita,'' 13.6<br />
|}<br />
<br />
Another important operation treated in the Tools and Views paper was the ''differential enlargement'' of propositions. This a mapping of the type,<br />
<br />
{| align="center" cellpadding="8"<br />
| <math>\mathrm{E} : (U \to \mathbb{B}) \to (U \times \mathrm{d}U \to \mathbb{B}),\!</math><br />
|}<br />
<br />
whose action on any proposition <math>f : U \to \mathbb{B}\!</math> is defined by an equation of the form,<br />
<br />
{| align="center" cellpadding="8"<br />
|<br />
<math>\begin{matrix}<br />
\mathrm{E}f(x_1, \ldots, x_n, \mathrm{d}x_1, \ldots, \mathrm{d}x_n)<br />
& = & f(x_1 + \mathrm{d}x_1, \ldots, x_n + \mathrm{d}x_n).<br />
\end{matrix}</math><br />
|}<br />
<br />
In the effort to comprehend what the differential enlargement of a proposition is telling us about the original proposition we usually parse it as a proposition field in either one of two obvious ways. If we factor the ordinary component out to the front we obtain the ''flow'',<br />
<br />
{| align="center" cellpadding="8"<br />
|<br />
<math>\begin{matrix}<br />
\Phi f & = & \boldsymbol\iota_U \mathrm{E}f : U \to (\mathrm{d}U \to \mathbb{B}),<br />
\end{matrix}</math><br />
|}<br />
<br />
If we factor the differential component out to the front we obtain ''fray'',<br />
<br />
{| align="center" cellpadding="8"<br />
|<br />
<math>\begin{matrix}<br />
\Psi f & = & \boldsymbol\iota_{\mathrm{d}U} \mathrm{E}f : \mathrm{d}U \to (U \to \mathbb{B}).<br />
\end{matrix}</math><br />
|}<br />
<br />
Evaluating the flow at points of the original universe, we discover the ''intentional attraction'' or ''influence'' of <math>f\!</math> at each point,<br />
<br />
{| align="center" cellpadding="8"<br />
|<br />
<math>\begin{matrix}<br />
T_u f<br />
& = & \boldsymbol\varphi_u \Phi f<br />
& = & \boldsymbol\varphi_u \boldsymbol\iota_U \mathrm{E}f : \mathrm{d}U \to \mathbb{B}.<br />
\end{matrix}</math><br />
|}<br />
<br />
Evaluating the fray along directions of the differential universe, we find the ''differential action'' or ''diffraction'' of <math>f\!</math> in each direction,<br />
<br />
{| align="center" cellpadding="8"<br />
|<br />
<math>\begin{matrix}<br />
T_v f<br />
& = & \boldsymbol\varphi_v \Psi f<br />
& = & \boldsymbol\varphi_v \boldsymbol\iota_{\mathrm{d}U} \mathrm{E}f : U \to \mathbb{B}.<br />
\end{matrix}</math><br />
|}<br />
<br />
''Diffraction'' = differential factorization (or differential fractionation):<br />
<br />
{| align="center" cellpadding="8"<br />
|<br />
<math>\begin{matrix}<br />
(\mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B})<br />
& \cong &<br />
(\mathbb{D}^n \to (\mathbb{B}^n \to \mathbb{B}).<br />
\end{matrix}</math><br />
|}<br />
<br />
Between the twin perspectives afforded by these two <math>\boldsymbol\iota\!</math>'s we have what we need to synthesize a stereotactic grasp of the situation represented in a proposition.<br />
<br />
==Differential Propositions and Transformation Groups==<br />
<br />
{| align="center" width="90%"<br />
| align="left" |<br />
Its hands and feet reach everywhere;<br><br />
its head and face see in every direction;<br><br />
hearing everything, it remains<br><br />
in the world, enveloping all.<br />
| align="right" valign="bottom" | &mdash; ''Bhagavad Gita,'' 13.13<br />
|}<br />
<br />
In this section we examine a number of relationships between differential operators and higher order propositions, together with the actions and characters of related transformation groups on the space of propositions.<br />
<br />
===Differential Expansions===<br />
<br />
{| align="center" width="90%"<br />
| align="left" |<br />
Lacking all the sense organs,<br><br />
it shines in their qualities;<br><br />
unattached, it supports everything;<br><br />
without qualities, it enjoys them.<br />
| align="right" valign="bottom" | &mdash; ''Bhagavad Gita,'' 13.14<br />
|}<br />
<br />
On analogy with usage in ordinary calculus, we introduce the following terminology. Given two sets of logical features <math>X \subseteq Y,\!</math> say<br />
<br />
{| align="center" cellpadding="8"<br />
| <math>X = \{ x_1, \ldots, x_n \}\!</math> and <math>Y = \{ x_1, \ldots, x_m \},\!</math><br />
|}<br />
<br />
a proposition <math>F : Y \to \mathbb{B}\!</math> is called an ''infinitesimal'' with respect to the universe <math>X,\!</math> written <math>F \in \Upsilon(X),\!</math> if <math>F\!</math> is false outside the region of the logical disjunction <math>((x_1)( \ldots )(x_n)),\!</math> in other words, if <math>F = 0\!</math> at the origin <math>(x_1)( \ldots )(x_n)\!</math> of <math>X.\!</math><br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Fields Of Inquiry 4.jpg|center|420px]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 6. Enlargement of Conjunction}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Fields Of Inquiry 5.jpg|center|420px]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 7. Diffraction of Conjunction}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Fields Of Inquiry 6.jpg|center|420px]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 8. Enlargement of Equality}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Fields Of Inquiry 7.jpg|center|420px]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 9. Diffraction of Equality}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Fields Of Inquiry 8.jpg|center|420px]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 10. Enlargement of Implication}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Fields Of Inquiry 9.jpg|center|420px]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure 11. Diffraction of Implication}\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Partial Evaluations===<br />
<br />
{| align="center" width="90%"<br />
| align="left" |<br />
Outside and within all creatures,<br><br />
inanimate but still animate,<br><br />
too subtle to be known,<br><br />
it is far distant, yet near.<br />
| align="right" valign="bottom" | &mdash; ''Bhagavad Gita,'' 13.15<br />
|}<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" style="border-spacing:0; text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 12.} ~ \mathrm{E}f ~ \text{Arranged by Principal Features}\!</math><br />
<br />
|- style="height:30px; background:ghostwhite"<br />
| style="border-top:0.75pt solid #000000; border-bottom:2.25pt double #000000; border-left:0.75pt solid #000000; border-right:none;" | &nbsp;<br />
| style="border-top:0.75pt solid #000000; border-bottom:2.25pt double #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>f\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:2.25pt double #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{E}f\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:2.25pt double #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{E}f\!</math> @ <br> <math>x \cdot y\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:2.25pt double #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{E}f\!</math> @ <br> <math>x \cdot (y)\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:2.25pt double #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{E}f\!</math> @ <br> <math>(x) \cdot y\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:2.25pt double #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>\mathrm{E}f\!</math> @ <br> <math>(x)(y)\!</math><br />
<br />
|-<br />
| style="border-top:2.25pt double #000000; border-bottom:1.5pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_0\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>()\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>()\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>0\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>0\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>0\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>0\!</math><br />
<br />
|-<br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_1\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x)(y)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((x,\mathrm{d}x))((y,\mathrm{d}y))\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}x ~ \mathrm{d}y\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}x (\mathrm{d}y)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(\mathrm{d}x) \mathrm{d}y\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>(\mathrm{d}x)(\mathrm{d}y)\!</math><br />
<br />
|-<br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_2\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x) y\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((x,\mathrm{d}x)) (y,\mathrm{d}y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}x (\mathrm{d}y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}x ~ \mathrm{d}y\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(\mathrm{d}x)(\mathrm{d}y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>(\mathrm{d}x) \mathrm{d}y\!</math><br />
<br />
|-<br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_4\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>x (y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x,\mathrm{d}x) ((y,\mathrm{d}y))\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(\mathrm{d}x) \mathrm{d}y\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(\mathrm{d}x)(\mathrm{d}y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}x ~ \mathrm{d}y\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>\mathrm{d}x (\mathrm{d}y)\!</math><br />
<br />
|-<br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_8\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>x y\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x,\mathrm{d}x) (y,\mathrm{d}y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(\mathrm{d}x)(\mathrm{d}y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(\mathrm{d}x) \mathrm{d}y\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}x (\mathrm{d}y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>\mathrm{d}x ~ \mathrm{d}y\!</math><br />
<br />
|-<br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_3\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((x,\mathrm{d}x))\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}x\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}x\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(\mathrm{d}x)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>(\mathrm{d}x)\!</math><br />
<br />
|-<br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_{12}\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>x\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x,\mathrm{d}x)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(\mathrm{d}x)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(\mathrm{d}x)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}x\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>\mathrm{d}x\!</math><br />
<br />
|-<br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_6\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x, y)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((x,\mathrm{d}x), (y,\mathrm{d}y))\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(\mathrm{d}x, \mathrm{d}y)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((\mathrm{d}x, \mathrm{d}y))\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((\mathrm{d}x, \mathrm{d}y))\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>(\mathrm{d}x, \mathrm{d}y)\!</math><br />
<br />
|-<br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_9\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((x, y))\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(((x,\mathrm{d}x), (y,\mathrm{d}y)))~\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((\mathrm{d}x, \mathrm{d}y))\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(\mathrm{d}x, \mathrm{d}y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(\mathrm{d}x, \mathrm{d}y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>((\mathrm{d}x, \mathrm{d}y))\!</math><br />
<br />
|-<br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_5\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(y)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((y,\mathrm{d}y))\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}y\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(\mathrm{d}y)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}y\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>(\mathrm{d}y)\!</math><br />
<br />
|-<br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_{10}\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>y\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(y,\mathrm{d}y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(\mathrm{d}y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}y\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(\mathrm{d}y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>\mathrm{d}y\!</math><br />
<br />
|-<br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_7\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x y)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((x,\mathrm{d}x) (y,\mathrm{d}y))\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((\mathrm{d}x)(\mathrm{d}y))\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((\mathrm{d}x) \mathrm{d}y)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(\mathrm{d}x (\mathrm{d}y))\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>(\mathrm{d}x ~ \mathrm{d}y)\!</math><br />
<br />
|-<br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_{11}\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x (y))\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((x,\mathrm{d}x) ((y,\mathrm{d}y)))\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((\mathrm{d}x) \mathrm{d}y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((\mathrm{d}x)(\mathrm{d}y))\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(\mathrm{d}x ~ \mathrm{d}y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>(\mathrm{d}x (\mathrm{d}y))\!</math><br />
<br />
|-<br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_{13}\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((x) y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(((x,\mathrm{d}x)) (y,\mathrm{d}y))~\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(\mathrm{d}x (\mathrm{d}y))\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(\mathrm{d}x ~ \mathrm{d}y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((\mathrm{d}x)(\mathrm{d}y))\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>((\mathrm{d}x) \mathrm{d}y)\!</math><br />
<br />
|-<br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_{14}\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((x)(y))\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(((x,\mathrm{d}x))((y,\mathrm{d}y)))\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(\mathrm{d}x ~ \mathrm{d}y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(\mathrm{d}x (\mathrm{d}y))\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((\mathrm{d}x) \mathrm{d}y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>((\mathrm{d}x)(\mathrm{d}y))\!</math><br />
<br />
|-<br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_{15}\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(())\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(())\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>1\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>1\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>1\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>1\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Group Actions===<br />
<br />
{| align="center" width="90%"<br />
| align="left" |<br />
Undivided, it seems divided<br><br />
among creatures;<br><br />
understood as their sustainer,<br><br />
it devours and creates them.<br />
| align="right" valign="bottom" | &mdash; ''Bhagavad Gita,'' 13.16<br />
|}<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" style="border-spacing:0; text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 13.} ~ \mathrm{E}f ~ \text{Arranged by Differential Features}\!</math><br />
<br />
|- style="height:30px; background:ghostwhite"<br />
| style="border-top:0.75pt solid #000000; border-bottom:none; border-left:0.75pt solid #000000; border-right:none;" | &nbsp;<br />
| style="border-top:0.75pt solid #000000; border-bottom:none; border-left:0.75pt solid #000000; border-right:none;" | <math>f\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:none; border-left:0.75pt solid #000000; border-right:none;" | <math>\mathrm{E}f\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:none; border-left:0.75pt solid #000000; border-right:none;" | <math>T_{11}f\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:none; border-left:0.75pt solid #000000; border-right:none;" | <math>T_{10}f\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:none; border-left:0.75pt solid #000000; border-right:none;" | <math>T_{01}f\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:none; border-left:0.75pt solid #000000; border-right:0.75pt solid #000000;" | <math>T_{00}f\!</math><br />
<br />
|- style="background:ghostwhite"<br />
| style="border-top:none; border-bottom:2.25pt double #000000; border-left:0.75pt solid #000000; border-right:none;" | &nbsp;<br />
| style="border-top:none; border-bottom:2.25pt double #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>x = x_1\!</math> <br> <math>y = x_2\!</math><br />
| style="border-top:none; border-bottom:2.25pt double #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\mathrm{sub}_i [(x_i + \mathrm{d}x_i)/x_i] F\!</math><br />
| style="border-top:none; border-bottom:2.25pt double #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\mathrm{E}f\!</math> @ <br> <math>\mathrm{d}x \cdot \mathrm{d}y\!</math><br />
| style="border-top:none; border-bottom:2.25pt double #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\mathrm{E}f\!</math> @ <br> <math>\mathrm{d}x \cdot (\mathrm{d}y)\!</math><br />
| style="border-top:none; border-bottom:2.25pt double #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\mathrm{E}f\!</math> @ <br> <math>(\mathrm{d}x) \cdot \mathrm{d}y\!</math><br />
| style="border-top:none; border-bottom:2.25pt double #000000; border-left:0.75pt solid #000000; border-right:0.75pt solid #000000;" | <math>\mathrm{E}f\!</math> @ <br> <math>(\mathrm{d}x)(\mathrm{d}y)\!</math><br />
<br />
|-<br />
| style="border-top:2.25pt double #000000; border-bottom:1.5pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_0\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>()\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>()\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>()\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>()\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>()\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>()\!</math><br />
<br />
|-<br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_1\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x)(y)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((x,\mathrm{d}x))((y,\mathrm{d}y))\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>x y\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>x (y)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x) y\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>(x)(y)\!</math><br />
<br />
|-<br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_2\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x) y\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((x,\mathrm{d}x))(y,\mathrm{d}y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>x (y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>x y\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x)(y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>(x) y\!</math><br />
<br />
|-<br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_4\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>x (y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x,\mathrm{d}x)((y,\mathrm{d}y))\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x) y\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x)(y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>x y\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>x (y)\!</math><br />
<br />
|-<br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_8\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>x y\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x,\mathrm{d}x)(y,\mathrm{d}y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x)(y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x) y\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>x (y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>x y\!</math><br />
<br />
|-<br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_3\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((x,\mathrm{d}x))\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>x\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>x\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>(x)\!</math><br />
<br />
|-<br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_{12}\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>x\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x,\mathrm{d}x)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>x\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>x\!</math><br />
<br />
|-<br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_6\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x, y)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((x,\mathrm{d}x), (y,\mathrm{d}y))\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x, y)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((x, y))\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((x, y))\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>(x, y)\!</math><br />
<br />
|-<br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_9\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((x, y))\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(((x,\mathrm{d}x), (y,\mathrm{d}y)))~\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((x, y))\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x, y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x, y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>((x, y))\!</math><br />
<br />
|-<br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_5\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(y)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((y,\mathrm{d}y))\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>y\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(y)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>y\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>(y)\!</math><br />
<br />
|-<br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_{10}\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>y\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(y,\mathrm{d}y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>y\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>y\!</math><br />
<br />
|-<br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_7\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x y)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((x,\mathrm{d}x)(y,\mathrm{d}y))\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((x)(y))\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((x) y)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x (y))\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>(x y)\!</math><br />
<br />
|-<br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_{11}\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x (y))\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((x,\mathrm{d}x)((y,\mathrm{d}y)))\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((x) y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((x)(y))\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>(x (y))\!</math><br />
<br />
|-<br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_{13}\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((x) y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(((x,\mathrm{d}x))(y,\mathrm{d}y))~\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x (y))\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((x)(y))\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>((x) y)\!</math><br />
<br />
|-<br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_{14}\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((x)(y))\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(((x,\mathrm{d}x))((y,\mathrm{d}y)))\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x (y))\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((x) y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>((x)(y))\!</math><br />
<br />
|-<br />
| style="border-top:1.5pt solid #000000; border-bottom:2.25pt double #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_{15}\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:2.25pt double #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(())\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:2.25pt double #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(())\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:2.25pt double #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(())\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:2.25pt double #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(())\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:2.25pt double #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(())\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:2.25pt double #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>(())\!</math><br />
<br />
|-<br />
| style="border-top:2.25pt double #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" align="left" colspan="3" | <math>\text{Total Number of Fixed Points:}\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:0.75pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>4\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:0.75pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>4\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:0.75pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>4\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:0.75pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>16\!</math><br />
|}<br />
<br />
<br><br />
<br />
===Derivations===<br />
<br />
{| align="center" width="90%"<br />
| align="left" |<br />
The light of lights<br><br />
beyond darkness it is called;<br><br />
knowledge attained by knowledge,<br><br />
fixed in the heart of everyone.<br />
| align="right" valign="bottom" | &mdash; ''Bhagavad Gita,'' 13.17<br />
|}<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" style="border-spacing:0; text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 14.} ~ \mathrm{D}f ~ \text{Arranged by Principal Features}~\!</math><br />
<br />
|- style="background:ghostwhite"<br />
| style="border-top:0.75pt solid #000000; border-bottom:2.25pt double #000000; border-left:0.75pt solid #000000; border-right:none;" | &nbsp;<br />
| style="border-top:0.75pt solid #000000; border-bottom:2.25pt double #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{D}f = f + \mathrm{E}f\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:2.25pt double #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{D}f~\!</math> @ <br> <math>x \cdot y\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:2.25pt double #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{D}f~\!</math> @ <br> <math>x \cdot (y)\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:2.25pt double #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{D}f~\!</math> @ <br> <math>(x) \cdot y\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:2.25pt double #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>\mathrm{D}f~\!</math> @ <br> <math>(x)(y)\!</math><br />
<br />
|-<br />
| style="border-top:2.25pt double #000000; border-bottom:1.5pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_0\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>() + ()\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>0\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>0\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>0\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>0\!</math><br />
<br />
|-<br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_1\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x)(y) + ((x,\mathrm{d}x))((y,\mathrm{d}y))\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}x ~ \mathrm{d}y\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}x (\mathrm{d}y)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(\mathrm{d}x) \mathrm{d}y\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>((\mathrm{d}x)(\mathrm{d}y))\!</math><br />
<br />
|-<br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_2\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x) y + ((x,\mathrm{d}x)) (y,\mathrm{d}y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}x (\mathrm{d}y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}x ~ \mathrm{d}y\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((\mathrm{d}x)(\mathrm{d}y))\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>(\mathrm{d}x) \mathrm{d}y\!</math><br />
<br />
|-<br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_4\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>x (y) + (x,\mathrm{d}x) ((y,\mathrm{d}y))\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(\mathrm{d}x) \mathrm{d}y\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((\mathrm{d}x)(\mathrm{d}y))\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}x ~ \mathrm{d}y\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>\mathrm{d}x (\mathrm{d}y)\!</math><br />
<br />
|-<br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_8\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>x y + (x,\mathrm{d}x) (y,\mathrm{d}y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((\mathrm{d}x)(\mathrm{d}y))\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(\mathrm{d}x) \mathrm{d}y\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}x (\mathrm{d}y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>\mathrm{d}x ~ \mathrm{d}y\!</math><br />
<br />
|-<br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_3\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x) + ((x,\mathrm{d}x))\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}x\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}x\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}x\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>\mathrm{d}x\!</math><br />
<br />
|-<br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_{12}\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>x + (x,\mathrm{d}x)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}x\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}x\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}x\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>\mathrm{d}x\!</math><br />
<br />
|-<br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_6\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x, y) + ((x,\mathrm{d}x), (y,\mathrm{d}y))\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(\mathrm{d}x, \mathrm{d}y)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(\mathrm{d}x, \mathrm{d}y)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(\mathrm{d}x, \mathrm{d}y)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>(\mathrm{d}x, \mathrm{d}y)\!</math><br />
<br />
|-<br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_9\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((x, y)) + (((x,\mathrm{d}x), (y,\mathrm{d}y)))\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(\mathrm{d}x, \mathrm{d}y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(\mathrm{d}x, \mathrm{d}y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(\mathrm{d}x, \mathrm{d}y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>(\mathrm{d}x, \mathrm{d}y)\!</math><br />
<br />
|-<br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_5\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(y) + ((y,\mathrm{d}y))\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}y\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}y\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}y\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>\mathrm{d}y\!</math><br />
<br />
|-<br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_{10}\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>y + (y,\mathrm{d}y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}y\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}y\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}y\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>\mathrm{d}y\!</math><br />
<br />
|-<br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_7\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x y) + ((x,\mathrm{d}x) (y,\mathrm{d}y))\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((\mathrm{d}x)(\mathrm{d}y))\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(\mathrm{d}x) \mathrm{d}y\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}x (\mathrm{d}y)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>\mathrm{d}x ~ \mathrm{d}y\!</math><br />
<br />
|-<br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_{11}\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(x (y)) + ((x,\mathrm{d}x) ((y,\mathrm{d}y)))\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(\mathrm{d}x) \mathrm{d}y\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((\mathrm{d}x)(\mathrm{d}y))\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}x ~ \mathrm{d}y\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>\mathrm{d}x (\mathrm{d}y)\!</math><br />
<br />
|-<br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_{13}\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((x) y) + (((x,\mathrm{d}x)) (y,\mathrm{d}y))\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}x (\mathrm{d}y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}x ~ \mathrm{d}y\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((\mathrm{d}x)(\mathrm{d}y))\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:0.25pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>(\mathrm{d}x) \mathrm{d}y\!</math><br />
<br />
|-<br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_{14}\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>((x)(y)) + (((x,\mathrm{d}x))((y,\mathrm{d}y)))\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}x ~ \mathrm{d}y\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>\mathrm{d}x (\mathrm{d}y)\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(\mathrm{d}x) \mathrm{d}y\!</math><br />
| style="border-top:0.25pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>((\mathrm{d}x)(\mathrm{d}y))\!</math><br />
<br />
|-<br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_{15}\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>(()) + (())\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>0\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>0\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.25pt solid #000000; border-right:none;" | <math>0\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.25pt solid #000000; border-right:0.75pt solid #000000;" | <math>0\!</math><br />
|}<br />
<br />
<br><br />
<br />
==Generalized Projections : Analytic Operators and Fields of Inquiry==<br />
<br />
{| align="center" width="90%"<br />
| align="left" |<br />
Arjuna, know that anything<br><br />
inanimate or alive with motion<br><br />
is born from the union<br><br />
of the field and its knower.<br />
| align="right" valign="bottom" | &mdash; ''Bhagavad Gita,'' 13.26<br />
|}<br />
<br />
Projections, Dissections, Distortions, Perspectives (Outlooks, Opinions, Local Views),<br />
<br />
Partial Derivatives, Fields of Inquiry, Surveys, Test Fields, Textures, Cultures, Biases, Warps.<br />
<br />
Relation to questions of value-free inquiry and inquiry into values, to what extent these projects are possible or approachable from the stance of a concrete interpreter.<br />
<br />
Analytic operators or derivations, all of type <math>q : (\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^n \to \mathbb{B}).\!</math> Includes projections and partial derivatives. Equivalence to ''proposition fields'', ''question fields'', ''fields of inquiry'', as follows:<br />
<br />
{| align="center" cellpadding="8"<br />
| <math>Q : \mathbb{B}^n \to (F : (\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}).\!</math><br />
|}<br />
<br />
Note relation between operator symmetry and dimension reduction. When the question field <math>Q\!</math> is based on a symmetric operation, <math>f(x, y) = f(y, x),\!</math> then the corresponding analytic operator <math>q\!</math> may be viewed under a dimension-reducing type as <math>q : (\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^{n-1} \to \mathbb{B}).\!</math><br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" style="border-spacing:0; text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table 15.} ~ \text{Fields of Inquiry :} ~ G(\varphi R, \varphi^\prime R)\!</math><br />
<br />
|- style="height:40px; background:ghostwhite"<br />
| style="border-top:0.75pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:1.5pt solid #000000; border-left:2.25pt double #000000; border-right:none;" | <math>\theta f_{xy1}\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\theta f_{xy0}\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:1.5pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>\theta f_{1yz}\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\theta f_{0yz}\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:1.5pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>\theta f_{x1z}\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.75pt solid #000000; border-right:0.75pt solid #000000;" | <math>\theta f_{x0z}\!</math><br />
<br />
|-<br />
| style="border-top:2.25pt double #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_0\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:0.75pt solid #000000; border-left:2.25pt double #000000; border-right:none;" | <math>0\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>1\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:0.75pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>(z)\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(z)\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:0.75pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>(z)\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:0.75pt solid #000000;" | <math>(z)\!</math><br />
<br />
|-<br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_1\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:2.25pt double #000000; border-right:none;" | <math>(x)(y)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>((x)(y))\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>(z)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(y, z)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>(z)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:0.75pt solid #000000;" | <math>(x, z)\!</math><br />
<br />
|-<br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_2\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:2.25pt double #000000; border-right:none;" | <math>(x) y\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>((x) y)\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>(z)\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>((y, z))\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>(x, z)\!</math><br />
| style="border:0.75pt solid #000000;" | <math>(z)\!</math><br />
<br />
|-<br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_4\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:2.25pt double #000000; border-right:none;" | <math>x (y)\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(x (y))\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>(y, z)\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(z)\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>(z)\!</math><br />
| style="border:0.75pt solid #000000;" | <math>((x, z))\!</math><br />
<br />
|-<br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_8\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:2.25pt double #000000; border-right:none;" | <math>x y\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(x y)\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>((y, z))\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(z)\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>((x, z))\!</math><br />
| style="border:0.75pt solid #000000;" | <math>(z)\!</math><br />
<br />
|-<br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_3\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:2.25pt double #000000; border-right:none;" | <math>(x)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>x\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>(z)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>z\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>(x, z)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:0.75pt solid #000000;" | <math>(x, z)\!</math><br />
<br />
|-<br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_{12}\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:2.25pt double #000000; border-right:none;" | <math>x\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(x)\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>z\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(z)\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>((x, z))\!</math><br />
| style="border:0.75pt solid #000000;" | <math>((x, z))\!</math><br />
<br />
|-<br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_6\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:2.25pt double #000000; border-right:none;" | <math>(x, y)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>((x, y))\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>(y, z)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>((y, z))\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>(x, z)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:0.75pt solid #000000;" | <math>((x, z))\!</math><br />
<br />
|-<br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_9\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:2.25pt double #000000; border-right:none;" | <math>((x, y))\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(x, y)\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>((y, z))\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(y, z)\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>((x, z))\!</math><br />
| style="border:0.75pt solid #000000;" | <math>(x, z)\!</math><br />
<br />
|-<br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_5\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:2.25pt double #000000; border-right:none;" | <math>(y)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>y\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>(y, z)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(y, z)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>(z)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:0.75pt solid #000000;" | <math>z\!</math><br />
<br />
|-<br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_{10}\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:2.25pt double #000000; border-right:none;" | <math>y\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(y)\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>((y, z))\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>((y, z))\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>z\!</math><br />
| style="border:0.75pt solid #000000;" | <math>(z)\!</math><br />
<br />
|-<br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_7\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:2.25pt double #000000; border-right:none;" | <math>(x y)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>x y\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>(y, z)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>z\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>(x, z)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:0.75pt solid #000000;" | <math>z\!</math><br />
<br />
|-<br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_{11}\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:2.25pt double #000000; border-right:none;" | <math>(x (y))\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>x (y)\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>((y, z))\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>z\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>z\!</math><br />
| style="border:0.75pt solid #000000;" | <math>(x, z)\!</math><br />
<br />
|-<br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_{13}\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:2.25pt double #000000; border-right:none;" | <math>((x) y)\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(x) y\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>z\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(y, z)\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>((x, z))\!</math><br />
| style="border:0.75pt solid #000000;" | <math>z\!</math><br />
<br />
|-<br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_{14}\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:2.25pt double #000000; border-right:none;" | <math>((x)(y))\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(x)(y)\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>z\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>((y, z))\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>z\!</math><br />
| style="border:0.75pt solid #000000;" | <math>((x, z))\!</math><br />
<br />
|-<br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_{15}\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:2.25pt double #000000; border-right:none;" | <math>1\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>0\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>z\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>z\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:1.5pt solid #000000; border-right:none;" | <math>z\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:0.75pt solid #000000;" | <math>z\!</math><br />
|}<br />
<br />
<br><br />
<br />
==References==<br />
<br />
<br><br />
<br />
[Bhag] ''The Bhagavad-Gita : Krishna's Counsel in Time of War'', B.S. Miller (trans.), Bantam Books, New York, NY, 1986.<br />
<br />
<br><br />
<br />
==Fragments==<br />
<br />
'''Note.''' This part needs work. It looks like I was trying to get <math>\frac{\mathrm{d}x}{\mathrm{d}y}\!</math> and <math>\frac{\mathrm{d}y}{\mathrm{d}x}\!</math> relative to the region where <math>f\!</math> is true. But the fragmentary notes I have found so far are bedeviled by a confused mix of conceptual, notational, and typographical errors. In ordinary real-value calculus ''implicit differentiation'' of <math>f(x, y) = 1\!</math> gives the following formulas:<br />
<br />
{| align="center" cellpadding="8"<br />
|<br />
<math>\begin{matrix}<br />
{\mathrm{d}_x y}<br />
= \frac{\mathrm{d}y}{\mathrm{d}x}<br />
= -\frac{f_x}{f_y}<br />
= -\frac{\partial_x f}{\partial_y f}<br />
= -\frac{{\partial f}/{\partial x}}{{\partial f}/{\partial y}}<br />
& \qquad &<br />
{\mathrm{d}_y x}<br />
= \frac{\mathrm{d}x}{\mathrm{d}y}<br />
= -\frac{f_y}{f_x}<br />
= -\frac{\partial_y f}{\partial_x f}<br />
= -\frac{{\partial f}/{\partial y}}{{\partial f}/{\partial x}}<br />
\end{matrix}\!</math><br />
|}<br />
<br />
The minus signs can be dropped in the boolean context but taking quotients is still a problem. If anything analogous to the preceding formulas does go through, it looks like I can correct the notation in the last two column heads to give the following table:<br />
<br />
<br><br />
<br />
{| align="center" cellpadding="6" style="border-spacing:0; text-align:center; width:75%"<br />
|+ style="height:30px" | <math>\text{Table A. Partial and Relative Differentials}\!</math><br />
<br />
|- style="height:60px; background:ghostwhite"<br />
| style="border-top:0.75pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.75pt solid #000000; border-right:none; " | &nbsp;<br />
| style="border-top:0.75pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.75pt solid #000000; border-right:none; " | <math>f\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.75pt solid #000000; border-right:none; " | <math>\frac{\partial f}{\partial x}\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.75pt solid #000000; border-right:none; " | <math>\frac{\partial f}{\partial y}\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.75pt solid #000000; border-right:none; " |<br />
<math>\begin{matrix}<br />
\mathrm{d}f =<br />
\\[2pt]<br />
\partial_x f \cdot \mathrm{d}x ~+~ \partial_y f \cdot \mathrm{d}y<br />
\end{matrix}\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.75pt solid #000000; border-right:none; " |<br />
<math>\begin{matrix}<br />
{\mathrm{d}_x y}|f =<br />
\\[2pt]<br />
{\partial_x f}/{\partial_y f}<br />
\end{matrix}\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:1.5pt solid #000000; border-left:0.75pt solid #000000; border-right:0.75pt solid #000000; " |<br />
<math>\begin{matrix}<br />
{\mathrm{d}_y x}|f =<br />
\\[2pt]<br />
{\partial_y f}/{\partial_x f}<br />
\end{matrix}\!</math><br />
<br />
|-<br />
| style="border-top:2.25pt double #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_0\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>()\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>0\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>0\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>0\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>0/0\!</math><br />
| style="border-top:2.25pt double #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:0.75pt solid #000000;" | <math>0/0\!</math><br />
<br />
|-<br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_1\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(x)(y)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(y)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(x)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(y) ~ \mathrm{d}x ~+~ (x) ~ \mathrm{d}y\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(y)/(x)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:0.75pt solid #000000;" | <math>(x)/(y)\!</math><br />
<br />
|-<br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_2\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(x) y\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>y\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(x)\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>y ~ \mathrm{d}x ~+~ (x) ~ \mathrm{d}y\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>y/(x)\!</math><br />
| style="border:0.75pt solid #000000;" | <math>(x)/y\!</math><br />
<br />
|-<br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_4\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>x (y)\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(y)\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>x\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(y) ~ \mathrm{d}x ~+~ x ~ \mathrm{d}y\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(y)/x\!</math><br />
| style="border:0.75pt solid #000000;" | <math>x/(y)\!</math><br />
<br />
|-<br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_8\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>x y\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>y\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>x\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>y ~ \mathrm{d}x ~+~ x ~ \mathrm{d}y\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>y/x\!</math><br />
| style="border:0.75pt solid #000000;" | <math>x/y\!</math><br />
<br />
|-<br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_3\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(x)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>1\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>0\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\mathrm{d}x\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>1/0\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:0.75pt solid #000000;" | <math>0\!</math><br />
<br />
|-<br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_{12}\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>x\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>1\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>0\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\mathrm{d}x\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>1/0\!</math><br />
| style="border:0.75pt solid #000000;" | <math>0\!</math><br />
<br />
|-<br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_6\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(x, y)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>1\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>1\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\mathrm{d}x + \mathrm{d}y\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>1\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:0.75pt solid #000000;" | <math>1\!</math><br />
<br />
|-<br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_9\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>((x, y))\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>1\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>1\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\mathrm{d}x + \mathrm{d}y\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>1\!</math><br />
| style="border:0.75pt solid #000000;" | <math>1\!</math><br />
<br />
|-<br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_5\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(y)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>0\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>1\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\mathrm{d}y\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>0\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:0.75pt solid #000000;" | <math>1/0\!</math><br />
<br />
|-<br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_{10}\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>y\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>0\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>1\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>\mathrm{d}y\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>0\!</math><br />
| style="border:0.75pt solid #000000;" | <math>1/0\!</math><br />
<br />
|-<br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_7\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(x y)\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>y\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>x\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>y ~ \mathrm{d}x ~+~ x ~ \mathrm{d}y\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>y/x\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:0.75pt solid #000000;" | <math>x/y\!</math><br />
<br />
|-<br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_{11}\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(x (y))\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(y)\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>x\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(y) ~ \mathrm{d}x ~+~ x ~ \mathrm{d}y\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(y)/x\!</math><br />
| style="border:0.75pt solid #000000;" | <math>x/(y)\!</math><br />
<br />
|-<br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_{13}\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>((x) y)\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>y\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(x)\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>y ~ \mathrm{d}x ~+~ (x) ~ \mathrm{d}y\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>y/(x)\!</math><br />
| style="border:0.75pt solid #000000;" | <math>(x)/y\!</math><br />
<br />
|-<br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_{14}\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>((x)(y))\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(y)\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(x)\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(y) ~ \mathrm{d}x ~+~ (x) ~ \mathrm{d}y\!</math><br />
| style="border-top:0.75pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(y)/(x)\!</math><br />
| style="border:0.75pt solid #000000;" | <math>(x)/(y)\!</math><br />
<br />
|-<br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>f_{15}\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>(())\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>0\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>0\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>0\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:none;" | <math>0/0\!</math><br />
| style="border-top:1.5pt solid #000000; border-bottom:0.75pt solid #000000; border-left:0.75pt solid #000000; border-right:0.75pt solid #000000;" | <math>0/0\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Fields Of Inquiry 10.jpg|center|500px]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure B. Enlargements of Conjunction}\!</math><br />
|}<br />
<br />
<br><br />
<br />
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"<br />
| [[Image:Fields Of Inquiry 11.jpg|center|500px]]<br />
|-<br />
| height="20px" valign="top" | <math>\text{Figure C. Differentials of Conjunction}\!</math><br />
|}<br />
<br />
<br><br />
<br />
==Work Area==<br />
<br />
'''Inquiry Driven System as Uncertainty Control System'''<br />
<br />
'''Types of Inquiry Arising from Types of Uncertainty'''<br />
<br />
1. Uncertainty about what is true. Logical.<br />
<br />
2. Uncertainty about what to do. Ethical.<br />
<br />
3. Uncertainty about what to hope. Esthetic.<br />
<br />
'''Do the Following Fall Under the Preceding?'''<br />
<br />
Uncertainty about what interpretant to form.<br><br />
Uncertainty about how to represent the present.<br><br />
Uncertainty about how to approach the moment.<br><br />
Uncertainty about what rheme to impose on the ration.<br><br />
Uncertainty about what role to take up and cast others in the play.<br><br />
Uncertainty about how to express, convey, carry, bear the moment.<br><br />
Uncertainty about what relation to form from elements of the situation.<br><br />
(Interpretant = equivalent, approximate, compensatory, substitute, or proxy sign.)<br />
<br />
<br><br />
<br />
==Sources==<br />
<br />
Sanjaya, tell me what my sons<br><br />
and the sons of Pandu did when they met,<br><br />
wanting to battle on the field of Kuru,<br><br />
on the field of sacred duty?<br><br />
&mdash; Dhritarashtra, ''Bhagavad Gita,'' 1.1<br />
<br />
Guarded by Bhishma, the strength<br><br />
of our army is without limit;<br><br />
but the strength of their army,<br><br />
guarded by Bhima, is limited.<br><br />
&mdash; Duryodhana, ''Bhagavad Gita,'' 1.10<br />
<br />
Conches and kettledrums,<br><br />
cymbals, tabors, and trumpets<br><br />
were sounded at once<br><br />
and the din of tumult arose.<br><br />
&mdash; Sanjaya, ''Bhagavad Gita,'' 1.13<br />
<br />
Arjuna, his war flag a rampant monkey,<br><br />
saw Dhritarashtra's sons assembled<br><br />
as weapons were ready to clash,<br><br />
and he lifted his bow.<br><br />
&mdash; Sanjaya, ''Bhagavad Gita,'' 1.20<br />
<br />
Nothing of nonbeing comes to be,<br><br />
nor does being cease to exist;<br><br />
the boundary between these two<br><br />
is seen by men who see reality.<br><br />
&mdash; Krishna, ''Bhagavad Gita,'' 2.16<br />
<br />
Indestructible is the presence<br><br />
that pervades all this;<br><br />
no one can destroy<br><br />
this unchanging reality.<br><br />
&mdash; ''Bhagavad Gita,'' 2.17<br />
<br />
Creatures are unmanifest in origin,<br><br />
manifest in the midst of life,<br><br />
and unmanifest again in the end.<br><br />
Since this is so, why do you lament?<br><br />
&mdash; ''Bhagavad Gita,'' 2.28<br />
<br />
Arjuna, the realm of sacred lore<br><br />
is nature &mdash; beyond its triad of qualities,<br><br />
dualities, and mundane rewards,<br><br />
be forever lucid, alive to your self.<br><br />
&mdash; ''Bhagavad Gita,'' 2.45<br />
<br />
For the discerning priest,<br><br />
all of sacred lore<br><br />
has no more value than a well<br><br />
when water flows everywhere.<br><br />
&mdash; ''Bhagavad Gita,'' 2.46<br />
<br />
Be intent on action,<br><br />
not on the fruits of action;<br><br />
avoid attraction to the fruits<br><br />
and attachment to inaction!<br><br />
&mdash; ''Bhagavad Gita,'' 2.47<br />
<br />
Perform actions, firm in discipline,<br><br />
relinquishing attachment;<br><br />
be impartial to failure and success &mdash;<br><br />
this equanimity is called discipline.<br><br />
&mdash; ''Bhagavad Gita,'' 2.48<br />
<br />
When it is night for all creatures,<br><br />
a master of restraint is awake;<br><br />
when they are awake, it is night<br><br />
for the sage who sees reality.<br><br />
&mdash; ''Bhagavad Gita,'' 2.69<br />
<br />
No wise man disturbs the understanding<br><br />
of ignorant men attached to action;<br><br />
he should inspire them,<br><br />
performing all actions with discipline.<br><br />
&mdash; ''Bhagavad Gita,'' 3.26<br />
<br />
Actions are all effected<br><br />
by the qualities of nature;<br><br />
but deluded by individuality,<br><br />
the self thinks, &ldquo;I am the actor.&rdquo;<br><br />
&mdash; ''Bhagavad Gita,'' 3.27<br />
<br />
When he can discriminate<br><br />
the actions of nature's qualities<br><br />
and think, &ldquo;The qualities depend<br><br />
on other qualities,&rdquo; he is detached.<br><br />
&mdash; ''Bhagavad Gita,'' 3.28<br />
<br />
Attraction and hatred are poised<br><br />
in the object of every sense experience;<br><br />
a man must not fall prey<br><br />
to these two brigands lurking on his path!<br><br />
&mdash; ''Bhagavad Gita,'' 3.34<br />
<br />
The senses, mind, and understanding<br><br />
are said to harbor desire;<br><br />
with these desire obscures knowledge<br><br />
and confounds the embodied self.<br><br />
&mdash; ''Bhagavad Gita,'' 3.40<br />
<br />
What is action? What is inaction?<br><br />
Even the poets were confused &mdash;<br><br />
what I shall teach you of action<br><br />
will free you from misfortune.<br><br />
&mdash; ''Bhagavad Gita,'' 4.16<br />
<br />
When ignorance is destroyed<br><br />
by knowledge of the self,<br><br />
then, like the sun, knowledge<br><br />
illumines ultimate reality.<br><br />
&mdash; ''Bhagavad Gita,'' 5.16<br />
<br />
That becomes their understanding,<br><br />
their self, their basis, and their goal,<br><br />
and they reach a state beyond return,<br><br />
their sin dispelled by knowledge.<br><br />
&mdash; ''Bhagavad Gita,'' 5.17<br />
<br />
Arming himself with discipline,<br><br />
seeing everything with an equal eye,<br><br />
he sees the self in all creatures<br><br />
and all creatures in the self.<br><br />
&mdash; ''Bhagavad Gita,'' 6.29<br />
<br />
When he sees identity in everything,<br><br />
whether joy or suffering,<br><br />
through analogy with the self,<br><br />
he is deemed a man of pure discipline.<br><br />
&mdash; ''Bhagavad Gita,'' 6.32<br />
<br />
Doomed by his double failure,<br><br />
is he not like a cloud split apart,<br><br />
unsettled, deluded on the path<br><br />
of the infinite spirit?<br><br />
&mdash; Arjuna, ''Bhagavad Gita,'' 6.38<br />
<br />
Arjuna, he does not suffer<br><br />
doom in this world or the next;<br><br />
any man who acts with honor<br><br />
cannot go the wrong way, my friend.<br><br />
&mdash; Krishna, ''Bhagavad Gita,'' 6.40<br />
<br />
Nothing is higher than I am;<br><br />
Arjuna, all that exists<br><br />
is woven on me,<br><br />
like a web of pearls on thread.<br><br />
&mdash; ''Bhagavad Gita,'' 7.7<br />
<br />
I am the taste in water, Arjuna,<br><br />
the light in the moon and sun,<br><br />
OM resonant in all sacred lore,<br><br />
the sound in space, valor in men.<br><br />
&mdash; ''Bhagavad Gita,'' 7.8<br />
<br />
Beyond this unmanifest nature<br><br />
is another unmanifest existence,<br><br />
a timeless being that does not perish<br><br />
when all creatures perish.<br><br />
&mdash; ''Bhagavad Gita,'' 8.20<br />
<br />
It is called eternal unmanifest nature,<br><br />
what men call the highest way,<br><br />
the goal from which they do not return;<br><br />
this highest realm is mine.<br><br />
&mdash; ''Bhagavad Gita,'' 8.21<br />
<br />
It is man's highest spirit,<br><br />
won by singular devotion, Arjuna,<br><br />
in whom creatures rest<br><br />
and the whole universe extends.<br><br />
&mdash; ''Bhagavad Gita,'' 8.22<br />
<br />
These bright and dark pathways<br><br />
are deemed constant for the universe;<br><br />
by one, a man escapes rebirth;<br><br />
by the other, he is born again.<br><br />
&mdash; ''Bhagavad Gita,'' 8.26<br />
<br />
The whole universe is pervaded<br><br />
by my unmanifest form;<br><br />
all creatures exist in me,<br><br />
but I do not exist in them.<br><br />
&mdash; ''Bhagavad Gita,'' 9.4<br />
<br />
Behold the power of my discipline;<br><br />
these creatures are really not in me;<br><br />
my self quickens creatures,<br><br />
sustaining them without being in them.<br><br />
&mdash; ''Bhagavad Gita,'' 9.5<br />
<br />
Just as the wide-moving wind<br><br />
is constantly present in space,<br><br />
so all creatures exist in me;<br><br />
understand it to be so!<br><br />
&mdash; ''Bhagavad Gita,'' 9.6<br />
<br />
I am heat that withholds<br><br />
and sends down the rains;<br><br />
I am immortality and death;<br><br />
both being and nonbeing am I.<br><br />
&mdash; ''Bhagavad Gita,'' 9.19<br />
<br />
I am the self abiding<br><br />
in the heart of all creatures;<br><br />
I am their beginning,<br><br />
their middle, and their end.<br><br />
&mdash; ''Bhagavad Gita,'' 10.20<br />
<br />
I am the endless cosmic serpent,<br><br />
the lord of all sea creatures;<br><br />
I am chief of the ancestral fathers;<br><br />
of restraints, I am death.<br><br />
&mdash; ''Bhagavad Gita,'' 10.29<br />
<br />
I am the pious son of demons;<br><br />
of measures, I am time;<br><br />
I am the lion among wild animals,<br><br />
the eagle among birds.<br><br />
&mdash; ''Bhagavad Gita,'' 10.30<br />
<br />
I am the vowel ''a'' of the syllabary,<br><br />
the pairing of words in a compound;<br><br />
I am indestructible time,<br><br />
the creator facing everywhere at once.<br><br />
&mdash; ''Bhagavad Gita,'' 10.33<br />
<br />
I am death the destroyer of all,<br><br />
the source of what will be,<br><br />
the feminine powers: fame, fortune, speech,<br><br />
memory, intelligence, resolve, patience.<br><br />
&mdash; ''Bhagavad Gita,'' 10.34<br />
<br />
Whatever is powerful, lucid,<br><br />
splendid, or invulnerable<br><br />
has its source in a fragment<br><br />
of my brilliance.<br><br />
&mdash; ''Bhagavad Gita,'' 10.41<br />
<br />
What use is so much knowledge<br><br />
to you, Arjuna?<br><br />
I stand sustaining this entire world<br><br />
with a fragment of my being.<br><br />
&mdash; ''Bhagavad Gita,'' 10.42<br />
<br />
Arjuna, see my forms<br><br />
in hundreds and thousands;<br><br />
diverse, divine,<br><br />
of many colors and shapes.<br><br />
&mdash; ''Bhagavad Gita,'' 11.5<br />
<br />
I am time grown old,<br><br />
creating world destruction,<br><br />
set in motion<br><br />
to annihilate the worlds; &hellip;<br><br />
&mdash; ''Bhagavad Gita,'' 11.32<br />
<br />
This form you have seen<br><br />
is rarely revealed;<br><br />
the gods are constantly craving<br><br />
for a vision of this form.<br><br />
&mdash; ''Bhagavad Gita,'' 11.52<br />
<br />
Knowledge is better than practice,<br><br />
meditation better than knowledge,<br><br />
rejecting fruits of action<br><br />
is better still &mdash; it brings peace.<br><br />
&mdash; ''Bhagavad Gita,'' 12.12<br />
<br />
The field denotes<br><br />
this body, and wise men<br><br />
call one who knows it<br><br />
the field-knower.<br><br />
&mdash; ''Bhagavad Gita,'' 13.1<br />
<br />
Know me as the field-knower<br><br />
in all fields &mdash; what I deem<br><br />
to be knowledge is knowledge<br><br />
of the field and its knower.<br><br />
&mdash; ''Bhagavad Gita,'' 13.2<br />
<br />
Hear from me in summary<br><br />
what the field is<br><br />
in its character and changes,<br><br />
and of the field-knower's power.<br><br />
&mdash; ''Bhagavad Gita,'' 13.3<br />
<br />
Ancient seers have sung of this<br><br />
in many ways, with varied meters<br><br />
and with aphorisms on the infinite spirit<br><br />
laced with logical arguments.<br><br />
&mdash; ''Bhagavad Gita,'' 13.4<br />
<br />
The field contains the great elements,<br><br />
individuality, understanding,<br><br />
unmanifest nature, the eleven senses,<br><br />
and the five sense realms.<br><br />
&mdash; ''Bhagavad Gita,'' 13.5<br />
<br />
Its hands and feet reach everywhere;<br><br />
its head and face see in every direction;<br><br />
hearing everything, it remains<br><br />
in the world, enveloping all.<br><br />
&mdash; ''Bhagavad Gita,'' 13.13<br />
<br />
Lacking all the sense organs,<br><br />
it shines in their qualities;<br><br />
unattached, it supports everything;<br><br />
without qualities, it enjoys them.<br><br />
&mdash; ''Bhagavad Gita,'' 13.14<br />
<br />
Outside and within all creatures,<br><br />
inanimate but still animate,<br><br />
too subtle to be known,<br><br />
it is far distant, yet near.<br><br />
&mdash; ''Bhagavad Gita,'' 13.15<br />
<br />
Undivided, it seems divided<br><br />
among creatures;<br><br />
understood as their sustainer,<br><br />
it devours and creates them.<br><br />
&mdash; ''Bhagavad Gita,'' 13.16<br />
<br />
The light of lights<br><br />
beyond darkness it is called;<br><br />
knowledge attained by knowledge,<br><br />
fixed in the heart of everyone.<br><br />
&mdash; ''Bhagavad Gita,'' 13.17<br />
<br />
So, in summary I have explained<br><br />
the field and knowledge of it;<br><br />
a man devoted to me, knowing this,<br><br />
enters into my being.<br><br />
&mdash; ''Bhagavad Gita,'' 13.18<br />
<br />
Arjuna, know that anything<br><br />
inanimate or alive with motion<br><br />
is born from the union<br><br />
of the field and its knower.<br><br />
&mdash; ''Bhagavad Gita,'' 13.26<br />
<br />
He really sees who sees<br><br />
that all actions are performed<br><br />
by nature alone and that the self<br><br />
is not an actor.<br><br />
&mdash; ''Bhagavad Gita,'' 13.29<br />
<br />
When he perceives the unity<br><br />
existing in separate creatures<br><br />
and how they expand from unity,<br><br />
he attains the infinite spirit.<br><br />
&mdash; ''Bhagavad Gita,'' 13.30<br />
<br />
Just as one sun<br><br />
illumines this entire world,<br><br />
so the master of the field<br><br />
illumines the entire field.<br><br />
&mdash; ''Bhagavad Gita,'' 13.33<br />
<br />
They reach the highest state<br><br />
who with the eye of knowledge know<br><br />
the boundary between the knower and its field,<br><br />
and the freedom creatures have from nature.<br><br />
&mdash; ''Bhagavad Gita,'' 13.34<br />
<br />
Lucidity, passion, dark inertia &mdash;<br><br />
these qualities inherent in nature<br><br />
bind the unchanging<br><br />
embodied self in the body.<br><br />
&mdash; ''Bhagavad Gita,'' 14.5<br />
<br />
Lucidity addicts one to joy,<br><br />
and passion to actions,<br><br />
but dark inertia obscures knowledge<br><br />
and addicts one to negligence.<br><br />
&mdash; ''Bhagavad Gita,'' 14.9<br />
<br />
When a man of vision sees<br><br />
nature's qualities as the agent<br><br />
of action and knows what lies beyond,<br><br />
he enters into my being.<br><br />
&mdash; ''Bhagavad Gita,'' 14.19<br />
<br />
Transcending the three qualities<br><br />
that are the body's source, the self<br><br />
achieves immortality, freed from the sorrows<br><br />
of birth, death, and old age.<br><br />
&mdash; ''Bhagavad Gita,'' 14.20<br />
<br />
Lord, what signs mark a man<br><br />
who passes beyond the three qualities?<br><br />
What does he do to cross<br><br />
beyond these qualities?<br><br />
&mdash; Arjuna, ''Bhagavad Gita,'' 14.21<br />
<br />
He does not dislike light<br><br />
or activity or delusion;<br><br />
when they cease to exist<br><br />
he does not desire them.<br><br />
&mdash; Krishna, ''Bhagavad Gita,'' 14.22<br />
<br />
He remains disinterested,<br><br />
unmoved by qualities of nature;<br><br />
he never wavers, knowing<br><br />
that only qualities are in motion.<br><br />
&mdash; ''Bhagavad Gita,'' 14.23<br />
<br />
I am the infinite spirit's foundation,<br><br />
immortal and immutable,<br><br />
the basis of eternal sacred duty<br><br />
and of perfect joy.<br><br />
&mdash; ''Bhagavad Gita,'' 14.27<br />
<br />
Roots in the air, branches below,<br><br />
the tree of life is unchanging,<br><br />
they say; its leaves are hymns,<br><br />
and he who knows it knows sacred lore.<br><br />
&mdash; ''Bhagavad Gita,'' 15.1<br />
<br />
Its branches<br><br />
stretch below and above,<br><br />
nourished by nature's qualities,<br><br />
budding with sense objects;<br><br />
aerial roots<br><br />
tangled in actions<br><br />
reach downward<br><br />
into the world of men.<br><br />
&mdash; ''Bhagavad Gita,'' 15.2<br />
<br />
Its form is unknown<br><br />
here in the world;<br><br />
unknown are its end,<br><br />
its beginning, its extent;<br><br />
cut down this tree<br><br />
that has such deep roots<br><br />
with the sharp ax<br><br />
of detachment.<br><br />
&mdash; ''Bhagavad Gita,'' 15.3<br />
<br />
Then search to find<br><br />
the realm<br><br />
that one enters<br><br />
without returning:<br><br />
&ldquo;I seek refuge<br><br />
in the original spirit of man,<br><br />
from which primordial<br><br />
activity extended.&rdquo;<br><br />
&mdash; ''Bhagavad Gita,'' 15.4<br />
<br />
A fragment of me in the living world<br><br />
is the timeless essence of life;<br><br />
it draws out the senses<br><br />
and the mind inherent in nature.<br><br />
&mdash; ''Bhagavad Gita,'' 15.7<br />
<br />
I dwell deep<br><br />
in the heart of everyone;<br><br />
memory, knowledge,<br><br />
and reasoning come from me;<br><br />
I am the object to be known<br><br />
through all sacred lore;<br><br />
and I am its knower,<br><br />
the creator of its final truth.<br><br />
&mdash; ''Bhagavad Gita,'' 15.15<br />
<br />
Demonic men cannot comprehend<br><br />
activity and rest;<br><br />
there exists no clarity,<br><br />
no morality, no truth in them.<br><br />
&mdash; ''Bhagavad Gita,'' 16.7<br />
<br />
They say that the world<br><br />
has no truth, no basis, no god,<br><br />
that no power of mutual dependence<br><br />
is its cause, but only desire.<br><br />
&mdash; ''Bhagavad Gita,'' 16.8<br />
<br />
Mired in this view, lost to themselves<br><br />
with their meager understanding,<br><br />
these fiends contrive terrible acts<br><br />
to destroy the world.<br><br />
&mdash; ''Bhagavad Gita,'' 16.9<br />
<br />
Subject to insatiable desire,<br><br />
drunk with hypocrisy and pride,<br><br />
holding false notions from delusion,<br><br />
they act with impure vows.<br><br />
&mdash; ''Bhagavad Gita,'' 16.10<br />
<br />
In their certainty that life<br><br />
consists in sating their desires,<br><br />
they suffer immeasurable anxiety<br><br />
that ends only with death.<br><br />
&mdash; ''Bhagavad Gita,'' 16.11<br />
<br />
Confused by endless thoughts,<br><br />
caught in the net of delusion,<br><br />
given to satisfying their desires,<br><br />
they fall into hell's foul abyss.<br><br />
&mdash; ''Bhagavad Gita,'' 16.16<br />
<br />
The three gates of hell<br><br />
that destroy the self<br><br />
are desire, anger, and greed;<br><br />
one must relinquish all three.<br><br />
&mdash; ''Bhagavad Gita,'' 16.21<br />
<br />
Let tradition be your standard<br><br />
in judging what to do or avoid;<br><br />
knowing the norms of tradition,<br><br />
perform your action here.<br><br />
&mdash; ''Bhagavad Gita,'' 16.24<br />
<br />
Given in due time and place<br><br />
to a fit recipient<br><br />
who can give no advantage,<br><br />
charity is remembered as lucid.<br><br />
&mdash; ''Bhagavad Gita,'' 17.20<br />
<br />
''OM TAT SAT:'' &ldquo;That Is the Real&rdquo; &mdash;<br><br />
this is the triple symbol of the infinite spirit<br><br />
that gave a primordial sanctity<br><br />
to priests, sacred lore, and sacrifice.<br><br />
&mdash; ''Bhagavad Gita,'' 17.23<br />
<br />
''SAT'' is steadfastness in sacrifice,<br><br />
in penance, in charity;<br><br />
any action of this order<br><br />
is denoted by ''SAT.''<br><br />
&mdash; ''Bhagavad Gita,'' 17.27<br />
<br />
Action in sacrifice, charity,<br><br />
and penance is to be performed,<br><br />
not relinquished &mdash; for wise men,<br><br />
they are acts of sanctity.<br><br />
&mdash; ''Bhagavad Gita,'' 18.5<br />
<br />
But even these actions<br><br />
should be done by relinquishing to me<br><br />
attachment and the fruit of action &mdash;<br><br />
this is my decisive idea.<br><br />
&mdash; ''Bhagavad Gita,'' 18.6<br />
<br />
Knowledge, its object, and its subject<br><br />
are the triple stimulus of action;<br><br />
instrument, act, and agent<br><br />
are the constituents of action.<br><br />
&mdash; ''Bhagavad Gita,'' 18.18<br />
<br />
Knowledge, action, agent are threefold,<br><br />
differentiated by qualities of nature;<br><br />
hear how this has been explained<br><br />
in the philosophical analysis of qualities.<br><br />
&mdash; ''Bhagavad Gita,'' 18.19<br />
<br />
Know that through lucid knowledge<br><br />
one sees in all creatures<br><br />
a single, unchanging existence,<br><br />
undivided within its divisions.<br><br />
&mdash; ''Bhagavad Gita,'' 18.20<br />
<br />
Action known for its lucidity<br><br />
is necessary, free of attachment,<br><br />
performed without attraction or hatred<br><br />
by one who seeks no fruit.<br><br />
&mdash; ''Bhagavad Gita,'' 18.23<br />
<br />
An agent called pure<br><br />
has no attachment or individualism,<br><br />
is resolute and energetic,<br><br />
unchanged in failure and success.<br><br />
&mdash; ''Bhagavad Gita,'' 18.26<br />
<br />
Listen as I tell you without reserve<br><br />
about understanding and resolve,<br><br />
each in three aspects,<br><br />
according to the qualities of nature.<br><br />
&mdash; ''Bhagavad Gita,'' 18.29<br />
<br />
In one who knows activity and rest,<br><br />
acts of right and wrong,<br><br />
bravery and fear, bondage and freedom,<br><br />
understanding is lucid.<br><br />
&mdash; ''Bhagavad Gita,'' 18.30<br />
<br />
When it sustains acts<br><br />
of mind, breath, and senses<br><br />
through discipline without wavering,<br><br />
resolve is lucid.<br><br />
&mdash; ''Bhagavad Gita,'' 18.33<br />
<br />
Arjuna, now hear about joy,<br><br />
the three ways of finding delight<br><br />
through practice<br><br />
that brings an end to suffering.<br><br />
&mdash; ''Bhagavad Gita,'' 18.36<br />
<br />
The joy of lucidity<br><br />
at first seems like poison<br><br />
but is in the end like ambrosia,<br><br />
from the calm of self-understanding.<br><br />
&mdash; ''Bhagavad Gita,'' 18.37<br />
<br />
The joy that is passion<br><br />
ateat first seems like ambrosia<br><br />
when senses encounter sense objects,<br><br />
but in the end it is like poison.<br><br />
&mdash; ''Bhagavad Gita,'' 18.38<br />
<br />
The joy arising from sleep,<br><br />
laziness, and negligence,<br><br />
self-deluding from beginning to end,<br><br />
is said to be darkly inert.<br><br />
&mdash; ''Bhagavad Gita,'' 18.39<br />
<br />
There is no being on earth<br><br />
or among the gods in heaven<br><br />
free from the triad of qualities<br><br />
that are born of nature.<br><br />
&mdash; ''Bhagavad Gita,'' 18.40<br />
<br />
Better to do one's own duty imperfectly<br><br />
than to do another man's well;<br><br />
doing action intrinsic to his being,<br><br />
a man avoids guilt.<br><br />
&mdash; ''Bhagavad Gita,'' 18.47<br />
<br />
Your resolve is futile<br><br />
if a sense of individuality<br><br />
makes you think, &ldquo;I shall not fight&rdquo; &mdash;<br><br />
nature will compel you to.<br><br />
&mdash; ''Bhagavad Gita,'' 18.59<br />
<br />
You are bound by your own action,<br><br />
intrinsic to your being, Arjuna;<br><br />
even against your will you must do<br><br />
what delusion now makes you refuse.<br><br />
&mdash; ''Bhagavad Gita,'' 18.60<br />
<br />
Arjuna, the lord resides<br><br />
in the heart of all creatures,<br><br />
making them reel magically,<br><br />
as if a machine moved them.<br><br />
&mdash; ''Bhagavad Gita,'' 18.61<br />
<br />
With your whole being, Arjuna,<br><br />
take refuge in him alone &mdash;<br><br />
from his grace you will attain<br><br />
the eternal place that is peace.<br><br />
&mdash; ''Bhagavad Gita,'' 18.62<br />
<br />
This knowledge I have taught<br><br />
is more arcane than any mystery &mdash;<br><br />
consider it completely,<br><br />
then act as you choose.<br><br />
&mdash; ''Bhagavad Gita,'' 18.63<br />
<br />
'''Key Words in the ''Bhagavad-Gita'''''<br />
<br />
[Compiled by the Translator, B.S. Miller, p. 162&ndash;168]<br />
<br />
Time (''k&atilde;la'') &mdash; a word that also means &ldquo;death&rdquo;. In Indian thought time is without beginning, endless, all-pervading. In the ''Gita,'' Krishna identifies himself as indestructible time that destroys the worlds (10.30, 10.33, 11.32). &mdash; B.S. Miller (translator of ''B-G,'' p.&nbsp;168)<br />
<br />
<br><br />
<br />
==Deletions==<br />
<br />
It is pleasing to contemplate the whole panorama of differential operations and group actions on a proposition as so many alternate scenes projected through the facets of its differential enlargement onto various perspectives taken up by the viewer.<br />
<br />
We hope to work toward a point where the reader finds it pleasing to contemplate the whole panorama of differential operations and group actions on a proposition as just so many facets of its differential enlargement as may be projected onto various perspectives.<br />
<br />
Let <math>X = \{ x_1, \ldots, x_n \}\!</math> and suppose that <math>U\!</math> is any subset of these variables, without loss of generality taking <math>U = \{ x_1, \ldots, x_j \},\!</math> for some <math>j \in [1, n].\!</math> For convenience, let us give a name to the remaining set of variables, say <math>V = \{ x_{j+1}, \ldots, x_n \}.\!</math><br />
<br />
Let <math>X = \{ x_1, \ldots, x_n \}\!</math> be our principal set of logical variables, and let <math>U\!</math> be any subset of these, assumed without loss of generality to form an initial segment <math>U = \{ x_1, \ldots, x_j \}\!</math> of cardinality <math>j.\!</math> For convenience, let the remaining variables be collected in the set <math>V = \{ x_{j+1}, \ldots, x_n \}\!</math> of cardinality <math>k = n - j.\!</math><br />
<br />
Given a proposition whose form of expression <math>F\!</math> is more or less difficult to comprehend it may occur to us that the matter could be simpler if we restrict our attention to a subset of the universe, say one described by the proposition <math>G,\!</math> expressed over the same set of variables as <math>F.\!</math><br />
<br />
{| align="center" cellpadding="8"<br />
| <math>F | G ~=~ \varphi_G F\!</math><br />
|}<br />
<br />
{| align="center" cellpadding="8" style="text-align:center"<br />
|<br />
<math>\begin{matrix}<br />
\mathbb{B}^j<br />
& \to &<br />
\mathbb{B}^n<br />
& \to &<br />
\mathbb{B}^k<br />
\\[2pt]<br />
(\mathbb{B}^j \to \mathbb{B})<br />
& \to &<br />
(\mathbb{B}^n \to \mathbb{B})<br />
& \to &<br />
(\mathbb{B}^k \to \mathbb{B})<br />
\end{matrix}</math><br />
|}<br />
<br />
{| align="center" cellpadding="8" style="text-align:center"<br />
|<br />
<math>\begin{matrix}<br />
U<br />
& \to &<br />
X<br />
& \to &<br />
V<br />
\\[2pt]<br />
(U \to \mathbb{B})<br />
& \to &<br />
(X \to \mathbb{B})<br />
& \to &<br />
(V \to \mathbb{B})<br />
\end{matrix}</math><br />
|}<br />
<br />
{| align="center" cellpadding="8" style="text-align:center"<br />
|<br />
<math>\begin{matrix}<br />
\mathbb{B}^j<br />
& \to &<br />
\mathbb{B}^n<br />
& \to &<br />
\mathbb{B}^j \times \mathbb{B}^k<br />
& \to &<br />
&&<br />
\\[2pt]<br />
(\mathbb{B}^j \to \mathbb{B})<br />
& \to &<br />
(\mathbb{B}^n \to \mathbb{B})<br />
& \to &<br />
(\mathbb{B}^j \times \mathbb{B}^k \to \mathbb{B})<br />
& \to &<br />
(\mathbb{B}^j \to (\mathbb{B}^k \to \mathbb{B}))<br />
\end{matrix}</math><br />
|}<br />
<br />
Diffraction = differential factorization (or differential fractionation):<br />
<br />
{| align="center" cellpadding="8"<br />
|<br />
<math>\begin{matrix}<br />
(\mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B})<br />
& \cong &<br />
(\mathbb{D}^n \to (\mathbb{B}^n \to \mathbb{B}))<br />
\end{matrix}</math><br />
|}<br />
<br />
We aim to combine these twin perspectives into a stereoscopic view of the situations represented in propositions.<br />
<br />
Between the contrasting perspectives afforded by these two <math>\iota\!</math>'s we have what we need to synthesize a stereologic conception of the situation represented in a proposition.<br />
<br />
Between the complementary perspectives afforded by these two <math>\iota\!</math>'s we aim to acquire / struggle to compose a stereoscopic vision of the situations represented in propositions.<br />
<br />
Between the complementary perspectives afforded by these two <math>\iota\!</math>'s we hope to fashion / strive to fasten a stereotactic grasp on the situations represented in propositions.<br />
<br />
==Document History==<br />
<br />
Author: Jon Awbrey, August 31, 1995<br><br />
Course: Engineering 690, Graduate Project<br><br />
Continued from Winter Term 1995<br><br />
Supervisors: M.A. Zohdy and F. Mili, Oakland University<br />
<br />
*</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=User:Jon_Awbrey/Exploratory_Qualitative_Analysis_of_Sequential_Observation_Data&diff=469865User:Jon Awbrey/Exploratory Qualitative Analysis of Sequential Observation Data2020-11-14T17:08:38Z<p>Jon Awbrey: ad user page</p>
<hr />
<div>'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''<br />
<br />
==Family Interaction Study &bull; Comments==<br />
<br />
===Note 1===<br />
<br />
Here is one vignette that comes to mind under the &ldquo;knowledge soup&rdquo; category, at least in the sense that one has to back up and explore the unstructured aspects of learning and reasoning processes, as they manifest themselves over time, for example, in sequential interaction data.<br />
<br />
I will give you the codebooks and a sample raw dataset first, just in case you want to do a bit of undirected exploratory<br />
data analysis, and later I'll explain the set-up and its implications in full detail.<br />
<br />
====Family Interaction Data &bull; Single Event Codes====<br />
<br />
<pre><br />
f#_family_member<br />
f1_child<br />
f2_father<br />
f3_mother<br />
f4_older_brother<br />
f5_older_sister<br />
f6_younger_brother<br />
f7_younger_sister<br />
f8_household_pets<br />
f9_multiple_recipients<br />
f0_objects<br />
c##_content<br />
c1#_conversation<br />
c11_positive_verbal<br />
c12_talk<br />
c13_negative_verbal<br />
c2#_affiliate/distance<br />
c21_endearment<br />
c22_tease<br />
c23_verbal_attack<br />
c3#_clear_directive<br />
c31_request<br />
c32_command<br />
c33_coerce<br />
c4#_ambiguous_directive<br />
c41_request_ambiguous<br />
c42_command_ambiguous<br />
c43_coerce_ambiguous<br />
c5#_response_to_directive<br />
c51_agree<br />
c52<br />
c53_refuse<br />
c6#_vocal_behavior<br />
c61<br />
c62_vocal<br />
c63<br />
c7#_nonverbal_behavior<br />
c71_positive_nonverbal<br />
c72_neutral_nonverbal<br />
c73_negative_nonverbal<br />
c8#_low_grade_physical_contact<br />
c81_touch<br />
c82<br />
c83_physical_aggression<br />
c9#_pronounced_physical_interaction<br />
c91_hold<br />
c92_physical_interact<br />
c93_physical_attack<br />
c0#_compliance_behavior<br />
c01_comply<br />
c02<br />
c03_noncomply<br />
v#_valence<br />
v1_exuberant_affect<br />
v2_positive_affect<br />
v3_neutral_affect<br />
v4_negative_affect<br />
v5_extreme_negative_affect<br />
</pre><br />
<br />
====Family Interaction Data &bull; Transition Event Codes====<br />
<br />
<pre><br />
j1_f*_family_member<br />
j1_f#_null_value<br />
j1_f1_child<br />
j1_f2_father<br />
j1_f3_mother<br />
j2_c*_content<br />
j2_c#_null_value<br />
j2_c1_conversation<br />
j2_c2_affiliate/distance<br />
j2_c3_clear_directive<br />
j2_c4_ambiguous_directive<br />
j2_c5_response_to_directive<br />
j2_c6_vocal_behavior<br />
j2_c7_nonverbal_behavior<br />
j2_c8_low_grade_physical_contact<br />
j2_c9_pronounced_physical_interaction<br />
j2_c0_compliance_behavior<br />
j3_q*_quality<br />
j3_q#_null_value<br />
j3_q1_positive<br />
j3_q2_neutral<br />
j3_q3_negative<br />
j4_v*_valence<br />
j4_v#_null_value<br />
j4_v1_exuberant_affect<br />
j4_v2_positive_affect<br />
j4_v3_neutral_affect<br />
j4_v4_negative_affect<br />
j4_v5_extreme_negative_affect<br />
k1_f*_family_member<br />
k1_f#_null_value<br />
k1_f1_child<br />
k1_f2_father<br />
k1_f3_mother<br />
k2_c*_content<br />
k2_c#_null_value<br />
k2_c1_conversation<br />
k2_c2_affiliate/distance<br />
k2_c3_clear_directive<br />
k2_c4_ambiguous_directive<br />
k2_c5_response_to_directive<br />
k2_c6_vocal_behavior<br />
k2_c7_nonverbal_behavior<br />
k2_c8_low_grade_physical_contact<br />
k2_c9_pronounced_physical_interaction<br />
k2_c0_compliance_behavior<br />
k3_q*_quality<br />
k3_q#_null_value<br />
k3_q1_positive<br />
k3_q2_neutral<br />
k3_q3_negative<br />
k4_v*_valence<br />
k4_v#_null_value<br />
k4_v1_exuberant_affect<br />
k4_v2_positive_affect<br />
k4_v3_neutral_affect<br />
k4_v4_negative_affect<br />
k4_v5_extreme_negative_affect<br />
</pre><br />
<br />
====Family Interaction Study &bull; Observational Dataset====<br />
<br />
<pre><br />
------------------------------------------------------------------------<br />
TRIAL FAMILY SESSION FOCUS OBSERVER FAMILY MEMBERS MO/DY/YR HR:MN<br />
1 MXYZ 1 1 22 13000000 12/03/86 12:58<br />
------------------------------------------------------------------------<br />
<br />
Family Interaction Transitions, Observational Data (FIT.OBS) File:<br />
<br />
0 -00100 0<br />
0 99999 0<br />
2 -00302 2<br />
2 31213 5<br />
2 11233 8<br />
2 34213 10<br />
2 33213 18<br />
2 10133 19<br />
2 14233 22<br />
2 30113 24<br />
2 99999 29<br />
2 31113 40<br />
2 11233 43<br />
2 31212 46<br />
2 11232 51<br />
2 31213 54<br />
2 11232 56<br />
2 99999 61<br />
2 11233 68<br />
2 36213 70<br />
2 33213 73<br />
2 16232 74<br />
2 33213 77<br />
2 16232 78<br />
2 31312 82<br />
2 99999 91<br />
2 11233 94<br />
2 31213 96<br />
2 99999 101<br />
2 31112 103<br />
2 11232 106<br />
2 31213 108<br />
2 11232 109<br />
2 36212 112<br />
2 16232 115<br />
2 11232 120<br />
2 31212 122<br />
2 11232 127<br />
2 16232 131<br />
2 11232 136<br />
2 16232 138<br />
2 31213 143<br />
2 11232 148<br />
2 16232 155<br />
2 31212 160<br />
2 16232 168<br />
2 31213 170<br />
2 33213 175<br />
2 10133 177<br />
2 35113 184<br />
2 31213 187<br />
2 30113 189<br />
2 11233 192<br />
2 31213 194<br />
2 11233 197<br />
2 31213 200<br />
2 11232 202<br />
2 16232 205<br />
2 99999 211<br />
2 11232 214<br />
2 31213 218<br />
2 13233 222<br />
2 35113 225<br />
2 31213 227<br />
2 11233 228<br />
2 30113 230<br />
2 11234 234<br />
2 31212 238<br />
2 33213 241<br />
2 10133 242<br />
2 31213 244<br />
2 11232 245<br />
2 31213 252<br />
2 11233 254<br />
2 31213 256<br />
2 16233 267<br />
2 11232 269<br />
2 11231 278<br />
2 34213 282<br />
2 10133 287<br />
2 31213 289<br />
2 11232 291<br />
2 31213 296<br />
2 13133 301<br />
2 35113 305<br />
2 11233 309<br />
2 31213 311<br />
2 11233 313<br />
2 31213 316<br />
2 11233 319<br />
2 31213 322<br />
2 11233 323<br />
2 31213 324<br />
2 11233 326<br />
2 33213 329<br />
2 10133 331<br />
2 11233 333<br />
2 31112 338<br />
2 11233 342<br />
2 31213 344<br />
2 11233 347<br />
2 31213 349<br />
2 11233 351<br />
2 31213 352<br />
2 16233 354<br />
2 36212 356<br />
2 11233 362<br />
2 31213 367<br />
2 11233 370<br />
2 31213 371<br />
2 11232 373<br />
2 31212 378<br />
2 11232 381<br />
2 31213 382<br />
2 11233 389<br />
2 31213 392<br />
2 34213 394<br />
2 10133 396<br />
2 31113 398<br />
2 11233 401<br />
2 31213 403<br />
2 11233 405<br />
2 31212 406<br />
2 11233 409<br />
2 31213 412<br />
2 11233 414<br />
2 31213 416<br />
2 36212 417<br />
2 16232 419<br />
2 31312 422<br />
2 33213 426<br />
2 33213 430<br />
2 10133 431<br />
2 10133 434<br />
2 31213 438<br />
2 11233 439<br />
2 31213 440<br />
2 11233 441<br />
2 31213 443<br />
2 11233 449<br />
2 31112 450<br />
2 31212 455<br />
2 16232 457<br />
2 31213 462<br />
2 11232 463<br />
2 31212 467<br />
2 16232 469<br />
2 31213 473<br />
2 11233 477<br />
2 31213 479<br />
2 11133 490<br />
2 31113 492<br />
2 31213 495<br />
2 16232 514<br />
2 31112 517<br />
2 16232 524<br />
2 11232 527<br />
2 31213 530<br />
2 16232 534<br />
2 36212 537<br />
2 32212 539<br />
2 16232 544<br />
2 31213 547<br />
2 11233 548<br />
2 31213 549<br />
2 11232 553<br />
2 99999 561<br />
2 16212 564<br />
2 31213 569<br />
2 11232 575<br />
2 31314 577<br />
2 31213 585<br />
2 16232 589<br />
2 33213 591<br />
2 10133 593<br />
2 11233 595<br />
2 31213 599<br />
2 88888 603<br />
0 -00500 605<br />
</pre><br />
<br />
====Family Interaction Transitions &bull; Logical Representation====<br />
<br />
File Type : FIT.LOG<br />
<br />
<pre><br />
(( j1_f# j2_c# j3_q# j4_v# k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c1 k3_q2 k4_v3<br />
)( j1_f1 j2_c1 j3_q2 j4_v3 k1_f3 k2_c4 k3_q2 k4_v3<br />
)( j1_f3 j2_c4 j3_q2 j4_v3 k1_f3 k2_c3 k3_q2 k4_v3<br />
)( j1_f3 j2_c3 j3_q2 j4_v3 k1_f1 k2_c0 k3_q1 k4_v3<br />
)( j1_f1 j2_c0 j3_q1 j4_v3 k1_f1 k2_c4 k3_q2 k4_v3<br />
)( j1_f1 j2_c4 j3_q2 j4_v3 k1_f3 k2_c0 k3_q1 k4_v3<br />
)( j1_f3 j2_c0 j3_q1 j4_v3 k1_f# k2_c# k3_q# k4_v#<br />
)( j1_f# j2_c# j3_q# j4_v# k1_f3 k2_c1 k3_q1 k4_v3<br />
)( j1_f3 j2_c1 j3_q1 j4_v3 k1_f1 k2_c1 k3_q2 k4_v3<br />
)( j1_f1 j2_c1 j3_q2 j4_v3 k1_f3 k2_c1 k3_q2 k4_v2<br />
)( j1_f3 j2_c1 j3_q2 j4_v2 k1_f1 k2_c1 k3_q2 k4_v2<br />
)( j1_f1 j2_c1 j3_q2 j4_v2 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c1 k3_q2 k4_v2<br />
)( j1_f1 j2_c1 j3_q2 j4_v2 k1_f# k2_c# k3_q# k4_v#<br />
)( j1_f# j2_c# j3_q# j4_v# k1_f1 k2_c1 k3_q2 k4_v3<br />
)( j1_f1 j2_c1 j3_q2 j4_v3 k1_f3 k2_c6 k3_q2 k4_v3<br />
)( j1_f3 j2_c6 j3_q2 j4_v3 k1_f3 k2_c3 k3_q2 k4_v3<br />
)( j1_f3 j2_c3 j3_q2 j4_v3 k1_f1 k2_c6 k3_q2 k4_v2<br />
)( j1_f1 j2_c6 j3_q2 j4_v2 k1_f3 k2_c3 k3_q2 k4_v3<br />
)( j1_f3 j2_c3 j3_q2 j4_v3 k1_f1 k2_c6 k3_q2 k4_v2<br />
)( j1_f1 j2_c6 j3_q2 j4_v2 k1_f3 k2_c1 k3_q3 k4_v2<br />
)( j1_f3 j2_c1 j3_q3 j4_v2 k1_f# k2_c# k3_q# k4_v#<br />
)( j1_f# j2_c# j3_q# j4_v# k1_f1 k2_c1 k3_q2 k4_v3<br />
)( j1_f1 j2_c1 j3_q2 j4_v3 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f# k2_c# k3_q# k4_v#<br />
)( j1_f# j2_c# j3_q# j4_v# k1_f3 k2_c1 k3_q1 k4_v2<br />
)( j1_f3 j2_c1 j3_q1 j4_v2 k1_f1 k2_c1 k3_q2 k4_v2<br />
)( j1_f1 j2_c1 j3_q2 j4_v2 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c1 k3_q2 k4_v2<br />
)( j1_f1 j2_c1 j3_q2 j4_v2 k1_f3 k2_c6 k3_q2 k4_v2<br />
)( j1_f3 j2_c6 j3_q2 j4_v2 k1_f1 k2_c6 k3_q2 k4_v2<br />
)( j1_f1 j2_c6 j3_q2 j4_v2 k1_f1 k2_c1 k3_q2 k4_v2<br />
)( j1_f1 j2_c1 j3_q2 j4_v2 k1_f3 k2_c1 k3_q2 k4_v2<br />
)( j1_f3 j2_c1 j3_q2 j4_v2 k1_f1 k2_c1 k3_q2 k4_v2<br />
)( j1_f1 j2_c1 j3_q2 j4_v2 k1_f1 k2_c6 k3_q2 k4_v2<br />
)( j1_f1 j2_c6 j3_q2 j4_v2 k1_f1 k2_c1 k3_q2 k4_v2<br />
)( j1_f1 j2_c1 j3_q2 j4_v2 k1_f1 k2_c6 k3_q2 k4_v2<br />
)( j1_f1 j2_c6 j3_q2 j4_v2 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c1 k3_q2 k4_v2<br />
)( j1_f1 j2_c1 j3_q2 j4_v2 k1_f1 k2_c6 k3_q2 k4_v2<br />
)( j1_f1 j2_c6 j3_q2 j4_v2 k1_f3 k2_c1 k3_q2 k4_v2<br />
)( j1_f3 j2_c1 j3_q2 j4_v2 k1_f1 k2_c6 k3_q2 k4_v2<br />
)( j1_f1 j2_c6 j3_q2 j4_v2 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f3 k2_c3 k3_q2 k4_v3<br />
)( j1_f3 j2_c3 j3_q2 j4_v3 k1_f1 k2_c0 k3_q1 k4_v3<br />
)( j1_f1 j2_c0 j3_q1 j4_v3 k1_f3 k2_c5 k3_q1 k4_v3<br />
)( j1_f3 j2_c5 j3_q1 j4_v3 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f3 k2_c0 k3_q1 k4_v3<br />
)( j1_f3 j2_c0 j3_q1 j4_v3 k1_f1 k2_c1 k3_q2 k4_v3<br />
)( j1_f1 j2_c1 j3_q2 j4_v3 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c1 k3_q2 k4_v3<br />
)( j1_f1 j2_c1 j3_q2 j4_v3 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c1 k3_q2 k4_v2<br />
)( j1_f1 j2_c1 j3_q2 j4_v2 k1_f1 k2_c6 k3_q2 k4_v2<br />
)( j1_f1 j2_c6 j3_q2 j4_v2 k1_f# k2_c# k3_q# k4_v#<br />
)( j1_f# j2_c# j3_q# j4_v# k1_f1 k2_c1 k3_q2 k4_v2<br />
)( j1_f1 j2_c1 j3_q2 j4_v2 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c3 k3_q2 k4_v3<br />
)( j1_f1 j2_c3 j3_q2 j4_v3 k1_f3 k2_c5 k3_q1 k4_v3<br />
)( j1_f3 j2_c5 j3_q1 j4_v3 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c1 k3_q2 k4_v3<br />
)( j1_f1 j2_c1 j3_q2 j4_v3 k1_f3 k2_c0 k3_q1 k4_v3<br />
)( j1_f3 j2_c0 j3_q1 j4_v3 k1_f1 k2_c1 k3_q2 k4_v4<br />
)( j1_f1 j2_c1 j3_q2 j4_v4 k1_f3 k2_c1 k3_q2 k4_v2<br />
)( j1_f3 j2_c1 j3_q2 j4_v2 k1_f3 k2_c3 k3_q2 k4_v3<br />
)( j1_f3 j2_c3 j3_q2 j4_v3 k1_f1 k2_c0 k3_q1 k4_v3<br />
)( j1_f1 j2_c0 j3_q1 j4_v3 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c1 k3_q2 k4_v2<br />
)( j1_f1 j2_c1 j3_q2 j4_v2 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c1 k3_q2 k4_v3<br />
)( j1_f1 j2_c1 j3_q2 j4_v3 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c6 k3_q2 k4_v3<br />
)( j1_f1 j2_c6 j3_q2 j4_v3 k1_f1 k2_c1 k3_q2 k4_v2<br />
)( j1_f1 j2_c1 j3_q2 j4_v2 k1_f1 k2_c1 k3_q2 k4_v1<br />
)( j1_f1 j2_c1 j3_q2 j4_v1 k1_f3 k2_c4 k3_q2 k4_v3<br />
)( j1_f3 j2_c4 j3_q2 j4_v3 k1_f1 k2_c0 k3_q1 k4_v3<br />
)( j1_f1 j2_c0 j3_q1 j4_v3 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c1 k3_q2 k4_v2<br />
)( j1_f1 j2_c1 j3_q2 j4_v2 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c3 k3_q1 k4_v3<br />
)( j1_f1 j2_c3 j3_q1 j4_v3 k1_f3 k2_c5 k3_q1 k4_v3<br />
)( j1_f3 j2_c5 j3_q1 j4_v3 k1_f1 k2_c1 k3_q2 k4_v3<br />
)( j1_f1 j2_c1 j3_q2 j4_v3 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c1 k3_q2 k4_v3<br />
)( j1_f1 j2_c1 j3_q2 j4_v3 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c1 k3_q2 k4_v3<br />
)( j1_f1 j2_c1 j3_q2 j4_v3 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c1 k3_q2 k4_v3<br />
)( j1_f1 j2_c1 j3_q2 j4_v3 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c1 k3_q2 k4_v3<br />
)( j1_f1 j2_c1 j3_q2 j4_v3 k1_f3 k2_c3 k3_q2 k4_v3<br />
)( j1_f3 j2_c3 j3_q2 j4_v3 k1_f1 k2_c0 k3_q1 k4_v3<br />
)( j1_f1 j2_c0 j3_q1 j4_v3 k1_f1 k2_c1 k3_q2 k4_v3<br />
)( j1_f1 j2_c1 j3_q2 j4_v3 k1_f3 k2_c1 k3_q1 k4_v2<br />
)( j1_f3 j2_c1 j3_q1 j4_v2 k1_f1 k2_c1 k3_q2 k4_v3<br />
)( j1_f1 j2_c1 j3_q2 j4_v3 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c1 k3_q2 k4_v3<br />
)( j1_f1 j2_c1 j3_q2 j4_v3 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c1 k3_q2 k4_v3<br />
)( j1_f1 j2_c1 j3_q2 j4_v3 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c6 k3_q2 k4_v3<br />
)( j1_f1 j2_c6 j3_q2 j4_v3 k1_f3 k2_c6 k3_q2 k4_v2<br />
)( j1_f3 j2_c6 j3_q2 j4_v2 k1_f1 k2_c1 k3_q2 k4_v3<br />
)( j1_f1 j2_c1 j3_q2 j4_v3 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c1 k3_q2 k4_v3<br />
)( j1_f1 j2_c1 j3_q2 j4_v3 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c1 k3_q2 k4_v2<br />
)( j1_f1 j2_c1 j3_q2 j4_v2 k1_f3 k2_c1 k3_q2 k4_v2<br />
)( j1_f3 j2_c1 j3_q2 j4_v2 k1_f1 k2_c1 k3_q2 k4_v2<br />
)( j1_f1 j2_c1 j3_q2 j4_v2 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c1 k3_q2 k4_v3<br />
)( j1_f1 j2_c1 j3_q2 j4_v3 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f3 k2_c4 k3_q2 k4_v3<br />
)( j1_f3 j2_c4 j3_q2 j4_v3 k1_f1 k2_c0 k3_q1 k4_v3<br />
)( j1_f1 j2_c0 j3_q1 j4_v3 k1_f3 k2_c1 k3_q1 k4_v3<br />
)( j1_f3 j2_c1 j3_q1 j4_v3 k1_f1 k2_c1 k3_q2 k4_v3<br />
)( j1_f1 j2_c1 j3_q2 j4_v3 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c1 k3_q2 k4_v3<br />
)( j1_f1 j2_c1 j3_q2 j4_v3 k1_f3 k2_c1 k3_q2 k4_v2<br />
)( j1_f3 j2_c1 j3_q2 j4_v2 k1_f1 k2_c1 k3_q2 k4_v3<br />
)( j1_f1 j2_c1 j3_q2 j4_v3 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c1 k3_q2 k4_v3<br />
)( j1_f1 j2_c1 j3_q2 j4_v3 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f3 k2_c6 k3_q2 k4_v2<br />
)( j1_f3 j2_c6 j3_q2 j4_v2 k1_f1 k2_c6 k3_q2 k4_v2<br />
)( j1_f1 j2_c6 j3_q2 j4_v2 k1_f3 k2_c1 k3_q3 k4_v2<br />
)( j1_f3 j2_c1 j3_q3 j4_v2 k1_f3 k2_c3 k3_q2 k4_v3<br />
)( j1_f3 j2_c3 j3_q2 j4_v3 k1_f3 k2_c3 k3_q2 k4_v3<br />
)( j1_f3 j2_c3 j3_q2 j4_v3 k1_f1 k2_c0 k3_q1 k4_v3<br />
)( j1_f1 j2_c0 j3_q1 j4_v3 k1_f1 k2_c0 k3_q1 k4_v3<br />
)( j1_f1 j2_c0 j3_q1 j4_v3 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c1 k3_q2 k4_v3<br />
)( j1_f1 j2_c1 j3_q2 j4_v3 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c1 k3_q2 k4_v3<br />
)( j1_f1 j2_c1 j3_q2 j4_v3 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c1 k3_q2 k4_v3<br />
)( j1_f1 j2_c1 j3_q2 j4_v3 k1_f3 k2_c1 k3_q1 k4_v2<br />
)( j1_f3 j2_c1 j3_q1 j4_v2 k1_f3 k2_c1 k3_q2 k4_v2<br />
)( j1_f3 j2_c1 j3_q2 j4_v2 k1_f1 k2_c6 k3_q2 k4_v2<br />
)( j1_f1 j2_c6 j3_q2 j4_v2 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c1 k3_q2 k4_v2<br />
)( j1_f1 j2_c1 j3_q2 j4_v2 k1_f3 k2_c1 k3_q2 k4_v2<br />
)( j1_f3 j2_c1 j3_q2 j4_v2 k1_f1 k2_c6 k3_q2 k4_v2<br />
)( j1_f1 j2_c6 j3_q2 j4_v2 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c1 k3_q2 k4_v3<br />
)( j1_f1 j2_c1 j3_q2 j4_v3 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c1 k3_q1 k4_v3<br />
)( j1_f1 j2_c1 j3_q1 j4_v3 k1_f3 k2_c1 k3_q1 k4_v3<br />
)( j1_f3 j2_c1 j3_q1 j4_v3 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c6 k3_q2 k4_v2<br />
)( j1_f1 j2_c6 j3_q2 j4_v2 k1_f3 k2_c1 k3_q1 k4_v2<br />
)( j1_f3 j2_c1 j3_q1 j4_v2 k1_f1 k2_c6 k3_q2 k4_v2<br />
)( j1_f1 j2_c6 j3_q2 j4_v2 k1_f1 k2_c1 k3_q2 k4_v2<br />
)( j1_f1 j2_c1 j3_q2 j4_v2 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c6 k3_q2 k4_v2<br />
)( j1_f1 j2_c6 j3_q2 j4_v2 k1_f3 k2_c6 k3_q2 k4_v2<br />
)( j1_f3 j2_c6 j3_q2 j4_v2 k1_f3 k2_c2 k3_q2 k4_v2<br />
)( j1_f3 j2_c2 j3_q2 j4_v2 k1_f1 k2_c6 k3_q2 k4_v2<br />
)( j1_f1 j2_c6 j3_q2 j4_v2 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c1 k3_q2 k4_v3<br />
)( j1_f1 j2_c1 j3_q2 j4_v3 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c1 k3_q2 k4_v2<br />
)( j1_f1 j2_c1 j3_q2 j4_v2 k1_f# k2_c# k3_q# k4_v#<br />
)( j1_f# j2_c# j3_q# j4_v# k1_f1 k2_c6 k3_q2 k4_v2<br />
)( j1_f1 j2_c6 j3_q2 j4_v2 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c1 k3_q2 k4_v2<br />
)( j1_f1 j2_c1 j3_q2 j4_v2 k1_f3 k2_c1 k3_q3 k4_v4<br />
)( j1_f3 j2_c1 j3_q3 j4_v4 k1_f3 k2_c1 k3_q2 k4_v3<br />
)( j1_f3 j2_c1 j3_q2 j4_v3 k1_f1 k2_c6 k3_q2 k4_v2<br />
)( j1_f1 j2_c6 j3_q2 j4_v2 k1_f3 k2_c3 k3_q2 k4_v3<br />
)( j1_f3 j2_c3 j3_q2 j4_v3 k1_f1 k2_c0 k3_q1 k4_v3<br />
)( j1_f1 j2_c0 j3_q1 j4_v3 k1_f1 k2_c1 k3_q2 k4_v3<br />
)( j1_f1 j2_c1 j3_q2 j4_v3 k1_f3 k2_c1 k3_q2 k4_v3<br />
))<br />
<br />
(( j1_f# ),( j1_f1 ),( j1_f3 ))<br />
(( j2_c# ),( j2_c1 ),( j2_c2 ),( j2_c3 ),( j2_c4 ),( j2_c5 ),<br />
( j2_c6 ),( j2_c7 ),( j2_c8 ),( j2_c9 ),( j2_c0 ))<br />
(( j3_q# ),( j3_q1 ),( j3_q2 ),( j3_q3 ))<br />
(( j4_v# ),( j4_v1 ),( j4_v2 ),( j4_v3 ),( j4_v4 ),( j4_v5 ))<br />
<br />
((( j2_c7 )( j2_c8 )( j2_c9 )( j4_v5 )))<br />
<br />
(( k1_f# ),( k1_f1 ),( k1_f3 ))<br />
(( k2_c# ),( k2_c1 ),( k2_c2 ),( k2_c3 ),( k2_c4 ),( k2_c5 ),<br />
( k2_c6 ),( k2_c7 ),( k2_c8 ),( k2_c9 ),( k2_c0 ))<br />
(( k3_q# ),( k3_q1 ),( k3_q2 ),( k3_q3 ))<br />
(( k4_v# ),( k4_v1 ),( k4_v2 ),( k4_v3 ),( k4_v4 ),( k4_v5 ))<br />
<br />
((( k2_c7 )( k2_c8 )( k2_c9 )( k4_v5 )))<br />
</pre><br />
<br />
===Note 2===<br />
<br />
<pre><br />
| Since 1987, I have been arguing that no ontology or<br />
| knowledge base can ever be adequate unless it comes<br />
| to grips with what I have called the "knowledge soup" --<br />
| the loosely organized, semi-structured mix of whatever<br />
| people have in their heads.<br />
|<br />
| John Sowa<br />
| http://suo.ieee.org/email/msg10558.html<br />
</pre><br />
<br />
====Sequential Interaction Data====<br />
<br />
I will try to explain why this dataset is interesting to me, and why it comes to mind whenever we turn to discussing the associative "prima materia" or semiotic "massa confusa" out of which our more rational knowledge precipitates, which is one of the things that I take John Sowa to be talking about under the label of "knowledge soup".<br />
<br />
The general type of data we are dealing with here is a timed sequence of categorical or qualitative codes. I think of it as a sequence of "words" or "sentences" that come from a very simple formal language, and the game afoot is to discover the grammar of the discourse, to find whatever patterns may be found in the data, whether they rule it in the manner of an absolute law or a likely constraint.<br />
<br />
====Family Interaction Dataset====<br />
<br />
An "episode" is coded as a 7-tuple x = (x_0, x_1, x_2, x_3, x_4, x_5, x_6).<br />
<br />
The components x_0, ..., x_5 are categorical codes that are described in the codebook.<br />
<br />
The component x_6 is a non-negative integer code that records the elapsed time in seconds at which the episode ends.<br />
<br />
Here is an initial segment of an individual data file, a set of records from a single session of observation:<br />
<br />
<pre><br />
| Family Interaction Data<br />
|<br />
| 0 -00100 0<br />
| 0 99999 0<br />
| 2 -00302 2<br />
| 2 31213 5<br />
| 2 11233 8<br />
| 2 34213 10<br />
| 2 33213 18<br />
| 2 10133 19<br />
| 2 14233 22<br />
| 2 30113 24<br />
| 2 99999 29<br />
| 2 31113 40<br />
| 2 11233 43<br />
| 2 31212 46<br />
| 2 11232 51<br />
| 2 31213 54<br />
| 2 11232 56<br />
| 2 99999 61<br />
| ... ... ...<br />
</pre><br />
<br />
Here is the meaning of the categorical data codes:<br />
<br />
<pre><br />
| Family Interaction Codes<br />
|<br />
| x_0 is a code denoting the overall session type:<br />
|<br />
| "1" means "work", a task-structured session<br />
| "2" means "play", a free-interaction session<br />
|<br />
| x_1 and x_4 are codes that record the two family members<br />
| that are present during the session, according to the<br />
| following scheme:<br />
|<br />
| f#_family_member<br />
| f1_child "1" means "child"<br />
| f2_father "2" means "father"<br />
| f3_mother "3" means "mother"<br />
| f4_older_brother etc.<br />
| f5_older_sister<br />
| f6_younger_brother<br />
| f7_younger_sister<br />
| f8_household_pets<br />
| f9_multiple_recipients<br />
| f0_objects<br />
|<br />
| x_2 and x_3 are codes for the "content type" and "content quality",<br />
| respectively. Together they form a composite code that denotes<br />
| a signed category of activity according to the following scheme,<br />
| where x_2 by itself denotes the "content type" and x_3 denotes<br />
| the "logical quality" of the episode:<br />
|<br />
| c##_content<br />
| c1#_conversation "1" means "conversation"<br />
| c11_positive_verbal<br />
| c12_talk<br />
| c13_negative_verbal<br />
| c2#_affiliate/distance "2" means "affiliate/distance"<br />
| c21_endearment<br />
| c22_tease<br />
| c23_verbal_attack<br />
| c3#_clear_directive "3" means "clear directive"<br />
| c31_request<br />
| c32_command<br />
| c33_coerce<br />
| c4#_ambiguous_directive "4" means "ambiguous directive"<br />
| c41_request_ambiguous<br />
| c42_command_ambiguous<br />
| c43_coerce_ambiguous<br />
| c5#_response_to_directive "5" means "response to directive"<br />
| c51_agree<br />
| c52<br />
| c53_refuse<br />
| c6#_vocal_behavior "6" means "vocal behavior"<br />
| c61<br />
| c62_vocal<br />
| c63<br />
| c7#_nonverbal_behavior "7" means "nonverbal behavior"<br />
| c71_positive_nonverbal<br />
| c72_neutral_nonverbal<br />
| c73_negative_nonverbal<br />
| c8#_low_grade_physical_contact "8" = "low grade physical contact"<br />
| c81_touch<br />
| c82<br />
| c83_physical_aggression<br />
| c9#_pronounced_physical_interaction "9" = "pronounced physical interaction"<br />
| c91_hold<br />
| c92_physical_interact<br />
| c93_physical_attack<br />
| c0#_compliance_behavior "0" = "compliance behavior"<br />
| c01_comply<br />
| c02<br />
| c03_noncomply<br />
|<br />
| x_5 is a code that records the "valence"<br />
| or the "affective quality" of the episode<br />
| according to the following scheme:<br />
|<br />
| v#_valence<br />
| v1_exuberant_affect "1" means "exuberant affect"<br />
| v2_positive_affect "2" means "positive affect"<br />
| v3_neutral_affect "3" means "neutral affect"<br />
| v4_negative_affect "4" means "negative affect"<br />
| v5_extreme_negative_affect "5" means "extreme negative affect"<br />
|<br />
| x_6 is a non-negative integer giving the elapsed time in seconds<br />
| at the end of the phase of activity coded in x_1 through x_5.<br />
|<br />
| Codes outside the above mentioned ranges indicate things like:<br />
| "beginning of session", "pause in activity", "end of session".<br />
</pre><br />
<br />
===Note 3===<br />
<br />
One of the things that makes this sort of categorical sequential data interesting to me is that it provides us with an array of intermediate cases or stepping stones in several different directions of increasing complexity or generality of data. In themselves, logical features can be coded by means of the simplest type of variable, with values in the boolean domain B = {0, 1}, but boolean data is used in connection with some of the richest formal languages for conveying information, namely, propositional and quantificational calculi. In another way, the types of finite domain categorical variables that we use in these settings are intermediate in complexity between boolean variables, that range over just two values, and quantitative variables, that range over integer or real domains. In addition, these temporal sequences of categorical codes will very often take on a simple sort of quasi-linguistic structure that provides us with a bridge to the more general types of linguistic data.<br />
<br />
For example, consider the following fragment of the family interaction data:<br />
<br />
<pre><br />
| 2 31213 5<br />
| 2 11233 8<br />
| 2 34213 10<br />
| 2 33213 18<br />
| 2 10133 19<br />
| 2 14233 22<br />
| 2 30113 24<br />
| 2 99999 29<br />
</pre><br />
<br />
This can be read according to the codebook in the following way:<br />
<br />
<pre><br />
| Begin play session.<br />
| Mother talks to child with neutral affect.<br />
| Child talks to mother with neutral affect.<br />
| Mother ambiguously commands child with neutral affect.<br />
| Mother commands child with neutral affect.<br />
| Child obeys mother with neutral affect.<br />
| Child ambiguously commands mother with neutral affect.<br />
| Mother obeys child with neutral affect.<br />
| Pause in activity.<br />
</pre><br />
<br />
===Note 4===<br />
<br />
One of the things that strikes you as soon as you look at anything like a real-world dataset is that you cannot engage in any form of data gathering, much less data analysis, without being involved in all sorts of enabling hypotheses, conscious or unconscious, that go far beyond the focal hypothesis of the study at hand. There are enabling hypotheses involved in the choice of a particular apparatus, codebook, or instrument. There are enabling hypotheses involved in the use of a particular method for arranging and analyzing the data collected. An enabling hypothesis may be something as simple as:<br />
<br />
# I guess this codebook will be a useful template for viewing this particular part of the world through.<br />
# I guess this way of looking at the data will disclose an aspect of the phenomenon or the reality that produced it.<br />
<br />
It is too much to expect that all of our enabling hypotheses can be made explicit. As Peirce observes, abduction shades off into perception and instinct. And yet, at any moment, it is the nature of recalcitrant experience that any one of our peripheral hypotheses, so long taken for granted that it may never have been part of our conscious reflection, can rise into prominence as the one that most acutely demands to be bought into focus.<br />
<br />
===Note 5. Quality Time===<br />
<br />
We need a better term for the type of mixed mode intermediate data model that we are looking at in the family interaction study, which is typical of the data that are gathered in a wide variety of observational fields, from animal ethography to human ethnography. Either one of the terms ''categorical time series'' or ''qualitative temporal data'' seems to fit, but the latter makes for a catchier acronym, and so I will refer to this whole family of data models as ''QT data'' from here on out.<br />
<br />
Given that guessing is a big part of the language game we play with Nature and our fellows, let us adopt a working assumption that suggests itself from the perspective of many observational studies: That it is from just such interactive settings that people slurp up &ldquo;loosely organized, semi-structured mix of whatever people have in their heads&rdquo;, what John Sowa calls &ldquo;[http://suo.ieee.org/email/msg10558.html knowledge soup]&rdquo;.<br />
<br />
Each session of observation in the family interaction study involves three participant observers: namely, the child ''C'', the parent ''P'', and the experimenter ''E''. The participant role of ''E'' is minimized by means of a 1-way mirror that masks ''E''&apos;s presence, allowing ''E'' to observe and record the interaction of ''C'' and ''P'' in a relatively unobtrusive manner.<br />
<br />
Each of the other participants more or less observes the interaction that takes place between them, within what is called a dyadic system.<br />
<br />
One of the interesting features of an observational set-up like this is that everyone involved is always naturally attempting to discover the patterns that are there to be found in the activity of this dyad. Thus each of the agents ''C'', ''P'', and ''E'' acts in some fashion as an agent of inquiry, accumulating experiences, organizing them under suitable heads, guessing what principles would serve to resolve their variety, and exposing these principles to the test of forthcoming experiences.<br />
<br />
===Note 6===<br />
<br />
Whenever we set about building a model of the reality that informs a phenomenon of interest, it is necessary to remind ourselves from time to time of these three contingencies of the modeling exercise:<br />
<br />
<ol style="list-style-type:upper-alpha"><br />
<br />
<li>The model is not the reality.</li><br />
<li>The model is not the appearances.</li><br />
<li>The appearances are not the reality.</li><br />
<br />
</ol><br />
<br />
As I mentioned, the codebook that is used to convert observations into a dataset proper reflects an enabling hypothesis about the nature of the reality that produces the phenomenon of interest. No datum is so raw that it has not been cooked to some degree according to the recipe of such a codebook. Moreover, each way of looking at the dataset embodies additional helpings of ancillary provisions and auxiliary hypotheses.<br />
<br />
I will next describe three closely related ways that I used to look at the family interaction data, all of which methods are generally useful for all sorts of QT data, such as we find in protocol analysis and in many other areas of qualitative research that involve similar species of ''intermediate data models'' (IDMs). These three paradigms are:<br />
<br />
<ol style="list-style-type:decimal"><br />
<br />
<li>Two-Level Formal Languages (2-FLs)</li><br />
<li>Finite State Transitions (FSTs)</li><br />
<li>Differential Logic (DLOG).</li><br />
<br />
</ol><br />
<br />
===Note 7===<br />
<br />
While we are reflecting on hypotheses that we run on, many of which are so refractory that we may not have reflected on them for a very long time, if ever, we might reflect that a biological species is, in its own right, a kind of hypothesis, a bit like this:<br />
<br />
* ''I guess this code is fitted to this niche.''<br />
<br />
Now I do not think anyone would imagine that the ''co-response'' between a genetic code and its environment is anything like the correspondence between an object and its mirror image, so let us not tie the word ''correspondence'' down to such literal and 2-dimensional reflections.<br />
<br />
In seeking to model the human mind, I have become convinced that we must find a way to integrate empirical and rational faculties, my senses of which I will explain as we go, but I am sensitive to a paradox in saying this, as I would not want to say that our empirical and rational faculties are ever yet as well-integrated as we might hope them to be.<br />
<br />
* An empirical faculty must deal with experience as it comes, toeing the line of the continuously updating data stream, and hewing closely to realtime processing constraints.<br />
<br />
* A rational faculty has tenure, as it were, and can afford to kick back and reflect on episodes beyond the immediate crisis, and even to speculate on images of things as they never were.<br />
<br />
In their bearing on the present example, these reflections tell us something about the sorts of methods that will tend to be more fitted to the empirical versus the rational tasks.<br />
<br />
Formal language theory, taken at the full, is generic enough to cover just about everything that we are thinking of here, but formal languages can also be used in a more literal way to code sequences of occurrences as they happen, and this way of using formal languages is very well suited to the constraints of the empirical task. By ''literal'' I mean that the codes in a formal language L can be read in a one-to-one or ''injective'' fashion as icons or indices of the objective occurrences that they thus denote.<br />
<br />
The next order of business will be to describe this more literal application of formal languages as an IDM (intermediate data model) for the kinds of QT data that we find in the family interaction study.<br />
<br />
===Note 8. Two-Level Formal Languages (2-FLs)===<br />
<br />
A ''k-level formal language'' (k-FL) is a selected set of finite sequences of a selected set of finite sequences of &hellip; a given finite set of symbols, called the ''alphabet'', that is provided to start, where the selection of sequences from the previous level is enacted k times.<br />
<br />
In particular, let L be a ''2-level formal language'' (2-FL) over an alphabet !A!. Then L is composed of a first level language L_1 c !A!* and a second level language L_2 c L_1*, where the "kleene-star" of a set X, written X*, is the set of all finite sequences over X. It is generally convenient to let the name L denote the whole structure L = <L_1, L_2>.<br />
<br />
It's an arbitrary matter what we call the levels of a 2-FL. Depending on what seems natural in a particular discussion, we may refer to L_1 and L_2 as "strings" and "strands", as "words" and "sentences", as "phrases" and "clauses", or as "sentences" and "paragraphs", respectively, just to name a few of the most common options that come to mind right off.<br />
<br />
===Note 9. Reading the Family Interaction Datasets as Two-Level Formal Languages : 1===<br />
<br />
Let's now develop a sense of how the family interaction dataset might look from the perspective of a 2-level formal language.<br />
<br />
For example, consider the following fragment of the data:<br />
<br />
<pre><br />
| 2 31213 5<br />
| 2 11233 8<br />
| 2 34213 10<br />
| 2 33213 18<br />
| 2 10133 19<br />
| 2 14233 22<br />
| 2 30113 24<br />
| 2 99999 29<br />
| 2 31113 40<br />
| 2 11233 43<br />
| 2 31212 46<br />
| 2 11232 51<br />
| 2 31213 54<br />
| 2 11232 56<br />
| 2 99999 61<br />
| 2 11233 68<br />
| 2 36213 70<br />
| 2 33213 73<br />
| 2 16232 74<br />
| 2 33213 77<br />
| 2 16232 78<br />
| 2 31312 82<br />
| 2 99999 91<br />
| 2 11233 94<br />
| 2 31213 96<br />
| 2 99999 101<br />
</pre><br />
<br />
Let's focus on the categorical variables in the middle columns, ignoring the session type and the elapsed time of each episode,<br />
using the time variable simply as a marker of sequential order, and reading the ''pause in activity'' code as a clause indicator.<br />
<br />
Just from the sample of this first hundred seconds we have the following information about the 2-level formal language L = (L_1, L_2) that depicts the interaction between C and P.<br />
<br />
L_1, the ''phrase-book'', or ''lexicon'', contains the following strings:<br />
<br />
<pre><br />
| 10133 = Child complies with Mother with neutral affect.<br />
| 11232 = Child talks to Mother with positive affect.<br />
| 11233 = Child talks to Mother with neutral affect.<br />
| 14233 = Child ambiguously commands Mother with neutral affect.<br />
| 16232 = Child vocalizes to Mother with positive affect.<br />
| 30113 = Mother complies with Child with neutral affect.<br />
| 31113 = Mother positively verbalizes to Child with neutral affect.<br />
| 31212 = Mother talks to Child with positive affect.<br />
| 31213 = Mother talks to Child with neutral affect.<br />
| 31312 = Mother negatively verbalizes to Child with positive affect.<br />
| 33213 = Mother commands Child with neutral affect.<br />
| 34213 = Mother ambiguously commands Child with neutral affect.<br />
| 36213 = Mother vocalizes to Child with neutral affect.<br />
| 99999 = Pause in activity.<br />
</pre><br />
<br />
L_2, the ''clause-book'', or ''liturgy'', contains the following strands:<br />
<br />
<pre><br />
| <11233, 31213, 99999><br />
|<br />
| <11233, 36213, 33213, 16232, 33213, 16232, 31312, 99999><br />
|<br />
| <31113, 11233, 31212, 11232, 31213, 11232, 99999><br />
|<br />
| <31213, 11233, 34213, 33213, 10133, 14233, 30113, 99999><br />
</pre><br />
<br />
===Note 10. Reading the Family Interaction Datasets as Two-Level Formal Languages : 2===<br />
<br />
Before I can discuss the formal language view of the family interaction dataset in more detail I will need to clear up a few discrepancies between the codebook given above, which was used by a sizable community of researchers for a variety of different studies over a period of many years, and the codes as they were adapted for use in the family interaction study. At the same time, I will focus on a reduced subset of the data that is adequate to illustrate the points of central interest here. The easiest way to accomplish this is simply to give the revised codebook and the reduced dataset as follows:<br />
<br />
====Family Interaction Analysis &bull; Abbreviated Codebook====<br />
<br />
<pre><br />
| Family Interaction Analysis. Abbreviated Codebook<br />
|<br />
| An "episode of activity" is coded as a 4-tuple y = <y_1, y_2, y_3, y_4>.<br />
|<br />
| The components y_1, y_2, y_3 are categorical codes<br />
| that are described in the codebook.<br />
|<br />
| The component y_4 is a non-negative integer code<br />
| that records the elapsed time in seconds at which<br />
| the episode ends.<br />
|<br />
| y_1 codes the agent of the activity,<br />
| according to the following scheme:<br />
|<br />
| f_family_member<br />
| f1_child<br />
| f2_father<br />
| f3_mother<br />
| f9_null<br />
|<br />
| y_2 is a two-digit code for the content of the activity,<br />
| where the first digit records the content type and the<br />
| second digit records the logical quality of the action,<br />
| according to the following scheme:<br />
|<br />
| c_content<br />
| c0_compliance_behavior<br />
| c01_comply<br />
| c02<br />
| c03_noncomply<br />
| c1_conversation<br />
| c11_positive_verbal<br />
| c12_talk<br />
| c13_negative_verbal<br />
| c2_affiliate/distance<br />
| c21_endearment<br />
| c22_tease<br />
| c23_verbal_attack<br />
| c3_clear_directive<br />
| c31_request<br />
| c32_command<br />
| c33_coerce<br />
| c4_ambiguous_directive<br />
| c41_request_ambiguous<br />
| c42_command_ambiguous<br />
| c43_coerce_ambiguous<br />
| c5_response_to_directive<br />
| c51_agree<br />
| c52<br />
| c53_refuse<br />
| c6_vocal_behavior<br />
| c61<br />
| c62_vocal<br />
| c63<br />
| c7_nonverbal_behavior<br />
| c71_positive_nonverbal<br />
| c72_neutral_nonverbal<br />
| c73_negative_nonverbal<br />
| c8_low_grade_physical_contact<br />
| c81_touch<br />
| c82<br />
| c83_physical_aggression<br />
| c9_pronounced_physical_interaction<br />
| c91_hold<br />
| c92_physical_interact<br />
| c93_physical_attack<br />
| c99_null<br />
|<br />
| y_3 is a code that records the "valence"<br />
| or the "affective quality" of the episode<br />
| according to the following scheme:<br />
|<br />
| v_valence<br />
| v1_exuberant_affect<br />
| v2_positive_affect<br />
| v3_neutral_affect<br />
| v4_negative_affect<br />
| v5_extreme_negative_affect<br />
| v9_null<br />
|<br />
| y_4 is a non-negative integer giving the elapsed time in seconds<br />
| at the end of the phase of activity coded in x_1 through x_5.<br />
</pre><br />
<br />
====Family Interaction Analysis &bull; Abbreviated Dataset 1====<br />
<br />
Listed next is the flat data file for y = (y_1, y_2, y_3, y_4). To the right of the numeric codes are written an equivalent set of alphanumeric codes that provide a bridge to several forms of logical representation that I plan to take up at a later point.<br />
<br />
<pre><br />
| 9 99 9 0 f9 c99 v9<br />
|<br />
| 3 12 3 5 f3 c12 v3<br />
| 1 12 3 8 f1 c12 v3<br />
| 3 42 3 10 f3 c42 v3<br />
| 3 32 3 18 f3 c32 v3<br />
| 1 01 3 19 f1 c01 v3<br />
| 1 42 3 22 f1 c42 v3<br />
| 3 01 3 24 f3 c01 v3<br />
| 9 99 9 29 f9 c99 v9<br />
|<br />
| 3 11 3 40 f3 c11 v3<br />
| 1 12 3 43 f1 c12 v3<br />
| 3 12 2 46 f3 c12 v2<br />
| 1 12 2 51 f1 c12 v2<br />
| 3 12 3 54 f3 c12 v3<br />
| 1 12 2 56 f1 c12 v2<br />
| 9 99 9 61 f9 c99 v9<br />
|<br />
| 1 12 3 68 f1 c12 v3<br />
| 3 62 3 70 f3 c62 v3<br />
| 3 32 3 73 f3 c32 v3<br />
| 1 62 2 74 f1 c62 v2<br />
| 3 32 3 77 f3 c32 v3<br />
| 1 62 2 78 f1 c62 v2<br />
| 3 13 2 82 f3 c13 v2<br />
| 9 99 9 91 f9 c99 v9<br />
|<br />
| 1 12 3 94 f1 c12 v3<br />
| 3 12 3 96 f3 c12 v3<br />
| 9 99 9 101 f9 c99 v9<br />
|<br />
| 3 11 2 103 f3 c11 v2<br />
| 1 12 2 106 f1 c12 v2<br />
| 3 12 3 108 f3 c12 v3<br />
| 1 12 2 109 f1 c12 v2<br />
| 3 62 2 112 f3 c62 v2<br />
| 1 62 2 115 f1 c62 v2<br />
| 1 12 2 120 f1 c12 v2<br />
| 3 12 2 122 f3 c12 v2<br />
| 1 12 2 127 f1 c12 v2<br />
| 1 62 2 131 f1 c62 v2<br />
| 1 12 2 136 f1 c12 v2<br />
| 1 62 2 138 f1 c62 v2<br />
| 3 12 3 143 f3 c12 v3<br />
| 1 12 2 148 f1 c12 v2<br />
| 1 62 2 155 f1 c62 v2<br />
| 3 12 2 160 f3 c12 v2<br />
| 1 62 2 168 f1 c62 v2<br />
| 3 12 3 170 f3 c12 v3<br />
| 3 32 3 175 f3 c32 v3<br />
| 1 01 3 177 f1 c01 v3<br />
| 3 51 3 184 f3 c51 v3<br />
| 3 12 3 187 f3 c12 v3<br />
| 3 01 3 189 f3 c01 v3<br />
| 1 12 3 192 f1 c12 v3<br />
| 3 12 3 194 f3 c12 v3<br />
| 1 12 3 197 f1 c12 v3<br />
| 3 12 3 200 f3 c12 v3<br />
| 1 12 2 202 f1 c12 v2<br />
| 1 62 2 205 f1 c62 v2<br />
| 9 99 9 211 f9 c99 v9<br />
|<br />
| 1 12 2 214 f1 c12 v2<br />
| 3 12 3 218 f3 c12 v3<br />
| 1 32 3 222 f1 c32 v3<br />
| 3 51 3 225 f3 c51 v3<br />
| 3 12 3 227 f3 c12 v3<br />
| 1 12 3 228 f1 c12 v3<br />
| 3 01 3 230 f3 c01 v3<br />
| 1 12 4 234 f1 c12 v4<br />
| 3 12 2 238 f3 c12 v2<br />
| 3 32 3 241 f3 c32 v3<br />
| 1 01 3 242 f1 c01 v3<br />
| 3 12 3 244 f3 c12 v3<br />
| 1 12 2 245 f1 c12 v2<br />
| 3 12 3 252 f3 c12 v3<br />
| 1 12 3 254 f1 c12 v3<br />
| 3 12 3 256 f3 c12 v3<br />
| 1 62 3 267 f1 c62 v3<br />
| 1 12 2 269 f1 c12 v2<br />
| 1 12 1 278 f1 c12 v1<br />
| 3 42 3 282 f3 c42 v3<br />
| 1 01 3 287 f1 c01 v3<br />
| 3 12 3 289 f3 c12 v3<br />
| 1 12 2 291 f1 c12 v2<br />
| 3 12 3 296 f3 c12 v3<br />
| 1 31 3 301 f1 c31 v3<br />
| 3 51 3 305 f3 c51 v3<br />
| 1 12 3 309 f1 c12 v3<br />
| 3 12 3 311 f3 c12 v3<br />
| 1 12 3 313 f1 c12 v3<br />
| 3 12 3 316 f3 c12 v3<br />
| 1 12 3 319 f1 c12 v3<br />
| 3 12 3 322 f3 c12 v3<br />
| 1 12 3 323 f1 c12 v3<br />
| 3 12 3 324 f3 c12 v3<br />
| 1 12 3 326 f1 c12 v3<br />
| 3 32 3 329 f3 c32 v3<br />
| 1 01 3 331 f1 c01 v3<br />
| 1 12 3 333 f1 c12 v3<br />
| 3 11 2 338 f3 c11 v2<br />
| 1 12 3 342 f1 c12 v3<br />
| 3 12 3 344 f3 c12 v3<br />
| 1 12 3 347 f1 c12 v3<br />
| 3 12 3 349 f3 c12 v3<br />
| 1 12 3 351 f1 c12 v3<br />
| 3 12 3 352 f3 c12 v3<br />
| 1 62 3 354 f1 c62 v3<br />
| 3 62 2 356 f3 c62 v2<br />
| 1 12 3 362 f1 c12 v3<br />
| 3 12 3 367 f3 c12 v3<br />
| 1 12 3 370 f1 c12 v3<br />
| 3 12 3 371 f3 c12 v3<br />
| 1 12 2 373 f1 c12 v2<br />
| 3 12 2 378 f3 c12 v2<br />
| 1 12 2 381 f1 c12 v2<br />
| 3 12 3 382 f3 c12 v3<br />
| 1 12 3 389 f1 c12 v3<br />
| 3 12 3 392 f3 c12 v3<br />
| 3 42 3 394 f3 c42 v3<br />
| 1 01 3 396 f1 c01 v3<br />
| 3 11 3 398 f3 c11 v3<br />
| 1 12 3 401 f1 c12 v3<br />
| 3 12 3 403 f3 c12 v3<br />
| 1 12 3 405 f1 c12 v3<br />
| 3 12 2 406 f3 c12 v2<br />
| 1 12 3 409 f1 c12 v3<br />
| 3 12 3 412 f3 c12 v3<br />
| 1 12 3 414 f1 c12 v3<br />
| 3 12 3 416 f3 c12 v3<br />
| 3 62 2 417 f3 c62 v2<br />
| 1 62 2 419 f1 c62 v2<br />
| 3 13 2 422 f3 c13 v2<br />
| 3 32 3 426 f3 c32 v3<br />
| 3 32 3 430 f3 c32 v3<br />
| 1 01 3 431 f1 c01 v3<br />
| 1 01 3 434 f1 c01 v3<br />
| 3 12 3 438 f3 c12 v3<br />
| 1 12 3 439 f1 c12 v3<br />
| 3 12 3 440 f3 c12 v3<br />
| 1 12 3 441 f1 c12 v3<br />
| 3 12 3 443 f3 c12 v3<br />
| 1 12 3 449 f1 c12 v3<br />
| 3 11 2 450 f3 c11 v2<br />
| 3 12 2 455 f3 c12 v2<br />
| 1 62 2 457 f1 c62 v2<br />
| 3 12 3 462 f3 c12 v3<br />
| 1 12 2 463 f1 c12 v2<br />
| 3 12 2 467 f3 c12 v2<br />
| 1 62 2 469 f1 c62 v2<br />
| 3 12 3 473 f3 c12 v3<br />
| 1 12 3 477 f1 c12 v3<br />
| 3 12 3 479 f3 c12 v3<br />
| 1 11 3 490 f1 c11 v3<br />
| 3 11 3 492 f3 c11 v3<br />
| 3 12 3 495 f3 c12 v3<br />
| 1 62 2 514 f1 c62 v2<br />
| 3 11 2 517 f3 c11 v2<br />
| 1 62 2 524 f1 c62 v2<br />
| 1 12 2 527 f1 c12 v2<br />
| 3 12 3 530 f3 c12 v3<br />
| 1 62 2 534 f1 c62 v2<br />
| 3 62 2 537 f3 c62 v2<br />
| 3 22 2 539 f3 c22 v2<br />
| 1 62 2 544 f1 c62 v2<br />
| 3 12 3 547 f3 c12 v3<br />
| 1 12 3 548 f1 c12 v3<br />
| 3 12 3 549 f3 c12 v3<br />
| 1 12 2 553 f1 c12 v2<br />
| 9 99 9 561 f9 c99 v9<br />
|<br />
| 1 62 2 564 f1 c62 v2<br />
| 3 12 3 569 f3 c12 v3<br />
| 1 12 2 575 f1 c12 v2<br />
| 3 13 4 577 f3 c13 v4<br />
| 3 12 3 585 f3 c12 v3<br />
| 1 62 2 589 f1 c62 v2<br />
| 3 32 3 591 f3 c32 v3<br />
| 1 01 3 593 f1 c01 v3<br />
| 1 12 3 595 f1 c12 v3<br />
| 3 12 3 599 f3 c12 v3<br />
| 9 99 9 603 f9 c99 v9<br />
</pre><br />
<br />
===Note 11. Reading the Family Interaction Datasets as Two-Level Formal Languages : 3===<br />
<br />
In the ideal situation a language ought to have a grammar. It has often been observed that a grammar is tantamount to a rational theory of the empirical language, subsuming the infinite variety of a linguistic corpus, passed or present or prospective, under the capitation of a finite mentality.<br />
<br />
In the empirical situation, however, always in the beginning far from ideal, the grammar of a language is a datum that awaits discovery at a future date.<br />
<br />
On the other hand, the experience that a finite mentality has actually experienced at any given time is of necessity a finite experience, and so it is possible to organize the experienced language in a finite way, even though it is likely to be far away from the ideal grammar, namely, in the form of ''finite state transition graphs'' (FST graphs). In this way, a k-level formal language can be recorded as it enters experience in the form of k FST graphs, one graph for each level of the language.<br />
<br />
In particular, it is especially feasible under real-time conditions to keep track of the language actually experienced by means of FST trees. A k-level formal language can be recorded as it comes into experience by means of a k-tuple of FST trees, using one tree to rule each level.<br />
<br />
Among the possible variety of data models, one has the option of using either ''fixed order'' or ''free order'' FST trees. In the case of a 2-FL we get a pair of FST trees that may be referred to as the ''lexical'' and the ''literal'' trees.<br />
<br />
At this point, though, it is probably best to return to our concrete datasets.<br />
<br />
I will next illustrate how the family interaction data looks when arranged as a pair of free order FST's. For the sake of comparison, and also because the first dataset is somewhat amorphous from this point of view, I will list here a second dataset, a father-child dyad from the same family, with what I think to be the same child, but I no longer have the corresponding demographic data.<br />
<br />
====Family Interaction Analysis &bull; Abbreviated Dataset 2====<br />
<br />
<pre><br />
Family Interaction Analysis. Abbreviated Dataset 2<br />
<br />
------------------------------------------------------------------------<br />
TRIAL FAMILY SESSION FOCUS OBSERVER FAMILY MEMBERS MO/DY/YR HR:MN<br />
1 MXYZ 5 1 22 12000000 12/03/86 13:34<br />
------------------------------------------------------------------------<br />
<br />
| 9 99 9 0<br />
|<br />
| 1 12 2 2<br />
| 2 12 3 4<br />
| 1 62 2 8<br />
| 2 12 3 11<br />
| 1 62 2 13<br />
| 1 12 3 15<br />
| 2 12 3 21<br />
| 2 32 3 24<br />
| 1 12 3 29<br />
| 1 01 3 31<br />
| 2 73 3 38<br />
| 2 12 3 41<br />
| 1 12 3 42<br />
| 2 12 3 44<br />
| 1 12 2 50<br />
| 2 12 3 52<br />
| 2 32 3 62<br />
| 1 53 3 65<br />
| 1 13 4 70<br />
| 1 62 3 73<br />
| 2 12 3 76<br />
| 9 99 9 83<br />
|<br />
| 1 62 2 85<br />
| 1 12 3 88<br />
| 2 12 3 90<br />
| 1 12 3 91<br />
| 2 12 3 93<br />
| 1 12 2 97<br />
| 2 12 2 99<br />
| 1 12 2 102<br />
| 9 99 9 109<br />
|<br />
| 1 12 2 112<br />
| 2 62 2 115<br />
| 1 12 2 117<br />
| 2 12 3 120<br />
| 2 13 4 123<br />
| 1 12 2 126<br />
| 1 62 2 128<br />
| 1 12 2 131<br />
| 2 12 3 133<br />
| 1 12 3 135<br />
| 2 12 3 139<br />
| 1 32 4 147<br />
| 2 01 3 152<br />
| 1 12 3 156<br />
| 2 12 3 158<br />
| 1 12 3 159<br />
| 2 12 3 162<br />
| 1 32 3 164<br />
| 2 12 3 167<br />
| 1 12 4 173<br />
| 2 12 3 179<br />
| 1 12 3 182<br />
| 2 32 3 187<br />
| 1 01 3 189<br />
| 2 32 3 193<br />
| 1 01 2 195<br />
| 1 12 2 199<br />
| 2 32 3 202<br />
| 1 01 3 205<br />
| 2 12 3 209<br />
| 1 62 2 210<br />
| 9 99 9 218<br />
|<br />
| 1 62 2 222<br />
| 1 12 2 225<br />
| 1 12 1 229<br />
| 2 32 3 235<br />
| 1 62 2 239<br />
| 2 13 4 244<br />
| 2 32 3 253<br />
| 1 12 3 254<br />
| 1 01 3 258<br />
| 2 11 3 259<br />
| 1 62 2 263<br />
| 2 12 3 265<br />
| 1 12 2 267<br />
| 2 32 3 270<br />
| 1 01 3 274<br />
| 1 12 3 278<br />
| 2 32 3 282<br />
| 1 01 3 286<br />
| 2 12 3 287<br />
| 1 12 2 289<br />
| 2 11 3 292<br />
| 1 12 3 295<br />
| 2 12 3 299<br />
| 1 12 3 303<br />
| 2 12 2 305<br />
| 1 73 4 313<br />
| 1 12 4 320<br />
| 2 12 3 322<br />
| 1 12 4 323<br />
| 2 12 3 326<br />
| 1 12 4 328<br />
| 2 62 3 332<br />
| 1 12 3 334<br />
| 2 12 3 338<br />
| 1 32 3 339<br />
| 2 12 3 346<br />
| 2 01 3 352<br />
| 1 62 2 355<br />
| 9 99 9 360<br />
|<br />
| 1 42 3 364<br />
| 2 12 2 369<br />
| 2 01 3 371<br />
| 1 12 3 375<br />
| 1 32 3 379<br />
| 2 32 3 382<br />
| 1 12 3 385<br />
| 2 12 3 388<br />
| 1 12 3 390<br />
| 2 12 3 392<br />
| 1 22 3 396<br />
| 2 32 3 405<br />
| 1 01 3 409<br />
| 2 12 3 412<br />
| 2 32 3 417<br />
| 1 01 3 419<br />
| 2 12 3 424<br />
| 2 32 3 427<br />
| 1 01 3 429<br />
| 9 99 9 436<br />
|<br />
| 1 62 2 437<br />
| 2 12 2 443<br />
| 1 12 2 445<br />
| 2 32 3 448<br />
| 1 01 3 449<br />
| 2 92 3 455<br />
| 1 62 2 462<br />
| 1 12 2 465<br />
| 2 12 3 467<br />
| 1 12 3 469<br />
| 2 12 3 471<br />
| 1 62 3 472<br />
| 2 12 3 474<br />
| 1 12 2 477<br />
| 9 99 9 485<br />
|<br />
| 1 62 2 489<br />
| 2 13 4 496<br />
| 1 12 3 499<br />
| 2 12 3 502<br />
| 1 12 2 509<br />
| 2 12 3 513<br />
| 1 42 3 514<br />
| 2 01 3 518<br />
| 1 12 3 523<br />
| 1 62 2 527<br />
| 1 12 2 531<br />
| 2 12 3 539<br />
| 1 62 2 541<br />
| 1 12 2 544<br />
| 2 12 3 547<br />
| 1 12 2 550<br />
| 2 12 3 554<br />
| 1 12 3 557<br />
| 2 12 2 564<br />
| 1 12 3 569<br />
| 2 12 3 573<br />
| 1 12 3 574<br />
| 2 12 3 575<br />
| 1 12 3 577<br />
| 2 12 3 579<br />
| 1 62 3 594<br />
| 9 99 9 597<br />
</pre><br />
<br />
Here is the literal level of Dataset 2 presented as a free order FST in outline form. The numbers at the right give the frequencies with which the corresponding nodes of the tree are observed or ''traversed'' during the session. Reading this display one can see that there are seven distinct strands at the second level of the corresponding 2-FL:<br />
<br />
* 4 strands begin with 1_62_2, Child vocalizes with positive affect.<br />
* 2 strands begin with 1_12_2, Child talks with positive affect.<br />
* 1 strand begins with 1_42_3, Child ambiguously commands with neutral affect.<br />
<br />
<pre><br />
| dataset_2 7<br />
| 1_62_2 4<br />
| 1_12_3 1<br />
| 2_12_3 1<br />
| 1_12_3 1<br />
| 2_12_3 1<br />
| 1_12_2 1<br />
| 2_12_2 1<br />
| 1_12_2 1<br />
| * 1<br />
| 1_12_2 1<br />
| 1_12_1 1<br />
| 2_32_3 1<br />
| 1_62_2 1<br />
| 2_13_4 1<br />
| 2_32_3 1<br />
| 1_12_3 1<br />
| 1_01_3 1<br />
| 2_11_3 1<br />
| 1_62_2 1<br />
| 2_12_3 1<br />
| 1_12_2 1<br />
| 2_32_3 1<br />
| 1_01_3 1<br />
| 1_12_3 1<br />
| 2_32_3 1<br />
| 1_01_3 1<br />
| 2_12_3 1<br />
| 1_12_2 1<br />
| 2_11_3 1<br />
| 1_12_3 1<br />
| 2_12_3 1<br />
| 1_12_3 1<br />
| 2_12_2 1<br />
| 1_73_4 1<br />
| 1_12_4 1<br />
| 2_12_3 1<br />
| 1_12_4 1<br />
| 2_12_3 1<br />
| 1_12_4 1<br />
| 2_62_3 1<br />
| 1_12_3 1<br />
| 2_12_3 1<br />
| 1_32_3 1<br />
| 2_12_3 1<br />
| 2_01_3 1<br />
| 1_62_2 1<br />
| * 1<br />
| 2_12_2 1<br />
| 1_12_2 1<br />
| 2_32_3 1<br />
| 1_01_3 1<br />
| 2_92_3 1<br />
| 1_62_2 1<br />
| 1_12_2 1<br />
| 2_12_3 1<br />
| 1_12_3 1<br />
| 2_12_3 1<br />
| 1_62_3 1<br />
| 2_12_3 1<br />
| 1_12_2 1<br />
| * 1<br />
| 2_13_4 1<br />
| 1_12_3 1<br />
| 2_12_3 1<br />
| 1_12_2 1<br />
| 2_12_3 1<br />
| 1_42_3 1<br />
| 2_01_3 1<br />
| 1_12_3 1<br />
| 1_62_2 1<br />
| 1_12_2 1<br />
| 2_12_3 1<br />
| 1_62_2 1<br />
| 1_12_2 1<br />
| 2_12_3 1<br />
| 1_12_2 1<br />
| 2_12_3 1<br />
| 1_12_3 1<br />
| 2_12_2 1<br />
| 1_12_3 1<br />
| 2_12_3 1<br />
| 1_12_3 1<br />
| 2_12_3 1<br />
| 1_12_3 1<br />
| 2_12_3 1<br />
| 1_62_3 1<br />
| * 1<br />
| 1_42_3 1<br />
| 2_12_2 1<br />
| 2_01_3 1<br />
| 1_12_3 1<br />
| 1_32_3 1<br />
| 2_32_3 1<br />
| 1_12_3 1<br />
| 2_12_3 1<br />
| 1_12_3 1<br />
| 2_12_3 1<br />
| 1_22_3 1<br />
| 2_32_3 1<br />
| 1_01_3 1<br />
| 2_12_3 1<br />
| 2_32_3 1<br />
| 1_01_3 1<br />
| 2_12_3 1<br />
| 2_32_3 1<br />
| 1_01_3 1<br />
| * 1<br />
| 1_12_2 2<br />
| 2_12_3 1<br />
| 1_62_2 1<br />
| 2_12_3 1<br />
| 1_62_2 1<br />
| 1_12_3 1<br />
| 2_12_3 1<br />
| 2_32_3 1<br />
| 1_12_3 1<br />
| 1_01_3 1<br />
| 2_73_3 1<br />
| 2_12_3 1<br />
| 1_12_3 1<br />
| 2_12_3 1<br />
| 1_12_2 1<br />
| 2_12_3 1<br />
| 2_32_3 1<br />
| 1_53_3 1<br />
| 1_13_4 1<br />
| 1_62_3 1<br />
| 2_12_3 1<br />
| * 1<br />
| 2_62_2 1<br />
| 1_12_2 1<br />
| 2_12_3 1<br />
| 2_13_4 1<br />
| 1_12_2 1<br />
| 1_62_2 1<br />
| 1_12_2 1<br />
| 2_12_3 1<br />
| 1_12_3 1<br />
| 2_12_3 1<br />
| 1_32_4 1<br />
| 2_01_3 1<br />
| 1_12_3 1<br />
| 2_12_3 1<br />
| 1_12_3 1<br />
| 2_12_3 1<br />
| 1_32_3 1<br />
| 2_12_3 1<br />
| 1_12_4 1<br />
| 2_12_3 1<br />
| 1_12_3 1<br />
| 2_32_3 1<br />
| 1_01_3 1<br />
| 2_32_3 1<br />
| 1_01_2 1<br />
| 1_12_2 1<br />
| 2_32_3 1<br />
| 1_01_3 1<br />
| 2_12_3 1<br />
| 1_62_2 1<br />
| * 1<br />
</pre><br />
<br />
===Note 12. Reading the Family Interaction Datasets as Two-Level Formal Languages : 4===<br />
<br />
Here is a suggestion of how the literal levels of the two family interaction datasets look when presented<br />
as free order finite state transition trees.<br />
<br />
====Dataset 1====<br />
<br />
<pre><br />
@--[1_12_3]--[3_62_3]--[3_32_3]--[1_62_2]--[3_32_3]--[1_62_2]--[3_13_2]<br />
| |<br />
| `-----[3_12_3]<br />
|<br />
`--[3_11_2]--[1_12_2]--[3_12_3]--[1_12_2]--[......]--[3_12_3]--[1_12_2]--[1_62_2]<br />
|<br />
`--[1_12_2]--[3_12_3]--[1_32_3]--[3_51_3]--[......]--[1_12_3]--[3_12_3]--[1_12_2]<br />
|<br />
`--[1_62_2]--[3_12_3]--[1_12_2]--[3_13_4]--[......]--[1_01_3]--[1_12_3]--[3_12_3]<br />
|<br />
`--[3_12_3]--[1_12_3]--[3_42_3]--[3_32_3]--[1_01_3]--[1_42_3]--[3_01_3]<br />
|<br />
`--[3_11_3]--[1_12_3]--[3_12_2]--[1_12_2]--[3_12_3]--[1_12_2]<br />
</pre><br />
<br />
====Dataset 2====<br />
<br />
<pre><br />
@--[1_62_2]--[1_12_3]--[2_12_3]--[1_12_3]--[2_12_3]--[1_12_2]--[2_12_2]--[1_12_2]<br />
| |<br />
| `-----[1_12_2]--[1_12_1]--[2_32_3]--[......]--[2_12_3]--[2_01_3]--[1_62_2]<br />
| |<br />
| `-----[2_12_2]--[1_12_2]--[2_32_3]--[......]--[1_62_3]--[2_12_3]--[1_12_2]<br />
| |<br />
| `-----[2_13_4]--[1_12_3]--[2_12_3]--[......]--[1_12_3]--[2_12_3]--[1_62_3]<br />
|<br />
`--[1_42_3]--[2_12_2]--[2_01_3]--[1_12_3]--[......]--[2_12_3]--[2_32_3]--[1_01_3]<br />
|<br />
`--[1_12_2]--[2_12_3]--[1_62_2]--[2_12_3]--[......]--[1_13_4]--[1_62_3]--[2_12_3]<br />
|<br />
`-----[2_62_2]--[1_12_2]--[2_12_3]--[......]--[1_01_3]--[2_12_3]--[1_62_2]<br />
</pre><br />
<br />
===Note 13. Reading the Family Interaction Datasets as Two-Level Formal Languages : 5===<br />
<br />
I have mentioned the fact that each way of looking at a dataset<br />
amounts to an enabling hypothesis to the effect that something<br />
interesting or useful might be seen by looking at the data in<br />
just that way.<br />
<br />
In parsing the family interaction datasets as 2-level formal languages,<br />
I made the pure hypothesis that the resting episodes were somehow more<br />
significant than others, in the sense that I interpreted them to mark <br />
the "commas", the "periods", or the "ends of strands" at the second<br />
level of the formal texts. One might speculate along the lines of<br />
classical learning theory, as in Thorndike's "law of effect", that<br />
these pauses are refreshing, rewarding, and thus reinforcing, or<br />
else along the lines of Berlyne's information hungry animal that<br />
the rests of the score requite the players with information about<br />
the structure of the interpersonal music in progress between them.<br />
<br />
But if one gets to thinking in this deliberate, goal-oriented,<br />
intentional, motivated, or purposive fashion, then one might<br />
just as well recognize many other subsets of episodic codes<br />
as representing the potential goals of either one or both<br />
players in the game. Similar considerations would apply<br />
to the choice of negative reinforcers, "deinforcements",<br />
or "disincentives", types of episodes that one or both<br />
participants seek to avoid, to escape, or to minimize.<br />
</pre><br />
<br />
===Note 14. Reading the Family Interaction Datasets as Two-Level Formal Languages : 6===<br />
<br />
'''Qualitative Sequential Datasets as Two-Level Formal Languages (cont.)'''<br />
<br />
I would like next to view the family interaction datasets as fixed order finite state transition trees. To begin, let's view Dataset&nbsp;1, one paradigm's way of codifying ten minutes from the life of a mother and a child, in respect of its 2-tuple, pairwise, or 2nd-order transitions from each episode of coded activity to the next.<br />
<br />
This time around, I will parse the code of each episode into parts that codify separately the agent, the content, and the affective valence of the activity, reducing the time data to simple indications of sequential order, the ''episode numbers'' of the coded activities that fill the observational sessions.<br />
<br />
====Family Interaction Analysis &bull; Dataset 1 &bull; Pairwise Transitions &bull; Numbered Episodes====<br />
<br />
Here are the beginning and end of Dataset&nbsp;1 under this view:<br />
<br />
<pre><br />
| Family Interaction Analysis<br />
| Abbreviated Dataset 1<br />
| Pairwise Transitions<br />
| Numbered Episodes<br />
|<br />
| F9 C99 V9 f3 c12 v3 n1<br />
| F3 C12 V3 f1 c12 v3 n2<br />
| F1 C12 V3 f3 c42 v3 n3<br />
| F3 C42 V3 f3 c32 v3 n4<br />
| F3 C32 V3 f1 c01 v3 n5<br />
| F1 C01 V3 f1 c42 v3 n6<br />
| F1 C42 V3 f3 c01 v3 n7<br />
| F3 C01 V3 f9 c99 v9 n8<br />
|<br />
| F9 C99 V9 f3 c11 v3 n9<br />
| F3 C11 V3 f1 c12 v3 n10<br />
| F1 C12 V3 f3 c12 v2 n11<br />
| F3 C12 V2 f1 c12 v2 n12<br />
| F1 C12 V2 f3 c12 v3 n13<br />
| F3 C12 V3 f1 c12 v2 n14<br />
| F1 C12 V2 f9 c99 v9 n15<br />
|<br />
| F9 C99 V9 f1 c12 v3 n16<br />
| F1 C12 V3 f3 c62 v3 n17<br />
| F3 C62 V3 f3 c32 v3 n18<br />
| F3 C32 V3 f1 c62 v2 n19<br />
| F1 C62 V2 f3 c32 v3 n20<br />
| F3 C32 V3 f1 c62 v2 n21<br />
| F1 C62 V2 f3 c13 v2 n22<br />
| F3 C13 V2 f9 c99 v9 n23<br />
|<br />
| F9 C99 V9 f1 c12 v3 n24<br />
| F1 C12 V3 f3 c12 v3 n25<br />
| F3 C12 V3 f9 c99 v9 n26<br />
|<br />
| ...<br />
|<br />
| F9 C99 V9 f1 c62 v2 n165<br />
| F1 C62 V2 f3 c12 v3 n166<br />
| F3 C12 V3 f1 c12 v2 n167<br />
| F1 C12 V2 f3 c13 v4 n168<br />
| F3 C13 V4 f3 c12 v3 n169<br />
| F3 C12 V3 f1 c62 v2 n170<br />
| F1 C62 V2 f3 c32 v3 n171<br />
| F3 C32 V3 f1 c01 v3 n172<br />
| F1 C01 V3 f1 c12 v3 n173<br />
| F1 C12 V3 f3 c12 v3 n174<br />
| F3 C12 V3 f9 c99 v9 n175<br />
</pre><br />
<br />
Here I am using capitalized code to distinguish the initial element of each pairwise transition.<br />
<br />
====Family Interaction Analysis &bull; Dataset 1 &bull; Pairwise Transition Tree in Outline Form====<br />
<br />
Folding in a dollop or three of syntactic syrup, the pairwise transition tree cooks up like this:<br />
<br />
<pre><br />
| Dataset 1. Pairwise Transition Tree in Outline Form<br />
|<br />
| pairwise_transitions 175 1.00 0.000<br />
| from 175 1.00 0.000<br />
| F3_mother 85 0.49 0.506<br />
| C12_talk 51 0.60 0.442<br />
| V3_neutral_affect 43 0.84 0.208<br />
| to 43 1.00 0.000<br />
| f1_child 37 0.86 0.187<br />
| c12_talk 29 0.78 0.275<br />
| v3_neutral_affect 19 0.66 0.400<br />
| at 19 1.00 0.000<br />
| n2 1 0.05 0.224<br />
| * 1 1.00 0.000<br />
| n52 1 0.05 0.224<br />
| * 1 1.00 0.000<br />
| n62 1 0.05 0.224<br />
| * 1 1.00 0.000<br />
| n71 1 0.05 0.224<br />
| * 1 1.00 0.000<br />
| n85 1 0.05 0.224<br />
| * 1 1.00 0.000<br />
| n87 1 0.05 0.224<br />
| * 1 1.00 0.000<br />
| n89 1 0.05 0.224<br />
| * 1 1.00 0.000<br />
| n91 1 0.05 0.224<br />
| * 1 1.00 0.000<br />
| n98 1 0.05 0.224<br />
| * 1 1.00 0.000<br />
| n100 1 0.05 0.224<br />
| * 1 1.00 0.000<br />
| n106 1 0.05 0.224<br />
| * 1 1.00 0.000<br />
| n112 1 0.05 0.224<br />
| * 1 1.00 0.000<br />
| n119 1 0.05 0.224<br />
| * 1 1.00 0.000<br />
| n123 1 0.05 0.224<br />
| * 1 1.00 0.000<br />
| n133 1 0.05 0.224<br />
| * 1 1.00 0.000<br />
| n135 1 0.05 0.224<br />
| * 1 1.00 0.000<br />
| n137 1 0.05 0.224<br />
| * 1 1.00 0.000<br />
| n146 1 0.05 0.224<br />
| * 1 1.00 0.000<br />
| n161 1 0.05 0.224<br />
| * 1 1.00 0.000<br />
| v2_positive_affect 10 0.34 0.530<br />
| at 10 1.00 0.000<br />
| n14 1 0.10 0.332<br />
| * 1 1.00 0.000<br />
| n30 1 0.10 0.332<br />
| * 1 1.00 0.000<br />
| n40 1 0.10 0.332<br />
| * 1 1.00 0.000<br />
| n54 1 0.10 0.332<br />
| * 1 1.00 0.000<br />
| n69 1 0.10 0.332<br />
| * 1 1.00 0.000<br />
| n79 1 0.10 0.332<br />
| * 1 1.00 0.000<br />
| n108 1 0.10 0.332<br />
| * 1 1.00 0.000<br />
| n142 1 0.10 0.332<br />
| * 1 1.00 0.000<br />
| n163 1 0.10 0.332<br />
| * 1 1.00 0.000<br />
| n167 1 0.10 0.332<br />
| * 1 1.00 0.000<br />
| c32_command 1 0.03 0.141<br />
| v3_neutral_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n59 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| c62_vocal 5 0.14 0.390<br />
| v2_positive_affect 3 0.60 0.442<br />
| at 3 1.00 0.000<br />
| n151 1 0.33 0.528<br />
| * 1 1.00 0.000<br />
| n156 1 0.33 0.528<br />
| * 1 1.00 0.000<br />
| n170 1 0.33 0.528<br />
| * 1 1.00 0.000<br />
| v3_neutral_affect 2 0.40 0.529<br />
| at 2 1.00 0.000<br />
| n73 1 0.50 0.500<br />
| * 1 1.00 0.000<br />
| n102 1 0.50 0.500<br />
| * 1 1.00 0.000<br />
| c31_request 1 0.03 0.141<br />
| v3_neutral_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n81 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| c11_positive_verbal 1 0.03 0.141<br />
| v3_neutral_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n148 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| f9_null 2 0.05 0.206<br />
| c99_null 2 1.00 0.000<br />
| v9_null 2 1.00 0.000<br />
| at 2 1.00 0.000<br />
| n26 1 0.50 0.500<br />
| * 1 1.00 0.000<br />
| n175 1 0.50 0.500<br />
| * 1 1.00 0.000<br />
| f3_mother 4 0.09 0.319<br />
| c32_command 1 0.25 0.500<br />
| v3_neutral_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n45 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| c01_comply 1 0.25 0.500<br />
| v3_neutral_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n49 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| c42_command_ambiguous 1 0.25 0.500<br />
| v3_neutral_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n114 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| c62_vocal 1 0.25 0.500<br />
| v2_positive_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n125 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| V2_positive_affect 8 0.16 0.419<br />
| to 8 1.00 0.000<br />
| f1_child 7 0.88 0.169<br />
| c12_talk 4 0.57 0.461<br />
| v2_positive_affect 3 0.75 0.311<br />
| at 3 1.00 0.000<br />
| n12 1 0.33 0.528<br />
| * 1 1.00 0.000<br />
| n35 1 0.33 0.528<br />
| * 1 1.00 0.000<br />
| n110 1 0.33 0.528<br />
| * 1 1.00 0.000<br />
| v3_neutral_affect 1 0.25 0.500<br />
| at 1 1.00 0.000<br />
| n121 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| c62_vocal 3 0.43 0.524<br />
| v2_positive_affect 3 1.00 0.000<br />
| at 3 1.00 0.000<br />
| n43 1 0.33 0.528<br />
| * 1 1.00 0.000<br />
| n140 1 0.33 0.528<br />
| * 1 1.00 0.000<br />
| n144 1 0.33 0.528<br />
| * 1 1.00 0.000<br />
| f3_mother 1 0.13 0.375<br />
| c32_command 1 1.00 0.000<br />
| v3_neutral_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n66 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| C42_command_ambiguous 3 0.04 0.170<br />
| V3_neutral_affect 3 1.00 0.000<br />
| to 3 1.00 0.000<br />
| f1_child 2 0.67 0.390<br />
| c01_comply 2 1.00 0.000<br />
| v3_neutral_affect 2 1.00 0.000<br />
| at 2 1.00 0.000<br />
| n77 1 0.50 0.500<br />
| * 1 1.00 0.000<br />
| n115 1 0.50 0.500<br />
| * 1 1.00 0.000<br />
| f3_mother 1 0.33 0.528<br />
| c32_command 1 1.00 0.000<br />
| v3_neutral_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n4 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| C32_command 9 0.11 0.343<br />
| V3_neutral_affect 9 1.00 0.000<br />
| to 9 1.00 0.000<br />
| f1_child 8 0.89 0.151<br />
| c01_comply 6 0.75 0.311<br />
| v3_neutral_affect 6 1.00 0.000<br />
| at 6 1.00 0.000<br />
| n5 1 0.17 0.431<br />
| * 1 1.00 0.000<br />
| n46 1 0.17 0.431<br />
| * 1 1.00 0.000<br />
| n67 1 0.17 0.431<br />
| * 1 1.00 0.000<br />
| n93 1 0.17 0.431<br />
| * 1 1.00 0.000<br />
| n130 1 0.17 0.431<br />
| * 1 1.00 0.000<br />
| n172 1 0.17 0.431<br />
| * 1 1.00 0.000<br />
| c62_vocal 2 0.25 0.500<br />
| v2_positive_affect 2 1.00 0.000<br />
| at 2 1.00 0.000<br />
| n19 1 0.50 0.500<br />
| * 1 1.00 0.000<br />
| n21 1 0.50 0.500<br />
| * 1 1.00 0.000<br />
| f3_mother 1 0.11 0.352<br />
| c32_command 1 1.00 0.000<br />
| v3_neutral_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n129 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| C01_comply 3 0.04 0.170<br />
| V3_neutral_affect 3 1.00 0.000<br />
| to 3 1.00 0.000<br />
| f1_child 2 0.67 0.390<br />
| c12_talk 2 1.00 0.000<br />
| v3_neutral_affect 1 0.50 0.500<br />
| at 1 1.00 0.000<br />
| n50 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| v4_negative_affect 1 0.50 0.500<br />
| at 1 1.00 0.000<br />
| n64 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| f9_null 1 0.33 0.528<br />
| c99_null 1 1.00 0.000<br />
| v9_null 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n8 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| C11_positive_verbal 7 0.08 0.297<br />
| V2_positive_affect 4 0.57 0.461<br />
| to 4 1.00 0.000<br />
| f1_child 3 0.75 0.311<br />
| c12_talk 2 0.67 0.390<br />
| v2_positive_affect 1 0.50 0.500<br />
| at 1 1.00 0.000<br />
| n28 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| v3_neutral_affect 1 0.50 0.500<br />
| at 1 1.00 0.000<br />
| n96 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| c62_vocal 1 0.33 0.528<br />
| v2_positive_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n153 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| f3_mother 1 0.25 0.500<br />
| c12_talk 1 1.00 0.000<br />
| v2_positive_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n139 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| V3_neutral_affect 3 0.43 0.524<br />
| to 3 1.00 0.000<br />
| f1_child 2 0.67 0.390<br />
| c12_talk 2 1.00 0.000<br />
| v3_neutral_affect 2 1.00 0.000<br />
| at 2 1.00 0.000<br />
| n10 1 0.50 0.500<br />
| * 1 1.00 0.000<br />
| n117 1 0.50 0.500<br />
| * 1 1.00 0.000<br />
| f3_mother 1 0.33 0.528<br />
| c12_talk 1 1.00 0.000<br />
| v3_neutral_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n150 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| C62_vocal 5 0.06 0.240<br />
| V2_positive_affect 4 0.80 0.258<br />
| to 4 1.00 0.000<br />
| f1_child 3 0.75 0.311<br />
| c62_vocal 2 0.67 0.390<br />
| v2_positive_affect 2 1.00 0.000<br />
| at 2 1.00 0.000<br />
| n32 1 0.50 0.500<br />
| * 1 1.00 0.000<br />
| n126 1 0.50 0.500<br />
| * 1 1.00 0.000<br />
| c12_talk 1 0.33 0.528<br />
| v3_neutral_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n104 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| f3_mother 1 0.25 0.500<br />
| c22_tease 1 1.00 0.000<br />
| v2_positive_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n158 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| V3_neutral_affect 1 0.20 0.464<br />
| to 1 1.00 0.000<br />
| f3_mother 1 1.00 0.000<br />
| c32_command 1 1.00 0.000<br />
| v3_neutral_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n18 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| C13_negative_verbal 3 0.04 0.170<br />
| V2_positive_affect 2 0.67 0.390<br />
| to 2 1.00 0.000<br />
| f9_null 1 0.50 0.500<br />
| c99_null 1 1.00 0.000<br />
| v9_null 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n23 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| f3_mother 1 0.50 0.500<br />
| c32_command 1 1.00 0.000<br />
| v3_neutral_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n128 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| V4_negative_affect 1 0.33 0.528<br />
| to 1 1.00 0.000<br />
| f3_mother 1 1.00 0.000<br />
| c12_talk 1 1.00 0.000<br />
| v3_neutral_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n169 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| C51_agree 3 0.04 0.170<br />
| V3_neutral_affect 3 1.00 0.000<br />
| to 3 1.00 0.000<br />
| f3_mother 2 0.67 0.390<br />
| c12_talk 2 1.00 0.000<br />
| v3_neutral_affect 2 1.00 0.000<br />
| at 2 1.00 0.000<br />
| n48 1 0.50 0.500<br />
| * 1 1.00 0.000<br />
| n61 1 0.50 0.500<br />
| * 1 1.00 0.000<br />
| f1_child 1 0.33 0.528<br />
| c12_talk 1 1.00 0.000<br />
| v3_neutral_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n83 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| C22_tease 1 0.01 0.075<br />
| V2_positive_affect 1 1.00 0.000<br />
| to 1 1.00 0.000<br />
| f1_child 1 1.00 0.000<br />
| c62_vocal 1 1.00 0.000<br />
| v2_positive_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n159 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| F1_child 83 0.47 0.510<br />
| C12_talk 51 0.61 0.432<br />
| V3_neutral_affect 30 0.59 0.450<br />
| to 30 1.00 0.000<br />
| f3_mother 30 1.00 0.000<br />
| c12_talk 24 0.80 0.258<br />
| v3_neutral_affect 22 0.92 0.115<br />
| at 22 1.00 0.000<br />
| n25 1 0.05 0.203<br />
| * 1 1.00 0.000<br />
| n51 1 0.05 0.203<br />
| * 1 1.00 0.000<br />
| n53 1 0.05 0.203<br />
| * 1 1.00 0.000<br />
| n72 1 0.05 0.203<br />
| * 1 1.00 0.000<br />
| n84 1 0.05 0.203<br />
| * 1 1.00 0.000<br />
| n86 1 0.05 0.203<br />
| * 1 1.00 0.000<br />
| n88 1 0.05 0.203<br />
| * 1 1.00 0.000<br />
| n90 1 0.05 0.203<br />
| * 1 1.00 0.000<br />
| n97 1 0.05 0.203<br />
| * 1 1.00 0.000<br />
| n99 1 0.05 0.203<br />
| * 1 1.00 0.000<br />
| n101 1 0.05 0.203<br />
| * 1 1.00 0.000<br />
| n105 1 0.05 0.203<br />
| * 1 1.00 0.000<br />
| n107 1 0.05 0.203<br />
| * 1 1.00 0.000<br />
| n113 1 0.05 0.203<br />
| * 1 1.00 0.000<br />
| n118 1 0.05 0.203<br />
| * 1 1.00 0.000<br />
| n122 1 0.05 0.203<br />
| * 1 1.00 0.000<br />
| n124 1 0.05 0.203<br />
| * 1 1.00 0.000<br />
| n134 1 0.05 0.203<br />
| * 1 1.00 0.000<br />
| n136 1 0.05 0.203<br />
| * 1 1.00 0.000<br />
| n147 1 0.05 0.203<br />
| * 1 1.00 0.000<br />
| n162 1 0.05 0.203<br />
| * 1 1.00 0.000<br />
| n174 1 0.05 0.203<br />
| * 1 1.00 0.000<br />
| v2_positive_affect 2 0.08 0.299<br />
| at 2 1.00 0.000<br />
| n11 1 0.50 0.500<br />
| * 1 1.00 0.000<br />
| n120 1 0.50 0.500<br />
| * 1 1.00 0.000<br />
| c62_vocal 1 0.03 0.164<br />
| v3_neutral_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n17 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| c01_comply 1 0.03 0.164<br />
| v3_neutral_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n63 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| c32_command 1 0.03 0.164<br />
| v3_neutral_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n92 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| c11_positive_verbal 2 0.07 0.260<br />
| v2_positive_affect 2 1.00 0.000<br />
| at 2 1.00 0.000<br />
| n95 1 0.50 0.500<br />
| * 1 1.00 0.000<br />
| n138 1 0.50 0.500<br />
| * 1 1.00 0.000<br />
| c42_command_ambiguous 1 0.03 0.164<br />
| v3_neutral_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n3 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| V2_positive_affect 19 0.37 0.531<br />
| to 19 1.00 0.000<br />
| f3_mother 12 0.63 0.419<br />
| c12_talk 10 0.83 0.219<br />
| v3_neutral_affect 7 0.70 0.360<br />
| at 7 1.00 0.000<br />
| n13 1 0.14 0.401<br />
| * 1 1.00 0.000<br />
| n29 1 0.14 0.401<br />
| * 1 1.00 0.000<br />
| n58 1 0.14 0.401<br />
| * 1 1.00 0.000<br />
| n70 1 0.14 0.401<br />
| * 1 1.00 0.000<br />
| n80 1 0.14 0.401<br />
| * 1 1.00 0.000<br />
| n111 1 0.14 0.401<br />
| * 1 1.00 0.000<br />
| n155 1 0.14 0.401<br />
| * 1 1.00 0.000<br />
| v2_positive_affect 3 0.30 0.521<br />
| at 3 1.00 0.000<br />
| n34 1 0.33 0.528<br />
| * 1 1.00 0.000<br />
| n109 1 0.33 0.528<br />
| * 1 1.00 0.000<br />
| n143 1 0.33 0.528<br />
| * 1 1.00 0.000<br />
| c62_vocal 1 0.08 0.299<br />
| v2_positive_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n31 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| c13_negative_verbal 1 0.08 0.299<br />
| v4_negative_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n168 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| f9_null 2 0.11 0.342<br />
| c99_null 2 1.00 0.000<br />
| v9_null 2 1.00 0.000<br />
| at 2 1.00 0.000<br />
| n15 1 0.50 0.500<br />
| * 1 1.00 0.000<br />
| n164 1 0.50 0.500<br />
| * 1 1.00 0.000<br />
| f1_child 5 0.26 0.507<br />
| c62_vocal 4 0.80 0.258<br />
| v2_positive_affect 4 1.00 0.000<br />
| at 4 1.00 0.000<br />
| n36 1 0.25 0.500<br />
| * 1 1.00 0.000<br />
| n38 1 0.25 0.500<br />
| * 1 1.00 0.000<br />
| n41 1 0.25 0.500<br />
| * 1 1.00 0.000<br />
| n55 1 0.25 0.500<br />
| * 1 1.00 0.000<br />
| c12_talk 1 0.20 0.464<br />
| v1_exuberant_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n75 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| V4_negative_affect 1 0.02 0.111<br />
| to 1 1.00 0.000<br />
| f3_mother 1 1.00 0.000<br />
| c12_talk 1 1.00 0.000<br />
| v2_positive_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n65 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| V1_exuberant_affect 1 0.02 0.111<br />
| to 1 1.00 0.000<br />
| f3_mother 1 1.00 0.000<br />
| c42_command_ambiguous 1 1.00 0.000<br />
| v3_neutral_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n76 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| C01_comply 9 0.11 0.348<br />
| V3_neutral_affect 9 1.00 0.000<br />
| to 9 1.00 0.000<br />
| f3_mother 5 0.56 0.471<br />
| c12_talk 3 0.60 0.442<br />
| v3_neutral_affect 3 1.00 0.000<br />
| at 3 1.00 0.000<br />
| n68 1 0.33 0.528<br />
| * 1 1.00 0.000<br />
| n78 1 0.33 0.528<br />
| * 1 1.00 0.000<br />
| n132 1 0.33 0.528<br />
| * 1 1.00 0.000<br />
| c11_positive_verbal 1 0.20 0.464<br />
| v3_neutral_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n116 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| c51_agree 1 0.20 0.464<br />
| v3_neutral_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n47 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| f1_child 4 0.44 0.520<br />
| c12_talk 2 0.50 0.500<br />
| v3_neutral_affect 2 1.00 0.000<br />
| at 2 1.00 0.000<br />
| n94 1 0.50 0.500<br />
| * 1 1.00 0.000<br />
| n173 1 0.50 0.500<br />
| * 1 1.00 0.000<br />
| c01_comply 1 0.25 0.500<br />
| v3_neutral_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n131 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| c42_command_ambiguous 1 0.25 0.500<br />
| v3_neutral_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n6 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| C42_command_ambiguous 1 0.01 0.077<br />
| V3_neutral_affect 1 1.00 0.000<br />
| to 1 1.00 0.000<br />
| f3_mother 1 1.00 0.000<br />
| c01_comply 1 1.00 0.000<br />
| v3_neutral_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n7 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| C62_vocal 19 0.23 0.487<br />
| V2_positive_affect 17 0.89 0.144<br />
| to 17 1.00 0.000<br />
| f3_mother 13 0.76 0.296<br />
| c12_talk 7 0.54 0.481<br />
| v3_neutral_affect 6 0.86 0.191<br />
| at 6 1.00 0.000<br />
| n39 1 0.17 0.431<br />
| * 1 1.00 0.000<br />
| n44 1 0.17 0.431<br />
| * 1 1.00 0.000<br />
| n141 1 0.17 0.431<br />
| * 1 1.00 0.000<br />
| n145 1 0.17 0.431<br />
| * 1 1.00 0.000<br />
| n160 1 0.17 0.431<br />
| * 1 1.00 0.000<br />
| n166 1 0.17 0.431<br />
| * 1 1.00 0.000<br />
| v2_positive_affect 1 0.14 0.401<br />
| at 1 1.00 0.000<br />
| n42 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| c11_positive_verbal 1 0.08 0.285<br />
| v2_positive_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n152 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| c62_vocal 1 0.08 0.285<br />
| v2_positive_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n157 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| c32_command 2 0.15 0.415<br />
| v3_neutral_affect 2 1.00 0.000<br />
| at 2 1.00 0.000<br />
| n20 1 0.50 0.500<br />
| * 1 1.00 0.000<br />
| n171 1 0.50 0.500<br />
| * 1 1.00 0.000<br />
| c13_negative_verbal 2 0.15 0.415<br />
| v2_positive_affect 2 1.00 0.000<br />
| at 2 1.00 0.000<br />
| n22 1 0.50 0.500<br />
| * 1 1.00 0.000<br />
| n127 1 0.50 0.500<br />
| * 1 1.00 0.000<br />
| f1_child 3 0.18 0.442<br />
| c12_talk 3 1.00 0.000<br />
| v2_positive_affect 3 1.00 0.000<br />
| at 3 1.00 0.000<br />
| n33 1 0.33 0.528<br />
| * 1 1.00 0.000<br />
| n37 1 0.33 0.528<br />
| * 1 1.00 0.000<br />
| n154 1 0.33 0.528<br />
| * 1 1.00 0.000<br />
| f9_null 1 0.06 0.240<br />
| c99_null 1 1.00 0.000<br />
| v9_null 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n56 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| V3_neutral_affect 2 0.11 0.342<br />
| to 2 1.00 0.000<br />
| f1_child 1 0.50 0.500<br />
| c12_talk 1 1.00 0.000<br />
| v2_positive_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n74 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| f3_mother 1 0.50 0.500<br />
| c62_vocal 1 1.00 0.000<br />
| v2_positive_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n103 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| C32_command 1 0.01 0.077<br />
| V3_neutral_affect 1 1.00 0.000<br />
| to 1 1.00 0.000<br />
| f3_mother 1 1.00 0.000<br />
| c51_agree 1 1.00 0.000<br />
| v3_neutral_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n60 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| C31_request 1 0.01 0.077<br />
| V3_neutral_affect 1 1.00 0.000<br />
| to 1 1.00 0.000<br />
| f3_mother 1 1.00 0.000<br />
| c51_agree 1 1.00 0.000<br />
| v3_neutral_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n82 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| C11_positive_verbal 1 0.01 0.077<br />
| V3_neutral_affect 1 1.00 0.000<br />
| to 1 1.00 0.000<br />
| f3_mother 1 1.00 0.000<br />
| c11_positive_verbal 1 1.00 0.000<br />
| v3_neutral_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n149 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| F9_null 7 0.04 0.186<br />
| C99_null 7 1.00 0.000<br />
| V9_null 7 1.00 0.000<br />
| to 7 1.00 0.000<br />
| f1_child 4 0.57 0.461<br />
| c12_talk 3 0.75 0.311<br />
| v3_neutral_affect 2 0.67 0.390<br />
| at 2 1.00 0.000<br />
| n16 1 0.50 0.500<br />
| * 1 1.00 0.000<br />
| n24 1 0.50 0.500<br />
| * 1 1.00 0.000<br />
| v2_positive_affect 1 0.33 0.528<br />
| at 1 1.00 0.000<br />
| n57 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| c62_vocal 1 0.25 0.500<br />
| v2_positive_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n165 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| f3_mother 3 0.43 0.524<br />
| c11_positive_verbal 2 0.67 0.390<br />
| v3_neutral_affect 1 0.50 0.500<br />
| at 1 1.00 0.000<br />
| n9 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| v2_positive_affect 1 0.50 0.500<br />
| at 1 1.00 0.000<br />
| n27 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
| c12_talk 1 0.33 0.528<br />
| v3_neutral_affect 1 1.00 0.000<br />
| at 1 1.00 0.000<br />
| n1 1 1.00 0.000<br />
| * 1 1.00 0.000<br />
</pre><br />
<br />
==Family Interaction Study &bull; Datasets==<br />
<br />
<pre><br />
| Family Interaction Analysis<br />
| Abbreviated Dataset 1<br />
| Pairwise Transitions<br />
| Numbered Episodes<br />
|<br />
| f9 c99 v9 f3 c12 v3 n1<br />
| f3 c12 v3 f1 c12 v3 n2<br />
| f1 c12 v3 f3 c42 v3 n3<br />
| f3 c42 v3 f3 c32 v3 n4<br />
| f3 c32 v3 f1 c01 v3 n5<br />
| f1 c01 v3 f1 c42 v3 n6<br />
| f1 c42 v3 f3 c01 v3 n7<br />
| f3 c01 v3 f9 c99 v9 n8<br />
|<br />
| f9 c99 v9 f3 c11 v3 n9<br />
| f3 c11 v3 f1 c12 v3 n10<br />
| f1 c12 v3 f3 c12 v2 n11<br />
| f3 c12 v2 f1 c12 v2 n12<br />
| f1 c12 v2 f3 c12 v3 n13<br />
| f3 c12 v3 f1 c12 v2 n14<br />
| f1 c12 v2 f9 c99 v9 n15<br />
|<br />
| f9 c99 v9 f1 c12 v3 n16<br />
| f1 c12 v3 f3 c62 v3 n17<br />
| f3 c62 v3 f3 c32 v3 n18<br />
| f3 c32 v3 f1 c62 v2 n19<br />
| f1 c62 v2 f3 c32 v3 n20<br />
| f3 c32 v3 f1 c62 v2 n21<br />
| f1 c62 v2 f3 c13 v2 n22<br />
| f3 c13 v2 f9 c99 v9 n23<br />
|<br />
| f9 c99 v9 f1 c12 v3 n24<br />
| f1 c12 v3 f3 c12 v3 n25<br />
| f3 c12 v3 f9 c99 v9 n26<br />
|<br />
| f9 c99 v9 f3 c11 v2 n27<br />
| f3 c11 v2 f1 c12 v2 n28<br />
| f1 c12 v2 f3 c12 v3 n29<br />
| f3 c12 v3 f1 c12 v2 n30<br />
| f1 c12 v2 f3 c62 v2 n31<br />
| f3 c62 v2 f1 c62 v2 n32<br />
| f1 c62 v2 f1 c12 v2 n33<br />
| f1 c12 v2 f3 c12 v2 n34<br />
| f3 c12 v2 f1 c12 v2 n35<br />
| f1 c12 v2 f1 c62 v2 n36<br />
| f1 c62 v2 f1 c12 v2 n37<br />
| f1 c12 v2 f1 c62 v2 n38<br />
| f1 c62 v2 f3 c12 v3 n39<br />
| f3 c12 v3 f1 c12 v2 n40<br />
| f1 c12 v2 f1 c62 v2 n41<br />
| f1 c62 v2 f3 c12 v2 n42<br />
| f3 c12 v2 f1 c62 v2 n43<br />
| f1 c62 v2 f3 c12 v3 n44<br />
| f3 c12 v3 f3 c32 v3 n45<br />
| f3 c32 v3 f1 c01 v3 n46<br />
| f1 c01 v3 f3 c51 v3 n47<br />
| f3 c51 v3 f3 c12 v3 n48<br />
| f3 c12 v3 f3 c01 v3 n49<br />
| f3 c01 v3 f1 c12 v3 n50<br />
| f1 c12 v3 f3 c12 v3 n51<br />
| f3 c12 v3 f1 c12 v3 n52<br />
| f1 c12 v3 f3 c12 v3 n53<br />
| f3 c12 v3 f1 c12 v2 n54<br />
| f1 c12 v2 f1 c62 v2 n55<br />
| f1 c62 v2 f9 c99 v9 n56<br />
|<br />
| f9 c99 v9 f1 c12 v2 n57<br />
| f1 c12 v2 f3 c12 v3 n58<br />
| f3 c12 v3 f1 c32 v3 n59<br />
| f1 c32 v3 f3 c51 v3 n60<br />
| f3 c51 v3 f3 c12 v3 n61<br />
| f3 c12 v3 f1 c12 v3 n62<br />
| f1 c12 v3 f3 c01 v3 n63<br />
| f3 c01 v3 f1 c12 v4 n64<br />
| f1 c12 v4 f3 c12 v2 n65<br />
| f3 c12 v2 f3 c32 v3 n66<br />
| f3 c32 v3 f1 c01 v3 n67<br />
| f1 c01 v3 f3 c12 v3 n68<br />
| f3 c12 v3 f1 c12 v2 n69<br />
| f1 c12 v2 f3 c12 v3 n70<br />
| f3 c12 v3 f1 c12 v3 n71<br />
| f1 c12 v3 f3 c12 v3 n72<br />
| f3 c12 v3 f1 c62 v3 n73<br />
| f1 c62 v3 f1 c12 v2 n74<br />
| f1 c12 v2 f1 c12 v1 n75<br />
| f1 c12 v1 f3 c42 v3 n76<br />
| f3 c42 v3 f1 c01 v3 n77<br />
| f1 c01 v3 f3 c12 v3 n78<br />
| f3 c12 v3 f1 c12 v2 n79<br />
| f1 c12 v2 f3 c12 v3 n80<br />
| f3 c12 v3 f1 c31 v3 n81<br />
| f1 c31 v3 f3 c51 v3 n82<br />
| f3 c51 v3 f1 c12 v3 n83<br />
| f1 c12 v3 f3 c12 v3 n84<br />
| f3 c12 v3 f1 c12 v3 n85<br />
| f1 c12 v3 f3 c12 v3 n86<br />
| f3 c12 v3 f1 c12 v3 n87<br />
| f1 c12 v3 f3 c12 v3 n88<br />
| f3 c12 v3 f1 c12 v3 n89<br />
| f1 c12 v3 f3 c12 v3 n90<br />
| f3 c12 v3 f1 c12 v3 n91<br />
| f1 c12 v3 f3 c32 v3 n92<br />
| f3 c32 v3 f1 c01 v3 n93<br />
| f1 c01 v3 f1 c12 v3 n94<br />
| f1 c12 v3 f3 c11 v2 n95<br />
| f3 c11 v2 f1 c12 v3 n96<br />
| f1 c12 v3 f3 c12 v3 n97<br />
| f3 c12 v3 f1 c12 v3 n98<br />
| f1 c12 v3 f3 c12 v3 n99<br />
| f3 c12 v3 f1 c12 v3 n100<br />
| f1 c12 v3 f3 c12 v3 n101<br />
| f3 c12 v3 f1 c62 v3 n102<br />
| f1 c62 v3 f3 c62 v2 n103<br />
| f3 c62 v2 f1 c12 v3 n104<br />
| f1 c12 v3 f3 c12 v3 n105<br />
| f3 c12 v3 f1 c12 v3 n106<br />
| f1 c12 v3 f3 c12 v3 n107<br />
| f3 c12 v3 f1 c12 v2 n108<br />
| f1 c12 v2 f3 c12 v2 n109<br />
| f3 c12 v2 f1 c12 v2 n110<br />
| f1 c12 v2 f3 c12 v3 n111<br />
| f3 c12 v3 f1 c12 v3 n112<br />
| f1 c12 v3 f3 c12 v3 n113<br />
| f3 c12 v3 f3 c42 v3 n114<br />
| f3 c42 v3 f1 c01 v3 n115<br />
| f1 c01 v3 f3 c11 v3 n116<br />
| f3 c11 v3 f1 c12 v3 n117<br />
| f1 c12 v3 f3 c12 v3 n118<br />
| f3 c12 v3 f1 c12 v3 n119<br />
| f1 c12 v3 f3 c12 v2 n120<br />
| f3 c12 v2 f1 c12 v3 n121<br />
| f1 c12 v3 f3 c12 v3 n122<br />
| f3 c12 v3 f1 c12 v3 n123<br />
| f1 c12 v3 f3 c12 v3 n124<br />
| f3 c12 v3 f3 c62 v2 n125<br />
| f3 c62 v2 f1 c62 v2 n126<br />
| f1 c62 v2 f3 c13 v2 n127<br />
| f3 c13 v2 f3 c32 v3 n128<br />
| f3 c32 v3 f3 c32 v3 n129<br />
| f3 c32 v3 f1 c01 v3 n130<br />
| f1 c01 v3 f1 c01 v3 n131<br />
| f1 c01 v3 f3 c12 v3 n132<br />
| f3 c12 v3 f1 c12 v3 n133<br />
| f1 c12 v3 f3 c12 v3 n134<br />
| f3 c12 v3 f1 c12 v3 n135<br />
| f1 c12 v3 f3 c12 v3 n136<br />
| f3 c12 v3 f1 c12 v3 n137<br />
| f1 c12 v3 f3 c11 v2 n138<br />
| f3 c11 v2 f3 c12 v2 n139<br />
| f3 c12 v2 f1 c62 v2 n140<br />
| f1 c62 v2 f3 c12 v3 n141<br />
| f3 c12 v3 f1 c12 v2 n142<br />
| f1 c12 v2 f3 c12 v2 n143<br />
| f3 c12 v2 f1 c62 v2 n144<br />
| f1 c62 v2 f3 c12 v3 n145<br />
| f3 c12 v3 f1 c12 v3 n146<br />
| f1 c12 v3 f3 c12 v3 n147<br />
| f3 c12 v3 f1 c11 v3 n148<br />
| f1 c11 v3 f3 c11 v3 n149<br />
| f3 c11 v3 f3 c12 v3 n150<br />
| f3 c12 v3 f1 c62 v2 n151<br />
| f1 c62 v2 f3 c11 v2 n152<br />
| f3 c11 v2 f1 c62 v2 n153<br />
| f1 c62 v2 f1 c12 v2 n154<br />
| f1 c12 v2 f3 c12 v3 n155<br />
| f3 c12 v3 f1 c62 v2 n156<br />
| f1 c62 v2 f3 c62 v2 n157<br />
| f3 c62 v2 f3 c22 v2 n158<br />
| f3 c22 v2 f1 c62 v2 n159<br />
| f1 c62 v2 f3 c12 v3 n160<br />
| f3 c12 v3 f1 c12 v3 n161<br />
| f1 c12 v3 f3 c12 v3 n162<br />
| f3 c12 v3 f1 c12 v2 n163<br />
| f1 c12 v2 f9 c99 v9 n164<br />
|<br />
| f9 c99 v9 f1 c62 v2 n165<br />
| f1 c62 v2 f3 c12 v3 n166<br />
| f3 c12 v3 f1 c12 v2 n167<br />
| f1 c12 v2 f3 c13 v4 n168<br />
| f3 c13 v4 f3 c12 v3 n169<br />
| f3 c12 v3 f1 c62 v2 n170<br />
| f1 c62 v2 f3 c32 v3 n171<br />
| f3 c32 v3 f1 c01 v3 n172<br />
| f1 c01 v3 f1 c12 v3 n173<br />
| f1 c12 v3 f3 c12 v3 n174<br />
| f3 c12 v3 f9 c99 v9 n175<br />
</pre><br />
<br />
<pre><br />
| Family Interaction Analysis<br />
| Abbreviated Dataset 1<br />
| Numbered Episodes<br />
|<br />
| 9 99 9 0 f9 c99 v9 n0<br />
|<br />
| 3 12 3 5 f3 c12 v3 n1<br />
| 1 12 3 8 f1 c12 v3 n2<br />
| 3 42 3 10 f3 c42 v3 n3<br />
| 3 32 3 18 f3 c32 v3 n4<br />
| 1 01 3 19 f1 c01 v3 n5<br />
| 1 42 3 22 f1 c42 v3 n6<br />
| 3 01 3 24 f3 c01 v3 n7<br />
| 9 99 9 29 f9 c99 v9 n8<br />
|<br />
| 3 11 3 40 f3 c11 v3 n9<br />
| 1 12 3 43 f1 c12 v3 n10<br />
| 3 12 2 46 f3 c12 v2 n11<br />
| 1 12 2 51 f1 c12 v2 n12<br />
| 3 12 3 54 f3 c12 v3 n13<br />
| 1 12 2 56 f1 c12 v2 n14<br />
| 9 99 9 61 f9 c99 v9 n15<br />
|<br />
| 1 12 3 68 f1 c12 v3 n16<br />
| 3 62 3 70 f3 c62 v3 n17<br />
| 3 32 3 73 f3 c32 v3 n18<br />
| 1 62 2 74 f1 c62 v2 n19<br />
| 3 32 3 77 f3 c32 v3 n20<br />
| 1 62 2 78 f1 c62 v2 n21<br />
| 3 13 2 82 f3 c13 v2 n22<br />
| 9 99 9 91 f9 c99 v9 n23<br />
|<br />
| 1 12 3 94 f1 c12 v3 n24<br />
| 3 12 3 96 f3 c12 v3 n25<br />
| 9 99 9 101 f9 c99 v9 n26<br />
|<br />
| 3 11 2 103 f3 c11 v2 n27<br />
| 1 12 2 106 f1 c12 v2 n28<br />
| 3 12 3 108 f3 c12 v3 n29<br />
| 1 12 2 109 f1 c12 v2 n30<br />
| 3 62 2 112 f3 c62 v2 n31<br />
| 1 62 2 115 f1 c62 v2 n32<br />
| 1 12 2 120 f1 c12 v2 n33<br />
| 3 12 2 122 f3 c12 v2 n34<br />
| 1 12 2 127 f1 c12 v2 n35<br />
| 1 62 2 131 f1 c62 v2 n36<br />
| 1 12 2 136 f1 c12 v2 n37<br />
| 1 62 2 138 f1 c62 v2 n38<br />
| 3 12 3 143 f3 c12 v3 n39<br />
| 1 12 2 148 f1 c12 v2 n40<br />
| 1 62 2 155 f1 c62 v2 n41<br />
| 3 12 2 160 f3 c12 v2 n42<br />
| 1 62 2 168 f1 c62 v2 n43<br />
| 3 12 3 170 f3 c12 v3 n44<br />
| 3 32 3 175 f3 c32 v3 n45<br />
| 1 01 3 177 f1 c01 v3 n46<br />
| 3 51 3 184 f3 c51 v3 n47<br />
| 3 12 3 187 f3 c12 v3 n48<br />
| 3 01 3 189 f3 c01 v3 n49<br />
| 1 12 3 192 f1 c12 v3 n50<br />
| 3 12 3 194 f3 c12 v3 n51<br />
| 1 12 3 197 f1 c12 v3 n52<br />
| 3 12 3 200 f3 c12 v3 n53<br />
| 1 12 2 202 f1 c12 v2 n54<br />
| 1 62 2 205 f1 c62 v2 n55<br />
| 9 99 9 211 f9 c99 v9 n56<br />
|<br />
| 1 12 2 214 f1 c12 v2 n57<br />
| 3 12 3 218 f3 c12 v3 n58<br />
| 1 32 3 222 f1 c32 v3 n59<br />
| 3 51 3 225 f3 c51 v3 n60<br />
| 3 12 3 227 f3 c12 v3 n61<br />
| 1 12 3 228 f1 c12 v3 n62<br />
| 3 01 3 230 f3 c01 v3 n63<br />
| 1 12 4 234 f1 c12 v4 n64<br />
| 3 12 2 238 f3 c12 v2 n65<br />
| 3 32 3 241 f3 c32 v3 n66<br />
| 1 01 3 242 f1 c01 v3 n67<br />
| 3 12 3 244 f3 c12 v3 n68<br />
| 1 12 2 245 f1 c12 v2 n69<br />
| 3 12 3 252 f3 c12 v3 n70<br />
| 1 12 3 254 f1 c12 v3 n71<br />
| 3 12 3 256 f3 c12 v3 n72<br />
| 1 62 3 267 f1 c62 v3 n73<br />
| 1 12 2 269 f1 c12 v2 n74<br />
| 1 12 1 278 f1 c12 v1 n75<br />
| 3 42 3 282 f3 c42 v3 n76<br />
| 1 01 3 287 f1 c01 v3 n77<br />
| 3 12 3 289 f3 c12 v3 n78<br />
| 1 12 2 291 f1 c12 v2 n79<br />
| 3 12 3 296 f3 c12 v3 n80<br />
| 1 31 3 301 f1 c31 v3 n81<br />
| 3 51 3 305 f3 c51 v3 n82<br />
| 1 12 3 309 f1 c12 v3 n83<br />
| 3 12 3 311 f3 c12 v3 n84<br />
| 1 12 3 313 f1 c12 v3 n85<br />
| 3 12 3 316 f3 c12 v3 n86<br />
| 1 12 3 319 f1 c12 v3 n87<br />
| 3 12 3 322 f3 c12 v3 n88<br />
| 1 12 3 323 f1 c12 v3 n89<br />
| 3 12 3 324 f3 c12 v3 n90<br />
| 1 12 3 326 f1 c12 v3 n91<br />
| 3 32 3 329 f3 c32 v3 n92<br />
| 1 01 3 331 f1 c01 v3 n93<br />
| 1 12 3 333 f1 c12 v3 n94<br />
| 3 11 2 338 f3 c11 v2 n95<br />
| 1 12 3 342 f1 c12 v3 n96<br />
| 3 12 3 344 f3 c12 v3 n97<br />
| 1 12 3 347 f1 c12 v3 n98<br />
| 3 12 3 349 f3 c12 v3 n99<br />
| 1 12 3 351 f1 c12 v3 n100<br />
| 3 12 3 352 f3 c12 v3 n101<br />
| 1 62 3 354 f1 c62 v3 n102<br />
| 3 62 2 356 f3 c62 v2 n103<br />
| 1 12 3 362 f1 c12 v3 n104<br />
| 3 12 3 367 f3 c12 v3 n105<br />
| 1 12 3 370 f1 c12 v3 n106<br />
| 3 12 3 371 f3 c12 v3 n107<br />
| 1 12 2 373 f1 c12 v2 n108<br />
| 3 12 2 378 f3 c12 v2 n109<br />
| 1 12 2 381 f1 c12 v2 n110<br />
| 3 12 3 382 f3 c12 v3 n111<br />
| 1 12 3 389 f1 c12 v3 n112<br />
| 3 12 3 392 f3 c12 v3 n113<br />
| 3 42 3 394 f3 c42 v3 n114<br />
| 1 01 3 396 f1 c01 v3 n115<br />
| 3 11 3 398 f3 c11 v3 n116<br />
| 1 12 3 401 f1 c12 v3 n117<br />
| 3 12 3 403 f3 c12 v3 n118<br />
| 1 12 3 405 f1 c12 v3 n119<br />
| 3 12 2 406 f3 c12 v2 n120<br />
| 1 12 3 409 f1 c12 v3 n121<br />
| 3 12 3 412 f3 c12 v3 n122<br />
| 1 12 3 414 f1 c12 v3 n123<br />
| 3 12 3 416 f3 c12 v3 n124<br />
| 3 62 2 417 f3 c62 v2 n125<br />
| 1 62 2 419 f1 c62 v2 n126<br />
| 3 13 2 422 f3 c13 v2 n127<br />
| 3 32 3 426 f3 c32 v3 n128<br />
| 3 32 3 430 f3 c32 v3 n129<br />
| 1 01 3 431 f1 c01 v3 n130<br />
| 1 01 3 434 f1 c01 v3 n131<br />
| 3 12 3 438 f3 c12 v3 n132<br />
| 1 12 3 439 f1 c12 v3 n133<br />
| 3 12 3 440 f3 c12 v3 n134<br />
| 1 12 3 441 f1 c12 v3 n135<br />
| 3 12 3 443 f3 c12 v3 n136<br />
| 1 12 3 449 f1 c12 v3 n137<br />
| 3 11 2 450 f3 c11 v2 n138<br />
| 3 12 2 455 f3 c12 v2 n139<br />
| 1 62 2 457 f1 c62 v2 n140<br />
| 3 12 3 462 f3 c12 v3 n141<br />
| 1 12 2 463 f1 c12 v2 n142<br />
| 3 12 2 467 f3 c12 v2 n143<br />
| 1 62 2 469 f1 c62 v2 n144<br />
| 3 12 3 473 f3 c12 v3 n145<br />
| 1 12 3 477 f1 c12 v3 n146<br />
| 3 12 3 479 f3 c12 v3 n147<br />
| 1 11 3 490 f1 c11 v3 n148<br />
| 3 11 3 492 f3 c11 v3 n149<br />
| 3 12 3 495 f3 c12 v3 n150<br />
| 1 62 2 514 f1 c62 v2 n151<br />
| 3 11 2 517 f3 c11 v2 n152<br />
| 1 62 2 524 f1 c62 v2 n153<br />
| 1 12 2 527 f1 c12 v2 n154<br />
| 3 12 3 530 f3 c12 v3 n155<br />
| 1 62 2 534 f1 c62 v2 n156<br />
| 3 62 2 537 f3 c62 v2 n157<br />
| 3 22 2 539 f3 c22 v2 n158<br />
| 1 62 2 544 f1 c62 v2 n159<br />
| 3 12 3 547 f3 c12 v3 n160<br />
| 1 12 3 548 f1 c12 v3 n161<br />
| 3 12 3 549 f3 c12 v3 n162<br />
| 1 12 2 553 f1 c12 v2 n163<br />
| 9 99 9 561 f9 c99 v9 n164<br />
|<br />
| 1 62 2 564 f1 c62 v2 n165<br />
| 3 12 3 569 f3 c12 v3 n166<br />
| 1 12 2 575 f1 c12 v2 n167<br />
| 3 13 4 577 f3 c13 v4 n168<br />
| 3 12 3 585 f3 c12 v3 n169<br />
| 1 62 2 589 f1 c62 v2 n170<br />
| 3 32 3 591 f3 c32 v3 n171<br />
| 1 01 3 593 f1 c01 v3 n172<br />
| 1 12 3 595 f1 c12 v3 n173<br />
| 3 12 3 599 f3 c12 v3 n174<br />
| 9 99 9 603 f9 c99 v9 n175<br />
</pre><br />
<br />
<pre><br />
| Family Interaction Analysis<br />
| Abbreviated Dataset 1<br />
| Timed Episodes<br />
|<br />
| 9 99 9 0 f9 c99 v9 t0<br />
|<br />
| 3 12 3 5 f3 c12 v3 t5<br />
| 1 12 3 8 f1 c12 v3 t8<br />
| 3 42 3 10 f3 c42 v3 t10<br />
| 3 32 3 18 f3 c32 v3 t18<br />
| 1 01 3 19 f1 c01 v3 t19<br />
| 1 42 3 22 f1 c42 v3 t22<br />
| 3 01 3 24 f3 c01 v3 t24<br />
| 9 99 9 29 f9 c99 v9 t29<br />
|<br />
| 3 11 3 40 f3 c11 v3 t40<br />
| 1 12 3 43 f1 c12 v3 t43<br />
| 3 12 2 46 f3 c12 v2 t46<br />
| 1 12 2 51 f1 c12 v2 t51<br />
| 3 12 3 54 f3 c12 v3 t54<br />
| 1 12 2 56 f1 c12 v2 t56<br />
| 9 99 9 61 f9 c99 v9 t61<br />
|<br />
| 1 12 3 68 f1 c12 v3 t68<br />
| 3 62 3 70 f3 c62 v3 t70<br />
| 3 32 3 73 f3 c32 v3 t73<br />
| 1 62 2 74 f1 c62 v2 t74<br />
| 3 32 3 77 f3 c32 v3 t77<br />
| 1 62 2 78 f1 c62 v2 t78<br />
| 3 13 2 82 f3 c13 v2 t82<br />
| 9 99 9 91 f9 c99 v9 t91<br />
|<br />
| 1 12 3 94 f1 c12 v3 t94<br />
| 3 12 3 96 f3 c12 v3 t96<br />
| 9 99 9 101 f9 c99 v9 t101<br />
|<br />
| 3 11 2 103 f3 c11 v2 t103<br />
| 1 12 2 106 f1 c12 v2 t106<br />
| 3 12 3 108 f3 c12 v3 t108<br />
| 1 12 2 109 f1 c12 v2 t109<br />
| 3 62 2 112 f3 c62 v2 t112<br />
| 1 62 2 115 f1 c62 v2 t115<br />
| 1 12 2 120 f1 c12 v2 t120<br />
| 3 12 2 122 f3 c12 v2 t122<br />
| 1 12 2 127 f1 c12 v2 t127<br />
| 1 62 2 131 f1 c62 v2 t131<br />
| 1 12 2 136 f1 c12 v2 t136<br />
| 1 62 2 138 f1 c62 v2 t138<br />
| 3 12 3 143 f3 c12 v3 t143<br />
| 1 12 2 148 f1 c12 v2 t148<br />
| 1 62 2 155 f1 c62 v2 t155<br />
| 3 12 2 160 f3 c12 v2 t160<br />
| 1 62 2 168 f1 c62 v2 t168<br />
| 3 12 3 170 f3 c12 v3 t170<br />
| 3 32 3 175 f3 c32 v3 t175<br />
| 1 01 3 177 f1 c01 v3 t177<br />
| 3 51 3 184 f3 c51 v3 t184<br />
| 3 12 3 187 f3 c12 v3 t187<br />
| 3 01 3 189 f3 c01 v3 t189<br />
| 1 12 3 192 f1 c12 v3 t192<br />
| 3 12 3 194 f3 c12 v3 t194<br />
| 1 12 3 197 f1 c12 v3 t197<br />
| 3 12 3 200 f3 c12 v3 t200<br />
| 1 12 2 202 f1 c12 v2 t202<br />
| 1 62 2 205 f1 c62 v2 t205<br />
| 9 99 9 211 f9 c99 v9 t211<br />
|<br />
| 1 12 2 214 f1 c12 v2 t214<br />
| 3 12 3 218 f3 c12 v3 t218<br />
| 1 32 3 222 f1 c32 v3 t222<br />
| 3 51 3 225 f3 c51 v3 t225<br />
| 3 12 3 227 f3 c12 v3 t227<br />
| 1 12 3 228 f1 c12 v3 t228<br />
| 3 01 3 230 f3 c01 v3 t230<br />
| 1 12 4 234 f1 c12 v4 t234<br />
| 3 12 2 238 f3 c12 v2 t238<br />
| 3 32 3 241 f3 c32 v3 t241<br />
| 1 01 3 242 f1 c01 v3 t242<br />
| 3 12 3 244 f3 c12 v3 t244<br />
| 1 12 2 245 f1 c12 v2 t245<br />
| 3 12 3 252 f3 c12 v3 t252<br />
| 1 12 3 254 f1 c12 v3 t254<br />
| 3 12 3 256 f3 c12 v3 t256<br />
| 1 62 3 267 f1 c62 v3 t267<br />
| 1 12 2 269 f1 c12 v2 t269<br />
| 1 12 1 278 f1 c12 v1 t278<br />
| 3 42 3 282 f3 c42 v3 t282<br />
| 1 01 3 287 f1 c01 v3 t287<br />
| 3 12 3 289 f3 c12 v3 t289<br />
| 1 12 2 291 f1 c12 v2 t291<br />
| 3 12 3 296 f3 c12 v3 t296<br />
| 1 31 3 301 f1 c31 v3 t301<br />
| 3 51 3 305 f3 c51 v3 t305<br />
| 1 12 3 309 f1 c12 v3 t309<br />
| 3 12 3 311 f3 c12 v3 t311<br />
| 1 12 3 313 f1 c12 v3 t313<br />
| 3 12 3 316 f3 c12 v3 t316<br />
| 1 12 3 319 f1 c12 v3 t319<br />
| 3 12 3 322 f3 c12 v3 t322<br />
| 1 12 3 323 f1 c12 v3 t323<br />
| 3 12 3 324 f3 c12 v3 t324<br />
| 1 12 3 326 f1 c12 v3 t326<br />
| 3 32 3 329 f3 c32 v3 t329<br />
| 1 01 3 331 f1 c01 v3 t331<br />
| 1 12 3 333 f1 c12 v3 t333<br />
| 3 11 2 338 f3 c11 v2 t338<br />
| 1 12 3 342 f1 c12 v3 t342<br />
| 3 12 3 344 f3 c12 v3 t344<br />
| 1 12 3 347 f1 c12 v3 t347<br />
| 3 12 3 349 f3 c12 v3 t349<br />
| 1 12 3 351 f1 c12 v3 t351<br />
| 3 12 3 352 f3 c12 v3 t352<br />
| 1 62 3 354 f1 c62 v3 t354<br />
| 3 62 2 356 f3 c62 v2 t356<br />
| 1 12 3 362 f1 c12 v3 t362<br />
| 3 12 3 367 f3 c12 v3 t367<br />
| 1 12 3 370 f1 c12 v3 t370<br />
| 3 12 3 371 f3 c12 v3 t371<br />
| 1 12 2 373 f1 c12 v2 t373<br />
| 3 12 2 378 f3 c12 v2 t378<br />
| 1 12 2 381 f1 c12 v2 t381<br />
| 3 12 3 382 f3 c12 v3 t382<br />
| 1 12 3 389 f1 c12 v3 t389<br />
| 3 12 3 392 f3 c12 v3 t392<br />
| 3 42 3 394 f3 c42 v3 t394<br />
| 1 01 3 396 f1 c01 v3 t396<br />
| 3 11 3 398 f3 c11 v3 t398<br />
| 1 12 3 401 f1 c12 v3 t401<br />
| 3 12 3 403 f3 c12 v3 t403<br />
| 1 12 3 405 f1 c12 v3 t405<br />
| 3 12 2 406 f3 c12 v2 t406<br />
| 1 12 3 409 f1 c12 v3 t409<br />
| 3 12 3 412 f3 c12 v3 t412<br />
| 1 12 3 414 f1 c12 v3 t414<br />
| 3 12 3 416 f3 c12 v3 t416<br />
| 3 62 2 417 f3 c62 v2 t417<br />
| 1 62 2 419 f1 c62 v2 t419<br />
| 3 13 2 422 f3 c13 v2 t422<br />
| 3 32 3 426 f3 c32 v3 t426<br />
| 3 32 3 430 f3 c32 v3 t430<br />
| 1 01 3 431 f1 c01 v3 t431<br />
| 1 01 3 434 f1 c01 v3 t434<br />
| 3 12 3 438 f3 c12 v3 t438<br />
| 1 12 3 439 f1 c12 v3 t439<br />
| 3 12 3 440 f3 c12 v3 t440<br />
| 1 12 3 441 f1 c12 v3 t441<br />
| 3 12 3 443 f3 c12 v3 t443<br />
| 1 12 3 449 f1 c12 v3 t449<br />
| 3 11 2 450 f3 c11 v2 t450<br />
| 3 12 2 455 f3 c12 v2 t455<br />
| 1 62 2 457 f1 c62 v2 t457<br />
| 3 12 3 462 f3 c12 v3 t462<br />
| 1 12 2 463 f1 c12 v2 t463<br />
| 3 12 2 467 f3 c12 v2 t467<br />
| 1 62 2 469 f1 c62 v2 t469<br />
| 3 12 3 473 f3 c12 v3 t473<br />
| 1 12 3 477 f1 c12 v3 t477<br />
| 3 12 3 479 f3 c12 v3 t479<br />
| 1 11 3 490 f1 c11 v3 t490<br />
| 3 11 3 492 f3 c11 v3 t492<br />
| 3 12 3 495 f3 c12 v3 t495<br />
| 1 62 2 514 f1 c62 v2 t514<br />
| 3 11 2 517 f3 c11 v2 t517<br />
| 1 62 2 524 f1 c62 v2 t524<br />
| 1 12 2 527 f1 c12 v2 t527<br />
| 3 12 3 530 f3 c12 v3 t530<br />
| 1 62 2 534 f1 c62 v2 t534<br />
| 3 62 2 537 f3 c62 v2 t537<br />
| 3 22 2 539 f3 c22 v2 t539<br />
| 1 62 2 544 f1 c62 v2 t544<br />
| 3 12 3 547 f3 c12 v3 t547<br />
| 1 12 3 548 f1 c12 v3 t548<br />
| 3 12 3 549 f3 c12 v3 t549<br />
| 1 12 2 553 f1 c12 v2 t553<br />
| 9 99 9 561 f9 c99 v9 t561<br />
|<br />
| 1 62 2 564 f1 c62 v2 t564<br />
| 3 12 3 569 f3 c12 v3 t569<br />
| 1 12 2 575 f1 c12 v2 t575<br />
| 3 13 4 577 f3 c13 v4 t577<br />
| 3 12 3 585 f3 c12 v3 t585<br />
| 1 62 2 589 f1 c62 v2 t589<br />
| 3 32 3 591 f3 c32 v3 t591<br />
| 1 01 3 593 f1 c01 v3 t593<br />
| 1 12 3 595 f1 c12 v3 t595<br />
| 3 12 3 599 f3 c12 v3 t599<br />
| 9 99 9 603 f9 c99 v9 t603<br />
</pre><br />
<br />
==Family Interaction Study &bull; Outputs==<br />
<br />
<pre><br />
FIT1.SUM<br />
<br />
from 175 1.00 0.000<br />
j1_f3_mother 85 0.49 0.506<br />
j2_c1_conversation 61 0.72 0.344<br />
j3_q2_neutral 51 0.84 0.216<br />
j4_v3_neutral_affect 43 0.84 0.208<br />
to 43 1.00 0.000<br />
k1_f1_child 37 0.86 0.187<br />
k2_c1_conversation 30 0.81 0.245<br />
k3_q2_neutral 29 0.97 0.047<br />
k4_v3_neutral_affect 19 0.66 0.400<br />
at 19 1.00 0.000<br />
n2 1 0.05 0.224<br />
* 1 1.00 0.000<br />
n52 1 0.05 0.224<br />
* 1 1.00 0.000<br />
n62 1 0.05 0.224<br />
* 1 1.00 0.000<br />
n71 1 0.05 0.224<br />
* 1 1.00 0.000<br />
n85 1 0.05 0.224<br />
* 1 1.00 0.000<br />
n87 1 0.05 0.224<br />
* 1 1.00 0.000<br />
n89 1 0.05 0.224<br />
* 1 1.00 0.000<br />
n91 1 0.05 0.224<br />
* 1 1.00 0.000<br />
n98 1 0.05 0.224<br />
* 1 1.00 0.000<br />
n100 1 0.05 0.224<br />
* 1 1.00 0.000<br />
n106 1 0.05 0.224<br />
* 1 1.00 0.000<br />
n112 1 0.05 0.224<br />
* 1 1.00 0.000<br />
n119 1 0.05 0.224<br />
* 1 1.00 0.000<br />
n123 1 0.05 0.224<br />
* 1 1.00 0.000<br />
n133 1 0.05 0.224<br />
* 1 1.00 0.000<br />
n135 1 0.05 0.224<br />
* 1 1.00 0.000<br />
n137 1 0.05 0.224<br />
* 1 1.00 0.000<br />
n146 1 0.05 0.224<br />
* 1 1.00 0.000<br />
n161 1 0.05 0.224<br />
* 1 1.00 0.000<br />
k4_v2_positive_affect 10 0.34 0.530<br />
at 10 1.00 0.000<br />
n14 1 0.10 0.332<br />
* 1 1.00 0.000<br />
n30 1 0.10 0.332<br />
* 1 1.00 0.000<br />
n40 1 0.10 0.332<br />
* 1 1.00 0.000<br />
n54 1 0.10 0.332<br />
* 1 1.00 0.000<br />
n69 1 0.10 0.332<br />
* 1 1.00 0.000<br />
n79 1 0.10 0.332<br />
* 1 1.00 0.000<br />
n108 1 0.10 0.332<br />
* 1 1.00 0.000<br />
n142 1 0.10 0.332<br />
* 1 1.00 0.000<br />
n163 1 0.10 0.332<br />
* 1 1.00 0.000<br />
n167 1 0.10 0.332<br />
* 1 1.00 0.000<br />
k3_q1_positive 1 0.03 0.164<br />
k4_v3_neutral_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n148 1 1.00 0.000<br />
* 1 1.00 0.000<br />
k2_c3_clear_directive 2 0.05 0.228<br />
k3_q2_neutral 1 0.50 0.500<br />
k4_v3_neutral_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n59 1 1.00 0.000<br />
* 1 1.00 0.000<br />
k3_q1_positive 1 0.50 0.500<br />
k4_v3_neutral_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n81 1 1.00 0.000<br />
* 1 1.00 0.000<br />
k2_c6_vocal_behavior 5 0.14 0.390<br />
k3_q2_neutral 5 1.00 0.000<br />
k4_v2_positive_affect 3 0.60 0.442<br />
at 3 1.00 0.000<br />
n151 1 0.33 0.528<br />
* 1 1.00 0.000<br />
n156 1 0.33 0.528<br />
* 1 1.00 0.000<br />
n170 1 0.33 0.528<br />
* 1 1.00 0.000<br />
k4_v3_neutral_affect 2 0.40 0.529<br />
at 2 1.00 0.000<br />
n73 1 0.50 0.500<br />
* 1 1.00 0.000<br />
n102 1 0.50 0.500<br />
* 1 1.00 0.000<br />
k1_f#_null_value 2 0.05 0.206<br />
k2_c#_null_value 2 1.00 0.000<br />
k3_q#_null_value 2 1.00 0.000<br />
k4_v#_null_value 2 1.00 0.000<br />
at 2 1.00 0.000<br />
n26 1 0.50 0.500<br />
* 1 1.00 0.000<br />
n175 1 0.50 0.500<br />
* 1 1.00 0.000<br />
k1_f3_mother 4 0.09 0.319<br />
k2_c3_clear_directive 1 0.25 0.500<br />
k3_q2_neutral 1 1.00 0.000<br />
k4_v3_neutral_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n45 1 1.00 0.000<br />
* 1 1.00 0.000<br />
k2_c0_compliance_behavior 1 0.25 0.500<br />
k3_q1_positive 1 1.00 0.000<br />
k4_v3_neutral_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n49 1 1.00 0.000<br />
* 1 1.00 0.000<br />
k2_c4_ambiguous_directive 1 0.25 0.500<br />
k3_q2_neutral 1 1.00 0.000<br />
k4_v3_neutral_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n114 1 1.00 0.000<br />
* 1 1.00 0.000<br />
k2_c6_vocal_behavior 1 0.25 0.500<br />
k3_q2_neutral 1 1.00 0.000<br />
k4_v2_positive_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n125 1 1.00 0.000<br />
* 1 1.00 0.000<br />
j4_v2_positive_affect 8 0.16 0.419<br />
to 8 1.00 0.000<br />
k1_f1_child 7 0.88 0.169<br />
k2_c1_conversation 4 0.57 0.461<br />
k3_q2_neutral 4 1.00 0.000<br />
k4_v2_positive_affect 3 0.75 0.311<br />
at 3 1.00 0.000<br />
n12 1 0.33 0.528<br />
* 1 1.00 0.000<br />
n35 1 0.33 0.528<br />
* 1 1.00 0.000<br />
n110 1 0.33 0.528<br />
* 1 1.00 0.000<br />
k4_v3_neutral_affect 1 0.25 0.500<br />
at 1 1.00 0.000<br />
n121 1 1.00 0.000<br />
* 1 1.00 0.000<br />
k2_c6_vocal_behavior 3 0.43 0.524<br />
k3_q2_neutral 3 1.00 0.000<br />
k4_v2_positive_affect 3 1.00 0.000<br />
at 3 1.00 0.000<br />
n43 1 0.33 0.528<br />
* 1 1.00 0.000<br />
n140 1 0.33 0.528<br />
* 1 1.00 0.000<br />
n144 1 0.33 0.528<br />
* 1 1.00 0.000<br />
k1_f3_mother 1 0.13 0.375<br />
k2_c3_clear_directive 1 1.00 0.000<br />
k3_q2_neutral 1 1.00 0.000<br />
k4_v3_neutral_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n66 1 1.00 0.000<br />
* 1 1.00 0.000<br />
j3_q1_positive 7 0.11 0.358<br />
j4_v2_positive_affect 4 0.57 0.461<br />
to 4 1.00 0.000<br />
k1_f1_child 3 0.75 0.311<br />
k2_c1_conversation 2 0.67 0.390<br />
k3_q2_neutral 2 1.00 0.000<br />
k4_v2_positive_affect 1 0.50 0.500<br />
at 1 1.00 0.000<br />
n28 1 1.00 0.000<br />
* 1 1.00 0.000<br />
k4_v3_neutral_affect 1 0.50 0.500<br />
at 1 1.00 0.000<br />
n96 1 1.00 0.000<br />
* 1 1.00 0.000<br />
k2_c6_vocal_behavior 1 0.33 0.528<br />
k3_q2_neutral 1 1.00 0.000<br />
k4_v2_positive_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n153 1 1.00 0.000<br />
* 1 1.00 0.000<br />
k1_f3_mother 1 0.25 0.500<br />
k2_c1_conversation 1 1.00 0.000<br />
k3_q2_neutral 1 1.00 0.000<br />
k4_v2_positive_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n139 1 1.00 0.000<br />
* 1 1.00 0.000<br />
j4_v3_neutral_affect 3 0.43 0.524<br />
to 3 1.00 0.000<br />
k1_f1_child 2 0.67 0.390<br />
k2_c1_conversation 2 1.00 0.000<br />
k3_q2_neutral 2 1.00 0.000<br />
k4_v3_neutral_affect 2 1.00 0.000<br />
at 2 1.00 0.000<br />
n10 1 0.50 0.500<br />
* 1 1.00 0.000<br />
n117 1 0.50 0.500<br />
* 1 1.00 0.000<br />
k1_f3_mother 1 0.33 0.528<br />
k2_c1_conversation 1 1.00 0.000<br />
k3_q2_neutral 1 1.00 0.000<br />
k4_v3_neutral_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n150 1 1.00 0.000<br />
* 1 1.00 0.000<br />
j3_q3_negative 3 0.05 0.214<br />
j4_v2_positive_affect 2 0.67 0.390<br />
to 2 1.00 0.000<br />
k1_f#_null_value 1 0.50 0.500<br />
k2_c#_null_value 1 1.00 0.000<br />
k3_q#_null_value 1 1.00 0.000<br />
k4_v#_null_value 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n23 1 1.00 0.000<br />
* 1 1.00 0.000<br />
k1_f3_mother 1 0.50 0.500<br />
k2_c3_clear_directive 1 1.00 0.000<br />
k3_q2_neutral 1 1.00 0.000<br />
k4_v3_neutral_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n128 1 1.00 0.000<br />
* 1 1.00 0.000<br />
j4_v4_negative_affect 1 0.33 0.528<br />
to 1 1.00 0.000<br />
k1_f3_mother 1 1.00 0.000<br />
k2_c1_conversation 1 1.00 0.000<br />
k3_q2_neutral 1 1.00 0.000<br />
k4_v3_neutral_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n169 1 1.00 0.000<br />
* 1 1.00 0.000<br />
j2_c4_ambiguous_directive 3 0.04 0.170<br />
j3_q2_neutral 3 1.00 0.000<br />
j4_v3_neutral_affect 3 1.00 0.000<br />
to 3 1.00 0.000<br />
k1_f1_child 2 0.67 0.390<br />
k2_c0_compliance_behavior 2 1.00 0.000<br />
k3_q1_positive 2 1.00 0.000<br />
k4_v3_neutral_affect 2 1.00 0.000<br />
at 2 1.00 0.000<br />
n77 1 0.50 0.500<br />
* 1 1.00 0.000<br />
n115 1 0.50 0.500<br />
* 1 1.00 0.000<br />
k1_f3_mother 1 0.33 0.528<br />
k2_c3_clear_directive 1 1.00 0.000<br />
k3_q2_neutral 1 1.00 0.000<br />
k4_v3_neutral_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n4 1 1.00 0.000<br />
* 1 1.00 0.000<br />
j2_c3_clear_directive 9 0.11 0.343<br />
j3_q2_neutral 9 1.00 0.000<br />
j4_v3_neutral_affect 9 1.00 0.000<br />
to 9 1.00 0.000<br />
k1_f1_child 8 0.89 0.151<br />
k2_c0_compliance_behavior 6 0.75 0.311<br />
k3_q1_positive 6 1.00 0.000<br />
k4_v3_neutral_affect 6 1.00 0.000<br />
at 6 1.00 0.000<br />
n5 1 0.17 0.431<br />
* 1 1.00 0.000<br />
n46 1 0.17 0.431<br />
* 1 1.00 0.000<br />
n67 1 0.17 0.431<br />
* 1 1.00 0.000<br />
n93 1 0.17 0.431<br />
* 1 1.00 0.000<br />
n130 1 0.17 0.431<br />
* 1 1.00 0.000<br />
n172 1 0.17 0.431<br />
* 1 1.00 0.000<br />
k2_c6_vocal_behavior 2 0.25 0.500<br />
k3_q2_neutral 2 1.00 0.000<br />
k4_v2_positive_affect 2 1.00 0.000<br />
at 2 1.00 0.000<br />
n19 1 0.50 0.500<br />
* 1 1.00 0.000<br />
n21 1 0.50 0.500<br />
* 1 1.00 0.000<br />
k1_f3_mother 1 0.11 0.352<br />
k2_c3_clear_directive 1 1.00 0.000<br />
k3_q2_neutral 1 1.00 0.000<br />
k4_v3_neutral_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n129 1 1.00 0.000<br />
* 1 1.00 0.000<br />
j2_c0_compliance_behavior 3 0.04 0.170<br />
j3_q1_positive 3 1.00 0.000<br />
j4_v3_neutral_affect 3 1.00 0.000<br />
to 3 1.00 0.000<br />
k1_f1_child 2 0.67 0.390<br />
k2_c1_conversation 2 1.00 0.000<br />
k3_q2_neutral 2 1.00 0.000<br />
k4_v3_neutral_affect 1 0.50 0.500<br />
at 1 1.00 0.000<br />
n50 1 1.00 0.000<br />
* 1 1.00 0.000<br />
k4_v4_negative_affect 1 0.50 0.500<br />
at 1 1.00 0.000<br />
n64 1 1.00 0.000<br />
* 1 1.00 0.000<br />
k1_f#_null_value 1 0.33 0.528<br />
k2_c#_null_value 1 1.00 0.000<br />
k3_q#_null_value 1 1.00 0.000<br />
k4_v#_null_value 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n8 1 1.00 0.000<br />
* 1 1.00 0.000<br />
j2_c6_vocal_behavior 5 0.06 0.240<br />
j3_q2_neutral 5 1.00 0.000<br />
j4_v2_positive_affect 4 0.80 0.258<br />
to 4 1.00 0.000<br />
k1_f1_child 3 0.75 0.311<br />
k2_c6_vocal_behavior 2 0.67 0.390<br />
k3_q2_neutral 2 1.00 0.000<br />
k4_v2_positive_affect 2 1.00 0.000<br />
at 2 1.00 0.000<br />
n32 1 0.50 0.500<br />
* 1 1.00 0.000<br />
n126 1 0.50 0.500<br />
* 1 1.00 0.000<br />
k2_c1_conversation 1 0.33 0.528<br />
k3_q2_neutral 1 1.00 0.000<br />
k4_v3_neutral_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n104 1 1.00 0.000<br />
* 1 1.00 0.000<br />
k1_f3_mother 1 0.25 0.500<br />
k2_c2_affiliate/distance 1 1.00 0.000<br />
k3_q2_neutral 1 1.00 0.000<br />
k4_v2_positive_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n158 1 1.00 0.000<br />
* 1 1.00 0.000<br />
j4_v3_neutral_affect 1 0.20 0.464<br />
to 1 1.00 0.000<br />
k1_f3_mother 1 1.00 0.000<br />
k2_c3_clear_directive 1 1.00 0.000<br />
k3_q2_neutral 1 1.00 0.000<br />
k4_v3_neutral_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n18 1 1.00 0.000<br />
* 1 1.00 0.000<br />
j2_c5_response_to_directive 3 0.04 0.170<br />
j3_q1_positive 3 1.00 0.000<br />
j4_v3_neutral_affect 3 1.00 0.000<br />
to 3 1.00 0.000<br />
k1_f3_mother 2 0.67 0.390<br />
k2_c1_conversation 2 1.00 0.000<br />
k3_q2_neutral 2 1.00 0.000<br />
k4_v3_neutral_affect 2 1.00 0.000<br />
at 2 1.00 0.000<br />
n48 1 0.50 0.500<br />
* 1 1.00 0.000<br />
n61 1 0.50 0.500<br />
* 1 1.00 0.000<br />
k1_f1_child 1 0.33 0.528<br />
k2_c1_conversation 1 1.00 0.000<br />
k3_q2_neutral 1 1.00 0.000<br />
k4_v3_neutral_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n83 1 1.00 0.000<br />
* 1 1.00 0.000<br />
j2_c2_affiliate/distance 1 0.01 0.075<br />
j3_q2_neutral 1 1.00 0.000<br />
j4_v2_positive_affect 1 1.00 0.000<br />
to 1 1.00 0.000<br />
k1_f1_child 1 1.00 0.000<br />
k2_c6_vocal_behavior 1 1.00 0.000<br />
k3_q2_neutral 1 1.00 0.000<br />
k4_v2_positive_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n159 1 1.00 0.000<br />
* 1 1.00 0.000<br />
j1_f1_child 83 0.47 0.510<br />
j2_c1_conversation 52 0.63 0.423<br />
j3_q2_neutral 51 0.98 0.027<br />
j4_v3_neutral_affect 30 0.59 0.450<br />
to 30 1.00 0.000<br />
k1_f3_mother 30 1.00 0.000<br />
k2_c1_conversation 26 0.87 0.179<br />
k3_q2_neutral 24 0.92 0.107<br />
k4_v3_neutral_affect 22 0.92 0.115<br />
at 22 1.00 0.000<br />
n25 1 0.05 0.203<br />
* 1 1.00 0.000<br />
n51 1 0.05 0.203<br />
* 1 1.00 0.000<br />
n53 1 0.05 0.203<br />
* 1 1.00 0.000<br />
n72 1 0.05 0.203<br />
* 1 1.00 0.000<br />
n84 1 0.05 0.203<br />
* 1 1.00 0.000<br />
n86 1 0.05 0.203<br />
* 1 1.00 0.000<br />
n88 1 0.05 0.203<br />
* 1 1.00 0.000<br />
n90 1 0.05 0.203<br />
* 1 1.00 0.000<br />
n97 1 0.05 0.203<br />
* 1 1.00 0.000<br />
n99 1 0.05 0.203<br />
* 1 1.00 0.000<br />
n101 1 0.05 0.203<br />
* 1 1.00 0.000<br />
n105 1 0.05 0.203<br />
* 1 1.00 0.000<br />
n107 1 0.05 0.203<br />
* 1 1.00 0.000<br />
n113 1 0.05 0.203<br />
* 1 1.00 0.000<br />
n118 1 0.05 0.203<br />
* 1 1.00 0.000<br />
n122 1 0.05 0.203<br />
* 1 1.00 0.000<br />
n124 1 0.05 0.203<br />
* 1 1.00 0.000<br />
n134 1 0.05 0.203<br />
* 1 1.00 0.000<br />
n136 1 0.05 0.203<br />
* 1 1.00 0.000<br />
n147 1 0.05 0.203<br />
* 1 1.00 0.000<br />
n162 1 0.05 0.203<br />
* 1 1.00 0.000<br />
n174 1 0.05 0.203<br />
* 1 1.00 0.000<br />
k4_v2_positive_affect 2 0.08 0.299<br />
at 2 1.00 0.000<br />
n11 1 0.50 0.500<br />
* 1 1.00 0.000<br />
n120 1 0.50 0.500<br />
* 1 1.00 0.000<br />
k3_q1_positive 2 0.08 0.285<br />
k4_v2_positive_affect 2 1.00 0.000<br />
at 2 1.00 0.000<br />
n95 1 0.50 0.500<br />
* 1 1.00 0.000<br />
n138 1 0.50 0.500<br />
* 1 1.00 0.000<br />
k2_c6_vocal_behavior 1 0.03 0.164<br />
k3_q2_neutral 1 1.00 0.000<br />
k4_v3_neutral_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n17 1 1.00 0.000<br />
* 1 1.00 0.000<br />
k2_c0_compliance_behavior 1 0.03 0.164<br />
k3_q1_positive 1 1.00 0.000<br />
k4_v3_neutral_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n63 1 1.00 0.000<br />
* 1 1.00 0.000<br />
k2_c3_clear_directive 1 0.03 0.164<br />
k3_q2_neutral 1 1.00 0.000<br />
k4_v3_neutral_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n92 1 1.00 0.000<br />
* 1 1.00 0.000<br />
k2_c4_ambiguous_directive 1 0.03 0.164<br />
k3_q2_neutral 1 1.00 0.000<br />
k4_v3_neutral_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n3 1 1.00 0.000<br />
* 1 1.00 0.000<br />
j4_v2_positive_affect 19 0.37 0.531<br />
to 19 1.00 0.000<br />
k1_f3_mother 12 0.63 0.419<br />
k2_c1_conversation 11 0.92 0.115<br />
k3_q2_neutral 10 0.91 0.125<br />
k4_v3_neutral_affect 7 0.70 0.360<br />
at 7 1.00 0.000<br />
n13 1 0.14 0.401<br />
* 1 1.00 0.000<br />
n29 1 0.14 0.401<br />
* 1 1.00 0.000<br />
n58 1 0.14 0.401<br />
* 1 1.00 0.000<br />
n70 1 0.14 0.401<br />
* 1 1.00 0.000<br />
n80 1 0.14 0.401<br />
* 1 1.00 0.000<br />
n111 1 0.14 0.401<br />
* 1 1.00 0.000<br />
n155 1 0.14 0.401<br />
* 1 1.00 0.000<br />
k4_v2_positive_affect 3 0.30 0.521<br />
at 3 1.00 0.000<br />
n34 1 0.33 0.528<br />
* 1 1.00 0.000<br />
n109 1 0.33 0.528<br />
* 1 1.00 0.000<br />
n143 1 0.33 0.528<br />
* 1 1.00 0.000<br />
k3_q3_negative 1 0.09 0.314<br />
k4_v4_negative_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n168 1 1.00 0.000<br />
* 1 1.00 0.000<br />
k2_c6_vocal_behavior 1 0.08 0.299<br />
k3_q2_neutral 1 1.00 0.000<br />
k4_v2_positive_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n31 1 1.00 0.000<br />
* 1 1.00 0.000<br />
k1_f#_null_value 2 0.11 0.342<br />
k2_c#_null_value 2 1.00 0.000<br />
k3_q#_null_value 2 1.00 0.000<br />
k4_v#_null_value 2 1.00 0.000<br />
at 2 1.00 0.000<br />
n15 1 0.50 0.500<br />
* 1 1.00 0.000<br />
n164 1 0.50 0.500<br />
* 1 1.00 0.000<br />
k1_f1_child 5 0.26 0.507<br />
k2_c6_vocal_behavior 4 0.80 0.258<br />
k3_q2_neutral 4 1.00 0.000<br />
k4_v2_positive_affect 4 1.00 0.000<br />
at 4 1.00 0.000<br />
n36 1 0.25 0.500<br />
* 1 1.00 0.000<br />
n38 1 0.25 0.500<br />
* 1 1.00 0.000<br />
n41 1 0.25 0.500<br />
* 1 1.00 0.000<br />
n55 1 0.25 0.500<br />
* 1 1.00 0.000<br />
k2_c1_conversation 1 0.20 0.464<br />
k3_q2_neutral 1 1.00 0.000<br />
k4_v1_exuberant_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n75 1 1.00 0.000<br />
* 1 1.00 0.000<br />
j4_v4_negative_affect 1 0.02 0.111<br />
to 1 1.00 0.000<br />
k1_f3_mother 1 1.00 0.000<br />
k2_c1_conversation 1 1.00 0.000<br />
k3_q2_neutral 1 1.00 0.000<br />
k4_v2_positive_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n65 1 1.00 0.000<br />
* 1 1.00 0.000<br />
j4_v1_exuberant_affect 1 0.02 0.111<br />
to 1 1.00 0.000<br />
k1_f3_mother 1 1.00 0.000<br />
k2_c4_ambiguous_directive 1 1.00 0.000<br />
k3_q2_neutral 1 1.00 0.000<br />
k4_v3_neutral_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n76 1 1.00 0.000<br />
* 1 1.00 0.000<br />
j3_q1_positive 1 0.02 0.110<br />
j4_v3_neutral_affect 1 1.00 0.000<br />
to 1 1.00 0.000<br />
k1_f3_mother 1 1.00 0.000<br />
k2_c1_conversation 1 1.00 0.000<br />
k3_q1_positive 1 1.00 0.000<br />
k4_v3_neutral_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n149 1 1.00 0.000<br />
* 1 1.00 0.000<br />
j2_c0_compliance_behavior 9 0.11 0.348<br />
j3_q1_positive 9 1.00 0.000<br />
j4_v3_neutral_affect 9 1.00 0.000<br />
to 9 1.00 0.000<br />
k1_f3_mother 5 0.56 0.471<br />
k2_c1_conversation 4 0.80 0.258<br />
k3_q2_neutral 3 0.75 0.311<br />
k4_v3_neutral_affect 3 1.00 0.000<br />
at 3 1.00 0.000<br />
n68 1 0.33 0.528<br />
* 1 1.00 0.000<br />
n78 1 0.33 0.528<br />
* 1 1.00 0.000<br />
n132 1 0.33 0.528<br />
* 1 1.00 0.000<br />
k3_q1_positive 1 0.25 0.500<br />
k4_v3_neutral_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n116 1 1.00 0.000<br />
* 1 1.00 0.000<br />
k2_c5_response_to_directive 1 0.20 0.464<br />
k3_q1_positive 1 1.00 0.000<br />
k4_v3_neutral_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n47 1 1.00 0.000<br />
* 1 1.00 0.000<br />
k1_f1_child 4 0.44 0.520<br />
k2_c1_conversation 2 0.50 0.500<br />
k3_q2_neutral 2 1.00 0.000<br />
k4_v3_neutral_affect 2 1.00 0.000<br />
at 2 1.00 0.000<br />
n94 1 0.50 0.500<br />
* 1 1.00 0.000<br />
n173 1 0.50 0.500<br />
* 1 1.00 0.000<br />
k2_c0_compliance_behavior 1 0.25 0.500<br />
k3_q1_positive 1 1.00 0.000<br />
k4_v3_neutral_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n131 1 1.00 0.000<br />
* 1 1.00 0.000<br />
k2_c4_ambiguous_directive 1 0.25 0.500<br />
k3_q2_neutral 1 1.00 0.000<br />
k4_v3_neutral_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n6 1 1.00 0.000<br />
* 1 1.00 0.000<br />
j2_c4_ambiguous_directive 1 0.01 0.077<br />
j3_q2_neutral 1 1.00 0.000<br />
j4_v3_neutral_affect 1 1.00 0.000<br />
to 1 1.00 0.000<br />
k1_f3_mother 1 1.00 0.000<br />
k2_c0_compliance_behavior 1 1.00 0.000<br />
k3_q1_positive 1 1.00 0.000<br />
k4_v3_neutral_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n7 1 1.00 0.000<br />
* 1 1.00 0.000<br />
j2_c6_vocal_behavior 19 0.23 0.487<br />
j3_q2_neutral 19 1.00 0.000<br />
j4_v2_positive_affect 17 0.89 0.144<br />
to 17 1.00 0.000<br />
k1_f3_mother 13 0.76 0.296<br />
k2_c1_conversation 10 0.77 0.291<br />
k3_q2_neutral 7 0.70 0.360<br />
k4_v3_neutral_affect 6 0.86 0.191<br />
at 6 1.00 0.000<br />
n39 1 0.17 0.431<br />
* 1 1.00 0.000<br />
n44 1 0.17 0.431<br />
* 1 1.00 0.000<br />
n141 1 0.17 0.431<br />
* 1 1.00 0.000<br />
n145 1 0.17 0.431<br />
* 1 1.00 0.000<br />
n160 1 0.17 0.431<br />
* 1 1.00 0.000<br />
n166 1 0.17 0.431<br />
* 1 1.00 0.000<br />
k4_v2_positive_affect 1 0.14 0.401<br />
at 1 1.00 0.000<br />
n42 1 1.00 0.000<br />
* 1 1.00 0.000<br />
k3_q1_positive 1 0.10 0.332<br />
k4_v2_positive_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n152 1 1.00 0.000<br />
* 1 1.00 0.000<br />
k3_q3_negative 2 0.20 0.464<br />
k4_v2_positive_affect 2 1.00 0.000<br />
at 2 1.00 0.000<br />
n22 1 0.50 0.500<br />
* 1 1.00 0.000<br />
n127 1 0.50 0.500<br />
* 1 1.00 0.000<br />
k2_c6_vocal_behavior 1 0.08 0.285<br />
k3_q2_neutral 1 1.00 0.000<br />
k4_v2_positive_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n157 1 1.00 0.000<br />
* 1 1.00 0.000<br />
k2_c3_clear_directive 2 0.15 0.415<br />
k3_q2_neutral 2 1.00 0.000<br />
k4_v3_neutral_affect 2 1.00 0.000<br />
at 2 1.00 0.000<br />
n20 1 0.50 0.500<br />
* 1 1.00 0.000<br />
n171 1 0.50 0.500<br />
* 1 1.00 0.000<br />
k1_f1_child 3 0.18 0.442<br />
k2_c1_conversation 3 1.00 0.000<br />
k3_q2_neutral 3 1.00 0.000<br />
k4_v2_positive_affect 3 1.00 0.000<br />
at 3 1.00 0.000<br />
n33 1 0.33 0.528<br />
* 1 1.00 0.000<br />
n37 1 0.33 0.528<br />
* 1 1.00 0.000<br />
n154 1 0.33 0.528<br />
* 1 1.00 0.000<br />
k1_f#_null_value 1 0.06 0.240<br />
k2_c#_null_value 1 1.00 0.000<br />
k3_q#_null_value 1 1.00 0.000<br />
k4_v#_null_value 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n56 1 1.00 0.000<br />
* 1 1.00 0.000<br />
j4_v3_neutral_affect 2 0.11 0.342<br />
to 2 1.00 0.000<br />
k1_f1_child 1 0.50 0.500<br />
k2_c1_conversation 1 1.00 0.000<br />
k3_q2_neutral 1 1.00 0.000<br />
k4_v2_positive_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n74 1 1.00 0.000<br />
* 1 1.00 0.000<br />
k1_f3_mother 1 0.50 0.500<br />
k2_c6_vocal_behavior 1 1.00 0.000<br />
k3_q2_neutral 1 1.00 0.000<br />
k4_v2_positive_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n103 1 1.00 0.000<br />
* 1 1.00 0.000<br />
j2_c3_clear_directive 2 0.02 0.130<br />
j3_q2_neutral 1 0.50 0.500<br />
j4_v3_neutral_affect 1 1.00 0.000<br />
to 1 1.00 0.000<br />
k1_f3_mother 1 1.00 0.000<br />
k2_c5_response_to_directive 1 1.00 0.000<br />
k3_q1_positive 1 1.00 0.000<br />
k4_v3_neutral_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n60 1 1.00 0.000<br />
* 1 1.00 0.000<br />
j3_q1_positive 1 0.50 0.500<br />
j4_v3_neutral_affect 1 1.00 0.000<br />
to 1 1.00 0.000<br />
k1_f3_mother 1 1.00 0.000<br />
k2_c5_response_to_directive 1 1.00 0.000<br />
k3_q1_positive 1 1.00 0.000<br />
k4_v3_neutral_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n82 1 1.00 0.000<br />
* 1 1.00 0.000<br />
j1_f#_null_value 7 0.04 0.186<br />
j2_c#_null_value 7 1.00 0.000<br />
j3_q#_null_value 7 1.00 0.000<br />
j4_v#_null_value 7 1.00 0.000<br />
to 7 1.00 0.000<br />
k1_f1_child 4 0.57 0.461<br />
k2_c1_conversation 3 0.75 0.311<br />
k3_q2_neutral 3 1.00 0.000<br />
k4_v3_neutral_affect 2 0.67 0.390<br />
at 2 1.00 0.000<br />
n16 1 0.50 0.500<br />
* 1 1.00 0.000<br />
n24 1 0.50 0.500<br />
* 1 1.00 0.000<br />
k4_v2_positive_affect 1 0.33 0.528<br />
at 1 1.00 0.000<br />
n57 1 1.00 0.000<br />
* 1 1.00 0.000<br />
k2_c6_vocal_behavior 1 0.25 0.500<br />
k3_q2_neutral 1 1.00 0.000<br />
k4_v2_positive_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n165 1 1.00 0.000<br />
* 1 1.00 0.000<br />
k1_f3_mother 3 0.43 0.524<br />
k2_c1_conversation 3 1.00 0.000<br />
k3_q1_positive 2 0.67 0.390<br />
k4_v3_neutral_affect 1 0.50 0.500<br />
at 1 1.00 0.000<br />
n9 1 1.00 0.000<br />
* 1 1.00 0.000<br />
k4_v2_positive_affect 1 0.50 0.500<br />
at 1 1.00 0.000<br />
n27 1 1.00 0.000<br />
* 1 1.00 0.000<br />
k3_q2_neutral 1 0.33 0.528<br />
k4_v3_neutral_affect 1 1.00 0.000<br />
at 1 1.00 0.000<br />
n1 1 1.00 0.000<br />
* 1 1.00 0.000<br />
</pre><br />
<br />
==Document History==<br />
<br />
===2003 &bull; Inquiry List &bull; Sequential Interactions Generating Hypotheses===<br />
<br />
* http://web.archive.org/web/20120518012303/http://stderr.org/pipermail/inquiry/2003-August/thread.html#753<br />
* http://web.archive.org/web/20120505135759/http://stderr.org/pipermail/inquiry/2003-September/thread.html#778<br />
# http://web.archive.org/web/20040906141818/http://stderr.org/pipermail/inquiry/2003-August/000753.html<br />
# http://web.archive.org/web/20040906141758/http://stderr.org/pipermail/inquiry/2003-August/000754.html<br />
# http://web.archive.org/web/20040906141706/http://stderr.org/pipermail/inquiry/2003-August/000755.html<br />
# http://web.archive.org/web/20040906141842/http://stderr.org/pipermail/inquiry/2003-August/000763.html<br />
# http://web.archive.org/web/20040906141742/http://stderr.org/pipermail/inquiry/2003-August/000764.html<br />
# http://web.archive.org/web/20040906141701/http://stderr.org/pipermail/inquiry/2003-August/000765.html<br />
# http://web.archive.org/web/20040906141745/http://stderr.org/pipermail/inquiry/2003-August/000766.html<br />
# http://web.archive.org/web/20040907185539/http://stderr.org/pipermail/inquiry/2003-August/000767.html<br />
# http://web.archive.org/web/20040907185559/http://stderr.org/pipermail/inquiry/2003-August/000768.html<br />
# http://web.archive.org/web/20040907185627/http://stderr.org/pipermail/inquiry/2003-August/000769.html<br />
# http://web.archive.org/web/20040907185620/http://stderr.org/pipermail/inquiry/2003-August/000770.html<br />
# http://web.archive.org/web/20040907185456/http://stderr.org/pipermail/inquiry/2003-August/000771.html<br />
# http://web.archive.org/web/20040907185500/http://stderr.org/pipermail/inquiry/2003-August/000772.html<br />
# http://web.archive.org/web/20061014001124/http://stderr.org/pipermail/inquiry/2003-September/000778.html<br />
<br />
===2003 &bull; Ontology List &bull; Sequential Interactions Generating Hypotheses===<br />
<br />
* http://web.archive.org/web/20070305021905/http://suo.ieee.org/ontology/thrd10.html#05003<br />
# http://web.archive.org/web/20070307044513/http://suo.ieee.org/ontology/msg05003.html<br />
# http://web.archive.org/web/20070313230835/http://suo.ieee.org/ontology/msg05004.html<br />
# http://web.archive.org/web/20070306151612/http://suo.ieee.org/ontology/msg05005.html<br />
# http://web.archive.org/web/20070313230845/http://suo.ieee.org/ontology/msg05013.html<br />
# http://web.archive.org/web/20070313230856/http://suo.ieee.org/ontology/msg05014.html<br />
# http://web.archive.org/web/20070313230905/http://suo.ieee.org/ontology/msg05015.html<br />
# http://web.archive.org/web/20070313230915/http://suo.ieee.org/ontology/msg05016.html<br />
# http://web.archive.org/web/20070313230925/http://suo.ieee.org/ontology/msg05017.html<br />
# http://web.archive.org/web/20070313230936/http://suo.ieee.org/ontology/msg05018.html<br />
# http://web.archive.org/web/20070308214231/http://suo.ieee.org/ontology/msg05020.html<br />
# http://web.archive.org/web/20070310141005/http://suo.ieee.org/ontology/msg05021.html<br />
# http://web.archive.org/web/20070313230946/http://suo.ieee.org/ontology/msg05022.html<br />
# http://web.archive.org/web/20070307044524/http://suo.ieee.org/ontology/msg05023.html<br />
# http://web.archive.org/web/20070307044535/http://suo.ieee.org/ontology/msg05025.html</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Semiotic_Information&diff=469864Semiotic Information2020-11-05T15:36:07Z<p>Jon Awbrey: add article</p>
<hr />
<div>'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''<br />
<br />
'''Semiotic information''' is the information content of signs as conceived within the [[semeiotic]] or [[sign-relational]] framework developed by [[Charles Sanders Peirce]].<br />
<br />
==Once over quickly==<br />
<br />
===What's it good for?===<br />
<br />
The good of information is its use in reducing our uncertainty about some issue that comes before us. Generally speaking, uncertainty comes in several flavors, and so the information that serves to reduce uncertainty can be applied in several different ways. The situations of uncertainty that human agents commonly find themselves facing have been investigated under many headings, literally for ages, and the classifications that subtle thinkers arrived at long before the dawn of modern information theory still have their uses in setting the stage of an introduction.<br />
<br />
Picking an example of a subtle thinker almost at random, the philosopher-scientist Immanuel Kant divided the principal questions of human existence into three parts:<br />
<br />
:* What's true?<br />
:* What's to do?<br />
:* What's to hope?<br />
<br />
The third question is a bit too subtle for the present frame of discussion, but the first and second are easily recognizable as staking out the two main axes of information theory, namely, the dual dimensions of ''information'' and ''control''. Roughly the same space of concerns is elsewhere spanned by the dual axes of ''competence'' and ''performance'', ''specification'' and ''optimization'', or just plain ''knowledge'' and ''skill''.<br />
<br />
A question of ''what's true'' is a ''descriptive question'', and there exist what are called ''[[descriptive science]]s'' devoted to answering descriptive questions about any domain of phenomena that one might care to name.<br />
<br />
A question of ''what's to do'', in other words, what must be done by way of achieving a given aim, is a ''normative question'', and there exist what are called ''[[normative science]]s'' devoted to answering normative questions about any domain of problems that one might care to address.<br />
<br />
Since information plays its role on a stage set by uncertainty, a big part of saying what information is will necessarily involve saying what uncertainty is. There is little chance that the vagueness of a word like ''uncertainty'', given the nuances of its ordinary, poetic, and technical uses, can be corralled by a single pen, but there do exist established models and formal theories that address definable aspects of uncertainty, and these have enough uses to make them worth looking into.<br />
<br />
===What is information that a sign may bear it?===<br />
<br />
Three more questions arise at this juncture:<br />
# How is a sign empowered to contain information?<br />
# What is the practical context of communication?<br />
# Why do we care about these bits of information?<br />
<br />
A very rough answer to these questions might begin as follows:<br />
<br />
Human beings are initially concerned solely with their own lives, but then a world obtrudes on their subjective existence, and so they find themselves forced to take an interest in the objective realities of its nature.<br />
<br />
In pragmatic terms our initial aim, concern, interest, object, or ''pragma'' is expressed by the verbal infinitive ''to live'', but the infinitive is soon reified into the derivative substantial forms of ''nature'', ''reality'', ''the world'', and so on. Against this backdrop we find ourselves cast as the protagonists on a ''scene of uncertainty''. The situation may be pictured as a juncture from which a manifold of options fan out before us. It may be an issue of ''truth'', ''duty'', or ''hope'', the last codifying a special type of uncertainty as to ''what regulative principle has any chance of success'', but the chief uncertainty is that we are called on to make a choice and find that we all too often have almost no clue as to which of the options is most fit to pick.<br />
<br />
Just to make up a discrete example, let us suppose that the cardinality of this choice is a finite ''n'', and just to make it fully concrete let us say that ''n''&nbsp;=&nbsp;5. Figure 1 affords a rough picture of the situation.<br />
<br />
{| align="center" cellspacing="6" style="text-align:center; width:60%"<br />
|<br />
<pre><br />
o-------------------------------------------------o<br />
| |<br />
| ? ? ? ? ? |<br />
| o o o o o |<br />
| |<br />
| o o o o o |<br />
| |<br />
| o o o o o |<br />
| |<br />
| o o o o o |<br />
| |<br />
| o o o o o |<br />
| |<br />
| ooooo |<br />
| |<br />
| O n = 5 |<br />
| |<br />
o-------------------------------------------------o<br />
Figure 1. Juncture of Degree 5<br />
</pre><br />
|}<br />
<br />
This pictures a juncture, represented by &ldquo;O&rdquo;, where there are ''n'' options for the outcome of a conduct, and we have no clue as to which it must be. In a sense, the degree of this node, in this case ''n''&nbsp;=&nbsp;5, measures the uncertainty that we have at this point.<br />
<br />
This is the minimal sort of setting in which a sign can make any sense at all. A sign has significance for an agent, interpreter, or observer because its actualization, its being given or its being present, serves to reduce the uncertainty of a decision that the agent has to make, whether it concerns the actions that the agent ought to take in order to achieve some objective of interest, or whether it concerns the predicates that the agent ought to treat as being true of some object in the world.<br />
<br />
The way that signs enter the scene is shown in Figure 2.<br />
<br />
{| align="center" cellspacing="6" style="text-align:center; width:60%"<br />
|<br />
<pre><br />
o-------------------------------------------------o<br />
| |<br />
| k_1 = 3 k_2 = 2 |<br />
| o-----o-----o o-----o |<br />
| "A" "B" |<br />
| o----o----o o----o |<br />
| |<br />
| o---o---o o---o |<br />
| |<br />
| o--o--o o--o |<br />
| |<br />
| o-o-o o-o |<br />
| |<br />
| ooooo |<br />
| |<br />
| O n = 5 |<br />
| |<br />
o-------------------------------------------------o<br />
Figure 2. Partition of Degrees 3 and 2<br />
</pre><br />
|}<br />
<br />
This illustrates a situation of uncertainty that has been augmented by a classification.<br />
<br />
In the particular pattern of classification that is shown here, the first three outcomes fall under the sign &ldquo;A&rdquo;, and the next two outcomes fall under the sign &ldquo;B&rdquo;. If the outcomes make up a set of ''things that might be true about an object'', then the signs could be read as nomens (terms) or notions (concepts) of a relevant empirical, ontological, taxonomical, or theoretical scheme, that is, as predicates and predictions of the outcomes. If the outcomes make up a set of ''things that might be good to do in order to achieve an objective'', then the signs could be read as bits of advice or other sorts of indicators that tell us what to do in the situation, relative to our active goals.<br />
<br />
This is the basic framework for talking about information and signs in regard to communication, decision, and the uncertainties thereof.<br />
<br />
Just to unpack some of the many things that may be getting glossed over in this little word ''sign'', it encompasses all of the ''data of the senses'' (DOTS) that we take as informing us about inner and outer worlds, plus all of the concepts and terms that we use to argue, to communicate, to inquire, or even to speculate, both about our ontologies for beings in the worlds and about our policies for action in the world.<br />
<br />
Here is one of the places where it is tempting to try to collapse the 3-adic sign relation into a 2-adic relation. For if these DOTS are so closely identified with objects that we can scarcely imagine how they might be discrepant, then it will appear to us that one role of beings can be eliminated from our picture of the world. In this event, the only things that we are required to inform ourselves about, via the inspection of these DOTS, are yet more DOTS, whether past, or present, or prospective, just more DOTS. This is the special form to which we frequently find the idea of an information channel being reduced, namely, to a ''source'' that has nothing more to tell us about than its own conceivable conducts or its own potential issues.<br />
<br />
As a matter of fact, at least in this discrete type of case, it would be possible to use the degree of the node as a measure of uncertainty, but it would operate as a multiplicative measure rather than the sort of additive measure that we would normally prefer. To illustrate how this would work out, let us consider an easier example, one where the degree of the choice point is 4.<br />
<br />
{| align="center" cellspacing="6" style="text-align:center; width:60%"<br />
|<br />
<pre><br />
o-------------------------------------------------o<br />
| |<br />
| ? ? ? ? |<br />
| o o o o |<br />
| |<br />
| o o o o |<br />
| |<br />
| o o o o |<br />
| |<br />
| o o o o |<br />
| |<br />
| o o o o |<br />
| |<br />
| oo oo |<br />
| |<br />
| O n = 4 |<br />
| |<br />
o-------------------------------------------------o<br />
Figure 3. Juncture of Degree 4<br />
</pre><br />
|}<br />
<br />
Suppose that we contemplate making another decision after the present issue has been decided, one that has a degree of 2 in every case. The compound situation is depicted in Figure&nbsp;4.<br />
<br />
{| align="center" cellspacing="6" style="text-align:center; width:60%"<br />
|<br />
<pre><br />
o-------------------------------------------------o<br />
| |<br />
| o o o o o o o o |<br />
| \ / \ / \ / \ / |<br />
| o o o o n_2 = 2 |<br />
| |<br />
| o o o o |<br />
| |<br />
| o o o o |<br />
| |<br />
| o o o o |<br />
| |<br />
| o o o o |<br />
| |<br />
| oo oo |<br />
| |<br />
| O n_1 = 4 |<br />
| |<br />
o-------------------------------------------------o<br />
Figure 4. Compound Junctures of Degrees 4 and 2<br />
</pre><br />
|}<br />
<br />
This illustrates the fact that the compound uncertainty, 8, is the product of the two component uncertainties, 4&nbsp;times&nbsp;2. To convert this to an additive measure, one simply takes the logarithms to a convenient base, say 2, and thus arrives at the not too astounding fact that the uncertainty of the first choice is 2 bits, the uncertainty of the next choice is 1 bit, and the compound uncertainty is 2&nbsp;+&nbsp;1&nbsp;=&nbsp;3 bits.<br />
<br />
In many ways, the provision of information, a process that reduces uncertainty, is the inverse process to the kind of uncertainty augmentation that occurs in compound decisions. By way of illustrating this relationship, let us return to our initial example.<br />
<br />
A set of signs enters on a setup like this as a system of ''middle terms'', a collection of signs that one may regard, aptly enough, as constellating a ''medium''.<br />
<br />
{| align="center" cellspacing="6" style="text-align:center; width:60%"<br />
|<br />
<pre><br />
o-------------------------------------------------o<br />
| |<br />
| k_1 = 3 k_2 = 2 |<br />
| o-----o-----o o-----o |<br />
| "A" "B" |<br />
| o----o----o o----o |<br />
| |<br />
| o---o---o o---o |<br />
| |<br />
| o--o--o o--o |<br />
| |<br />
| o-o-o o-o |<br />
| |<br />
| ooooo |<br />
| |<br />
| O n = 5 |<br />
| |<br />
o-------------------------------------------------o<br />
Figure 5. Partition of Degrees 3 and 2<br />
</pre><br />
|}<br />
<br />
The ''language'' or ''medium'' here is the set of signs {&ldquo;A&rdquo;,&nbsp;&ldquo;B&rdquo;}. On the assumption that the initial 5 outcomes are equally likely, one may associate a frequency distribution (''k''<sub>1</sub>, ''k''<sub>2</sub>) = (3, 2) and thus a probability distribution (''p''<sub>1</sub>, ''p''<sub>2</sub>) = (3/5, 2/5) = (0.6, 0.4) with this language, and thus define a communication ''channel''.<br />
<br />
The most important thing here is really just to get a handle on the ''conditions for the possibility of signs making sense'', but once we have this much of a setup we find that we can begin to construct some rough and ready bits of information-theoretic furniture, like measures of uncertainty, channel capacity, and the amount of information that can be associated with the reception or the recognition of a single sign. Still, before we get into all of this, it needs to be emphasized that, even when these measures are too ad&nbsp;hoc and insufficient to be of much use per&nbsp;se, the significance of the setup that it takes to support them is not at all diminished.<br />
<br />
Consider the classification-augmented or sign-enhanced situation of uncertainty that was depicted above. What happens if one or the other of the two signs, &ldquo;A&rdquo; or &ldquo;B&rdquo;, is observed or received, on the constant assumption that its significance is recognized on receipt?<br />
<br />
:* If we receive &ldquo;A&rdquo; our uncertainty is reduced from <math>\log 5</math> to <math>\log 3.</math><br />
<br />
:* If we receive &ldquo;B&rdquo; our uncertainty is reduced from <math>\log 5</math> to <math>\log 2.</math><br />
<br />
It is from these characteristics that the ''information capacity'' of a communication channel can be defined, specifically, as the ''average uncertainty reduction on receiving a sign'', a formula with the splendid mnemonic &ldquo;AURORAS&rdquo;.<br />
<br />
:* On receiving the message &ldquo;A&rdquo;, the additive measure of uncertainty is reduced from <math>\log 5</math> to <math>\log 3</math>, so the net reduction is <math>(\log 5 - \log 3).</math><br />
<br />
:* On receiving the message &ldquo;B&rdquo;, the additive measure of uncertainty is reduced from <math>\log 5</math> to <math>\log 2</math>, so the net reduction is <math>(\log 5 - \log 2).</math><br />
<br />
The average uncertainty reduction per sign of the language is computed by taking a ''weighted average'' of the reductions that occur in the channel, where the weight of each reduction is the number of options or outcomes that fall under the associated sign.<br />
<br />
:* The uncertainty reduction of <math>(\log 5 - \log 3)\!</math> gets a weight of 3.<br />
<br />
:* The uncertainty reduction of <math>(\log 5 - \log 2)\!</math> gets a weight of 2.<br />
<br />
Finally, the weighted average of these two reductions is:<br />
<br />
: <math>{1 \over {2 + 3}}(3(\log 5 - \log 3) + 2(\log 5 - \log 2))\!</math><br />
<br />
Extracting the general pattern of this calculation yields the following worksheet for computing the capacity of a 2-symbol channel with frequencies that partition as <math>n = k_1 + k_2.\!</math><br />
<br />
{| cellspacing="6" <br />
| Capacity<br />
| of a channel {&ldquo;A&rdquo;, &ldquo;B&rdquo;} that bears the odds of 60 &ldquo;A&rdquo; to 40 &ldquo;B&rdquo;<br />
|-<br />
| &nbsp;<br />
| <math>=\quad {1 \over n}(k_1(\log n - \log k_1) + k_2(\log n - \log k_2))\!</math><br />
|-<br />
| &nbsp;<br />
| <math>=\quad {k_1 \over n}(\log n - \log k_1) + {k_2 \over n}(\log n - \log k_2)\!</math><br />
|-<br />
| &nbsp;<br />
| <math>=\quad -{k_1 \over n}(\log k_1 - \log n) -{k_2 \over n}(\log k_2 - \log n)\!</math><br />
|-<br />
| &nbsp;<br />
| <math>=\quad -{k_1 \over n}(\log {k_1 \over n}) - {k_2 \over n}(\log {k_2 \over n})\!</math><br />
|-<br />
| &nbsp;<br />
| <math>=\quad -(p_1 \log p_1 + p_2 \log p_2)\!</math><br />
|-<br />
| &nbsp;<br />
| <math>=\quad - (0.6 \log 0.6 + 0.4 \log 0.4)\!</math><br />
|-<br />
| &nbsp;<br />
| <math>=\quad 0.971\!</math><br />
|}<br />
<br />
In other words, the capacity of this channel is slightly under 1 bit. This makes intuitive sense, since 3 against 2 is a near-even split of 5, and the measure of the channel capacity or the ''entropy'' is supposed to attain its maximum of 1 bit whenever a two-way partition is 50-50, that is to say, when it's as ''uniform'' a distribution as it can be.<br />
<br />
==Bibliography==<br />
<br />
* [[Charles Sanders Peirce (Bibliography)]]<br />
<br />
* Peirce, C.S. (1867), &ldquo;Upon Logical Comprehension and Extension&rdquo;, [http://www.iupui.edu/~peirce/writings/v2/w2/w2_06/v2_06.htm Online].<br />
<br />
==See also==<br />
<br />
===Related topics===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [[Ampheck]]<br />
* [[Boolean domain]]<br />
* [[Boolean function]]<br />
* [[Boolean-valued function]]<br />
{{col-break}}<br />
* [[Logical graph]]<br />
* [[Logical matrix]]<br />
* [[Logical NAND]]<br />
* [[Logical NNOR]]<br />
{{col-break}}<br />
* [[Minimal negation operator]]<br />
* [[Peirce's law]]<br />
* [[Propositional calculus]]<br />
* [[Semeiotic]]<br />
{{col-break}}<br />
* [[Sign relation]]<br />
* [[Triadic relation]]<br />
* [[Truth table]]<br />
* [[Zeroth order logic]]<br />
{{col-end}}<br />
<br />
===Related articles===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Peirce%27s_Logic_Of_Information Peirce's Logic Of Information]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Information_%3D_Comprehension_%C3%97_Extension Information = Comprehension × Extension]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]<br />
{{col-break}}<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]<br />
{{col-end}}<br />
<br />
[[Category:Information Theory]]<br />
[[Category:Inquiry]]<br />
[[Category:Inquiry Driven Systems]]<br />
[[Category:Logic]]<br />
[[Category:Peirce, Charles Sanders]]<br />
[[Category:Pragmatism]]<br />
[[Category:Scientific Method]]<br />
[[Category:Semiotics]]<br />
[[Category:Sign Relations]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Semiotic_Information&diff=469863Semiotic Information2020-11-05T15:32:08Z<p>Jon Awbrey: remove redirect to directory space</p>
<hr />
<div></div>Jon Awbreyhttps://mywikibiz.com/index.php?title=User:Jon_Awbrey&diff=469862User:Jon Awbrey2020-11-04T16:34:33Z<p>Jon Awbrey: /* Recent Sightings */ update links</p>
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<p><font face="lucida calligraphy" size="7">Jon Awbrey</font></p><br />
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[https://oeis.org/wiki/Differential_Propositional_Calculus Differential Propositional Calculus]<br />
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[https://oeis.org/wiki/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]<br />
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[https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overview Differential Logic and Dynamic Systems]<br />
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<br />
===Articles===<br />
<br />
[[Ampheck]]<br />
[[Boolean domain]]<br />
[[Boolean function]]<br />
[[Boolean-valued function]]<br />
[[Charles Sanders Peirce]]<br />
[[Charles Sanders Peirce (Bibliography)]]<br />
[[Comprehension (logic)]]<br />
[[Continuous predicate]]<br />
[[Correspondence theory of truth]]<br />
[[Cybernetics]]<br />
[[Descriptive science]]<br />
[[Differential logic]]<br />
[[Dynamics of inquiry]]<br />
[[Entitative graph]]<br />
[[Exclusive disjunction]]<br />
[[Formal science]]<br />
[[Graph (mathematics)]]<br />
[[Graph theory]]<br />
[[Grounded relation]]<br />
[[Inquiry]]<br />
[[Inquiry driven system]]<br />
[[Integer sequence]]<br />
[[Hypostatic abstraction]]<br />
[[Hypostatic object]]<br />
[[Kaina Stoicheia]]<br />
[[Logic]]<br />
[[Logic of information]]<br />
[[Logic of relatives]]<br />
[[Logic of Relatives (1870)]]<br />
[[Logic of Relatives (1883)]]<br />
[[Logical conjunction]]<br />
[[Logical disjunction]]<br />
[[Logical equality]]<br />
[[Logical graph]]<br />
[[Logical implication]]<br />
[[Logical matrix]]<br />
[[Logical NAND]]<br />
[[Logical negation]]<br />
[[Logical NNOR]]<br />
[[Minimal negation operator]]<br />
[[Multigrade operator]]<br />
[[Normative science]]<br />
[[Null graph]]<br />
[[On a New List of Categories]]<br />
[[Parametric operator]]<br />
[[Peirce's law]]<br />
[[Philosophy of mathematics]]<br />
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[[Pragmatic maxim]]<br />
[[Pragmatic theory of truth]]<br />
[[Pragmaticism]] <br />
[[Pragmatism]]<br />
[[Prescisive abstraction]]<br />
[[Propositional calculus]]<br />
[[Relation (mathematics)]]<br />
[[Relation composition]]<br />
[[Relation construction]]<br />
[[Relation reduction]]<br />
[[Relation theory]]<br />
[[Relation type]]<br />
[[Relative term]]<br />
[[Semeiotic]]<br />
[[Semiotic information]]<br />
[[Semiotics]]<br />
[[Sign relation]]<br />
[[Sign relational complex]]<br />
[[Sole sufficient operator]]<br />
[[Tacit extension]]<br />
[[The Simplest Mathematics]]<br />
[[Triadic relation]]<br />
[[Truth table]]<br />
[[Truth theory]]<br />
[[Universe of discourse]]<br />
[[What we've got here is (a) failure to communicate]]<br />
[[Zeroth order logic]]<br />
<br />
===Notes===<br />
<br />
* [[Directory:Jon Awbrey/Notes/Factorization Issues|Factorization Issues]]<br />
<br />
* [[Directory:Jon Awbrey/Notes/Factorization And Reification|Factorization And Reification]]<br />
<br />
===Papers===<br />
<br />
====Functional Logic====<br />
<br />
* [[Directory:Jon Awbrey/Papers/Functional Logic : Higher Order Propositions|Functional Logic : Higher Order Propositions]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Functional Logic : Inquiry and Analogy|Functional Logic : Inquiry and Analogy]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Functional Logic : Quantification Theory|Functional Logic : Quantification Theory]]<br />
<br />
====Differential Logic====<br />
<br />
* [[Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction|Differential Logic : Introduction]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Differential Propositional Calculus|Differential Propositional Calculus]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems|Differential Logic and Dynamic Systems 1.0]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0|Differential Logic and Dynamic Systems 2.0]]<br />
<br />
====Logic and Semiotics====<br />
<br />
* [[Directory:Jon Awbrey/Papers/Futures Of Logical Graphs|Futures Of Logical Graphs]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives|Peirce's 1870 Logic Of Relatives]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Peirce's Logic Of Information|Peirce's Logic Of Information]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Propositional Equation Reasoning Systems|Propositional Equation Reasoning Systems]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Semiotic Information|Semiotic Information]]<br />
<br />
====Inquiry Driven Systems====<br />
<br />
* [[Directory:Jon Awbrey/Essays/Prospects For Inquiry Driven Systems|Prospects for Inquiry Driven Systems]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Introduction to Inquiry Driven Systems|Introduction to Inquiry Driven Systems]]<br />
<br />
* [[Directory:Jon Awbrey/Essays/Inquiry Driven Systems : Fields Of Inquiry|Inquiry Driven Systems : Fields Of Inquiry]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems|Inquiry Driven Systems : Inquiry Into Inquiry]]<br />
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===Projects===<br />
<br />
* [[Directory:Jon Awbrey/Projects/Cactus Language|Cactus Language]]<br />
<br />
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<br />
* [[Directory:Jon Awbrey/Projects/Logic Of Information|Logic Of Information]]<br />
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<br />
* [[Directory:Jon Awbrey/Projects/Notes And Queries|Notes And Queries]]<br />
<br />
* [[Directory:Jon Awbrey/Projects/Peircean Pragmata|Peircean Pragmata]]<br />
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<br />
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<br />
==Presentations and Publications==<br />
<br />
* Awbrey, S.M., and Awbrey, J.L. (May 2001), &ldquo;Conceptual Barriers to Creating Integrative Universities&rdquo;, ''Organization : The Interdisciplinary Journal of Organization, Theory, and Society'' 8(2), Sage Publications, London, UK, pp. 269&ndash;284. [http://org.sagepub.com/cgi/content/abstract/8/2/269 Abstract].<br />
<br />
* Awbrey, S.M., and Awbrey, J.L. (September 1999), &ldquo;Organizations of Learning or Learning Organizations : The Challenge of Creating Integrative Universities for the Next Century&rdquo;, ''Second International Conference of the Journal &lsquo;Organization&rsquo;'', ''Re-Organizing Knowledge, Trans-Forming Institutions : Knowing, Knowledge, and the University in the 21st Century'', University of Massachusetts, Amherst, MA. [http://cspeirce.com/menu/library/aboutcsp/awbrey/integrat.htm Online].<br />
<br />
* Awbrey, J.L., and Awbrey, S.M. (Autumn 1995), &ldquo;Interpretation as Action : The Risk of Inquiry&rdquo;, ''Inquiry : Critical Thinking Across the Disciplines'' 15(1), pp. 40&ndash;52. [https://web.archive.org/web/20001210162300/http://chss.montclair.edu/inquiry/fall95/awbrey.html Archive]. [https://www.pdcnet.org/inquiryct/content/inquiryct_1995_0015_0001_0040_0052 Journal]. [https://independent.academia.edu/JonAwbrey/Papers/1302117/Interpretation_as_Action_The_Risk_of_Inquiry Online].<br />
<br />
* Awbrey, J.L., and Awbrey, S.M. (June 1992), &ldquo;Interpretation as Action : The Risk of Inquiry&rdquo;, ''The Eleventh International Human Science Research Conference'', Oakland University, Rochester, Michigan.<br />
<br />
* Awbrey, S.M., and Awbrey, J.L. (May 1991), &ldquo;An Architecture for Inquiry : Building Computer Platforms for Discovery&rdquo;, ''Proceedings of the Eighth International Conference on Technology and Education'', Toronto, Canada, pp. 874&ndash;875. [http://www.academia.edu/1270327/An_Architecture_for_Inquiry_Building_Computer_Platforms_for_Discovery Online].<br />
<br />
* Awbrey, J.L., and Awbrey, S.M. (January 1991), &ldquo;Exploring Research Data Interactively : Developing a Computer Architecture for Inquiry&rdquo;, Poster presented at the ''Annual Sigma Xi Research Forum'', University of Texas Medical Branch, Galveston, TX.<br />
<br />
* Awbrey, J.L., and Awbrey, S.M. (August 1990), &ldquo;Exploring Research Data Interactively. Theme One : A Program of Inquiry&rdquo;, ''Proceedings of the Sixth Annual Conference on Applications of Artificial Intelligence and CD-ROM in Education and Training'', Society for Applied Learning Technology, Washington, DC, pp. 9&ndash;15. [http://academia.edu/1272839/Exploring_Research_Data_Interactively._Theme_One_A_Program_of_Inquiry Online].<br />
<br />
==Education==<br />
<br />
* 1993&ndash;2003. [http://web.archive.org/web/20120202222443/http://www2.oakland.edu/secs/dispprofile.asp?Fname=Fatma&Lname=Mili Graduate Study], [http://web.archive.org/web/20120203004703/http://www2.oakland.edu/secs/dispprofile.asp?Fname=Mohamed&Lname=Zohdy Systems Engineering], [http://meadowbrookhall.org/explore/history/meadowbrookhall Oakland University].<br />
<br />
* '''1989. [http://web.archive.org/web/20001206101800/http://www.msu.edu/dig/msumap/psychology.html M.A. Psychology]''', [http://web.archive.org/web/20000902103838/http://www.msu.edu/dig/msumap/beaumont.html Michigan State University].<br />
<br />
* 1985&ndash;1986. [http://quod.lib.umich.edu/cgi/i/image/image-idx?id=S-BHL-X-BL001808%5DBL001808 Graduate Study, Mathematics], [http://www.umich.edu/ University of Michigan].<br />
<br />
* 1985. [http://web.archive.org/web/20061230174652/http://www.uiuc.edu/navigation/buildings/altgeld.top.html Graduate Study, Mathematics], [http://www.uiuc.edu/ University of Illinois at Urbana&ndash;Champaign].<br />
<br />
* 1984. [http://www.psych.uiuc.edu/graduate/ Graduate Study, Psychology], [http://www.uiuc.edu/ University of Illinois at Champaign&ndash;Urbana].<br />
<br />
* '''1980. [http://www.mth.msu.edu/images/wells_medium.jpg M.A. Mathematics]''', [http://www.msu.edu/~hvac/survey/BeaumontTower.html Michigan State University].<br />
<br />
* '''1976. [http://web.archive.org/web/20001206050600/http://www.msu.edu/dig/msumap/phillips.html B.A. Mathematical and Philosophical Method]''', <br> [http://www.enolagaia.com/JMC.html Justin Morrill College], [http://www.msu.edu/ Michigan State University].<br />
<br />
==Category and Subject Interests==<br />
<br />
[[Category:Artificial Intelligence]]<br />
[[Category:Automata Theory]]<br />
[[Category:Category Theory]]<br />
[[Category:Charles Sanders Peirce]]<br />
[[Category:Cognitive Science]] <br />
[[Category:Combinatorics]]<br />
[[Category:Computer Science]]<br />
[[Category:Critical Thinking]]<br />
[[Category:Cybernetics]]<br />
[[Category:Differential Logic]]<br />
[[Category:Education]]<br />
[[Category:Formal Languages]]<br />
[[Category:Formal Sciences]]<br />
[[Category:Graph Theory]]<br />
[[Category:Group Theory]]<br />
[[Category:Hermeneutics]]<br />
[[Category:Information Systems]]<br />
[[Category:Information Theory]]<br />
[[Category:Inquiry]]<br />
[[Category:Inquiry Driven Systems]]<br />
[[Category:Integer Sequences]]<br />
[[Category:Intelligence Amplification]]<br />
[[Category:Learning Organizations]]<br />
[[Category:Linguistics]]<br />
[[Category:Knowledge Representation]]<br />
[[Category:Logic]]<br />
[[Category:Mathematics]]<br />
[[Category:Natural Languages]]<br />
[[Category:Philosophy]]<br />
[[Category:Pragmatics]]<br />
[[Category:Psychology]]<br />
[[Category:Science]]<br />
[[Category:Semantics]]<br />
[[Category:Semiotics]]<br />
[[Category:Statistics]]<br />
[[Category:Systems Science]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=User:Jon_Awbrey&diff=469861User:Jon Awbrey2020-11-04T16:26:06Z<p>Jon Awbrey: /* Recent Sightings */ update</p>
<hr />
<div><br><br />
<p><font face="lucida calligraphy" size="7">Jon Awbrey</font></p><br />
<br><br />
__NOTOC__<br />
==Presently &hellip;==<br />
<br />
<center><br />
<br />
[https://oeis.org/wiki/Riffs_and_Rotes Riffs and Rotes]<br />
<br />
[https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Overview Cactus Language]<br />
<br />
[https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Overview Theme One Program]<br />
<br />
[https://oeis.org/wiki/Propositions_As_Types Propositions As Types]<br />
<br />
[https://oeis.org/wiki/Futures_Of_Logical_Graphs Futures Of Logical Graphs]<br />
<br />
[https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Overview Peirce's Logic Of Relatives]<br />
<br />
[https://oeis.org/wiki/Logical_Graphs Logical Graphs] [https://inquiryintoinquiry.com/2008/07/29/logical-graphs-1/ One] [https://inquiryintoinquiry.com/2008/09/19/logical-graphs-2/ Two]<br />
<br />
[https://oeis.org/wiki/Differential_Propositional_Calculus Differential Propositional Calculus]<br />
<br />
[https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Overview Differential Logic] [https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1 (1)] [https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_2 (2)] [https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3 (3)]<br />
<br />
[https://oeis.org/wiki/Differential_Analytic_Turing_Automata Differential Analytic Turing Automata]<br />
<br />
[https://oeis.org/wiki/User:Jon_Awbrey/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]<br />
<br />
[https://oeis.org/wiki/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]<br />
<br />
[https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overview Differential Logic and Dynamic Systems]<br />
<br />
[https://oeis.org/wiki/Information_%3D_Comprehension_%C3%97_Extension Information = Comprehension &times; Extension]<br />
<br />
[https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]<br />
<br />
[https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Overview Inquiry Driven Systems &bull; Inquiry Into Inquiry]<br />
<br />
</center><br />
<br />
==Recent Sightings==<br />
<br />
{| align="center" style="text-align:center" width="100%"<br />
|-<br />
| [http://mywikibiz.com/Inquiry_Live Inquiry Live]<br />
| [http://mywikibiz.com/Logic_Live Logic Live]<br />
|-<br />
| [http://stderr.org/cgi-bin/mailman/listinfo/inquiry Inquiry Project]<br />
| [http://web.archive.org/web/20150301180400/http://stderr.org/pipermail/inquiry/ Inquiry Archive]<br />
|-<br />
| [http://inquiryintoinquiry.com/ Inquiry Into Inquiry]<br />
| [http://jonawbrey.wordpress.com/ Jon Awbrey &bull; Blog]<br />
|-<br />
| [https://web.archive.org/web/20191124081501/http://intersci.ss.uci.edu/wiki/index.php/User:Jon_Awbrey InterSciWiki User Page]<br />
| [https://web.archive.org/web/20191025121620/http://intersci.ss.uci.edu/wiki/index.php/User_talk:Jon_Awbrey InterSciWiki Talk Page]<br />
|-<br />
| [http://mywikibiz.com/Directory:Jon_Awbrey MyWikiBiz Directory]<br />
| [http://mywikibiz.com/Directory_talk:Jon_Awbrey MyWikiBiz Discussion]<br />
|-<br />
| [http://mywikibiz.com/User:Jon_Awbrey MyWikiBiz User Page]<br />
| [http://mywikibiz.com/User_talk:Jon_Awbrey MyWikiBiz Talk Page]<br />
|-<br />
| [http://planetmath.org/ PlanetMath Project]<br />
| [http://planetmath.org/users/Jon-Awbrey PlanetMath Profile]<br />
|-<br />
| [http://forum.wolframscience.com/ NKS Forum]<br />
| [http://forum.wolframscience.com/member.php?s=&action=getinfo&userid=336 NKS Profile]<br />
|-<br />
| [https://oeis.org/wiki/Welcome OEIS Land]<br />
| [https://oeis.org/search?q=Awbrey Bolgia Mia]<br />
|-<br />
| [https://oeis.org/wiki/User:Jon_Awbrey OEIS Wiki Page]<br />
| [https://oeis.org/wiki/User_talk:Jon_Awbrey OEIS Wiki Talk]<br />
|-<br />
| [http://list.seqfan.eu/cgi-bin/mailman/listinfo/seqfan Fantasia Sequentia]<br />
| [http://list.seqfan.eu/pipermail/seqfan/ SeqFan Archive]<br />
|-<br />
| [http://mathforum.org/kb/ Math Forum Project]<br />
| [http://mathforum.org/kb/accountView.jspa?userID=99854 Math Forum Profile]<br />
|-<br />
| [http://www.mathweb.org/wiki/User:Jon_Awbrey MathWeb Page]<br />
| [http://www.mathweb.org/wiki/User_talk:Jon_Awbrey MathWeb Talk]<br />
|-<br />
| [http://www.proofwiki.org/wiki/User:Jon_Awbrey ProofWiki Page]<br />
| [http://www.proofwiki.org/wiki/User_talk:Jon_Awbrey ProofWiki Talk]<br />
|-<br />
| [http://mathoverflow.net/ MathOverFlow]<br />
| [http://mathoverflow.net/users/1636/jon-awbrey MOFler Profile]<br />
|-<br />
| [http://p2pfoundation.net/User:JonAwbrey P2P Wiki Page]<br />
| [http://p2pfoundation.net/User_talk:JonAwbrey P2P Wiki Talk]<br />
|-<br />
| [http://vectors.usc.edu/ Vectors Project]<br />
| [http://vectors.usc.edu/thoughtmesh/ ThoughtMesh]<br />
|-<br />
| [http://ontolog.cim3.net/ OntoLog Project]<br />
| [http://ontolog.cim3.net/cgi-bin/wiki.pl?JonAwbrey OntoLog Profile]<br />
|-<br />
| [http://web.archive.org/web/20150127223035/http://semanticweb.org/wiki/User:Jon_Awbrey SemanticWeb Page]<br />
| [http://web.archive.org/web/20100716202618/http://semanticweb.org/wiki/User_talk:Jon_Awbrey SemanticWeb Talk]<br />
|-<br />
| [http://zh.wikipedia.org/wiki/User:Jon_Awbrey 维基百科 : 用户页面]<br />
| [http://zh.wikipedia.org/wiki/User_talk:Jon_Awbrey 维基百科 : 讨论]<br />
|}<br />
<br />
==Contributions==<br />
<br />
===Articles===<br />
<br />
[[Ampheck]]<br />
[[Boolean domain]]<br />
[[Boolean function]]<br />
[[Boolean-valued function]]<br />
[[Charles Sanders Peirce]]<br />
[[Charles Sanders Peirce (Bibliography)]]<br />
[[Comprehension (logic)]]<br />
[[Continuous predicate]]<br />
[[Correspondence theory of truth]]<br />
[[Cybernetics]]<br />
[[Descriptive science]]<br />
[[Differential logic]]<br />
[[Dynamics of inquiry]]<br />
[[Entitative graph]]<br />
[[Exclusive disjunction]]<br />
[[Formal science]]<br />
[[Graph (mathematics)]]<br />
[[Graph theory]]<br />
[[Grounded relation]]<br />
[[Inquiry]]<br />
[[Inquiry driven system]]<br />
[[Integer sequence]]<br />
[[Hypostatic abstraction]]<br />
[[Hypostatic object]]<br />
[[Kaina Stoicheia]]<br />
[[Logic]]<br />
[[Logic of information]]<br />
[[Logic of relatives]]<br />
[[Logic of Relatives (1870)]]<br />
[[Logic of Relatives (1883)]]<br />
[[Logical conjunction]]<br />
[[Logical disjunction]]<br />
[[Logical equality]]<br />
[[Logical graph]]<br />
[[Logical implication]]<br />
[[Logical matrix]]<br />
[[Logical NAND]]<br />
[[Logical negation]]<br />
[[Logical NNOR]]<br />
[[Minimal negation operator]]<br />
[[Multigrade operator]]<br />
[[Normative science]]<br />
[[Null graph]]<br />
[[On a New List of Categories]]<br />
[[Parametric operator]]<br />
[[Peirce's law]]<br />
[[Philosophy of mathematics]]<br />
[[Pragmatic information]]<br />
[[Pragmatic maxim]]<br />
[[Pragmatic theory of truth]]<br />
[[Pragmaticism]] <br />
[[Pragmatism]]<br />
[[Prescisive abstraction]]<br />
[[Propositional calculus]]<br />
[[Relation (mathematics)]]<br />
[[Relation composition]]<br />
[[Relation construction]]<br />
[[Relation reduction]]<br />
[[Relation theory]]<br />
[[Relation type]]<br />
[[Relative term]]<br />
[[Semeiotic]]<br />
[[Semiotic information]]<br />
[[Semiotics]]<br />
[[Sign relation]]<br />
[[Sign relational complex]]<br />
[[Sole sufficient operator]]<br />
[[Tacit extension]]<br />
[[The Simplest Mathematics]]<br />
[[Triadic relation]]<br />
[[Truth table]]<br />
[[Truth theory]]<br />
[[Universe of discourse]]<br />
[[What we've got here is (a) failure to communicate]]<br />
[[Zeroth order logic]]<br />
<br />
===Notes===<br />
<br />
* [[Directory:Jon Awbrey/Notes/Factorization Issues|Factorization Issues]]<br />
<br />
* [[Directory:Jon Awbrey/Notes/Factorization And Reification|Factorization And Reification]]<br />
<br />
===Papers===<br />
<br />
====Functional Logic====<br />
<br />
* [[Directory:Jon Awbrey/Papers/Functional Logic : Higher Order Propositions|Functional Logic : Higher Order Propositions]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Functional Logic : Inquiry and Analogy|Functional Logic : Inquiry and Analogy]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Functional Logic : Quantification Theory|Functional Logic : Quantification Theory]]<br />
<br />
====Differential Logic====<br />
<br />
* [[Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction|Differential Logic : Introduction]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Differential Propositional Calculus|Differential Propositional Calculus]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems|Differential Logic and Dynamic Systems 1.0]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0|Differential Logic and Dynamic Systems 2.0]]<br />
<br />
====Logic and Semiotics====<br />
<br />
* [[Directory:Jon Awbrey/Papers/Futures Of Logical Graphs|Futures Of Logical Graphs]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives|Peirce's 1870 Logic Of Relatives]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Peirce's Logic Of Information|Peirce's Logic Of Information]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Propositional Equation Reasoning Systems|Propositional Equation Reasoning Systems]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Semiotic Information|Semiotic Information]]<br />
<br />
====Inquiry Driven Systems====<br />
<br />
* [[Directory:Jon Awbrey/Essays/Prospects For Inquiry Driven Systems|Prospects for Inquiry Driven Systems]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Introduction to Inquiry Driven Systems|Introduction to Inquiry Driven Systems]]<br />
<br />
* [[Directory:Jon Awbrey/Essays/Inquiry Driven Systems : Fields Of Inquiry|Inquiry Driven Systems : Fields Of Inquiry]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems|Inquiry Driven Systems : Inquiry Into Inquiry]]<br />
<br />
===Projects===<br />
<br />
* [[Directory:Jon Awbrey/Projects/Cactus Language|Cactus Language]]<br />
<br />
* [[Directory:Jon Awbrey/Projects/Differential Logic|Differential Logic]]<br />
<br />
* [[Directory:Jon Awbrey/Projects/Inquiry|Inquiry]]<br />
** [[Directory:Jon Awbrey/Projects/Architecture For Inquiry|Architecture For Inquiry]]<br />
** [[Directory:Jon Awbrey/Projects/Inquiry Driven Systems|Inquiry Driven Systems]]<br />
<br />
* [[Directory:Jon Awbrey/Projects/Logic Of Information|Logic Of Information]]<br />
** [[Directory:Jon Awbrey/Projects/Pragmatic Theory Of Information|Pragmatic Theory Of Information]]<br />
** [[Directory:Jon Awbrey/Projects/Semiotic Theory Of Information|Semiotic Theory Of Information]]<br />
<br />
* [[Directory:Jon Awbrey/Projects/Notes And Queries|Notes And Queries]]<br />
<br />
* [[Directory:Jon Awbrey/Projects/Peircean Pragmata|Peircean Pragmata]]<br />
<br />
* [[Directory:Jon Awbrey/Projects/Theme One Program|Theme One Program]]<br />
<br />
* [[Directory:Jon Awbrey/Projects/Theory Of Relations|Theory Of Relations]]<br />
<br />
===Poetry===<br />
<br />
* [[Directory:Jon Awbrey/Poetry/Past All Reckoning|Past All Reckoning]]<br />
<br />
* [[Directory:Jon Awbrey/Poetry/Poems Of Emediate Moment|Poems Of Emediate Moment]]<br />
<br />
* [[Directory:Jon Awbrey/Poetry/Questionable Verses|Questionable Verses]]<br />
<br />
* [[Directory:Jon_Awbrey/Poetry/Iconoclast|Iconoclast]]<br />
<br />
===User Pages===<br />
<br />
* [[Directory:Jon Awbrey/EXCERPTS|Collection Of Source Materials]]<br />
* [[User:Jon Awbrey/Examples Of Inquiry|Examples Of Inquiry]]<br />
* [[User:Jon Awbrey/Mathematical Notes|Mathematical Notes]]<br />
* [[User:Jon Awbrey/Philosophical Notes|Philosophical Notes]]<br />
<br />
* [http://mywikibiz.com/index.php?title=Special%3APrefixIndex&prefix=Jon+Awbrey&namespace=2 MyWikiBiz User Pages]<br />
* [http://intersci.ss.uci.edu/wiki/index.php?title=Special%3APrefixIndex&prefix=Jon+Awbrey&namespace=2 InterSciWiki User Pages]<br />
* [http://mywikibiz.com/index.php?title=Special%3APrefixIndex&prefix=Jon+Awbrey&namespace=110 MyWikiBiz Directory Pages]<br />
<br />
==Presentations and Publications==<br />
<br />
* Awbrey, S.M., and Awbrey, J.L. (May 2001), &ldquo;Conceptual Barriers to Creating Integrative Universities&rdquo;, ''Organization : The Interdisciplinary Journal of Organization, Theory, and Society'' 8(2), Sage Publications, London, UK, pp. 269&ndash;284. [http://org.sagepub.com/cgi/content/abstract/8/2/269 Abstract].<br />
<br />
* Awbrey, S.M., and Awbrey, J.L. (September 1999), &ldquo;Organizations of Learning or Learning Organizations : The Challenge of Creating Integrative Universities for the Next Century&rdquo;, ''Second International Conference of the Journal &lsquo;Organization&rsquo;'', ''Re-Organizing Knowledge, Trans-Forming Institutions : Knowing, Knowledge, and the University in the 21st Century'', University of Massachusetts, Amherst, MA. [http://cspeirce.com/menu/library/aboutcsp/awbrey/integrat.htm Online].<br />
<br />
* Awbrey, J.L., and Awbrey, S.M. (Autumn 1995), &ldquo;Interpretation as Action : The Risk of Inquiry&rdquo;, ''Inquiry : Critical Thinking Across the Disciplines'' 15(1), pp. 40&ndash;52. [https://web.archive.org/web/20001210162300/http://chss.montclair.edu/inquiry/fall95/awbrey.html Archive]. [https://www.pdcnet.org/inquiryct/content/inquiryct_1995_0015_0001_0040_0052 Journal]. [https://independent.academia.edu/JonAwbrey/Papers/1302117/Interpretation_as_Action_The_Risk_of_Inquiry Online].<br />
<br />
* Awbrey, J.L., and Awbrey, S.M. (June 1992), &ldquo;Interpretation as Action : The Risk of Inquiry&rdquo;, ''The Eleventh International Human Science Research Conference'', Oakland University, Rochester, Michigan.<br />
<br />
* Awbrey, S.M., and Awbrey, J.L. (May 1991), &ldquo;An Architecture for Inquiry : Building Computer Platforms for Discovery&rdquo;, ''Proceedings of the Eighth International Conference on Technology and Education'', Toronto, Canada, pp. 874&ndash;875. [http://www.academia.edu/1270327/An_Architecture_for_Inquiry_Building_Computer_Platforms_for_Discovery Online].<br />
<br />
* Awbrey, J.L., and Awbrey, S.M. (January 1991), &ldquo;Exploring Research Data Interactively : Developing a Computer Architecture for Inquiry&rdquo;, Poster presented at the ''Annual Sigma Xi Research Forum'', University of Texas Medical Branch, Galveston, TX.<br />
<br />
* Awbrey, J.L., and Awbrey, S.M. (August 1990), &ldquo;Exploring Research Data Interactively. Theme One : A Program of Inquiry&rdquo;, ''Proceedings of the Sixth Annual Conference on Applications of Artificial Intelligence and CD-ROM in Education and Training'', Society for Applied Learning Technology, Washington, DC, pp. 9&ndash;15. [http://academia.edu/1272839/Exploring_Research_Data_Interactively._Theme_One_A_Program_of_Inquiry Online].<br />
<br />
==Education==<br />
<br />
* 1993&ndash;2003. [http://web.archive.org/web/20120202222443/http://www2.oakland.edu/secs/dispprofile.asp?Fname=Fatma&Lname=Mili Graduate Study], [http://web.archive.org/web/20120203004703/http://www2.oakland.edu/secs/dispprofile.asp?Fname=Mohamed&Lname=Zohdy Systems Engineering], [http://meadowbrookhall.org/explore/history/meadowbrookhall Oakland University].<br />
<br />
* '''1989. [http://web.archive.org/web/20001206101800/http://www.msu.edu/dig/msumap/psychology.html M.A. Psychology]''', [http://web.archive.org/web/20000902103838/http://www.msu.edu/dig/msumap/beaumont.html Michigan State University].<br />
<br />
* 1985&ndash;1986. [http://quod.lib.umich.edu/cgi/i/image/image-idx?id=S-BHL-X-BL001808%5DBL001808 Graduate Study, Mathematics], [http://www.umich.edu/ University of Michigan].<br />
<br />
* 1985. [http://web.archive.org/web/20061230174652/http://www.uiuc.edu/navigation/buildings/altgeld.top.html Graduate Study, Mathematics], [http://www.uiuc.edu/ University of Illinois at Urbana&ndash;Champaign].<br />
<br />
* 1984. [http://www.psych.uiuc.edu/graduate/ Graduate Study, Psychology], [http://www.uiuc.edu/ University of Illinois at Champaign&ndash;Urbana].<br />
<br />
* '''1980. [http://www.mth.msu.edu/images/wells_medium.jpg M.A. Mathematics]''', [http://www.msu.edu/~hvac/survey/BeaumontTower.html Michigan State University].<br />
<br />
* '''1976. [http://web.archive.org/web/20001206050600/http://www.msu.edu/dig/msumap/phillips.html B.A. Mathematical and Philosophical Method]''', <br> [http://www.enolagaia.com/JMC.html Justin Morrill College], [http://www.msu.edu/ Michigan State University].<br />
<br />
==Category and Subject Interests==<br />
<br />
[[Category:Artificial Intelligence]]<br />
[[Category:Automata Theory]]<br />
[[Category:Category Theory]]<br />
[[Category:Charles Sanders Peirce]]<br />
[[Category:Cognitive Science]] <br />
[[Category:Combinatorics]]<br />
[[Category:Computer Science]]<br />
[[Category:Critical Thinking]]<br />
[[Category:Cybernetics]]<br />
[[Category:Differential Logic]]<br />
[[Category:Education]]<br />
[[Category:Formal Languages]]<br />
[[Category:Formal Sciences]]<br />
[[Category:Graph Theory]]<br />
[[Category:Group Theory]]<br />
[[Category:Hermeneutics]]<br />
[[Category:Information Systems]]<br />
[[Category:Information Theory]]<br />
[[Category:Inquiry]]<br />
[[Category:Inquiry Driven Systems]]<br />
[[Category:Integer Sequences]]<br />
[[Category:Intelligence Amplification]]<br />
[[Category:Learning Organizations]]<br />
[[Category:Linguistics]]<br />
[[Category:Knowledge Representation]]<br />
[[Category:Logic]]<br />
[[Category:Mathematics]]<br />
[[Category:Natural Languages]]<br />
[[Category:Philosophy]]<br />
[[Category:Pragmatics]]<br />
[[Category:Psychology]]<br />
[[Category:Science]]<br />
[[Category:Semantics]]<br />
[[Category:Semiotics]]<br />
[[Category:Statistics]]<br />
[[Category:Systems Science]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Descriptive_science&diff=469860Descriptive science2020-11-04T16:15:16Z<p>Jon Awbrey: waybak links</p>
<hr />
<div><font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].<br />
<br />
A '''descriptive science''', also called a '''special science''', is a form of [[inquiry]], typically involving a community of inquiry and its accumulated body of provisional knowledge, that seeks to discover what is true about a recognized domain of phenomena.<br />
<br />
==Syllabus==<br />
<br />
===Focal nodes===<br />
<br />
* [[Inquiry Live]]<br />
* [[Logic Live]]<br />
<br />
===Peer nodes===<br />
<br />
* [https://web.archive.org/web/20191209233842/http://intersci.ss.uci.edu/wiki/index.php/Descriptive_science Descriptive Science @ InterSciWiki]<br />
* [http://mywikibiz.com/Descriptive_science Descriptive Science @ MyWikiBiz]<br />
* [http://ref.subwiki.org/wiki/Descriptive_science Descriptive Science @ Subject Wikis]<br />
* [http://en.wikiversity.org/wiki/Descriptive_science Descriptive Science @ Wikiversity]<br />
* [http://beta.wikiversity.org/wiki/Descriptive_science Descriptive Science @ Wikiversity Beta]<br />
<br />
===Logical operators===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [[Exclusive disjunction]]<br />
* [[Logical conjunction]]<br />
* [[Logical disjunction]]<br />
* [[Logical equality]]<br />
{{col-break}}<br />
* [[Logical implication]]<br />
* [[Logical NAND]]<br />
* [[Logical NNOR]]<br />
* [[Logical negation|Negation]]<br />
{{col-end}}<br />
<br />
===Related topics===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [[Ampheck]]<br />
* [[Boolean domain]]<br />
* [[Boolean function]]<br />
* [[Boolean-valued function]]<br />
* [[Differential logic]]<br />
{{col-break}}<br />
* [[Logical graph]]<br />
* [[Minimal negation operator]]<br />
* [[Multigrade operator]]<br />
* [[Parametric operator]]<br />
* [[Peirce's law]]<br />
{{col-break}}<br />
* [[Propositional calculus]]<br />
* [[Sole sufficient operator]]<br />
* [[Truth table]]<br />
* [[Universe of discourse]]<br />
* [[Zeroth order logic]]<br />
{{col-end}}<br />
<br />
===Relational concepts===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [[Continuous predicate]]<br />
* [[Hypostatic abstraction]]<br />
* [[Logic of relatives]]<br />
* [[Logical matrix]]<br />
{{col-break}}<br />
* [[Relation (mathematics)|Relation]]<br />
* [[Relation composition]]<br />
* [[Relation construction]]<br />
* [[Relation reduction]]<br />
{{col-break}}<br />
* [[Relation theory]]<br />
* [[Relative term]]<br />
* [[Sign relation]]<br />
* [[Triadic relation]]<br />
{{col-end}}<br />
<br />
===Information, Inquiry===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [[Inquiry]]<br />
* [[Dynamics of inquiry]]<br />
{{col-break}}<br />
* [[Semeiotic]]<br />
* [[Logic of information]]<br />
{{col-break}}<br />
* [[Descriptive science]]<br />
* [[Normative science]]<br />
{{col-break}}<br />
* [[Pragmatic maxim]]<br />
* [[Truth theory]]<br />
{{col-end}}<br />
<br />
===Related articles===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Semiotic_Information Semiotic Information]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]<br />
{{col-break}}<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]<br />
{{col-break}}<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]<br />
{{col-end}}<br />
<br />
==Document history==<br />
<br />
Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.<br />
<br />
* [https://web.archive.org/web/20191209233842/http://intersci.ss.uci.edu/wiki/index.php/Descriptive_science Descriptive Science], [https://web.archive.org/web/20191124081501/http://intersci.ss.uci.edu/wiki/index.php/User:Jon_Awbrey InterSciWiki]<br />
* [http://mywikibiz.com/Descriptive_science Descriptive Science], [http://mywikibiz.com/ MyWikiBiz]<br />
* [http://semanticweb.org/wiki/Descriptive_science Descriptive Science], [http://semanticweb.org/ Semantic Web]<br />
* [http://wikinfo.org/w/index.php/Descriptive_science Descriptive Science], [http://wikinfo.org/w/ Wikinfo]<br />
* [http://en.wikiversity.org/wiki/Descriptive_science Descriptive Science], [http://en.wikiversity.org/ Wikiversity]<br />
* [http://beta.wikiversity.org/wiki/Descriptive_science Descriptive Science], [http://beta.wikiversity.org/ Wikiversity Beta]<br />
* [http://en.wikipedia.org/w/index.php?title=Descriptive_science&oldid=51990248 Descriptive Science], [http://en.wikipedia.org/ Wikipedia]<br />
<br />
[[Category:Descriptive Sciences]]<br />
[[Category:Inquiry]]<br />
[[Category:Philosophy]]<br />
[[Category:Philosophy of Science]]<br />
[[Category:Science]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Normative_science&diff=469859Normative science2020-11-04T16:10:32Z<p>Jon Awbrey: waybak links</p>
<hr />
<div><font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].<br />
<br />
A '''normative science''' is a form of [[inquiry]], typically involving a community of inquiry and its accumulated body of provisional knowledge, that seeks to discover good ways of achieving recognized aims, ends, goals, objectives, or purposes.<br />
<br />
The three '''normative sciences''', according to traditional conceptions in philosophy, are ''aesthetics'', ''ethics'', and ''logic''.<br />
<br />
==Syllabus==<br />
<br />
===Focal nodes===<br />
<br />
* [[Inquiry Live]]<br />
* [[Logic Live]]<br />
<br />
===Peer nodes===<br />
<br />
* [https://web.archive.org/web/20191209141945/http://intersci.ss.uci.edu/wiki/index.php/Normative_science Normative Science @ InterSciWiki]<br />
* [http://mywikibiz.com/Normative_science Normative Science @ MyWikiBiz]<br />
* [http://ref.subwiki.org/wiki/Normative_science Normative Science @ Subject Wikis]<br />
* [http://en.wikiversity.org/wiki/Normative_science Normative Science @ Wikiversity]<br />
* [http://beta.wikiversity.org/wiki/Normative_science Normative Science @ Wikiversity Beta]<br />
<br />
===Logical operators===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [[Exclusive disjunction]]<br />
* [[Logical conjunction]]<br />
* [[Logical disjunction]]<br />
* [[Logical equality]]<br />
{{col-break}}<br />
* [[Logical implication]]<br />
* [[Logical NAND]]<br />
* [[Logical NNOR]]<br />
* [[Logical negation|Negation]]<br />
{{col-end}}<br />
<br />
===Related topics===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [[Ampheck]]<br />
* [[Boolean domain]]<br />
* [[Boolean function]]<br />
* [[Boolean-valued function]]<br />
* [[Differential logic]]<br />
{{col-break}}<br />
* [[Logical graph]]<br />
* [[Minimal negation operator]]<br />
* [[Multigrade operator]]<br />
* [[Parametric operator]]<br />
* [[Peirce's law]]<br />
{{col-break}}<br />
* [[Propositional calculus]]<br />
* [[Sole sufficient operator]]<br />
* [[Truth table]]<br />
* [[Universe of discourse]]<br />
* [[Zeroth order logic]]<br />
{{col-end}}<br />
<br />
===Relational concepts===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [[Continuous predicate]]<br />
* [[Hypostatic abstraction]]<br />
* [[Logic of relatives]]<br />
* [[Logical matrix]]<br />
{{col-break}}<br />
* [[Relation (mathematics)|Relation]]<br />
* [[Relation composition]]<br />
* [[Relation construction]]<br />
* [[Relation reduction]]<br />
{{col-break}}<br />
* [[Relation theory]]<br />
* [[Relative term]]<br />
* [[Sign relation]]<br />
* [[Triadic relation]]<br />
{{col-end}}<br />
<br />
===Information, Inquiry===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [[Inquiry]]<br />
* [[Dynamics of inquiry]]<br />
{{col-break}}<br />
* [[Semeiotic]]<br />
* [[Logic of information]]<br />
{{col-break}}<br />
* [[Descriptive science]]<br />
* [[Normative science]]<br />
{{col-break}}<br />
* [[Pragmatic maxim]]<br />
* [[Truth theory]]<br />
{{col-end}}<br />
<br />
===Related articles===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Semiotic_Information Semiotic Information]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]<br />
{{col-break}}<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]<br />
{{col-break}}<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]<br />
{{col-end}}<br />
<br />
==Document history==<br />
<br />
Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.<br />
<br />
* [https://web.archive.org/web/20191209141945/http://intersci.ss.uci.edu/wiki/index.php/Normative_science Normative Science], [https://web.archive.org/web/20191124081501/http://intersci.ss.uci.edu/wiki/index.php/User:Jon_Awbrey InterSciWiki]<br />
* [http://mywikibiz.com/Normative_science Normative Science], [http://mywikibiz.com/ MyWikiBiz]<br />
* [http://semanticweb.org/wiki/Normative_science Normative Science], [http://semanticweb.org/ Semantic Web]<br />
* [http://wikinfo.org/w/index.php/Normative_science Normative Science], [http://wikinfo.org/w/ Wikinfo]<br />
* [http://en.wikiversity.org/wiki/Normative_science Normative Science], [http://en.wikiversity.org/ Wikiversity]<br />
* [http://beta.wikiversity.org/wiki/Normative_science Normative Science], [http://beta.wikiversity.org/ Wikiversity Beta]<br />
* [http://en.wikipedia.org/w/index.php?title=Normative_science&oldid=51993011 Normative Science], [http://en.wikipedia.org/ Wikipedia]<br />
<br />
[[Category:Inquiry]]<br />
[[Category:Normative Sciences]]<br />
[[Category:Philosophy]]<br />
[[Category:Philosophy of Science]]<br />
[[Category:Science]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=User:Jon_Awbrey&diff=469858User:Jon Awbrey2020-11-04T15:40:04Z<p>Jon Awbrey: /* Presently &hellip; */ update</p>
<hr />
<div><br><br />
<p><font face="lucida calligraphy" size="7">Jon Awbrey</font></p><br />
<br><br />
__NOTOC__<br />
==Presently &hellip;==<br />
<br />
<center><br />
<br />
[https://oeis.org/wiki/Riffs_and_Rotes Riffs and Rotes]<br />
<br />
[https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Overview Cactus Language]<br />
<br />
[https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Overview Theme One Program]<br />
<br />
[https://oeis.org/wiki/Propositions_As_Types Propositions As Types]<br />
<br />
[https://oeis.org/wiki/Futures_Of_Logical_Graphs Futures Of Logical Graphs]<br />
<br />
[https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Overview Peirce's Logic Of Relatives]<br />
<br />
[https://oeis.org/wiki/Logical_Graphs Logical Graphs] [https://inquiryintoinquiry.com/2008/07/29/logical-graphs-1/ One] [https://inquiryintoinquiry.com/2008/09/19/logical-graphs-2/ Two]<br />
<br />
[https://oeis.org/wiki/Differential_Propositional_Calculus Differential Propositional Calculus]<br />
<br />
[https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Overview Differential Logic] [https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1 (1)] [https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_2 (2)] [https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3 (3)]<br />
<br />
[https://oeis.org/wiki/Differential_Analytic_Turing_Automata Differential Analytic Turing Automata]<br />
<br />
[https://oeis.org/wiki/User:Jon_Awbrey/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]<br />
<br />
[https://oeis.org/wiki/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]<br />
<br />
[https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overview Differential Logic and Dynamic Systems]<br />
<br />
[https://oeis.org/wiki/Information_%3D_Comprehension_%C3%97_Extension Information = Comprehension &times; Extension]<br />
<br />
[https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]<br />
<br />
[https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Overview Inquiry Driven Systems &bull; Inquiry Into Inquiry]<br />
<br />
</center><br />
<br />
==Recent Sightings==<br />
<br />
{| align="center" style="text-align:center" width="100%"<br />
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|-<br />
| [http://stderr.org/cgi-bin/mailman/listinfo/inquiry Inquiry Project]<br />
| [http://stderr.org/pipermail/inquiry/ Inquiry Archive]<br />
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| [http://inquiryintoinquiry.com/ Inquiry Into Inquiry]<br />
| [http://jonawbrey.wordpress.com/ Jon Awbrey • Blog]<br />
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| [http://intersci.ss.uci.edu/wiki/index.php/User:Jon_Awbrey InterSciWiki User Page]<br />
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| [http://mywikibiz.com/Directory:Jon_Awbrey MyWikiBiz Directory]<br />
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| [http://planetmath.org/users/Jon-Awbrey PlanetMath Profile]<br />
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| [http://forum.wolframscience.com/ NKS Forum]<br />
| [http://forum.wolframscience.com/member.php?s=&action=getinfo&userid=336 NKS Profile]<br />
|-<br />
| [https://oeis.org/wiki/Welcome OEIS Land]<br />
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|-<br />
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| [http://list.seqfan.eu/pipermail/seqfan/ SeqFan Archive]<br />
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| [http://web.archive.org/web/20100716202618/http://semanticweb.org/wiki/User_talk:Jon_Awbrey SemanticWeb Talk]<br />
|}<br />
<br />
==Contributions==<br />
<br />
===Articles===<br />
<br />
[[Ampheck]]<br />
[[Boolean domain]]<br />
[[Boolean function]]<br />
[[Boolean-valued function]]<br />
[[Charles Sanders Peirce]]<br />
[[Charles Sanders Peirce (Bibliography)]]<br />
[[Comprehension (logic)]]<br />
[[Continuous predicate]]<br />
[[Correspondence theory of truth]]<br />
[[Cybernetics]]<br />
[[Descriptive science]]<br />
[[Differential logic]]<br />
[[Dynamics of inquiry]]<br />
[[Entitative graph]]<br />
[[Exclusive disjunction]]<br />
[[Formal science]]<br />
[[Graph (mathematics)]]<br />
[[Graph theory]]<br />
[[Grounded relation]]<br />
[[Inquiry]]<br />
[[Inquiry driven system]]<br />
[[Integer sequence]]<br />
[[Hypostatic abstraction]]<br />
[[Hypostatic object]]<br />
[[Kaina Stoicheia]]<br />
[[Logic]]<br />
[[Logic of information]]<br />
[[Logic of relatives]]<br />
[[Logic of Relatives (1870)]]<br />
[[Logic of Relatives (1883)]]<br />
[[Logical conjunction]]<br />
[[Logical disjunction]]<br />
[[Logical equality]]<br />
[[Logical graph]]<br />
[[Logical implication]]<br />
[[Logical matrix]]<br />
[[Logical NAND]]<br />
[[Logical negation]]<br />
[[Logical NNOR]]<br />
[[Minimal negation operator]]<br />
[[Multigrade operator]]<br />
[[Normative science]]<br />
[[Null graph]]<br />
[[On a New List of Categories]]<br />
[[Parametric operator]]<br />
[[Peirce's law]]<br />
[[Philosophy of mathematics]]<br />
[[Pragmatic information]]<br />
[[Pragmatic maxim]]<br />
[[Pragmatic theory of truth]]<br />
[[Pragmaticism]] <br />
[[Pragmatism]]<br />
[[Prescisive abstraction]]<br />
[[Propositional calculus]]<br />
[[Relation (mathematics)]]<br />
[[Relation composition]]<br />
[[Relation construction]]<br />
[[Relation reduction]]<br />
[[Relation theory]]<br />
[[Relation type]]<br />
[[Relative term]]<br />
[[Semeiotic]]<br />
[[Semiotic information]]<br />
[[Semiotics]]<br />
[[Sign relation]]<br />
[[Sign relational complex]]<br />
[[Sole sufficient operator]]<br />
[[Tacit extension]]<br />
[[The Simplest Mathematics]]<br />
[[Triadic relation]]<br />
[[Truth table]]<br />
[[Truth theory]]<br />
[[Universe of discourse]]<br />
[[What we've got here is (a) failure to communicate]]<br />
[[Zeroth order logic]]<br />
<br />
===Notes===<br />
<br />
* [[Directory:Jon Awbrey/Notes/Factorization Issues|Factorization Issues]]<br />
<br />
* [[Directory:Jon Awbrey/Notes/Factorization And Reification|Factorization And Reification]]<br />
<br />
===Papers===<br />
<br />
====Functional Logic====<br />
<br />
* [[Directory:Jon Awbrey/Papers/Functional Logic : Higher Order Propositions|Functional Logic : Higher Order Propositions]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Functional Logic : Inquiry and Analogy|Functional Logic : Inquiry and Analogy]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Functional Logic : Quantification Theory|Functional Logic : Quantification Theory]]<br />
<br />
====Differential Logic====<br />
<br />
* [[Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction|Differential Logic : Introduction]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Differential Propositional Calculus|Differential Propositional Calculus]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems|Differential Logic and Dynamic Systems 1.0]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0|Differential Logic and Dynamic Systems 2.0]]<br />
<br />
====Logic and Semiotics====<br />
<br />
* [[Directory:Jon Awbrey/Papers/Futures Of Logical Graphs|Futures Of Logical Graphs]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives|Peirce's 1870 Logic Of Relatives]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Peirce's Logic Of Information|Peirce's Logic Of Information]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Propositional Equation Reasoning Systems|Propositional Equation Reasoning Systems]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Semiotic Information|Semiotic Information]]<br />
<br />
====Inquiry Driven Systems====<br />
<br />
* [[Directory:Jon Awbrey/Essays/Prospects For Inquiry Driven Systems|Prospects for Inquiry Driven Systems]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Introduction to Inquiry Driven Systems|Introduction to Inquiry Driven Systems]]<br />
<br />
* [[Directory:Jon Awbrey/Essays/Inquiry Driven Systems : Fields Of Inquiry|Inquiry Driven Systems : Fields Of Inquiry]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems|Inquiry Driven Systems : Inquiry Into Inquiry]]<br />
<br />
===Projects===<br />
<br />
* [[Directory:Jon Awbrey/Projects/Cactus Language|Cactus Language]]<br />
<br />
* [[Directory:Jon Awbrey/Projects/Differential Logic|Differential Logic]]<br />
<br />
* [[Directory:Jon Awbrey/Projects/Inquiry|Inquiry]]<br />
** [[Directory:Jon Awbrey/Projects/Architecture For Inquiry|Architecture For Inquiry]]<br />
** [[Directory:Jon Awbrey/Projects/Inquiry Driven Systems|Inquiry Driven Systems]]<br />
<br />
* [[Directory:Jon Awbrey/Projects/Logic Of Information|Logic Of Information]]<br />
** [[Directory:Jon Awbrey/Projects/Pragmatic Theory Of Information|Pragmatic Theory Of Information]]<br />
** [[Directory:Jon Awbrey/Projects/Semiotic Theory Of Information|Semiotic Theory Of Information]]<br />
<br />
* [[Directory:Jon Awbrey/Projects/Notes And Queries|Notes And Queries]]<br />
<br />
* [[Directory:Jon Awbrey/Projects/Peircean Pragmata|Peircean Pragmata]]<br />
<br />
* [[Directory:Jon Awbrey/Projects/Theme One Program|Theme One Program]]<br />
<br />
* [[Directory:Jon Awbrey/Projects/Theory Of Relations|Theory Of Relations]]<br />
<br />
===Poetry===<br />
<br />
* [[Directory:Jon Awbrey/Poetry/Past All Reckoning|Past All Reckoning]]<br />
<br />
* [[Directory:Jon Awbrey/Poetry/Poems Of Emediate Moment|Poems Of Emediate Moment]]<br />
<br />
* [[Directory:Jon Awbrey/Poetry/Questionable Verses|Questionable Verses]]<br />
<br />
* [[Directory:Jon_Awbrey/Poetry/Iconoclast|Iconoclast]]<br />
<br />
===User Pages===<br />
<br />
* [[Directory:Jon Awbrey/EXCERPTS|Collection Of Source Materials]]<br />
* [[User:Jon Awbrey/Examples Of Inquiry|Examples Of Inquiry]]<br />
* [[User:Jon Awbrey/Mathematical Notes|Mathematical Notes]]<br />
* [[User:Jon Awbrey/Philosophical Notes|Philosophical Notes]]<br />
<br />
* [http://mywikibiz.com/index.php?title=Special%3APrefixIndex&prefix=Jon+Awbrey&namespace=2 MyWikiBiz User Pages]<br />
* [http://intersci.ss.uci.edu/wiki/index.php?title=Special%3APrefixIndex&prefix=Jon+Awbrey&namespace=2 InterSciWiki User Pages]<br />
* [http://mywikibiz.com/index.php?title=Special%3APrefixIndex&prefix=Jon+Awbrey&namespace=110 MyWikiBiz Directory Pages]<br />
<br />
==Presentations and Publications==<br />
<br />
* Awbrey, S.M., and Awbrey, J.L. (May 2001), &ldquo;Conceptual Barriers to Creating Integrative Universities&rdquo;, ''Organization : The Interdisciplinary Journal of Organization, Theory, and Society'' 8(2), Sage Publications, London, UK, pp. 269&ndash;284. [http://org.sagepub.com/cgi/content/abstract/8/2/269 Abstract].<br />
<br />
* Awbrey, S.M., and Awbrey, J.L. (September 1999), &ldquo;Organizations of Learning or Learning Organizations : The Challenge of Creating Integrative Universities for the Next Century&rdquo;, ''Second International Conference of the Journal &lsquo;Organization&rsquo;'', ''Re-Organizing Knowledge, Trans-Forming Institutions : Knowing, Knowledge, and the University in the 21st Century'', University of Massachusetts, Amherst, MA. [http://cspeirce.com/menu/library/aboutcsp/awbrey/integrat.htm Online].<br />
<br />
* Awbrey, J.L., and Awbrey, S.M. (Autumn 1995), &ldquo;Interpretation as Action : The Risk of Inquiry&rdquo;, ''Inquiry : Critical Thinking Across the Disciplines'' 15(1), pp. 40&ndash;52. [https://web.archive.org/web/20001210162300/http://chss.montclair.edu/inquiry/fall95/awbrey.html Archive]. [https://www.pdcnet.org/inquiryct/content/inquiryct_1995_0015_0001_0040_0052 Journal]. [https://independent.academia.edu/JonAwbrey/Papers/1302117/Interpretation_as_Action_The_Risk_of_Inquiry Online].<br />
<br />
* Awbrey, J.L., and Awbrey, S.M. (June 1992), &ldquo;Interpretation as Action : The Risk of Inquiry&rdquo;, ''The Eleventh International Human Science Research Conference'', Oakland University, Rochester, Michigan.<br />
<br />
* Awbrey, S.M., and Awbrey, J.L. (May 1991), &ldquo;An Architecture for Inquiry : Building Computer Platforms for Discovery&rdquo;, ''Proceedings of the Eighth International Conference on Technology and Education'', Toronto, Canada, pp. 874&ndash;875. [http://www.academia.edu/1270327/An_Architecture_for_Inquiry_Building_Computer_Platforms_for_Discovery Online].<br />
<br />
* Awbrey, J.L., and Awbrey, S.M. (January 1991), &ldquo;Exploring Research Data Interactively : Developing a Computer Architecture for Inquiry&rdquo;, Poster presented at the ''Annual Sigma Xi Research Forum'', University of Texas Medical Branch, Galveston, TX.<br />
<br />
* Awbrey, J.L., and Awbrey, S.M. (August 1990), &ldquo;Exploring Research Data Interactively. Theme One : A Program of Inquiry&rdquo;, ''Proceedings of the Sixth Annual Conference on Applications of Artificial Intelligence and CD-ROM in Education and Training'', Society for Applied Learning Technology, Washington, DC, pp. 9&ndash;15. [http://academia.edu/1272839/Exploring_Research_Data_Interactively._Theme_One_A_Program_of_Inquiry Online].<br />
<br />
==Education==<br />
<br />
* 1993&ndash;2003. [http://web.archive.org/web/20120202222443/http://www2.oakland.edu/secs/dispprofile.asp?Fname=Fatma&Lname=Mili Graduate Study], [http://web.archive.org/web/20120203004703/http://www2.oakland.edu/secs/dispprofile.asp?Fname=Mohamed&Lname=Zohdy Systems Engineering], [http://meadowbrookhall.org/explore/history/meadowbrookhall Oakland University].<br />
<br />
* '''1989. [http://web.archive.org/web/20001206101800/http://www.msu.edu/dig/msumap/psychology.html M.A. Psychology]''', [http://web.archive.org/web/20000902103838/http://www.msu.edu/dig/msumap/beaumont.html Michigan State University].<br />
<br />
* 1985&ndash;1986. [http://quod.lib.umich.edu/cgi/i/image/image-idx?id=S-BHL-X-BL001808%5DBL001808 Graduate Study, Mathematics], [http://www.umich.edu/ University of Michigan].<br />
<br />
* 1985. [http://web.archive.org/web/20061230174652/http://www.uiuc.edu/navigation/buildings/altgeld.top.html Graduate Study, Mathematics], [http://www.uiuc.edu/ University of Illinois at Urbana&ndash;Champaign].<br />
<br />
* 1984. [http://www.psych.uiuc.edu/graduate/ Graduate Study, Psychology], [http://www.uiuc.edu/ University of Illinois at Champaign&ndash;Urbana].<br />
<br />
* '''1980. [http://www.mth.msu.edu/images/wells_medium.jpg M.A. Mathematics]''', [http://www.msu.edu/~hvac/survey/BeaumontTower.html Michigan State University].<br />
<br />
* '''1976. [http://web.archive.org/web/20001206050600/http://www.msu.edu/dig/msumap/phillips.html B.A. Mathematical and Philosophical Method]''', <br> [http://www.enolagaia.com/JMC.html Justin Morrill College], [http://www.msu.edu/ Michigan State University].<br />
<br />
==Category and Subject Interests==<br />
<br />
[[Category:Artificial Intelligence]]<br />
[[Category:Automata Theory]]<br />
[[Category:Category Theory]]<br />
[[Category:Charles Sanders Peirce]]<br />
[[Category:Cognitive Science]] <br />
[[Category:Combinatorics]]<br />
[[Category:Computer Science]]<br />
[[Category:Critical Thinking]]<br />
[[Category:Cybernetics]]<br />
[[Category:Differential Logic]]<br />
[[Category:Education]]<br />
[[Category:Formal Languages]]<br />
[[Category:Formal Sciences]]<br />
[[Category:Graph Theory]]<br />
[[Category:Group Theory]]<br />
[[Category:Hermeneutics]]<br />
[[Category:Information Systems]]<br />
[[Category:Information Theory]]<br />
[[Category:Inquiry]]<br />
[[Category:Inquiry Driven Systems]]<br />
[[Category:Integer Sequences]]<br />
[[Category:Intelligence Amplification]]<br />
[[Category:Learning Organizations]]<br />
[[Category:Linguistics]]<br />
[[Category:Knowledge Representation]]<br />
[[Category:Logic]]<br />
[[Category:Mathematics]]<br />
[[Category:Natural Languages]]<br />
[[Category:Philosophy]]<br />
[[Category:Pragmatics]]<br />
[[Category:Psychology]]<br />
[[Category:Science]]<br />
[[Category:Semantics]]<br />
[[Category:Semiotics]]<br />
[[Category:Statistics]]<br />
[[Category:Systems Science]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Descriptive_science&diff=469857Descriptive science2020-11-04T15:30:05Z<p>Jon Awbrey: /* Document history */ waybak links</p>
<hr />
<div><font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].<br />
<br />
A '''descriptive science''', also called a '''special science''', is a form of [[inquiry]], typically involving a community of inquiry and its accumulated body of provisional knowledge, that seeks to discover what is true about a recognized domain of phenomena.<br />
<br />
==Syllabus==<br />
<br />
===Focal nodes===<br />
<br />
* [[Inquiry Live]]<br />
* [[Logic Live]]<br />
<br />
===Peer nodes===<br />
<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Descriptive_science Descriptive Science @ InterSciWiki]<br />
* [http://mywikibiz.com/Descriptive_science Descriptive Science @ MyWikiBiz]<br />
* [http://ref.subwiki.org/wiki/Descriptive_science Descriptive Science @ Subject Wikis]<br />
* [http://en.wikiversity.org/wiki/Descriptive_science Descriptive Science @ Wikiversity]<br />
* [http://beta.wikiversity.org/wiki/Descriptive_science Descriptive Science @ Wikiversity Beta]<br />
<br />
===Logical operators===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [[Exclusive disjunction]]<br />
* [[Logical conjunction]]<br />
* [[Logical disjunction]]<br />
* [[Logical equality]]<br />
{{col-break}}<br />
* [[Logical implication]]<br />
* [[Logical NAND]]<br />
* [[Logical NNOR]]<br />
* [[Logical negation|Negation]]<br />
{{col-end}}<br />
<br />
===Related topics===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [[Ampheck]]<br />
* [[Boolean domain]]<br />
* [[Boolean function]]<br />
* [[Boolean-valued function]]<br />
* [[Differential logic]]<br />
{{col-break}}<br />
* [[Logical graph]]<br />
* [[Minimal negation operator]]<br />
* [[Multigrade operator]]<br />
* [[Parametric operator]]<br />
* [[Peirce's law]]<br />
{{col-break}}<br />
* [[Propositional calculus]]<br />
* [[Sole sufficient operator]]<br />
* [[Truth table]]<br />
* [[Universe of discourse]]<br />
* [[Zeroth order logic]]<br />
{{col-end}}<br />
<br />
===Relational concepts===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [[Continuous predicate]]<br />
* [[Hypostatic abstraction]]<br />
* [[Logic of relatives]]<br />
* [[Logical matrix]]<br />
{{col-break}}<br />
* [[Relation (mathematics)|Relation]]<br />
* [[Relation composition]]<br />
* [[Relation construction]]<br />
* [[Relation reduction]]<br />
{{col-break}}<br />
* [[Relation theory]]<br />
* [[Relative term]]<br />
* [[Sign relation]]<br />
* [[Triadic relation]]<br />
{{col-end}}<br />
<br />
===Information, Inquiry===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [[Inquiry]]<br />
* [[Dynamics of inquiry]]<br />
{{col-break}}<br />
* [[Semeiotic]]<br />
* [[Logic of information]]<br />
{{col-break}}<br />
* [[Descriptive science]]<br />
* [[Normative science]]<br />
{{col-break}}<br />
* [[Pragmatic maxim]]<br />
* [[Truth theory]]<br />
{{col-end}}<br />
<br />
===Related articles===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Semiotic_Information Semiotic Information]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]<br />
{{col-break}}<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]<br />
{{col-break}}<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]<br />
{{col-end}}<br />
<br />
==Document history==<br />
<br />
Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.<br />
<br />
* [https://web.archive.org/web/20191209233842/http://intersci.ss.uci.edu/wiki/index.php/Descriptive_science Descriptive Science], [https://web.archive.org/web/20191124081501/http://intersci.ss.uci.edu/wiki/index.php/User:Jon_Awbrey InterSciWiki]<br />
* [http://mywikibiz.com/Descriptive_science Descriptive Science], [http://mywikibiz.com/ MyWikiBiz]<br />
* [http://semanticweb.org/wiki/Descriptive_science Descriptive Science], [http://semanticweb.org/ Semantic Web]<br />
* [http://wikinfo.org/w/index.php/Descriptive_science Descriptive Science], [http://wikinfo.org/w/ Wikinfo]<br />
* [http://en.wikiversity.org/wiki/Descriptive_science Descriptive Science], [http://en.wikiversity.org/ Wikiversity]<br />
* [http://beta.wikiversity.org/wiki/Descriptive_science Descriptive Science], [http://beta.wikiversity.org/ Wikiversity Beta]<br />
* [http://en.wikipedia.org/w/index.php?title=Descriptive_science&oldid=51990248 Descriptive Science], [http://en.wikipedia.org/ Wikipedia]<br />
<br />
[[Category:Descriptive Sciences]]<br />
[[Category:Inquiry]]<br />
[[Category:Philosophy]]<br />
[[Category:Philosophy of Science]]<br />
[[Category:Science]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Logic_Live&diff=469845Logic Live2020-09-17T19:32:10Z<p>Jon Awbrey: /* Blog */ + [https://inquiryintoinquiry.com/2012/05/17/inquiry-live-and-logic-live/ Inquiry Live &amp; Logic Live]</p>
<hr />
<div><font size="3">&#9758;</font> This page serves as a '''focal node''' for a collection of related resources.<br />
<br />
==Blog==<br />
<br />
* [https://inquiryintoinquiry.com/ Inquiry Into Inquiry]<br />
* [https://inquiryintoinquiry.com/2012/05/17/inquiry-live-and-logic-live/ Inquiry Live &amp; Logic Live]<br />
<br />
==Syllabus==<br />
<br />
===Focal nodes===<br />
<br />
* [[Inquiry Live]]<br />
* [[Logic Live]]<br />
<br />
===Peer nodes===<br />
<br />
* [http://mywikibiz.com/Logic_Live Logic Live @ MyWikiBiz]<br />
* [http://ref.subwiki.org/wiki/Logic_Live Logic Live @ Subject Wikis]<br />
* [http://en.wikiversity.org/wiki/Logic_Live Logic Live @ Wikiversity]<br />
* [http://beta.wikiversity.org/wiki/Logic_Live Logic Live @ Wikiversity Beta]<br />
<br />
===Logical operators===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [[Exclusive disjunction]]<br />
* [[Logical conjunction]]<br />
* [[Logical disjunction]]<br />
* [[Logical equality]]<br />
{{col-break}}<br />
* [[Logical implication]]<br />
* [[Logical NAND]]<br />
* [[Logical NNOR]]<br />
* [[Logical negation|Negation]]<br />
{{col-end}}<br />
<br />
===Related topics===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [[Ampheck]]<br />
* [[Boolean domain]]<br />
* [[Boolean function]]<br />
* [[Boolean-valued function]]<br />
* [[Differential logic]]<br />
{{col-break}}<br />
* [[Logical graph]]<br />
* [[Minimal negation operator]]<br />
* [[Multigrade operator]]<br />
* [[Parametric operator]]<br />
* [[Peirce's law]]<br />
{{col-break}}<br />
* [[Propositional calculus]]<br />
* [[Sole sufficient operator]]<br />
* [[Truth table]]<br />
* [[Universe of discourse]]<br />
* [[Zeroth order logic]]<br />
{{col-end}}<br />
<br />
===Relational concepts===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [[Continuous predicate]]<br />
* [[Hypostatic abstraction]]<br />
* [[Logic of relatives]]<br />
* [[Logical matrix]]<br />
{{col-break}}<br />
* [[Relation (mathematics)|Relation]]<br />
* [[Relation composition]]<br />
* [[Relation construction]]<br />
* [[Relation reduction]]<br />
{{col-break}}<br />
* [[Relation theory]]<br />
* [[Relative term]]<br />
* [[Sign relation]]<br />
* [[Triadic relation]]<br />
{{col-end}}<br />
<br />
===Information, Inquiry===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [[Inquiry]]<br />
* [[Dynamics of inquiry]]<br />
{{col-break}}<br />
* [[Semeiotic]]<br />
* [[Logic of information]]<br />
{{col-break}}<br />
* [[Descriptive science]]<br />
* [[Normative science]]<br />
{{col-break}}<br />
* [[Pragmatic maxim]]<br />
* [[Truth theory]]<br />
{{col-end}}<br />
<br />
===Related articles===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]<br />
{{col-break}}<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]<br />
{{col-break}}<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]<br />
{{col-end}}<br />
<br />
==Archive links==<br />
<br />
* [http://web.archive.org/web/20191209181958/http://intersci.ss.uci.edu/wiki/index.php/Logic_Live Logic Live @ InterSciWiki]<br />
<br />
==Participants==<br />
<br />
* Interested parties may add their names on [[Logic Live/Participants|this page]].<br />
<br />
[[Category:Inquiry]]<br />
[[Category:Logic]]<br />
[[Category:Open Educational Resource]]<br />
[[Category:Peer Educational Resource]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Inquiry_Live&diff=469844Inquiry Live2020-09-17T19:30:13Z<p>Jon Awbrey: /* Blog */</p>
<hr />
<div><font size="3">&#9758;</font> This page serves as a '''focal node''' for a collection of related resources.<br />
<br />
==Blog==<br />
<br />
* [https://inquiryintoinquiry.com/ Inquiry Into Inquiry]<br />
* [https://inquiryintoinquiry.com/2012/05/17/inquiry-live-and-logic-live/ Inquiry Live &amp; Logic Live]<br />
<br />
==Focal nodes==<br />
<br />
'''Focal nodes''' serve as hubs for collections of related resources, in particular, activity sites and article contents.<br />
<br />
* [[Inquiry Live]]<br />
* [[Logic Live]]<br />
<br />
==Peer nodes==<br />
<br />
'''Peer nodes''' are roughly parallel pages on different sites but are not necessarily identical in content &mdash; especially as they develop in time across different environments through interaction with diverse populations &mdash; but they should preserve enough information to reproduce each other, more or less, in case of damage or loss.<br />
<br />
* [http://mywikibiz.com/Inquiry_Live Inquiry Live @ MyWikiBiz]<br />
* [http://ref.subwiki.org/wiki/Inquiry_Live Inquiry Live @ Subject Wikis]<br />
* [http://en.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity]<br />
* [http://beta.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity Beta]<br />
<br />
==Archive links==<br />
<br />
* [http://web.archive.org/web/20191024043046/http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Live Inquiry Live @ InterSciWiki]<br />
<br />
==Participants==<br />
<br />
Interested parties may add their names on [[Inquiry Live/Participants|this page]].<br />
<br />
[[Category:Inquiry]]<br />
[[Category:Logic]]<br />
[[Category:Open Educational Resource]]<br />
[[Category:Peer Educational Resource]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Inquiry_Live&diff=469843Inquiry Live2020-09-17T19:28:33Z<p>Jon Awbrey: /* Blog */ + [https://inquiryintoinquiry.com/2012/05/17/inquiry-live-and-logic-live/ Inquiry Live & Logic Live]</p>
<hr />
<div><font size="3">&#9758;</font> This page serves as a '''focal node''' for a collection of related resources.<br />
<br />
==Blog==<br />
<br />
* [https://inquiryintoinquiry.com/ Inquiry Into Inquiry]<br />
* [https://inquiryintoinquiry.com/2012/05/17/inquiry-live-and-logic-live/ Inquiry Live & Logic Live]<br />
<br />
==Focal nodes==<br />
<br />
'''Focal nodes''' serve as hubs for collections of related resources, in particular, activity sites and article contents.<br />
<br />
* [[Inquiry Live]]<br />
* [[Logic Live]]<br />
<br />
==Peer nodes==<br />
<br />
'''Peer nodes''' are roughly parallel pages on different sites but are not necessarily identical in content &mdash; especially as they develop in time across different environments through interaction with diverse populations &mdash; but they should preserve enough information to reproduce each other, more or less, in case of damage or loss.<br />
<br />
* [http://mywikibiz.com/Inquiry_Live Inquiry Live @ MyWikiBiz]<br />
* [http://ref.subwiki.org/wiki/Inquiry_Live Inquiry Live @ Subject Wikis]<br />
* [http://en.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity]<br />
* [http://beta.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity Beta]<br />
<br />
==Archive links==<br />
<br />
* [http://web.archive.org/web/20191024043046/http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Live Inquiry Live @ InterSciWiki]<br />
<br />
==Participants==<br />
<br />
Interested parties may add their names on [[Inquiry Live/Participants|this page]].<br />
<br />
[[Category:Inquiry]]<br />
[[Category:Logic]]<br />
[[Category:Open Educational Resource]]<br />
[[Category:Peer Educational Resource]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Logic_Live&diff=469842Logic Live2020-09-17T16:50:20Z<p>Jon Awbrey: </p>
<hr />
<div><font size="3">&#9758;</font> This page serves as a '''focal node''' for a collection of related resources.<br />
<br />
==Blog==<br />
<br />
* [https://inquiryintoinquiry.com/ Inquiry Into Inquiry]<br />
* [https://inquiryintoinquiry.com/2012/05/17/inquiry-live-and-logic-live/ Inquiry Live and Logic Live]<br />
<br />
==Syllabus==<br />
<br />
===Focal nodes===<br />
<br />
* [[Inquiry Live]]<br />
* [[Logic Live]]<br />
<br />
===Peer nodes===<br />
<br />
* [http://mywikibiz.com/Logic_Live Logic Live @ MyWikiBiz]<br />
* [http://ref.subwiki.org/wiki/Logic_Live Logic Live @ Subject Wikis]<br />
* [http://en.wikiversity.org/wiki/Logic_Live Logic Live @ Wikiversity]<br />
* [http://beta.wikiversity.org/wiki/Logic_Live Logic Live @ Wikiversity Beta]<br />
<br />
===Logical operators===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [[Exclusive disjunction]]<br />
* [[Logical conjunction]]<br />
* [[Logical disjunction]]<br />
* [[Logical equality]]<br />
{{col-break}}<br />
* [[Logical implication]]<br />
* [[Logical NAND]]<br />
* [[Logical NNOR]]<br />
* [[Logical negation|Negation]]<br />
{{col-end}}<br />
<br />
===Related topics===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [[Ampheck]]<br />
* [[Boolean domain]]<br />
* [[Boolean function]]<br />
* [[Boolean-valued function]]<br />
* [[Differential logic]]<br />
{{col-break}}<br />
* [[Logical graph]]<br />
* [[Minimal negation operator]]<br />
* [[Multigrade operator]]<br />
* [[Parametric operator]]<br />
* [[Peirce's law]]<br />
{{col-break}}<br />
* [[Propositional calculus]]<br />
* [[Sole sufficient operator]]<br />
* [[Truth table]]<br />
* [[Universe of discourse]]<br />
* [[Zeroth order logic]]<br />
{{col-end}}<br />
<br />
===Relational concepts===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [[Continuous predicate]]<br />
* [[Hypostatic abstraction]]<br />
* [[Logic of relatives]]<br />
* [[Logical matrix]]<br />
{{col-break}}<br />
* [[Relation (mathematics)|Relation]]<br />
* [[Relation composition]]<br />
* [[Relation construction]]<br />
* [[Relation reduction]]<br />
{{col-break}}<br />
* [[Relation theory]]<br />
* [[Relative term]]<br />
* [[Sign relation]]<br />
* [[Triadic relation]]<br />
{{col-end}}<br />
<br />
===Information, Inquiry===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [[Inquiry]]<br />
* [[Dynamics of inquiry]]<br />
{{col-break}}<br />
* [[Semeiotic]]<br />
* [[Logic of information]]<br />
{{col-break}}<br />
* [[Descriptive science]]<br />
* [[Normative science]]<br />
{{col-break}}<br />
* [[Pragmatic maxim]]<br />
* [[Truth theory]]<br />
{{col-end}}<br />
<br />
===Related articles===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]<br />
{{col-break}}<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]<br />
{{col-break}}<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]<br />
{{col-end}}<br />
<br />
==Archive links==<br />
<br />
* [http://web.archive.org/web/20191209181958/http://intersci.ss.uci.edu/wiki/index.php/Logic_Live Logic Live @ InterSciWiki]<br />
<br />
==Participants==<br />
<br />
* Interested parties may add their names on [[Logic Live/Participants|this page]].<br />
<br />
[[Category:Inquiry]]<br />
[[Category:Logic]]<br />
[[Category:Open Educational Resource]]<br />
[[Category:Peer Educational Resource]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Inquiry_Live&diff=469841Inquiry Live2020-09-17T16:50:10Z<p>Jon Awbrey: </p>
<hr />
<div><font size="3">&#9758;</font> This page serves as a '''focal node''' for a collection of related resources.<br />
<br />
==Blog==<br />
<br />
* [https://inquiryintoinquiry.com/ Inquiry Into Inquiry]<br />
* [https://inquiryintoinquiry.com/2012/05/17/inquiry-live-and-logic-live/ Inquiry Live and Logic Live]<br />
<br />
==Focal nodes==<br />
<br />
'''Focal nodes''' serve as hubs for collections of related resources, in particular, activity sites and article contents.<br />
<br />
* [[Inquiry Live]]<br />
* [[Logic Live]]<br />
<br />
==Peer nodes==<br />
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'''Peer nodes''' are roughly parallel pages on different sites but are not necessarily identical in content &mdash; especially as they develop in time across different environments through interaction with diverse populations &mdash; but they should preserve enough information to reproduce each other, more or less, in case of damage or loss.<br />
<br />
* [http://mywikibiz.com/Inquiry_Live Inquiry Live @ MyWikiBiz]<br />
* [http://ref.subwiki.org/wiki/Inquiry_Live Inquiry Live @ Subject Wikis]<br />
* [http://en.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity]<br />
* [http://beta.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity Beta]<br />
<br />
==Archive links==<br />
<br />
* [http://web.archive.org/web/20191024043046/http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Live Inquiry Live @ InterSciWiki]<br />
<br />
==Participants==<br />
<br />
Interested parties may add their names on [[Inquiry Live/Participants|this page]].<br />
<br />
[[Category:Inquiry]]<br />
[[Category:Logic]]<br />
[[Category:Open Educational Resource]]<br />
[[Category:Peer Educational Resource]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Inquiry_Live&diff=469840Inquiry Live2020-09-17T16:42:12Z<p>Jon Awbrey: de-cap headings</p>
<hr />
<div><font size="3">&#9758;</font> This page serves as a '''focal node''' for a collection of related resources.<br />
<br />
==Blog==<br />
<br />
* [https://inquiryintoinquiry.com/ Inquiry Into Inquiry]<br />
* [https://inquiryintoinquiry.com/2012/05/17/inquiry-live-and-logic-live/ Inquiry Live and Logic Live]<br />
<br />
==Focal nodes==<br />
<br />
'''Focal nodes''' serve as hubs for collections of related resources, in particular, activity sites and article contents.<br />
<br />
* [[Inquiry Live]]<br />
* [[Logic Live]]<br />
<br />
==Peer nodes==<br />
<br />
'''Peer nodes''' are roughly parallel pages on different sites but are not necessarily identical in content &mdash; especially as they develop in time across different environments through interaction with diverse populations &mdash; but they should preserve enough information to reproduce each other, more or less, in case of damage or loss.<br />
<br />
* [http://mywikibiz.com/Inquiry_Live Inquiry Live @ MyWikiBiz]<br />
* [http://ref.subwiki.org/wiki/Inquiry_Live Inquiry Live @ Subject Wikis]<br />
* [http://en.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity]<br />
* [http://beta.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity Beta]<br />
<br />
==Web archive==<br />
<br />
* [http://web.archive.org/web/20191024043046/http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Live Inquiry Live @ InterSciWiki]<br />
<br />
==Participants==<br />
<br />
Interested parties may add their names on [[Inquiry Live/Participants|this page]].<br />
<br />
[[Category:Inquiry]]<br />
[[Category:Logic]]<br />
[[Category:Open Educational Resource]]<br />
[[Category:Peer Educational Resource]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Logic_Live&diff=469839Logic Live2020-09-17T16:40:12Z<p>Jon Awbrey: re-organize</p>
<hr />
<div><font size="3">&#9758;</font> This page serves as a '''focal node''' for a collection of related resources.<br />
<br />
==Blog==<br />
<br />
* [https://inquiryintoinquiry.com/ Inquiry Into Inquiry]<br />
* [https://inquiryintoinquiry.com/2012/05/17/inquiry-live-and-logic-live/ Inquiry Live and Logic Live]<br />
<br />
==Syllabus==<br />
<br />
===Focal nodes===<br />
<br />
* [[Inquiry Live]]<br />
* [[Logic Live]]<br />
<br />
===Peer nodes===<br />
<br />
* [http://mywikibiz.com/Logic_Live Logic Live @ MyWikiBiz]<br />
* [http://ref.subwiki.org/wiki/Logic_Live Logic Live @ Subject Wikis]<br />
* [http://en.wikiversity.org/wiki/Logic_Live Logic Live @ Wikiversity]<br />
* [http://beta.wikiversity.org/wiki/Logic_Live Logic Live @ Wikiversity Beta]<br />
<br />
===Logical operators===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [[Exclusive disjunction]]<br />
* [[Logical conjunction]]<br />
* [[Logical disjunction]]<br />
* [[Logical equality]]<br />
{{col-break}}<br />
* [[Logical implication]]<br />
* [[Logical NAND]]<br />
* [[Logical NNOR]]<br />
* [[Logical negation|Negation]]<br />
{{col-end}}<br />
<br />
===Related topics===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [[Ampheck]]<br />
* [[Boolean domain]]<br />
* [[Boolean function]]<br />
* [[Boolean-valued function]]<br />
* [[Differential logic]]<br />
{{col-break}}<br />
* [[Logical graph]]<br />
* [[Minimal negation operator]]<br />
* [[Multigrade operator]]<br />
* [[Parametric operator]]<br />
* [[Peirce's law]]<br />
{{col-break}}<br />
* [[Propositional calculus]]<br />
* [[Sole sufficient operator]]<br />
* [[Truth table]]<br />
* [[Universe of discourse]]<br />
* [[Zeroth order logic]]<br />
{{col-end}}<br />
<br />
===Relational concepts===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [[Continuous predicate]]<br />
* [[Hypostatic abstraction]]<br />
* [[Logic of relatives]]<br />
* [[Logical matrix]]<br />
{{col-break}}<br />
* [[Relation (mathematics)|Relation]]<br />
* [[Relation composition]]<br />
* [[Relation construction]]<br />
* [[Relation reduction]]<br />
{{col-break}}<br />
* [[Relation theory]]<br />
* [[Relative term]]<br />
* [[Sign relation]]<br />
* [[Triadic relation]]<br />
{{col-end}}<br />
<br />
===Information, Inquiry===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [[Inquiry]]<br />
* [[Dynamics of inquiry]]<br />
{{col-break}}<br />
* [[Semeiotic]]<br />
* [[Logic of information]]<br />
{{col-break}}<br />
* [[Descriptive science]]<br />
* [[Normative science]]<br />
{{col-break}}<br />
* [[Pragmatic maxim]]<br />
* [[Truth theory]]<br />
{{col-end}}<br />
<br />
===Related articles===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]<br />
{{col-break}}<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]<br />
{{col-break}}<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]<br />
{{col-end}}<br />
<br />
==Web archive==<br />
<br />
* [http://web.archive.org/web/20191209181958/http://intersci.ss.uci.edu/wiki/index.php/Logic_Live Logic Live @ InterSciWiki]<br />
<br />
<br />
==Participants==<br />
<br />
* Interested parties may add their names on [[Logic Live/Participants|this page]].<br />
<br />
[[Category:Inquiry]]<br />
[[Category:Logic]]<br />
[[Category:Open Educational Resource]]<br />
[[Category:Peer Educational Resource]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Inquiry_Live&diff=469838Inquiry Live2020-09-17T16:34:59Z<p>Jon Awbrey: re-organize</p>
<hr />
<div><font size="3">&#9758;</font> This page serves as a '''focal node''' for a collection of related resources.<br />
<br />
==Blog==<br />
<br />
* [https://inquiryintoinquiry.com/ Inquiry Into Inquiry]<br />
* [https://inquiryintoinquiry.com/2012/05/17/inquiry-live-and-logic-live/ Inquiry Live and Logic Live]<br />
<br />
==Focal Nodes==<br />
<br />
'''Focal nodes''' serve as hubs for collections of related resources, in particular, activity sites and article contents.<br />
<br />
* [[Inquiry Live]]<br />
* [[Logic Live]]<br />
<br />
==Peer Nodes==<br />
<br />
'''Peer nodes''' are roughly parallel pages on different sites but are not necessarily identical in content &mdash; especially as they develop in time across different environments through interaction with diverse populations &mdash; but they should preserve enough information to reproduce each other, more or less, in case of damage or loss.<br />
<br />
* [http://mywikibiz.com/Inquiry_Live Inquiry Live @ MyWikiBiz]<br />
* [http://ref.subwiki.org/wiki/Inquiry_Live Inquiry Live @ Subject Wikis]<br />
* [http://en.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity]<br />
* [http://beta.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity Beta]<br />
<br />
==Web Archive==<br />
<br />
* [http://web.archive.org/web/20191024043046/http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Live Inquiry Live @ InterSciWiki]<br />
<br />
==Participants==<br />
<br />
Interested parties may add their names on [[Inquiry Live/Participants|this page]].<br />
<br />
[[Category:Inquiry]]<br />
[[Category:Logic]]<br />
[[Category:Open Educational Resource]]<br />
[[Category:Peer Educational Resource]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Inquiry_Live&diff=469837Inquiry Live2020-09-17T16:20:08Z<p>Jon Awbrey: /* Organization */</p>
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<br />
==Participants==<br />
<br />
Interested parties may add their names on [[Inquiry Live/Participants|this page]].<br />
<br />
==Organization==<br />
<br />
===Blog===<br />
<br />
* [https://inquiryintoinquiry.com/ Inquiry Into Inquiry]<br />
* [https://inquiryintoinquiry.com/2012/05/17/inquiry-live-and-logic-live/ Inquiry Live and Logic Live]<br />
<br />
===Focal Nodes===<br />
<br />
'''Focal nodes''' serve as hubs for collections of related resources, in particular, activity sites and article contents.<br />
<br />
* [[Inquiry Live]]<br />
* [[Logic Live]]<br />
<br />
===Peer Nodes===<br />
<br />
'''Peer nodes''' are roughly parallel pages on different sites but are not necessarily identical in content &mdash; especially as they develop in time across different environments through interaction with diverse populations &mdash; but they should preserve enough information to reproduce each other, more or less, in case of damage or loss.<br />
<br />
* [http://mywikibiz.com/Inquiry_Live Inquiry Live @ MyWikiBiz]<br />
* [http://ref.subwiki.org/wiki/Inquiry_Live Inquiry Live @ Subject Wikis]<br />
* [http://en.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity]<br />
* [http://beta.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity Beta]<br />
<br />
===Archive===<br />
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* [http://web.archive.org/web/20191024043046/http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Live Inquiry Live @ InterSciWiki]<br />
<br />
[[Category:Inquiry]]<br />
[[Category:Logic]]<br />
[[Category:Open Educational Resource]]<br />
[[Category:Peer Educational Resource]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Inquiry_Live&diff=469836Inquiry Live2020-09-17T15:48:05Z<p>Jon Awbrey: /* Organization */</p>
<hr />
<div><font size="3">&#9758;</font> This page serves as a '''focal node''' for a collection of related resources.<br />
<br />
==Participants==<br />
<br />
Interested parties may add their names on [[Inquiry Live/Participants|this page]].<br />
<br />
==Organization==<br />
<br />
===Focal nodes===<br />
<br />
'''Focal nodes''' serve as hubs for collections of related resources, in particular, activity sites and article contents.<br />
<br />
* [[Inquiry Live]]<br />
* [[Logic Live]]<br />
<br />
===Peer nodes===<br />
<br />
'''Peer nodes''' are roughly parallel pages on different sites but are not necessarily identical in content &mdash; especially as they develop in time across different environments through interaction with diverse populations &mdash; but they should preserve enough information to reproduce each other, more or less, in case of damage or loss.<br />
<br />
* [http://mywikibiz.com/Inquiry_Live Inquiry Live @ MyWikiBiz]<br />
* [http://ref.subwiki.org/wiki/Inquiry_Live Inquiry Live @ Subject Wikis]<br />
* [http://en.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity]<br />
* [http://beta.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity Beta]<br />
<br />
===Archive===<br />
<br />
* [http://web.archive.org/web/20191024043046/http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Live Inquiry Live @ InterSciWiki]<br />
<br />
[[Category:Inquiry]]<br />
[[Category:Logic]]<br />
[[Category:Open Educational Resource]]<br />
[[Category:Peer Educational Resource]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Inquiry_Live&diff=469835Inquiry Live2020-09-17T15:40:03Z<p>Jon Awbrey: /* Focal nodes */</p>
<hr />
<div><font size="3">&#9758;</font> This page serves as a '''focal node''' for a collection of related resources.<br />
<br />
==Participants==<br />
<br />
Interested parties may add their names on [[Inquiry Live/Participants|this page]].<br />
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<br />
===Focal nodes===<br />
<br />
'''Focal nodes''' serve as hubs for collections of related resources, in particular, activity sites and article contents.<br />
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* [[Inquiry Live]]<br />
* [[Logic Live]]<br />
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===Peer nodes===<br />
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'''Peer nodes''' are roughly parallel pages on different sites but are not necessarily identical in content &mdash; especially as they develop in time across different environments through interaction with diverse populations &mdash; but they should preserve enough information to reproduce each other, more or less, in case of damage or loss.<br />
<br />
* [http://mywikibiz.com/Inquiry_Live Inquiry Live @ MyWikiBiz]<br />
* [http://ref.subwiki.org/wiki/Inquiry_Live Inquiry Live @ Subject Wikis]<br />
* [http://en.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity]<br />
* [http://beta.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity Beta]<br />
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===Archive===<br />
<br />
* [http://web.archive.org/web/20191024043046/http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Live Inquiry Live @ InterSciWiki]<br />
<br />
[[Category:Inquiry]]<br />
[[Category:Logic]]<br />
[[Category:Open Educational Resource]]<br />
[[Category:Peer Educational Resource]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Inquiry_Live&diff=469834Inquiry Live2020-09-17T15:38:04Z<p>Jon Awbrey: /* Peer nodes */</p>
<hr />
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<br />
==Participants==<br />
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Interested parties may add their names on [[Inquiry Live/Participants|this page]].<br />
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==Rudiments of organization==<br />
<br />
===Focal nodes===<br />
<br />
* [[Inquiry Live]]<br />
* [[Logic Live]]<br />
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===Peer nodes===<br />
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'''Peer nodes''' are roughly parallel pages on different sites but are not necessarily identical in content &mdash; especially as they develop in time across different environments through interaction with diverse populations &mdash; but they should preserve enough information to reproduce each other, more or less, in case of damage or loss.<br />
<br />
* [http://mywikibiz.com/Inquiry_Live Inquiry Live @ MyWikiBiz]<br />
* [http://ref.subwiki.org/wiki/Inquiry_Live Inquiry Live @ Subject Wikis]<br />
* [http://en.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity]<br />
* [http://beta.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity Beta]<br />
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===Archive===<br />
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* [http://web.archive.org/web/20191024043046/http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Live Inquiry Live @ InterSciWiki]<br />
<br />
[[Category:Inquiry]]<br />
[[Category:Logic]]<br />
[[Category:Open Educational Resource]]<br />
[[Category:Peer Educational Resource]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=User:Jon_Awbrey&diff=469833User:Jon Awbrey2020-09-17T15:32:11Z<p>Jon Awbrey: /* Recent Sightings */ update</p>
<hr />
<div><br><br />
<p><font face="lucida calligraphy" size="7">Jon Awbrey</font></p><br />
<br><br />
__NOTOC__<br />
==Presently &hellip;==<br />
<br />
<center><br />
<br />
[https://oeis.org/wiki/Riffs_and_Rotes Riffs and Rotes]<br />
<br />
[http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]<br />
<br />
[http://intersci.ss.uci.edu/wiki/index.php/Theme_One_Program Theme One Program]<br />
<br />
[http://intersci.ss.uci.edu/wiki/index.php/Propositions_As_Types Propositions As Types]<br />
<br />
[http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems]<br />
<br />
[http://intersci.ss.uci.edu/wiki/index.php/Peirce%27s_1870_Logic_Of_Relatives Peirce's Logic Of Relatives]<br />
<br />
Logical Graphs : [http://inquiryintoinquiry.com/2008/07/29/logical-graphs-1/ One] [http://inquiryintoinquiry.com/2008/09/19/logical-graphs-2/ Two]<br />
<br />
[http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]<br />
<br />
[http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]<br />
<br />
[http://intersci.ss.uci.edu/wiki/index.php/Differential_Analytic_Turing_Automata Differential Analytic Turing Automata]<br />
<br />
[http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]<br />
<br />
[http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]<br />
<br />
[http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]<br />
<br />
[http://intersci.ss.uci.edu/wiki/index.php/Information_%3D_Comprehension_%C3%97_Extension Information = Comprehension &times; Extension]<br />
<br />
[http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]<br />
<br />
</center><br />
<br />
==Recent Sightings==<br />
<br />
{| align="center" style="text-align:center" width="100%"<br />
|-<br />
| [http://mywikibiz.com/Inquiry_Live Inquiry Live]<br />
| [http://mywikibiz.com/Logic_Live Logic Live]<br />
|-<br />
| [http://stderr.org/cgi-bin/mailman/listinfo/inquiry Inquiry Project]<br />
| [http://stderr.org/pipermail/inquiry/ Inquiry Archive]<br />
|-<br />
| [http://inquiryintoinquiry.com/ Inquiry Into Inquiry]<br />
| [http://jonawbrey.wordpress.com/ Jon Awbrey • Blog]<br />
|-<br />
| [http://intersci.ss.uci.edu/wiki/index.php/User:Jon_Awbrey InterSciWiki User Page]<br />
| [http://intersci.ss.uci.edu/wiki/index.php/User_talk:Jon_Awbrey InterSciWiki Talk Page]<br />
|-<br />
| [http://mywikibiz.com/Directory:Jon_Awbrey MyWikiBiz Directory]<br />
| [http://mywikibiz.com/Directory_talk:Jon_Awbrey MyWikiBiz Discussion]<br />
|-<br />
| [http://mywikibiz.com/User:Jon_Awbrey MyWikiBiz User Page]<br />
| [http://mywikibiz.com/User_talk:Jon_Awbrey MyWikiBiz Talk Page]<br />
|-<br />
| [http://planetmath.org/ PlanetMath Project]<br />
| [http://planetmath.org/users/Jon-Awbrey PlanetMath Profile]<br />
|-<br />
| [http://forum.wolframscience.com/ NKS Forum]<br />
| [http://forum.wolframscience.com/member.php?s=&action=getinfo&userid=336 NKS Profile]<br />
|-<br />
| [https://oeis.org/wiki/Welcome OEIS Land]<br />
| [https://oeis.org/search?q=Awbrey Bolgia Mia]<br />
|-<br />
| [https://oeis.org/wiki/User:Jon_Awbrey OEIS Wiki Page]<br />
| [https://oeis.org/wiki/User_talk:Jon_Awbrey OEIS Wiki Talk]<br />
|-<br />
| [http://list.seqfan.eu/cgi-bin/mailman/listinfo/seqfan Fantasia Sequentia]<br />
| [http://list.seqfan.eu/pipermail/seqfan/ SeqFan Archive]<br />
|-<br />
| [http://mathforum.org/kb/ Math Forum Project]<br />
| [http://mathforum.org/kb/accountView.jspa?userID=99854 Math Forum Profile]<br />
|-<br />
| [http://www.mathweb.org/wiki/User:Jon_Awbrey MathWeb Page]<br />
| [http://www.mathweb.org/wiki/User_talk:Jon_Awbrey MathWeb Talk]<br />
|-<br />
| [http://www.proofwiki.org/wiki/User:Jon_Awbrey ProofWiki Page]<br />
| [http://www.proofwiki.org/wiki/User_talk:Jon_Awbrey ProofWiki Talk]<br />
|-<br />
| [http://mathoverflow.net/ MathOverFlow]<br />
| [http://mathoverflow.net/users/1636/jon-awbrey MOFler Profile]<br />
|-<br />
| [http://p2pfoundation.net/User:JonAwbrey P2P Wiki Page]<br />
| [http://p2pfoundation.net/User_talk:JonAwbrey P2P Wiki Talk]<br />
|-<br />
| [http://vectors.usc.edu/ Vectors Project]<br />
| [http://vectors.usc.edu/thoughtmesh/ ThoughtMesh]<br />
|-<br />
| [http://ontolog.cim3.net/ OntoLog Project]<br />
| [http://ontolog.cim3.net/cgi-bin/wiki.pl?JonAwbrey OntoLog Profile]<br />
|-<br />
| [http://web.archive.org/web/20150127223035/http://semanticweb.org/wiki/User:Jon_Awbrey SemanticWeb Page]<br />
| [http://web.archive.org/web/20100716202618/http://semanticweb.org/wiki/User_talk:Jon_Awbrey SemanticWeb Talk]<br />
|}<br />
<br />
==Contributions==<br />
<br />
===Articles===<br />
<br />
[[Ampheck]]<br />
[[Boolean domain]]<br />
[[Boolean function]]<br />
[[Boolean-valued function]]<br />
[[Charles Sanders Peirce]]<br />
[[Charles Sanders Peirce (Bibliography)]]<br />
[[Comprehension (logic)]]<br />
[[Continuous predicate]]<br />
[[Correspondence theory of truth]]<br />
[[Cybernetics]]<br />
[[Descriptive science]]<br />
[[Differential logic]]<br />
[[Dynamics of inquiry]]<br />
[[Entitative graph]]<br />
[[Exclusive disjunction]]<br />
[[Formal science]]<br />
[[Graph (mathematics)]]<br />
[[Graph theory]]<br />
[[Grounded relation]]<br />
[[Inquiry]]<br />
[[Inquiry driven system]]<br />
[[Integer sequence]]<br />
[[Hypostatic abstraction]]<br />
[[Hypostatic object]]<br />
[[Kaina Stoicheia]]<br />
[[Logic]]<br />
[[Logic of information]]<br />
[[Logic of relatives]]<br />
[[Logic of Relatives (1870)]]<br />
[[Logic of Relatives (1883)]]<br />
[[Logical conjunction]]<br />
[[Logical disjunction]]<br />
[[Logical equality]]<br />
[[Logical graph]]<br />
[[Logical implication]]<br />
[[Logical matrix]]<br />
[[Logical NAND]]<br />
[[Logical negation]]<br />
[[Logical NNOR]]<br />
[[Minimal negation operator]]<br />
[[Multigrade operator]]<br />
[[Normative science]]<br />
[[Null graph]]<br />
[[On a New List of Categories]]<br />
[[Parametric operator]]<br />
[[Peirce's law]]<br />
[[Philosophy of mathematics]]<br />
[[Pragmatic information]]<br />
[[Pragmatic maxim]]<br />
[[Pragmatic theory of truth]]<br />
[[Pragmaticism]] <br />
[[Pragmatism]]<br />
[[Prescisive abstraction]]<br />
[[Propositional calculus]]<br />
[[Relation (mathematics)]]<br />
[[Relation composition]]<br />
[[Relation construction]]<br />
[[Relation reduction]]<br />
[[Relation theory]]<br />
[[Relation type]]<br />
[[Relative term]]<br />
[[Semeiotic]]<br />
[[Semiotic information]]<br />
[[Semiotics]]<br />
[[Sign relation]]<br />
[[Sign relational complex]]<br />
[[Sole sufficient operator]]<br />
[[Tacit extension]]<br />
[[The Simplest Mathematics]]<br />
[[Triadic relation]]<br />
[[Truth table]]<br />
[[Truth theory]]<br />
[[Universe of discourse]]<br />
[[What we've got here is (a) failure to communicate]]<br />
[[Zeroth order logic]]<br />
<br />
===Notes===<br />
<br />
* [[Directory:Jon Awbrey/Notes/Factorization Issues|Factorization Issues]]<br />
<br />
* [[Directory:Jon Awbrey/Notes/Factorization And Reification|Factorization And Reification]]<br />
<br />
===Papers===<br />
<br />
====Functional Logic====<br />
<br />
* [[Directory:Jon Awbrey/Papers/Functional Logic : Higher Order Propositions|Functional Logic : Higher Order Propositions]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Functional Logic : Inquiry and Analogy|Functional Logic : Inquiry and Analogy]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Functional Logic : Quantification Theory|Functional Logic : Quantification Theory]]<br />
<br />
====Differential Logic====<br />
<br />
* [[Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction|Differential Logic : Introduction]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Differential Propositional Calculus|Differential Propositional Calculus]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems|Differential Logic and Dynamic Systems 1.0]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0|Differential Logic and Dynamic Systems 2.0]]<br />
<br />
====Logic and Semiotics====<br />
<br />
* [[Directory:Jon Awbrey/Papers/Futures Of Logical Graphs|Futures Of Logical Graphs]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives|Peirce's 1870 Logic Of Relatives]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Peirce's Logic Of Information|Peirce's Logic Of Information]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Propositional Equation Reasoning Systems|Propositional Equation Reasoning Systems]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Semiotic Information|Semiotic Information]]<br />
<br />
====Inquiry Driven Systems====<br />
<br />
* [[Directory:Jon Awbrey/Essays/Prospects For Inquiry Driven Systems|Prospects for Inquiry Driven Systems]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Introduction to Inquiry Driven Systems|Introduction to Inquiry Driven Systems]]<br />
<br />
* [[Directory:Jon Awbrey/Essays/Inquiry Driven Systems : Fields Of Inquiry|Inquiry Driven Systems : Fields Of Inquiry]]<br />
<br />
* [[Directory:Jon Awbrey/Papers/Inquiry Driven Systems|Inquiry Driven Systems : Inquiry Into Inquiry]]<br />
<br />
===Projects===<br />
<br />
* [[Directory:Jon Awbrey/Projects/Cactus Language|Cactus Language]]<br />
<br />
* [[Directory:Jon Awbrey/Projects/Differential Logic|Differential Logic]]<br />
<br />
* [[Directory:Jon Awbrey/Projects/Inquiry|Inquiry]]<br />
** [[Directory:Jon Awbrey/Projects/Architecture For Inquiry|Architecture For Inquiry]]<br />
** [[Directory:Jon Awbrey/Projects/Inquiry Driven Systems|Inquiry Driven Systems]]<br />
<br />
* [[Directory:Jon Awbrey/Projects/Logic Of Information|Logic Of Information]]<br />
** [[Directory:Jon Awbrey/Projects/Pragmatic Theory Of Information|Pragmatic Theory Of Information]]<br />
** [[Directory:Jon Awbrey/Projects/Semiotic Theory Of Information|Semiotic Theory Of Information]]<br />
<br />
* [[Directory:Jon Awbrey/Projects/Notes And Queries|Notes And Queries]]<br />
<br />
* [[Directory:Jon Awbrey/Projects/Peircean Pragmata|Peircean Pragmata]]<br />
<br />
* [[Directory:Jon Awbrey/Projects/Theme One Program|Theme One Program]]<br />
<br />
* [[Directory:Jon Awbrey/Projects/Theory Of Relations|Theory Of Relations]]<br />
<br />
===Poetry===<br />
<br />
* [[Directory:Jon Awbrey/Poetry/Past All Reckoning|Past All Reckoning]]<br />
<br />
* [[Directory:Jon Awbrey/Poetry/Poems Of Emediate Moment|Poems Of Emediate Moment]]<br />
<br />
* [[Directory:Jon Awbrey/Poetry/Questionable Verses|Questionable Verses]]<br />
<br />
* [[Directory:Jon_Awbrey/Poetry/Iconoclast|Iconoclast]]<br />
<br />
===User Pages===<br />
<br />
* [[Directory:Jon Awbrey/EXCERPTS|Collection Of Source Materials]]<br />
* [[User:Jon Awbrey/Examples Of Inquiry|Examples Of Inquiry]]<br />
* [[User:Jon Awbrey/Mathematical Notes|Mathematical Notes]]<br />
* [[User:Jon Awbrey/Philosophical Notes|Philosophical Notes]]<br />
<br />
* [http://mywikibiz.com/index.php?title=Special%3APrefixIndex&prefix=Jon+Awbrey&namespace=2 MyWikiBiz User Pages]<br />
* [http://intersci.ss.uci.edu/wiki/index.php?title=Special%3APrefixIndex&prefix=Jon+Awbrey&namespace=2 InterSciWiki User Pages]<br />
* [http://mywikibiz.com/index.php?title=Special%3APrefixIndex&prefix=Jon+Awbrey&namespace=110 MyWikiBiz Directory Pages]<br />
<br />
==Presentations and Publications==<br />
<br />
* Awbrey, S.M., and Awbrey, J.L. (May 2001), &ldquo;Conceptual Barriers to Creating Integrative Universities&rdquo;, ''Organization : The Interdisciplinary Journal of Organization, Theory, and Society'' 8(2), Sage Publications, London, UK, pp. 269&ndash;284. [http://org.sagepub.com/cgi/content/abstract/8/2/269 Abstract].<br />
<br />
* Awbrey, S.M., and Awbrey, J.L. (September 1999), &ldquo;Organizations of Learning or Learning Organizations : The Challenge of Creating Integrative Universities for the Next Century&rdquo;, ''Second International Conference of the Journal &lsquo;Organization&rsquo;'', ''Re-Organizing Knowledge, Trans-Forming Institutions : Knowing, Knowledge, and the University in the 21st Century'', University of Massachusetts, Amherst, MA. [http://cspeirce.com/menu/library/aboutcsp/awbrey/integrat.htm Online].<br />
<br />
* Awbrey, J.L., and Awbrey, S.M. (Autumn 1995), &ldquo;Interpretation as Action : The Risk of Inquiry&rdquo;, ''Inquiry : Critical Thinking Across the Disciplines'' 15(1), pp. 40&ndash;52. [https://web.archive.org/web/20001210162300/http://chss.montclair.edu/inquiry/fall95/awbrey.html Archive]. [https://www.pdcnet.org/inquiryct/content/inquiryct_1995_0015_0001_0040_0052 Journal]. [https://independent.academia.edu/JonAwbrey/Papers/1302117/Interpretation_as_Action_The_Risk_of_Inquiry Online].<br />
<br />
* Awbrey, J.L., and Awbrey, S.M. (June 1992), &ldquo;Interpretation as Action : The Risk of Inquiry&rdquo;, ''The Eleventh International Human Science Research Conference'', Oakland University, Rochester, Michigan.<br />
<br />
* Awbrey, S.M., and Awbrey, J.L. (May 1991), &ldquo;An Architecture for Inquiry : Building Computer Platforms for Discovery&rdquo;, ''Proceedings of the Eighth International Conference on Technology and Education'', Toronto, Canada, pp. 874&ndash;875. [http://www.academia.edu/1270327/An_Architecture_for_Inquiry_Building_Computer_Platforms_for_Discovery Online].<br />
<br />
* Awbrey, J.L., and Awbrey, S.M. (January 1991), &ldquo;Exploring Research Data Interactively : Developing a Computer Architecture for Inquiry&rdquo;, Poster presented at the ''Annual Sigma Xi Research Forum'', University of Texas Medical Branch, Galveston, TX.<br />
<br />
* Awbrey, J.L., and Awbrey, S.M. (August 1990), &ldquo;Exploring Research Data Interactively. Theme One : A Program of Inquiry&rdquo;, ''Proceedings of the Sixth Annual Conference on Applications of Artificial Intelligence and CD-ROM in Education and Training'', Society for Applied Learning Technology, Washington, DC, pp. 9&ndash;15. [http://academia.edu/1272839/Exploring_Research_Data_Interactively._Theme_One_A_Program_of_Inquiry Online].<br />
<br />
==Education==<br />
<br />
* 1993&ndash;2003. [http://web.archive.org/web/20120202222443/http://www2.oakland.edu/secs/dispprofile.asp?Fname=Fatma&Lname=Mili Graduate Study], [http://web.archive.org/web/20120203004703/http://www2.oakland.edu/secs/dispprofile.asp?Fname=Mohamed&Lname=Zohdy Systems Engineering], [http://meadowbrookhall.org/explore/history/meadowbrookhall Oakland University].<br />
<br />
* '''1989. [http://web.archive.org/web/20001206101800/http://www.msu.edu/dig/msumap/psychology.html M.A. Psychology]''', [http://web.archive.org/web/20000902103838/http://www.msu.edu/dig/msumap/beaumont.html Michigan State University].<br />
<br />
* 1985&ndash;1986. [http://quod.lib.umich.edu/cgi/i/image/image-idx?id=S-BHL-X-BL001808%5DBL001808 Graduate Study, Mathematics], [http://www.umich.edu/ University of Michigan].<br />
<br />
* 1985. [http://web.archive.org/web/20061230174652/http://www.uiuc.edu/navigation/buildings/altgeld.top.html Graduate Study, Mathematics], [http://www.uiuc.edu/ University of Illinois at Urbana&ndash;Champaign].<br />
<br />
* 1984. [http://www.psych.uiuc.edu/graduate/ Graduate Study, Psychology], [http://www.uiuc.edu/ University of Illinois at Champaign&ndash;Urbana].<br />
<br />
* '''1980. [http://www.mth.msu.edu/images/wells_medium.jpg M.A. Mathematics]''', [http://www.msu.edu/~hvac/survey/BeaumontTower.html Michigan State University].<br />
<br />
* '''1976. [http://web.archive.org/web/20001206050600/http://www.msu.edu/dig/msumap/phillips.html B.A. Mathematical and Philosophical Method]''', <br> [http://www.enolagaia.com/JMC.html Justin Morrill College], [http://www.msu.edu/ Michigan State University].<br />
<br />
==Category and Subject Interests==<br />
<br />
[[Category:Artificial Intelligence]]<br />
[[Category:Automata Theory]]<br />
[[Category:Category Theory]]<br />
[[Category:Charles Sanders Peirce]]<br />
[[Category:Cognitive Science]] <br />
[[Category:Combinatorics]]<br />
[[Category:Computer Science]]<br />
[[Category:Critical Thinking]]<br />
[[Category:Cybernetics]]<br />
[[Category:Differential Logic]]<br />
[[Category:Education]]<br />
[[Category:Formal Languages]]<br />
[[Category:Formal Sciences]]<br />
[[Category:Graph Theory]]<br />
[[Category:Group Theory]]<br />
[[Category:Hermeneutics]]<br />
[[Category:Information Systems]]<br />
[[Category:Information Theory]]<br />
[[Category:Inquiry]]<br />
[[Category:Inquiry Driven Systems]]<br />
[[Category:Integer Sequences]]<br />
[[Category:Intelligence Amplification]]<br />
[[Category:Learning Organizations]]<br />
[[Category:Linguistics]]<br />
[[Category:Knowledge Representation]]<br />
[[Category:Logic]]<br />
[[Category:Mathematics]]<br />
[[Category:Natural Languages]]<br />
[[Category:Philosophy]]<br />
[[Category:Pragmatics]]<br />
[[Category:Psychology]]<br />
[[Category:Science]]<br />
[[Category:Semantics]]<br />
[[Category:Semiotics]]<br />
[[Category:Statistics]]<br />
[[Category:Systems Science]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Logic_Live&diff=469832Logic Live2020-09-17T15:25:08Z<p>Jon Awbrey: /* Peer nodes */ waybak llink</p>
<hr />
<div><font size="3">&#9758;</font> This page serves as a '''focal node''' for a collection of related resources.<br />
<br />
==Participants==<br />
<br />
* Interested parties may add their names on [[Logic Live/Participants|this page]].<br />
<br />
==Syllabus==<br />
<br />
===Focal nodes===<br />
<br />
* [[Inquiry Live]]<br />
* [[Logic Live]]<br />
<br />
===Peer nodes===<br />
<br />
* [http://mywikibiz.com/Logic_Live Logic Live @ MyWikiBiz]<br />
* [http://ref.subwiki.org/wiki/Logic_Live Logic Live @ Subject Wikis]<br />
* [http://en.wikiversity.org/wiki/Logic_Live Logic Live @ Wikiversity]<br />
* [http://beta.wikiversity.org/wiki/Logic_Live Logic Live @ Wikiversity Beta]<br />
<br />
===Archive===<br />
<br />
* [http://web.archive.org/web/20191209181958/http://intersci.ss.uci.edu/wiki/index.php/Logic_Live Logic Live @ InterSciWiki]<br />
<br />
===Logical operators===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [[Exclusive disjunction]]<br />
* [[Logical conjunction]]<br />
* [[Logical disjunction]]<br />
* [[Logical equality]]<br />
{{col-break}}<br />
* [[Logical implication]]<br />
* [[Logical NAND]]<br />
* [[Logical NNOR]]<br />
* [[Logical negation|Negation]]<br />
{{col-end}}<br />
<br />
===Related topics===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [[Ampheck]]<br />
* [[Boolean domain]]<br />
* [[Boolean function]]<br />
* [[Boolean-valued function]]<br />
* [[Differential logic]]<br />
{{col-break}}<br />
* [[Logical graph]]<br />
* [[Minimal negation operator]]<br />
* [[Multigrade operator]]<br />
* [[Parametric operator]]<br />
* [[Peirce's law]]<br />
{{col-break}}<br />
* [[Propositional calculus]]<br />
* [[Sole sufficient operator]]<br />
* [[Truth table]]<br />
* [[Universe of discourse]]<br />
* [[Zeroth order logic]]<br />
{{col-end}}<br />
<br />
===Relational concepts===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [[Continuous predicate]]<br />
* [[Hypostatic abstraction]]<br />
* [[Logic of relatives]]<br />
* [[Logical matrix]]<br />
{{col-break}}<br />
* [[Relation (mathematics)|Relation]]<br />
* [[Relation composition]]<br />
* [[Relation construction]]<br />
* [[Relation reduction]]<br />
{{col-break}}<br />
* [[Relation theory]]<br />
* [[Relative term]]<br />
* [[Sign relation]]<br />
* [[Triadic relation]]<br />
{{col-end}}<br />
<br />
===Information, Inquiry===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [[Inquiry]]<br />
* [[Dynamics of inquiry]]<br />
{{col-break}}<br />
* [[Semeiotic]]<br />
* [[Logic of information]]<br />
{{col-break}}<br />
* [[Descriptive science]]<br />
* [[Normative science]]<br />
{{col-break}}<br />
* [[Pragmatic maxim]]<br />
* [[Truth theory]]<br />
{{col-end}}<br />
<br />
===Related articles===<br />
<br />
{{col-begin}}<br />
{{col-break}}<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]<br />
{{col-break}}<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]<br />
{{col-break}}<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]<br />
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]<br />
{{col-end}}<br />
<br />
[[Category:Inquiry]]<br />
[[Category:Logic]]<br />
[[Category:Open Educational Resource]]<br />
[[Category:Peer Educational Resource]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Inquiry_Live&diff=469831Inquiry Live2020-09-17T13:05:53Z<p>Jon Awbrey: /* Rudiments of organization */ move waybak link to archive section</p>
<hr />
<div><font size="3">&#9758;</font> This page serves as a '''focal node''' for a collection of related resources.<br />
<br />
==Participants==<br />
<br />
Interested parties may add their names on [[Inquiry Live/Participants|this page]].<br />
<br />
==Rudiments of organization==<br />
<br />
===Focal nodes===<br />
<br />
* [[Inquiry Live]]<br />
* [[Logic Live]]<br />
<br />
'''Focal nodes''' serve as hubs for collections of related resources, in particular, activity sites and article contents.<br />
<br />
===Peer nodes===<br />
<br />
* [http://mywikibiz.com/Inquiry_Live Inquiry Live @ MyWikiBiz]<br />
* [http://ref.subwiki.org/wiki/Inquiry_Live Inquiry Live @ Subject Wikis]<br />
* [http://en.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity]<br />
* [http://beta.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity Beta]<br />
<br />
'''Peer nodes''' are roughly parallel pages on different sites that are not necessarily identical in content &mdash; especially as they develop in time across different environments through interaction with diverse populations &mdash; but they should preserve enough information to reproduce each other, more or less, in case of damage or loss.<br />
<br />
===Archive===<br />
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* [http://web.archive.org/web/20191024043046/http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Live Inquiry Live @ InterSciWiki]<br />
<br />
[[Category:Inquiry]]<br />
[[Category:Logic]]<br />
[[Category:Open Educational Resource]]<br />
[[Category:Peer Educational Resource]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Inquiry_Live&diff=469830Inquiry Live2020-09-17T13:04:27Z<p>Jon Awbrey: /* Rudiments of organization */ + /* Archive */</p>
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<div><font size="3">&#9758;</font> This page serves as a '''focal node''' for a collection of related resources.<br />
<br />
==Participants==<br />
<br />
Interested parties may add their names on [[Inquiry Live/Participants|this page]].<br />
<br />
==Rudiments of organization==<br />
<br />
===Focal nodes===<br />
<br />
* [[Inquiry Live]]<br />
* [[Logic Live]]<br />
<br />
'''Focal nodes''' serve as hubs for collections of related resources, in particular, activity sites and article contents.<br />
<br />
===Peer nodes===<br />
<br />
* [http://web.archive.org/web/20191024043046/http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Live Inquiry Live @ InterSciWiki]<br />
* [http://mywikibiz.com/Inquiry_Live Inquiry Live @ MyWikiBiz]<br />
* [http://ref.subwiki.org/wiki/Inquiry_Live Inquiry Live @ Subject Wikis]<br />
* [http://en.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity]<br />
* [http://beta.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity Beta]<br />
<br />
'''Peer nodes''' are roughly parallel pages on different sites that are not necessarily identical in content &mdash; especially as they develop in time across different environments through interaction with diverse populations &mdash; but they should preserve enough information to reproduce each other, more or less, in case of damage or loss.<br />
<br />
===Archive===<br />
<br />
[[Category:Inquiry]]<br />
[[Category:Logic]]<br />
[[Category:Open Educational Resource]]<br />
[[Category:Peer Educational Resource]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Inquiry_Live&diff=469829Inquiry Live2020-09-17T13:00:08Z<p>Jon Awbrey: /* Peer nodes */ waybak link</p>
<hr />
<div><font size="3">&#9758;</font> This page serves as a '''focal node''' for a collection of related resources.<br />
<br />
==Participants==<br />
<br />
Interested parties may add their names on [[Inquiry Live/Participants|this page]].<br />
<br />
==Rudiments of organization==<br />
<br />
===Focal nodes===<br />
<br />
* [[Inquiry Live]]<br />
* [[Logic Live]]<br />
<br />
'''Focal nodes''' serve as hubs for collections of related resources, in particular, activity sites and article contents.<br />
<br />
===Peer nodes===<br />
<br />
* [http://web.archive.org/web/20191024043046/http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Live Inquiry Live @ InterSciWiki]<br />
* [http://mywikibiz.com/Inquiry_Live Inquiry Live @ MyWikiBiz]<br />
* [http://ref.subwiki.org/wiki/Inquiry_Live Inquiry Live @ Subject Wikis]<br />
* [http://en.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity]<br />
* [http://beta.wikiversity.org/wiki/Inquiry_Live Inquiry Live @ Wikiversity Beta]<br />
<br />
'''Peer nodes''' are roughly parallel pages on different sites that are not necessarily identical in content &mdash; especially as they develop in time across different environments through interaction with diverse populations &mdash; but they should preserve enough information to reproduce each other, more or less, in case of damage or loss.<br />
<br />
[[Category:Inquiry]]<br />
[[Category:Logic]]<br />
[[Category:Open Educational Resource]]<br />
[[Category:Peer Educational Resource]]</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Category:Systems_theory&diff=469756Category:Systems theory2020-06-05T13:45:09Z<p>Jon Awbrey: '''{{PAGENAME}}'''</p>
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<div>'''{{PAGENAME}}'''</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Category:Set_theory&diff=469755Category:Set theory2020-06-05T13:42:52Z<p>Jon Awbrey: '''{{PAGENAME}}'''</p>
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<div>'''{{PAGENAME}}'''</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Category:Propositional_calculus&diff=469754Category:Propositional calculus2020-06-05T13:42:05Z<p>Jon Awbrey: '''{{PAGENAME}}'''</p>
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<div>'''{{PAGENAME}}'''</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Category:Proof_theory&diff=469753Category:Proof theory2020-06-05T13:40:40Z<p>Jon Awbrey: '''{{PAGENAME}}'''</p>
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<div>'''{{PAGENAME}}'''</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Category:Model_theory&diff=469752Category:Model theory2020-06-05T13:36:57Z<p>Jon Awbrey: '''{{PAGENAME}}'''</p>
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<div>'''{{PAGENAME}}'''</div>Jon Awbreyhttps://mywikibiz.com/index.php?title=Category:Logical_graphs&diff=469751Category:Logical graphs2020-06-05T13:36:32Z<p>Jon Awbrey: '''{{PAGENAME}}'''</p>
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<div>'''{{PAGENAME}}'''</div>Jon Awbrey